lean/OAI/Computability/BinPacking/Computation/MachineExpanderTable.lean

import Mathlib.Tactic.FinCases
import OAI.Computability.BinPacking.Computation.MachineExpanderRowEmit
import OAI.Computability.BinPacking.Computation.MachineExpanderTableReverse

namespace OAI

namespace BinPackingGames.Foundations.Complexity.MachineExpanderRow

section

open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords
open MachineComposition

def lookupPhaseLabel {d : Nat} (second : Bool)
    (l : MachinePreservingLookupClean.Label) : Label d :=
  if second then .secondLookup l else .firstLookup l

def scanPhaseLabel {d : Nat} (second : Bool) : Label d :=
  if second then .secondScan else .firstScan

def reversePhaseLabel {d : Nat} (second : Bool) : Label d :=
  if second then .secondReverse else .firstReverse

def lookupPhasePort {d : Nat} (second : Bool) (control : Control d) : Fin (degree d) :=
  if second then secondOffset control else firstOffset control

def lookupPhaseAddress {d : Nat} (second : Bool) (vertex : Nat) (control : Control d) : Nat :=
  if second then secondAddress vertex control else firstAddress vertex control

theorem lookupPhaseAddress_eq_rowIndex {v d : Nat} (second : Bool)
    (vertex : Fin v) (control : Control d) :
    lookupPhaseAddress second vertex.val control =
      (rowIndex v (degree d) (vertex, lookupPhasePort second control)).val := by
  cases second
  · exact firstAddress_eq_rowIndex vertex control
  · exact secondAddress_eq_rowIndex vertex control

theorem lookupTape_injective : Function.Injective lookupTape := by decide

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

def lookupResultTapes (ports : Tape → K) (base : K → List Bool)
    (value : Nat) (indexSuffix : List Bool) : K → List Bool :=
  MachinePreservingLookup.initialTapes (ports ∘ lookupTape) base 0 indexSuffix []
    (encodeWord value ++ base (ports .lookupOutput))

theorem lookupResultTapes_index (ports : Tape → K) (distinct : Function.Injective ports)
    (base : K → List Bool) (value : Nat) (indexSuffix : List Bool) :
    lookupResultTapes ports base value indexSuffix (ports .queryIndex) =
      encodeWord 0 ++ indexSuffix := by
  apply MachineLookup.tapes_index
  · exact fun h => (by decide : Tape.queryIndex ≠ .lookupWork) (distinct h)
  · exact fun h => (by decide : Tape.queryIndex ≠ .lookupOutput) (distinct h)

theorem lookupResultTapes_work (ports : Tape → K) (distinct : Function.Injective ports)
    (base : K → List Bool) (value : Nat) (indexSuffix : List Bool) :
    lookupResultTapes ports base value indexSuffix (ports .lookupWork) = [] := by
  apply MachineLookup.tapes_source
  exact fun h => (by decide : Tape.lookupWork ≠ .lookupOutput) (distinct h)

theorem lookupResultTapes_output (ports : Tape → K)
    (base : K → List Bool) (value : Nat) (indexSuffix : List Bool) :
    lookupResultTapes ports base value indexSuffix (ports .lookupOutput) =
      encodeWord value ++ base (ports .lookupOutput) := by
  apply MachineLookup.tapes_destination

theorem lookupResultTapes_other (ports : Tape → K) (base : K → List Bool)
    (value : Nat) (indexSuffix : List Bool) (p : K)
    (hi : p ≠ ports .queryIndex) (hw : p ≠ ports .lookupWork)
    (ho : p ≠ ports .lookupOutput) :
    lookupResultTapes ports base value indexSuffix p = base p := by
  exact MachineLookup.tapes_other _ _ _ p hi hw ho _ _ _ _

theorem lookupResultTapes_table (ports : Tape → K) (distinct : Function.Injective ports)
    (base : K → List Bool) (value : Nat) (indexSuffix : List Bool) :
    lookupResultTapes ports base value indexSuffix (ports .table) = base (ports .table) := by
  apply lookupResultTapes_other
  all_goals intro h; have := distinct h; cases this

theorem lookupResultTapes_restore (ports : Tape → K) (distinct : Function.Injective ports)
    (base : K → List Bool) (value : Nat) (indexSuffix : List Bool) :
    lookupResultTapes ports base value indexSuffix (ports .lookupRestore) =
      base (ports .lookupRestore) := by
  apply lookupResultTapes_other
  all_goals intro h; have := distinct h; cases this

section Lookup

variable {v d : Nat} (second : Bool) (positive : 0 < d)
    (H : Table (cloudSize d) d) (ports : Tape → K)
    (distinct : Function.Injective ports) (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ l, target (labels l) = statement positive H ports labels exit l)
    (base : K → List Bool) (G : Table v (degree d))
    (tableWord : base (ports .table) = encodeWords (rotationWords G))
    (scratchEmpty : base (ports .lookupRestore) = [])
    (x : Fin v × Fin (degree d)) (indexSuffix : List Bool)
    (counterWord : base (ports .queryIndex) =
      encodeWord (rowIndex v (degree d) x).val ++ indexSuffix)
    (workEmpty : base (ports .lookupWork) = [])
    (ambient : (Ambient ρ d × Fin (degree d)) × Unit) (register : Option Bool)

include positive H distinct exit code tableWord scratchEmpty counterWord workEmpty

theorem lookupPhaseTrace :
    (advance (TM2.step target))^[MachinePreservingLookupClean.steps
        (rotationWords G) (rowIndex v (degree d) x).val]
      (some ⟨some (labels (lookupPhaseLabel second (.run .copyFirst))),
        (ambient, register), base⟩) =
      some ⟨some (labels (scanPhaseLabel second)), (ambient, none),
        lookupResultTapes ports base (reverseIndex G (rowIndex v (degree d) x)).val
          indexSuffix⟩ := by
  apply MachinePreservingLookupClean.traceAt_fromTapes (ports ∘ lookupTape)
    (distinct.comp lookupTape_injective)
    (fun l => labels (lookupPhaseLabel second l)) (some (labels (scanPhaseLabel second)))
    target
  · intro l
    cases second
    · exact code (.firstLookup l)
    · exact code (.secondLookup l)
  · exact tableWord
  · exact scratchEmpty
  · exact rotationWords_getElem? G (rowIndex v (degree d) x)
  · exact counterWord
  · exact workEmpty

def lookupPhaseInTime :
    StateTransition.EvalsToInTime (TM2.step target)
      ⟨some (labels (lookupPhaseLabel second (.run .copyFirst))),
        (ambient, register), base⟩
      (some ⟨some (labels (scanPhaseLabel second)), (ambient, none),
        lookupResultTapes ports base (reverseIndex G (rowIndex v (degree d) x)).val
          indexSuffix⟩)
      (6 * (encodeWords (rotationWords G)).length + 4) where
  steps := MachinePreservingLookupClean.steps (rotationWords G) (rowIndex v (degree d) x).val
  evals_in_steps := lookupPhaseTrace second positive H ports distinct labels exit target code
    base G tableWord scratchEmpty x indexSuffix counterWord workEmpty ambient register
  steps_le_m := MachinePreservingLookupClean.steps_le (rotationWords G) _ _
    (rotationWords_getElem? G (rowIndex v (degree d) x))

end Lookup

variable {d : Nat} (second : Bool) (positive : 0 < d)
    (H : Table (cloudSize d) d) (ports : Tape → K)
    (distinct : Function.Injective ports) (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ l, target (labels l) = statement positive H ports labels exit l)
    (base : K → List Bool)
    (ambient : (Ambient ρ d × Fin (degree d)) × Unit) (register : Option Bool)

include positive H distinct exit code

theorem reversePhaseTrace :
    (advance (TM2.step target))^[(base (ports .queryReverse)).length + 1]
      (some ⟨some (labels (reversePhaseLabel second)), (ambient, register), base⟩) =
      some ⟨some (labels (lookupPhaseLabel second (.run .copyFirst))), (ambient, none),
        Reduction.MachineTransfer.tapesAt (ports .queryReverse) (ports .queryIndex) base []
          ((base (ports .queryReverse)).reverse ++ base (ports .queryIndex))⟩ := by
  have h := Reduction.MachineTransfer.transferAt_fromTapes
    (ports .queryReverse) (ports .queryIndex)
    (fun h => (by decide : Tape.queryReverse ≠ .queryIndex) (distinct h))
    id false (labels (reversePhaseLabel second))
    (some (labels (lookupPhaseLabel second (.run .copyFirst)))) target
    (by cases second <;> exact code _) base ambient register
  unfold Reduction.MachineTransfer.nextAt at h
  unfold advance
  simpa only [List.map_id] using h

def reversePhaseInTime :
    StateTransition.EvalsToInTime (TM2.step target)
      ⟨some (labels (reversePhaseLabel second)), (ambient, register), base⟩
      (some ⟨some (labels (lookupPhaseLabel second (.run .copyFirst))), (ambient, none),
        Reduction.MachineTransfer.tapesAt (ports .queryReverse) (ports .queryIndex) base []
          ((base (ports .queryReverse)).reverse ++ base (ports .queryIndex))⟩)
      ((base (ports .queryReverse)).length + 1) where
  steps := (base (ports .queryReverse)).length + 1
  evals_in_steps := reversePhaseTrace second positive H ports distinct labels exit target code
    base ambient register
  steps_le_m := Nat.le_refl _

theorem reversePhaseTrace_unary (index : Nat) (suffix : List Bool)
    (queryWord : base (ports .queryReverse) = (encodeWord index).reverse)
    (indexWord : base (ports .queryIndex) = suffix) :
    (advance (TM2.step target))^[index + 2]
      (some ⟨some (labels (reversePhaseLabel second)), (ambient, register), base⟩) =
      some ⟨some (labels (lookupPhaseLabel second (.run .copyFirst))), (ambient, none),
        Reduction.MachineTransfer.tapesAt (ports .queryReverse) (ports .queryIndex) base []
          (encodeWord index ++ suffix)⟩ := by
  have h := reversePhaseTrace second positive H ports distinct labels exit target code
    base ambient register
  simpa only [queryWord, indexWord, List.length_reverse, encodeWord_length,
    List.reverse_reverse, Nat.add_assoc] using h

end

section

open Turing MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl

@[simp] private theorem dirtyTape_zero : dirtyTape 0 = .queryIndex := rfl
@[simp] private theorem dirtyTape_one : dirtyTape 1 = .lookupOutput := rfl
@[simp] private theorem dirtyTape_two : dirtyTape 2 = .quotientFirst := rfl
@[simp] private theorem dirtyTape_three : dirtyTape 3 = .quotientSecond := rfl
@[simp] private theorem dirtyTape_four : dirtyTape 4 = .remainderFirst := rfl
@[simp] private theorem dirtyTape_five : dirtyTape 5 = .remainderSecond := rfl

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

def cleanupTapes (ports : Tape → K) (base : K → List Bool) : K → List Bool :=
  Function.update
    (Function.update
      (Function.update
        (Function.update
          (Function.update
            (Function.update base (ports .queryIndex) [])
            (ports .lookupOutput) [])
          (ports .quotientFirst) [])
        (ports .quotientSecond) [])
      (ports .remainderFirst) [])
    (ports .remainderSecond) []

def cleanupSteps (ports : Tape → K) (base : K → List Bool) : Nat :=
  ((base (ports .queryIndex)).length + 1) +
  ((base (ports .lookupOutput)).length + 1) +
  ((base (ports .quotientFirst)).length + 1) +
  ((base (ports .quotientSecond)).length + 1) +
  ((base (ports .remainderFirst)).length + 1) +
  ((base (ports .remainderSecond)).length + 1)

omit [DecidableEq K] in
theorem cleanupSteps_eq_sum (ports : Tape → K) (base : K → List Bool) :
    cleanupSteps ports base = ∑ i : Fin 6, ((base (ports (dirtyTape i))).length + 1) := by
  simp [cleanupSteps, Fin.sum_univ_succ, Nat.add_assoc]

@[simp] theorem cleanupTapes_dirty (ports : Tape → K) (distinct : Function.Injective ports)
    (base : K → List Bool) (i : Fin 6) :
    cleanupTapes ports base (ports (dirtyTape i)) = [] := by
  fin_cases i <;>
    simp [cleanupTapes, distinct.eq_iff]

theorem cleanupTapes_other (ports : Tape → K) (base : K → List Bool) (k : K)
    (outside : ∀ i : Fin 6, k ≠ ports (dirtyTape i)) :
    cleanupTapes ports base k = base k := by
  have h0 := outside 0
  have h1 := outside 1
  have h2 := outside 2
  have h3 := outside 3
  have h4 := outside 4
  have h5 := outside 5
  change k ≠ ports .queryIndex at h0
  change k ≠ ports .lookupOutput at h1
  change k ≠ ports .quotientFirst at h2
  change k ≠ ports .quotientSecond at h3
  change k ≠ ports .remainderFirst at h4
  change k ≠ ports .remainderSecond at h5
  simp [cleanupTapes, h0, h1, h2, h3, h4, h5]

@[simp] theorem cleanupTapes_table (ports : Tape → K) (distinct : Function.Injective ports)
    (base : K → List Bool) :
    cleanupTapes ports base (ports .table) = base (ports .table) := by
  simp [cleanupTapes, distinct.eq_iff]

@[simp] theorem cleanupTapes_inputVertex (ports : Tape → K)
    (distinct : Function.Injective ports) (base : K → List Bool) :
    cleanupTapes ports base (ports .inputVertex) = base (ports .inputVertex) := by
  simp [cleanupTapes, distinct.eq_iff]

@[simp] theorem cleanupTapes_output (ports : Tape → K) (distinct : Function.Injective ports)
    (base : K → List Bool) :
    cleanupTapes ports base (ports .output) = base (ports .output) := by
  simp [cleanupTapes, distinct.eq_iff]

private theorem join_trace {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]

variable [Fintype ρ] {d : Nat}

theorem cleanupOneTraceAt (positive : 0 < d) (H : Table (cloudSize d) d)
    (ports : Tape → K) (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (i : Fin 6) (base : K → List Bool) (state : State ρ d) :
    (advance (TM2.step target))^[((base (ports (dirtyTape i))).length + 1)]
      (some ⟨some (labels (.cleanup i)), state, base⟩) =
      some ⟨some (if hi : i.val + 1 < 6 then labels (.cleanup ⟨i.val + 1, hi⟩)
          else labels .done),
        (state.1, none), Function.update base (ports (dirtyTape i)) []⟩ := by
  have run := MachineDrain.drainTrace (ports (dirtyTape i)) (labels (.cleanup i))
    (some (if hi : i.val + 1 < 6 then labels (.cleanup ⟨i.val + 1, hi⟩)
      else labels .done)) target
    (by simpa only [statement] using code (.cleanup i))
    base (base (ports (dirtyTape i))) state.1 state.2
  simpa only [Function.update_eq_self] using run

theorem cleanupTraceAt (positive : 0 < d) (H : Table (cloudSize d) d)
    (ports : Tape → K) (distinct : Function.Injective ports)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (state : State ρ d) :
    (advance (TM2.step target))^[cleanupSteps ports base]
      (some ⟨some (labels (.cleanup 0)), state, base⟩) =
      some ⟨some (labels .done), (state.1, none), cleanupTapes ports base⟩ := by
  let t1 := Function.update base (ports .queryIndex) []
  let t2 := Function.update t1 (ports .lookupOutput) []
  let t3 := Function.update t2 (ports .quotientFirst) []
  let t4 := Function.update t3 (ports .quotientSecond) []
  let t5 := Function.update t4 (ports .remainderFirst) []
  have h0 := cleanupOneTraceAt positive H ports labels exit target code 0 base state
  have h1 := cleanupOneTraceAt positive H ports labels exit target code 1 t1 (state.1, none)
  have h2 := cleanupOneTraceAt positive H ports labels exit target code 2 t2 (state.1, none)
  have h3 := cleanupOneTraceAt positive H ports labels exit target code 3 t3 (state.1, none)
  have h4 := cleanupOneTraceAt positive H ports labels exit target code 4 t4 (state.1, none)
  have h5 := cleanupOneTraceAt positive H ports labels exit target code 5 t5 (state.1, none)
  simp [t1, t2, t3, t4, t5, distinct.eq_iff,
    -Function.iterate_succ] at h0 h1 h2 h3 h4 h5
  have finished := join_trace (join_trace (join_trace (join_trace (join_trace h0 h1) h2) h3) h4) h5
  simpa only [cleanupSteps, cleanupTapes] using finished

theorem doneStepAt (positive : 0 < d) (H : Table (cloudSize d) d)
    (ports : Tape → K) (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (state : State ρ d) :
    TM2.step target ⟨some (labels .done), state, base⟩ = some ⟨exit, state, base⟩ := by
  change some (TM2.stepAux (target (labels .done)) state base) = _
  rw [code]
  cases exit <;> rfl

theorem cleanupExitTraceAt (positive : 0 < d) (H : Table (cloudSize d) d)
    (ports : Tape → K) (distinct : Function.Injective ports)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (state : State ρ d) :
    (advance (TM2.step target))^[cleanupSteps ports base + 1]
      (some ⟨some (labels (.cleanup 0)), state, base⟩) =
      some ⟨exit, (state.1, none), cleanupTapes ports base⟩ := by
  have cleaned := cleanupTraceAt positive H ports distinct labels exit target code base state
  have done : (advance (TM2.step target))^[1]
      (some ⟨some (labels .done), (state.1, none), cleanupTapes ports base⟩) =
      some ⟨exit, (state.1, none), cleanupTapes ports base⟩ := by
    simpa only [Function.iterate_one, advance_some] using
      doneStepAt positive H ports labels exit target code (cleanupTapes ports base) (state.1, none)
  exact join_trace cleaned done

def initializedTapes (ports : Tape → K) (base : K → List Bool) : K → List Bool :=
  Function.update
    (Function.update
      (Function.update
        (Function.update base (ports .quotientFirst) (false :: base (ports .quotientFirst)))
        (ports .quotientSecond) (false :: base (ports .quotientSecond)))
      (ports .remainderFirst) (false :: base (ports .remainderFirst)))
    (ports .remainderSecond) (false :: base (ports .remainderSecond))

theorem initializeStepAt (positive : 0 < d) (H : Table (cloudSize d) d)
    (ports : Tape → K) (distinct : Function.Injective ports)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (state : State ρ d) :
    TM2.step target ⟨some (labels .initialize), state, base⟩ =
      some ⟨some (labels (.firstEmit (PCP.AlphabetTable.Emitter.labelAt 3 _ 0 .entry))),
        (state.1, none), initializedTapes ports base⟩ := by
  change some (TM2.stepAux (target (labels .initialize)) state base) = _
  rw [code]
  simp [statement, TM2.stepAux, initializedTapes, distinct.eq_iff]

def clearedQueryTapes (ports : Tape → K) (base : K → List Bool) : K → List Bool :=
  Function.update
    (Function.update base (ports .queryIndex) (base (ports .queryIndex)).tail)
    (ports .lookupOutput) (base (ports .lookupOutput)).tail

theorem clearQueryStepAt (positive : 0 < d) (H : Table (cloudSize d) d)
    (ports : Tape → K) (distinct : Function.Injective ports)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
    (code : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (state : State ρ d) :
    TM2.step target ⟨some (labels .clearQuery), state, base⟩ =
      some ⟨some (labels (.secondEmit (PCP.AlphabetTable.Emitter.labelAt 3 _ 0 .entry))),
        state, clearedQueryTapes ports base⟩ := by
  change some (TM2.stepAux (target (labels .clearQuery)) state base) = _
  rw [code]
  simp [statement, TM2.stepAux, clearedQueryTapes, distinct.eq_iff]

end

section

open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords

structure RowData (v d : Nat) where
  oldTable : Table v (degree d)
  smallTable : Table (cloudSize d) d
  inputVertex : Fin v
  inputCloud : Fin (cloudSize d)
  inputPort : Fin (degree d)

def rowData {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) : RowData v d :=
  ⟨G, H, vertex, cloud, port⟩

namespace RowData

variable {v d : Nat} (r : RowData v d)

def control0 : Control d := start r.smallTable r.inputCloud r.inputPort
def firstRow : Fin (v * degree d) := rowIndex v (degree d) (r.inputVertex, firstOffset r.control0)
def firstValue : Nat := (reverseIndex r.oldTable r.firstRow).val
def firstPair : Fin v × Fin (degree d) := lookup r.oldTable (r.inputVertex, firstOffset r.control0)
def firstVertex : Fin v := r.firstPair.1
def firstReturn : Fin (degree d) := r.firstPair.2
def control1 : Control d := receiveFirst r.control0 r.firstReturn
def secondRow : Fin (v * degree d) := rowIndex v (degree d) (r.firstVertex, secondOffset r.control1)
def secondValue : Nat := (reverseIndex r.oldTable r.secondRow).val
def secondPair : Fin v × Fin (degree d) := lookup r.oldTable (r.firstVertex, secondOffset r.control1)
def secondVertex : Fin v := r.secondPair.1
def secondReturn : Fin (degree d) := r.secondPair.2
def control2 : Control d := receiveSecond r.smallTable r.control1 r.secondReturn
def query1 : Nat := firstAddress r.inputVertex.val r.control0
def query2 : Nat := secondAddress r.firstVertex.val r.control1
def finalValue : Nat := outputAddress r.secondVertex.val r.control2
def tableLength : Nat := (encodeWords (rotationWords r.oldTable)).length

theorem query1_eq_firstRow : r.query1 = r.firstRow.val :=
  firstAddress_eq_rowIndex r.inputVertex r.control0

theorem query2_eq_secondRow : r.query2 = r.secondRow.val :=
  secondAddress_eq_rowIndex r.firstVertex r.control1

@[simp] theorem firstValue_div_degree : r.firstValue / degree d = r.firstVertex.val := rfl
@[simp] theorem firstValue_mod_degree : r.firstValue % degree d = r.firstReturn.val := rfl
@[simp] theorem secondValue_div_degree : r.secondValue / degree d = r.secondVertex.val := rfl
@[simp] theorem secondValue_mod_degree : r.secondValue % degree d = r.secondReturn.val := rfl

theorem firstReturn_eq_residue (positive : 0 < d) :
    r.firstReturn = MachineFixedDivMod.residue (degree d) (Nat.mul_pos positive positive)
      r.firstValue := Fin.ext rfl

theorem secondReturn_eq_residue (positive : 0 < d) :
    r.secondReturn = MachineFixedDivMod.residue (degree d) (Nat.mul_pos positive positive)
      r.secondValue := Fin.ext rfl

theorem receiveFirst_residue_eq_control1 (positive : 0 < d) :
    receiveFirst r.control0 (MachineFixedDivMod.residue (degree d)
      (Nat.mul_pos positive positive) r.firstValue) = r.control1 := by
  rw [← r.firstReturn_eq_residue positive]
  rfl

theorem receiveSecond_residue_eq_control2 (positive : 0 < d) :
    receiveSecond r.smallTable r.control1 (MachineFixedDivMod.residue (degree d)
      (Nat.mul_pos positive positive) r.secondValue) = r.control2 := by
  rw [← r.secondReturn_eq_residue positive]
  rfl

theorem firstSelected : (rotationWords r.oldTable)[r.query1]? = some r.firstValue := by
  rw [query1_eq_firstRow]
  exact rotationWords_getElem? r.oldTable r.firstRow

theorem secondSelected : (rotationWords r.oldTable)[r.query2]? = some r.secondValue := by
  rw [query2_eq_secondRow]
  exact rotationWords_getElem? r.oldTable r.secondRow

theorem evaluateRow_eq : evaluateRow r.oldTable r.smallTable r.inputVertex r.inputCloud r.inputPort =
    (r.secondVertex, r.control2) := rfl

theorem finalValue_eq_step_reverseIndex :
    r.finalValue = (reverseIndex (step r.oldTable r.smallTable)
      (rowIndex (v * cloudSize d) (degree d)
        (rowIndex v (cloudSize d) (r.inputVertex, r.inputCloud), r.inputPort))).val := by
  simpa only [evaluateRow_eq, finalValue] using outputAddress_eq_step_reverseIndex
    r.oldTable r.smallTable r.inputVertex r.inputCloud r.inputPort

theorem rows_le_tableLength : v * degree d ≤ r.tableLength := by
  simp only [tableLength, encodeWords_length, rotationWords_length]
  omega

theorem query1_le_tableLength : r.query1 ≤ r.tableLength := by
  rw [query1_eq_firstRow]
  exact r.firstRow.isLt.le.trans r.rows_le_tableLength

theorem query2_le_tableLength : r.query2 ≤ r.tableLength := by
  rw [query2_eq_secondRow]
  exact r.secondRow.isLt.le.trans r.rows_le_tableLength

theorem firstValue_le_tableLength : r.firstValue ≤ r.tableLength :=
  (reverseIndex r.oldTable r.firstRow).isLt.le.trans r.rows_le_tableLength

theorem secondValue_le_tableLength : r.secondValue ≤ r.tableLength :=
  (reverseIndex r.oldTable r.secondRow).isLt.le.trans r.rows_le_tableLength

theorem inputVertex_le_tableLength : r.inputVertex.val ≤ r.tableLength := by
  have hq : 1 ≤ degree d := Nat.zero_lt_of_lt r.inputPort.isLt
  have hv : v ≤ v * degree d := by simpa using Nat.mul_le_mul_left v hq
  exact r.inputVertex.isLt.le.trans (hv.trans r.rows_le_tableLength)

theorem firstVertex_le_tableLength : r.firstVertex.val ≤ r.tableLength := by
  rw [← firstValue_div_degree]
  exact (Nat.div_le_self r.firstValue (degree d)).trans r.firstValue_le_tableLength

theorem secondVertex_le_tableLength : r.secondVertex.val ≤ r.tableLength := by
  rw [← secondValue_div_degree]
  exact (Nat.div_le_self r.secondValue (degree d)).trans r.secondValue_le_tableLength

theorem firstReturn_le_tableLength : r.firstReturn.val ≤ r.tableLength := by
  rw [← firstValue_mod_degree]
  exact (Nat.mod_le r.firstValue (degree d)).trans r.firstValue_le_tableLength

theorem secondReturn_le_tableLength : r.secondReturn.val ≤ r.tableLength := by
  rw [← secondValue_mod_degree]
  exact (Nat.mod_le r.secondValue (degree d)).trans r.secondValue_le_tableLength

end RowData

variable {v d : Nat}

def frameContents (r : RowData v d) (output : List Bool)
    (qr qi lo q1 q2 r1 r2 : List Bool) : Tape → List Bool
  | .inputVertex => encodeWord r.inputVertex.val
  | .table => encodeWords (rotationWords r.oldTable)
  | .output => output
  | .queryReverse => qr
  | .queryIndex => qi
  | .lookupOutput => lo
  | .quotientFirst => q1
  | .quotientSecond => q2
  | .remainderFirst => r1
  | .remainderSecond => r2
  | _ => []

def frame0 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  frameContents r output [] [] [] [] [] [] []
def frame1 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  initializedTapes id (frame0 r output)
def frame2 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  emittedWord .queryReverse (frame1 r output) r.query1
def frame3 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  Reduction.MachineTransfer.tapesAt .queryReverse .queryIndex (frame2 r output) []
    (encodeWord r.query1)
def frame4 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  lookupResultTapes id (frame3 r output) r.firstValue []
def frame5 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  divisionOutput d id .quotientFirst .remainderFirst (frame4 r output) r.firstValue [] [] []
def frame6 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  clearedQueryTapes id (frame5 r output)
def frame7 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  emittedWord .queryReverse (frame6 r output) r.query2
def frame8 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  Reduction.MachineTransfer.tapesAt .queryReverse .queryIndex (frame7 r output) []
    (encodeWord r.query2)
def frame9 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  lookupResultTapes id (frame8 r output) r.secondValue []
def frame10 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  divisionOutput d id .quotientSecond .remainderSecond (frame9 r output) r.secondValue [] [] []
def frame11 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  emittedWord .output (frame10 r output) r.finalValue
def frame12 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  cleanupTapes id (frame11 r output)

private theorem lookupResultTapes_id_eq (base : Tape → List Bool)
    (value : Nat) (suffix : List Bool) :
    lookupResultTapes id base value suffix =
      Function.update
        (Function.update (Function.update base .queryIndex (encodeWord 0 ++ suffix))
          .lookupWork [])
        .lookupOutput (encodeWord value ++ base .lookupOutput) := rfl

@[simp] theorem frame1_eq (r : RowData v d) (output : List Bool) :
    frame1 r output = frameContents r output [] [] []
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame1, frame0, frameContents, initializedTapes, encodeWord]

@[simp] theorem frame2_eq (r : RowData v d) (output : List Bool) :
    frame2 r output = frameContents r output (encodeWord r.query1).reverse [] []
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame2, frameContents, emittedWord]

@[simp] theorem frame3_eq (r : RowData v d) (output : List Bool) :
    frame3 r output = frameContents r output [] (encodeWord r.query1) []
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame3, frameContents, Reduction.MachineTransfer.tapesAt]

@[simp] theorem frame4_eq (r : RowData v d) (output : List Bool) :
    frame4 r output = frameContents r output [] (encodeWord 0) (encodeWord r.firstValue)
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame4, lookupResultTapes_id_eq, frameContents]

@[simp] theorem frame5_eq (r : RowData v d) (output : List Bool) :
    frame5 r output = frameContents r output [] (encodeWord 0) (encodeWord 0)
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame5, divisionOutput, MachineFixedDivMod.unaryTapes,
    MachineFixedDivMod.tapes, MachineCopy.forkTapes, frameContents]

@[simp] theorem frame6_eq (r : RowData v d) (output : List Bool) :
    frame6 r output = frameContents r output [] [] [] (encodeWord r.firstVertex.val)
      (encodeWord 0) (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame6, clearedQueryTapes, frameContents, encodeWord]

@[simp] theorem frame7_eq (r : RowData v d) (output : List Bool) :
    frame7 r output = frameContents r output (encodeWord r.query2).reverse [] []
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame7, emittedWord, frameContents]

@[simp] theorem frame8_eq (r : RowData v d) (output : List Bool) :
    frame8 r output = frameContents r output [] (encodeWord r.query2) []
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame8, Reduction.MachineTransfer.tapesAt, frameContents]

@[simp] theorem frame9_eq (r : RowData v d) (output : List Bool) :
    frame9 r output = frameContents r output [] (encodeWord 0) (encodeWord r.secondValue)
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame9, lookupResultTapes_id_eq, frameContents]

@[simp] theorem frame10_eq (r : RowData v d) (output : List Bool) :
    frame10 r output = frameContents r output [] (encodeWord 0) (encodeWord 0)
      (encodeWord r.firstVertex.val) (encodeWord r.secondVertex.val)
      (encodeWord r.firstReturn.val) (encodeWord r.secondReturn.val) := by
  funext tape
  cases tape <;> simp [frame10, divisionOutput, MachineFixedDivMod.unaryTapes,
    MachineFixedDivMod.tapes, MachineCopy.forkTapes, frameContents]

@[simp] theorem frame11_eq (r : RowData v d) (output : List Bool) :
    frame11 r output = frameContents r ((encodeWord r.finalValue).reverse ++ output)
      [] (encodeWord 0) (encodeWord 0)
      (encodeWord r.firstVertex.val) (encodeWord r.secondVertex.val)
      (encodeWord r.firstReturn.val) (encodeWord r.secondReturn.val) := by
  funext tape
  cases tape <;> simp [frame11, emittedWord, frameContents]

@[simp] theorem frame12_eq (r : RowData v d) (output : List Bool) :
    frame12 r output = frameContents r ((encodeWord r.finalValue).reverse ++ output)
      [] [] [] [] [] [] [] := by
  funext tape
  cases tape <;> simp [frame12, cleanupTapes, frameContents]

theorem final_frame_eq (r : RowData v d) (output : List Bool) :
    frame12 r output = emittedWord .output (frame0 r output) r.finalValue := by
  funext tape
  cases tape <;> simp [frame0, emittedWord, frameContents]

@[simp] theorem frame0_inputVertex (r : RowData v d) (output : List Bool) :
    frame0 r output .inputVertex = encodeWord r.inputVertex.val := rfl

@[simp] theorem frame0_table (r : RowData v d) (output : List Bool) :
    frame0 r output .table = encodeWords (rotationWords r.oldTable) := rfl

@[simp] theorem frame0_output (r : RowData v d) (output : List Bool) :
    frame0 r output .output = output := rfl

@[simp] theorem frame1_inputVertex (r : RowData v d) (output : List Bool) :
    frame1 r output .inputVertex = encodeWord r.inputVertex.val := by simp [frameContents]

@[simp] theorem frame1_emitScratch (r : RowData v d) (output : List Bool) :
    frame1 r output .emitScratch = [] := by simp [frameContents]

@[simp] theorem frame2_queryReverse (r : RowData v d) (output : List Bool) :
    frame2 r output .queryReverse = (encodeWord r.query1).reverse := by simp [frameContents]

@[simp] theorem frame2_queryIndex (r : RowData v d) (output : List Bool) :
    frame2 r output .queryIndex = [] := by simp [frameContents]

@[simp] theorem frame3_table (r : RowData v d) (output : List Bool) :
    frame3 r output .table = encodeWords (rotationWords r.oldTable) := by simp [frameContents]

@[simp] theorem frame3_lookupRestore (r : RowData v d) (output : List Bool) :
    frame3 r output .lookupRestore = [] := by simp [frameContents]

@[simp] theorem frame3_lookupWork (r : RowData v d) (output : List Bool) :
    frame3 r output .lookupWork = [] := by simp [frameContents]

@[simp] theorem frame3_lookupOutput (r : RowData v d) (output : List Bool) :
    frame3 r output .lookupOutput = [] := by simp [frameContents]

@[simp] theorem frame3_queryIndex (r : RowData v d) (output : List Bool) :
    frame3 r output .queryIndex = encodeWord r.query1 := by simp [frameContents]

@[simp] theorem frame4_lookupOutput (r : RowData v d) (output : List Bool) :
    frame4 r output .lookupOutput = encodeWord r.firstValue := by simp [frameContents]

@[simp] theorem frame4_quotientFirst (r : RowData v d) (output : List Bool) :
    frame4 r output .quotientFirst = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame4_remainderFirst (r : RowData v d) (output : List Bool) :
    frame4 r output .remainderFirst = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame6_quotientFirst (r : RowData v d) (output : List Bool) :
    frame6 r output .quotientFirst = encodeWord r.firstVertex.val := by simp [frameContents]

@[simp] theorem frame6_emitScratch (r : RowData v d) (output : List Bool) :
    frame6 r output .emitScratch = [] := by simp [frameContents]

@[simp] theorem frame7_queryReverse (r : RowData v d) (output : List Bool) :
    frame7 r output .queryReverse = (encodeWord r.query2).reverse := by simp [frameContents]

@[simp] theorem frame7_queryIndex (r : RowData v d) (output : List Bool) :
    frame7 r output .queryIndex = [] := by simp [frameContents]

@[simp] theorem frame8_table (r : RowData v d) (output : List Bool) :
    frame8 r output .table = encodeWords (rotationWords r.oldTable) := by simp [frameContents]

@[simp] theorem frame8_lookupRestore (r : RowData v d) (output : List Bool) :
    frame8 r output .lookupRestore = [] := by simp [frameContents]

@[simp] theorem frame8_lookupWork (r : RowData v d) (output : List Bool) :
    frame8 r output .lookupWork = [] := by simp [frameContents]

@[simp] theorem frame8_lookupOutput (r : RowData v d) (output : List Bool) :
    frame8 r output .lookupOutput = [] := by simp [frameContents]

@[simp] theorem frame8_queryIndex (r : RowData v d) (output : List Bool) :
    frame8 r output .queryIndex = encodeWord r.query2 := by simp [frameContents]

@[simp] theorem frame9_lookupOutput (r : RowData v d) (output : List Bool) :
    frame9 r output .lookupOutput = encodeWord r.secondValue := by simp [frameContents]

@[simp] theorem frame9_quotientSecond (r : RowData v d) (output : List Bool) :
    frame9 r output .quotientSecond = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame9_remainderSecond (r : RowData v d) (output : List Bool) :
    frame9 r output .remainderSecond = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame10_quotientSecond (r : RowData v d) (output : List Bool) :
    frame10 r output .quotientSecond = encodeWord r.secondVertex.val := by simp [frameContents]

@[simp] theorem frame10_emitScratch (r : RowData v d) (output : List Bool) :
    frame10 r output .emitScratch = [] := by simp [frameContents]

@[simp] theorem frame10_output (r : RowData v d) (output : List Bool) :
    frame10 r output .output = output := by simp [frameContents]

theorem final_dirty_word_length_le (r : RowData v d) (output : List Bool) (i : Fin 6) :
    ((frame11 r output) (dirtyTape i)).length ≤ r.tableLength + 1 := by
  have h1 := r.firstVertex_le_tableLength
  have h2 := r.secondVertex_le_tableLength
  have h3 := r.firstReturn_le_tableLength
  have h4 := r.secondReturn_le_tableLength
  fin_cases i <;> simp only [frame11_eq, dirtyTape, frameContents, encodeWord_length] <;> omega

theorem cleanupSteps_le_tableLength (r : RowData v d) (output : List Bool) :
    cleanupSteps id (frame11 r output) ≤ 6 * (r.tableLength + 1) + 6 := by
  have h1 := r.firstVertex_le_tableLength
  have h2 := r.secondVertex_le_tableLength
  have h3 := r.firstReturn_le_tableLength
  have h4 := r.secondReturn_le_tableLength
  simp only [cleanupSteps, frame11_eq, id_eq, frameContents, encodeWord_length]
  omega

end

open Turing MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords PCP.AlphabetTable

private theorem concatenate {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]

def rowSteps {v d : Nat} (r : RowData v d) (output : List Bool) : Nat :=
  1 + emitSteps r.inputVertex.val + (r.query1 + 2) +
    MachinePreservingLookupClean.steps (rotationWords r.oldTable) r.firstRow.val +
    (r.firstValue + 2) + 1 + emitSteps r.firstVertex.val + (r.query2 + 2) +
    MachinePreservingLookupClean.steps (rotationWords r.oldTable) r.secondRow.val +
    (r.secondValue + 2) + emitSteps r.secondVertex.val +
    (cleanupSteps id (frame11 r output) + 1)

section Execution

variable {v d : Nat} {ρ Λ : Type} [Fintype ρ]
    (positive : 0 < d) (r : RowData v d) (output : List Bool) (ambient : ρ)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : Tape => Bool) Λ (State ρ d))
    (code : ∀ l, target (labels l) = statement positive r.smallTable id labels exit l)

include code

theorem rowTraceAt :
    (advance (TM2.step target))^[rowSteps r output]
      (some ⟨some (labels .initialize), divisionState positive ambient r.control0 none,
        frame0 r output⟩) =
      some ⟨exit, divisionState positive ambient r.control2 none,
        emittedWord .output (frame0 r output) r.finalValue⟩ := by
  let zero := MachineFixedDivMod.residue (degree d) (Nat.mul_pos positive positive) 0
  have h0 : (advance (TM2.step target))^[1]
      (some ⟨some (labels .initialize), divisionState positive ambient r.control0 none,
        frame0 r output⟩) =
      some ⟨some (labels (.firstEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control0 none, frame1 r output⟩ := by
    simpa only [Function.iterate_one, advance_some, divisionState, frame1] using
      initializeStepAt positive r.smallTable id Function.injective_id labels exit target code
        (frame0 r output) (divisionState positive ambient r.control0 none)
  have h1 := firstEmitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code r.inputVertex.val (frame1 r output)
    (by simp) (by simp) ((ambient,r.control0),zero)
  change (advance (TM2.step target))^[emitSteps r.inputVertex.val]
      (some ⟨some (labels (.firstEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control0 none, frame1 r output⟩) =
      some ⟨some (labels .firstReverse), divisionState positive ambient r.control0 none,
        frame2 r output⟩ at h1
  have h2 := reversePhaseTrace_unary false positive r.smallTable id Function.injective_id
    labels exit target code (frame2 r output) (((ambient,r.control0),zero),()) none
    r.query1 [] (by simp) (by simp)
  simp only [List.append_nil] at h2
  change (advance (TM2.step target))^[r.query1 + 2]
      (some ⟨some (labels .firstReverse), divisionState positive ambient r.control0 none,
        frame2 r output⟩) =
      some ⟨some (labels (.firstLookup (.run .copyFirst))),
        divisionState positive ambient r.control0 none, frame3 r output⟩ at h2
  have h3 := lookupPhaseTrace false positive r.smallTable id Function.injective_id
    labels exit target code (frame3 r output) r.oldTable
    (by simp) (by simp)
    (r.inputVertex, firstOffset r.control0) []
    (by simpa [frameContents, RowData.firstRow] using
      congrArg encodeWord r.query1_eq_firstRow)
    (by simp) (((ambient,r.control0),zero),()) none
  change (advance (TM2.step target))^[MachinePreservingLookupClean.steps
      (rotationWords r.oldTable) r.firstRow.val]
      (some ⟨some (labels (.firstLookup (.run .copyFirst))),
        divisionState positive ambient r.control0 none, frame3 r output⟩) =
      some ⟨some (labels .firstScan), divisionState positive ambient r.control0 none,
        frame4 r output⟩ at h3
  have h4 := firstDivisionTrace positive r.smallTable id Function.injective_id labels exit
    target code (frame4 r output) r.firstValue [] [] []
    (by simp) (by simp) (by simp)
    ambient r.control0 none
  rw [← r.firstReturn_eq_residue positive] at h4
  change (advance (TM2.step target))^[r.firstValue + 2]
      (some ⟨some (labels .firstScan), divisionState positive ambient r.control0 none,
        frame4 r output⟩) =
      some ⟨some (labels .clearQuery), divisionState positive ambient r.control1 none,
        frame5 r output⟩ at h4
  have h5 : (advance (TM2.step target))^[1]
      (some ⟨some (labels .clearQuery), divisionState positive ambient r.control1 none,
        frame5 r output⟩) =
      some ⟨some (labels (.secondEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control1 none, frame6 r output⟩ := by
    simpa only [Function.iterate_one, advance_some, frame6] using
      clearQueryStepAt positive r.smallTable id Function.injective_id labels exit target code
        (frame5 r output) (divisionState positive ambient r.control1 none)
  have h6 := secondEmitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code r.firstVertex.val (frame6 r output)
    (by simp) (by simp) ((ambient,r.control1),zero)
  change (advance (TM2.step target))^[emitSteps r.firstVertex.val]
      (some ⟨some (labels (.secondEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control1 none, frame6 r output⟩) =
      some ⟨some (labels .secondReverse), divisionState positive ambient r.control1 none,
        frame7 r output⟩ at h6
  have h7 := reversePhaseTrace_unary true positive r.smallTable id Function.injective_id
    labels exit target code (frame7 r output) (((ambient,r.control1),zero),()) none
    r.query2 [] (by simp) (by simp)
  simp only [List.append_nil] at h7
  change (advance (TM2.step target))^[r.query2 + 2]
      (some ⟨some (labels .secondReverse), divisionState positive ambient r.control1 none,
        frame7 r output⟩) =
      some ⟨some (labels (.secondLookup (.run .copyFirst))),
        divisionState positive ambient r.control1 none, frame8 r output⟩ at h7
  have h8 := lookupPhaseTrace true positive r.smallTable id Function.injective_id
    labels exit target code (frame8 r output) r.oldTable
    (by simp) (by simp)
    (r.firstVertex, secondOffset r.control1) []
    (by simpa [frameContents, RowData.secondRow] using
      congrArg encodeWord r.query2_eq_secondRow)
    (by simp) (((ambient,r.control1),zero),()) none
  change (advance (TM2.step target))^[MachinePreservingLookupClean.steps
      (rotationWords r.oldTable) r.secondRow.val]
      (some ⟨some (labels (.secondLookup (.run .copyFirst))),
        divisionState positive ambient r.control1 none, frame8 r output⟩) =
      some ⟨some (labels .secondScan), divisionState positive ambient r.control1 none,
        frame9 r output⟩ at h8
  have h9 := secondDivisionTrace positive r.smallTable id Function.injective_id labels exit
    target code (frame9 r output) r.secondValue [] [] []
    (by simp) (by simp) (by simp)
    ambient r.control1 none
  rw [← r.secondReturn_eq_residue positive] at h9
  change (advance (TM2.step target))^[r.secondValue + 2]
      (some ⟨some (labels .secondScan), divisionState positive ambient r.control1 none,
        frame9 r output⟩) =
      some ⟨some (labels (.outputEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control2 none, frame10 r output⟩ at h9
  have h10 := outputEmitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code r.secondVertex.val (frame10 r output)
    (by simp) (by simp) ((ambient,r.control2),zero)
  change (advance (TM2.step target))^[emitSteps r.secondVertex.val]
      (some ⟨some (labels (.outputEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control2 none, frame10 r output⟩) =
      some ⟨some (labels (.cleanup 0)), divisionState positive ambient r.control2 none,
        frame11 r output⟩ at h10
  have h11 := cleanupExitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code (frame11 r output) (divisionState positive ambient r.control2 none)
  change (advance (TM2.step target))^[cleanupSteps id (frame11 r output) + 1]
      (some ⟨some (labels (.cleanup 0)), divisionState positive ambient r.control2 none,
        frame11 r output⟩) =
      some ⟨exit, divisionState positive ambient r.control2 none, frame12 r output⟩ at h11
  rw [final_frame_eq] at h11
  exact concatenate (concatenate (concatenate (concatenate (concatenate
    (concatenate (concatenate (concatenate (concatenate (concatenate
      (concatenate h0 h1) h2) h3) h4) h5) h6) h7) h8) h9) h10) h11

end Execution

theorem rowSteps_le {v d : Nat} (r : RowData v d) (output : List Bool) :
    rowSteps r output ≤ 80 * (r.tableLength + 1) := by
  have lookup1 := MachinePreservingLookupClean.steps_le (rotationWords r.oldTable)
    r.firstRow.val r.firstValue (rotationWords_getElem? r.oldTable r.firstRow)
  have lookup2 := MachinePreservingLookupClean.steps_le (rotationWords r.oldTable)
    r.secondRow.val r.secondValue (rotationWords_getElem? r.oldTable r.secondRow)
  have h0 := r.inputVertex_le_tableLength
  have h1 := r.query1_le_tableLength
  have h2 := r.firstValue_le_tableLength
  have h3 := r.firstVertex_le_tableLength
  have h4 := r.query2_le_tableLength
  have h5 := r.secondValue_le_tableLength
  have h6 := r.secondVertex_le_tableLength
  have clean := cleanupSteps_le_tableLength r output
  change _ ≤ 6 * r.tableLength + 4 at lookup1 lookup2
  simp only [rowSteps, emitSteps]
  omega

def rowInTimeAt {v d : Nat} {ρ Λ : Type} [Fintype ρ]
    (positive : 0 < d) (r : RowData v d) (output : List Bool) (ambient : ρ)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : Tape => Bool) Λ (State ρ d))
    (code : ∀ l, target (labels l) = statement positive r.smallTable id labels exit l) :
    StateTransition.EvalsToInTime (TM2.step target)
      ⟨some (labels .initialize), divisionState positive ambient r.control0 none,
        frame0 r output⟩
      (some ⟨exit, divisionState positive ambient r.control2 none,
        emittedWord .output (frame0 r output) r.finalValue⟩)
      (80 * (r.tableLength + 1)) where
  steps := rowSteps r output
  evals_in_steps := rowTraceAt positive r output ambient labels exit target code
  steps_le_m := rowSteps_le r output

def rowInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (r : RowData v d) (output : List Bool) (ambient : ρ) :
    StateTransition.EvalsToInTime (TM2.step (program positive r.smallTable))
      ⟨some .initialize, divisionState positive ambient r.control0 none, frame0 r output⟩
      (some ⟨none, divisionState positive ambient r.control2 none,
        emittedWord .output (frame0 r output) r.finalValue⟩)
      (80 * (r.tableLength + 1)) :=
  rowInTimeAt positive r output ambient id none (program positive r.smallTable) (fun _ => rfl)

end BinPackingGames.Foundations.Complexity.MachineExpanderRow

namespace BinPackingGames.Foundations.Complexity.MachineExpanderTable

section

open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords
open PCP.ExpanderTableEnumeration

private def singleStep {α : Type*} (step : α → Option α) (a b : α)
    (h : step a = some b) : StateTransition.EvalsToInTime step a (some b) 1 where
  steps := 1
  evals_in_steps := by change step a = some b; exact h
  steps_le_m := Nat.le_refl _

private def sequence {α : Type*} {step : α → Option α} {a b c : α} {m n : Nat}
    (first : StateTransition.EvalsToInTime step a (some b) m)
    (second : StateTransition.EvalsToInTime step b (some c) n) :
    StateTransition.EvalsToInTime step a (some c) (m + n) := by
  simpa only [Nat.add_comm n m] using
    StateTransition.EvalsToInTime.trans step m n a b (some c) first second

private def widen {α : Type*} {step : α → Option α} {a : α} {b : Option α} {m n : Nat}
    (run : StateTransition.EvalsToInTime step a b m) (bound : m ≤ n) :
    StateTransition.EvalsToInTime step a b n where
  toEvalsTo := run.toEvalsTo
  steps_le_m := run.steps_le_m.trans bound

def tableRowData {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (vertex : Fin v) (position : Position d) : MachineExpanderRow.RowData v d :=
  MachineExpanderRow.rowData G H vertex (positionPair position).1 (positionPair position).2

theorem tableRowData_finalValue {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (position : Position d) :
    (tableRowData G H vertex position).finalValue = rowValue G H vertex position := rfl

theorem tableRowData_frame0 {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (position : Position d) (output : List Bool) :
    MachineExpanderRow.frame0 (tableRowData G H vertex position) output =
      rowTapes vertex.val (encodeWords (rotationWords G)) output := by
  funext tape
  cases tape <;> rfl

theorem tableRowData_finalFrame {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (position : Position d) (output : List Bool) :
    MachineExpanderRow.emittedWord .output
      (MachineExpanderRow.frame0 (tableRowData G H vertex position) output)
      (tableRowData G H vertex position).finalValue =
      rowTapes vertex.val (encodeWords (rotationWords G))
        ((encodeWord (rowValue G H vertex position)).reverse ++ output) := by
  rw [tableRowData_frame0, tableRowData_finalValue]
  funext tape
  cases tape <;> simp [MachineExpanderRow.emittedWord, rowTapes]

variable {ρ : Type} [Fintype ρ] {v d : Nat}

def rowAfterState (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d) : State ρ d :=
  (MachineExpanderRow.divisionState positive (caller state)
    (tableRowData G H vertex state.2).control2 none, state.2)

omit [Fintype ρ] in
@[simp] theorem rowAfterState_position (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d) :
    (rowAfterState positive G H vertex state).2 = state.2 := rfl

omit [Fintype ρ] in
@[simp] theorem caller_rowAfterState (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d) :
    caller (rowAfterState positive G H vertex state) = caller state := rfl

def preparedRowInTime (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d)
    (remaining : Nat) (output countSuffix result : List Bool) :
    StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inr .prepareRow), state,
        boundaryTapes vertex.val remaining (encodeWords (rotationWords G)) output countSuffix result⟩
      (some ⟨some (.inr .afterRow), rowAfterState positive G H vertex state,
        boundaryTapes vertex.val remaining (encodeWords (rotationWords G))
          ((encodeWord (rowValue G H vertex state.2)).reverse ++ output) countSuffix result⟩)
      (1 + 80 * ((encodeWords (rotationWords G)).length + 1)) := by
  have raw := rowExecution positive H state.2 (extraTapes remaining countSuffix result)
    (MachineExpanderRow.rowInTime positive (tableRowData G H vertex state.2) output (caller state))
  have run : StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inl .initialize), prepareState positive H state,
        boundaryTapes vertex.val remaining (encodeWords (rotationWords G)) output countSuffix result⟩
      (some ⟨some (.inr .afterRow), rowAfterState positive G H vertex state,
        boundaryTapes vertex.val remaining (encodeWords (rotationWords G))
          ((encodeWord (rowValue G H vertex state.2)).reverse ++ output) countSuffix result⟩)
      (80 * ((encodeWords (rotationWords G)).length + 1)) := by
    rw [tableRowData_finalFrame, tableRowData_frame0] at raw
    simpa only [MachineEmbedding.configuration, MachineEmbedding.label, rowReturn,
      prepareState, boundaryState, rowAfterState, boundaryTapes,
      MachineExpanderRow.RowData.tableLength, MachineExpanderRow.RowData.control0,
      tableRowData, MachineExpanderRow.rowData] using raw
  exact sequence
    (singleStep _ _ _ (prepareRowStep positive H
      (boundaryTapes vertex.val remaining (encodeWords (rotationWords G)) output countSuffix result)
      state)) run

structure VertexSuffixRun (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d)
    (remaining : Nat) (output countSuffix result : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  position_zero : finalState.2 = zeroPosition positive
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H))
    ⟨some (.inr .prepareRow), state,
      boundaryTapes vertex.val remaining (encodeWords (rotationWords G)) output countSuffix result⟩
    (some ⟨some (.inr .vertexGuard), finalState,
      boundaryTapes (vertex.val + 1) remaining (encodeWords (rotationWords G))
        (accumulate ((vertexWords G H vertex).drop state.2.val) output) countSuffix result⟩)
    ((rowFactor d - state.2.val) * (80 * ((encodeWords (rotationWords G)).length + 1) + 2))

private def suffixRun_aux (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (remaining : Nat)
    (countSuffix result : List Bool) (n : Nat) :
    (state : State ρ d) → state.2.val + n = rowFactor d → (output : List Bool) →
      VertexSuffixRun positive G H vertex state remaining output countSuffix result := by
  induction n with
  | zero =>
      intro state count output
      have hp := state.2.isLt
      exact False.elim (by omega)
  | succ n ih =>
      intro state count output
      let after := rowAfterState positive G H vertex state
      let nextOutput := (encodeWord (rowValue G H vertex state.2)).reverse ++ output
      have row := preparedRowInTime positive G H vertex state remaining output countSuffix result
      by_cases last : n = 0
      · have hp : state.2.val + 1 = rowFactor d := by omega
        have finish := singleStep _ _ _ (afterRow_last_boundaryStep positive H vertex.val remaining
          (encodeWords (rotationWords G)) nextOutput countSuffix result after (by
            change ¬ state.2.val + 1 < rowFactor d
            omega))
        refine ⟨resetPositionState positive after, ?_, rfl, ?_⟩
        · rfl
        · have all := sequence row finish
          have words : accumulate ((vertexWords G H vertex).drop state.2.val) output =
              nextOutput := by
            rw [vertexWords_drop_last G H vertex state.2 hp]
            rfl
          have budget : rowFactor d - state.2.val = 1 := by omega
          rw [words, budget, Nat.one_mul]
          exact widen all (by omega)
      · have more : state.2.val + 1 < rowFactor d := by omega
        let next := advancePositionState positive after
        have nextPositionValue : next.2.val = state.2.val + 1 :=
          nextPosition_val_of_lt positive state.2 more
        have nextCount : next.2.val + n = rowFactor d := by omega
        let tail := ih next nextCount nextOutput
        have advance := singleStep _ _ _ (afterRow_moreStep positive H
          (boundaryTapes vertex.val remaining (encodeWords (rotationWords G))
            nextOutput countSuffix result) after more)
        have all := sequence (sequence row advance) tail.execution
        refine ⟨tail.finalState, ?_, tail.position_zero, ?_⟩
        · exact tail.caller_preserved
        · have words : accumulate ((vertexWords G H vertex).drop state.2.val) output =
              accumulate ((vertexWords G H vertex).drop next.2.val) nextOutput := by
            rw [vertexWords_drop_succ G H vertex state.2, nextPositionValue]
            rfl
          rw [words]
          apply widen all
          have hcount : rowFactor d - state.2.val = n + 1 := by omega
          have hnext : rowFactor d - next.2.val = n := by omega
          rw [hcount, hnext, Nat.add_mul, Nat.one_mul]
          omega

def suffixInTime (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d)
    (remaining : Nat) (output countSuffix result : List Bool) :
    VertexSuffixRun positive G H vertex state remaining output countSuffix result :=
  suffixRun_aux positive G H vertex remaining countSuffix result
    (rowFactor d - state.2.val) state (by have h := state.2.isLt; omega) output

structure VertexRun (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d)
    (remaining : Nat) (output countSuffix result : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  position_zero : finalState.2 = zeroPosition positive
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H))
    ⟨some (.inr .prepareRow), state,
      boundaryTapes vertex.val remaining (encodeWords (rotationWords G)) output countSuffix result⟩
    (some ⟨some (.inr .vertexGuard), finalState,
      boundaryTapes (vertex.val + 1) remaining (encodeWords (rotationWords G))
        (accumulate (vertexWords G H vertex) output) countSuffix result⟩)
    (rowFactor d * (80 * ((encodeWords (rotationWords G)).length + 1) + 2))

def vertexInTime (positive : 0 < d) (G : Table v (degree d))
    (H : Table (cloudSize d) d) (vertex : Fin v) (state : State ρ d)
    (position : state.2 = zeroPosition positive)
    (remaining : Nat) (output countSuffix result : List Bool) :
    VertexRun positive G H vertex state remaining output countSuffix result := by
  let run := suffixInTime positive G H vertex state remaining output countSuffix result
  refine ⟨run.finalState, run.caller_preserved, run.position_zero, ?_⟩
  have zero : state.2.val = 0 := congrArg Fin.val position
  simpa only [zero, List.drop_zero, Nat.sub_zero] using run.execution

end

open Turing MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords
open PCP.ExpanderTableEnumeration

def oldTableWord {v d : Nat} (G : Table v (degree d)) : List Bool :=
  encodeWords (rotationWords G)

def accumulatedVertices {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (current : Nat) : List Bool :=
  accumulate (priorVertices G H current) []

@[simp] theorem accumulatedVertices_zero {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) : accumulatedVertices G H 0 = [] := by
  simp [accumulatedVertices, accumulate]

theorem accumulatedVertices_succ {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (current : Nat) (valid : current < v) :
    accumulatedVertices G H (current + 1) =
      accumulate (vertexWords G H ⟨current, valid⟩) (accumulatedVertices G H current) := by
  simp only [accumulatedVertices, priorVertices_succ G H current valid, accumulate_append]

theorem accumulatedVertices_full {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) : accumulatedVertices G H v =
      (encodeWords (rotationWords (step G H))).reverse := by
  simp [accumulatedVertices, accumulate_generatedWords]

def vertexBodyBudget {v d : Nat} (G : Table v (degree d)) : Nat :=
  rowFactor d * (80 * ((oldTableWord G).length + 1) + 2)

def vertexLoopBudget {v d : Nat} (G : Table v (degree d)) (remaining : Nat) : Nat :=
  remaining * (vertexBodyBudget G + 1) + 1

def vertexLoopTapes {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (current remaining : Nat) (suffix : List Bool) : ∀ tape, List (Alphabet tape) :=
  boundaryTapes current remaining (oldTableWord G) (accumulatedVertices G H current) suffix []

structure VertexLoopRun {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (current remaining : Nat) (state : State ρ d) (suffix : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  position_zero : finalState.2 = zeroPosition positive
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H))
    ⟨some (.inr .vertexGuard), state, vertexLoopTapes G H current remaining suffix⟩
    (some ⟨some (.inr .reverseOutput), finalState, vertexLoopTapes G H v 0 suffix⟩)
    (vertexLoopBudget G remaining)

private def boundarySingleStep {S : Type*} (f : S → Option S) (a b : S)
    (h : f a = some b) : StateTransition.EvalsToInTime f a (some b) 1 where
  steps := 1
  evals_in_steps := by change f a = some b; exact h
  steps_le_m := Nat.le_refl _

def vertexLoopInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (current remaining : Nat) (state : State ρ d) (suffix : List Bool)
    (count : current + remaining = v) (position : state.2 = zeroPosition positive) :
    VertexLoopRun positive G H current remaining state suffix := by
  induction remaining generalizing current state with
  | zero =>
      have hc : current = v := by omega
      subst current
      refine ⟨clearRegister state, caller_clearRegister state, ?_, ?_⟩
      · simpa only [clearRegister] using position
      · simpa only [vertexLoopBudget, Nat.zero_mul, Nat.zero_add, vertexLoopTapes] using
          boundarySingleStep _ _ _ (vertexGuard_boundary_zeroStep positive H v (oldTableWord G)
            (accumulatedVertices G H v) suffix [] state)
  | succ remaining ih =>
      have valid : current < v := by omega
      let vertex : Fin v := ⟨current, valid⟩
      let body := vertexInTime positive G H vertex (clearRegister state)
        (by simpa only [clearRegister] using position) remaining
        (accumulatedVertices G H current) suffix []
      have nextCount : current + 1 + remaining = v := by omega
      let rest := ih (current + 1) body.finalState nextCount body.position_zero
      have output_eq : accumulate (vertexWords G H vertex)
          (accumulatedVertices G H current) = accumulatedVertices G H (current + 1) :=
        (accumulatedVertices_succ G H current valid).symm
      let guard := boundarySingleStep _ _ _
        (vertexGuard_boundary_succStep positive H current remaining (oldTableWord G)
          (accumulatedVertices G H current) suffix [] state)
      have bodyRun : StateTransition.EvalsToInTime (TM2.step (program positive H))
          ⟨some (.inr .prepareRow), clearRegister state,
            boundaryTapes current remaining (oldTableWord G)
              (accumulatedVertices G H current) suffix []⟩
          (some ⟨some (.inr .vertexGuard), body.finalState,
            vertexLoopTapes G H (current + 1) remaining suffix⟩)
          (vertexBodyBudget G) := by
        simpa only [vertexLoopTapes, vertexBodyBudget, oldTableWord, output_eq] using body.execution
      let first := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
        _ _ _ _ _ guard bodyRun
      let all := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
        _ _ _ _ _ first rest.execution
      refine ⟨rest.finalState, ?_, rest.position_zero, ?_⟩
      · exact rest.caller_preserved.trans
          (body.caller_preserved.trans (caller_clearRegister state))
      · refine { toEvalsTo := all.toEvalsTo, steps_le_m := ?_ }
        have bound := all.steps_le_m
        simp only [vertexLoopBudget, Nat.succ_mul] at bound ⊢
        omega

def completeBudget {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d) : Nat :=
  1 + vertexLoopBudget G v + ((encodeWords (rotationWords (step G H))).length + 2)

def timeCoefficient (d : Nat) : Nat :=
  rowFactor d * (rowFactor d + 1) + 82 * rowFactor d + 5

noncomputable def timePolynomial (d : Nat) : Polynomial Nat :=
  Polynomial.C (timeCoefficient d) * (Polynomial.X + 1) ^ 2

theorem vertices_le_word_length {v d : Nat} (positive : 0 < d)
    (G : Table v (degree d)) : v ≤ (oldTableWord G).length := by
  have hq : 1 ≤ degree d := Nat.mul_pos positive positive
  have hv : v ≤ v * degree d := by simpa using Nat.mul_le_mul_left v hq
  have hr : v * degree d ≤ (oldTableWord G).length := by
    simp only [oldTableWord, encodeWords_length, rotationWords_length]
    omega
  exact hv.trans hr

theorem completeBudget_le {v d : Nat} (positive : 0 < d)
    (G : Table v (degree d)) (H : Table (cloudSize d) d) :
    completeBudget G H ≤ (timePolynomial d).eval (oldTableWord G).length := by
  let L := (oldTableWord G).length
  let M := rowFactor d
  have vBound : v ≤ L + 1 := (vertices_le_word_length positive G).trans (Nat.le_succ L)
  have emitBound : 80 * (L + 1) + 2 ≤ 82 * (L + 1) := by omega
  have bodyBound : vertexBodyBudget G + 1 ≤ (82 * M + 1) * (L + 1) := by
    have mulBound := Nat.mul_le_mul_left M emitBound
    change M * (80 * (L + 1) + 2) + 1 ≤ _
    nlinarith
  have iterations := Nat.mul_le_mul vBound bodyBound
  have loopBound : v * (vertexBodyBudget G + 1) ≤ (82 * M + 1) * (L + 1)^2 := by
    calc
      _ ≤ (L + 1) * ((82 * M + 1) * (L + 1)) := iterations
      _ = _ := by ring
  have rowBound : v * M ≤ M * (L + 1) := by
    simpa only [Nat.mul_comm v M] using Nat.mul_le_mul_left M vBound
  have rowSuccBound : v * M + 1 ≤ (M + 1) * (L + 1) := by nlinarith
  have productBound := Nat.mul_le_mul rowBound rowSuccBound
  have lengthBound : (encodeWords (rotationWords (step G H))).length ≤
      M * (M + 1) * (L + 1)^2 := by
    calc
      _ ≤ (v * M) * (v * M + 1) := by
        simpa only [generatedWords_eq_rotationWords] using encode_generatedWords_length_le G H
      _ ≤ (M * (L + 1)) * ((M + 1) * (L + 1)) := productBound
      _ = _ := by ring
  have oneBound : 1 ≤ (L + 1)^2 := by nlinarith
  have combined := Nat.add_le_add loopBound lengthBound
  simp only [timePolynomial, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_pow,
    Polynomial.eval_add, Polynomial.eval_X, Polynomial.eval_one]
  change completeBudget G H ≤ (M * (M + 1) + 82 * M + 5) * (L + 1)^2
  unfold completeBudget vertexLoopBudget
  nlinarith

structure TableRun {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (state : State ρ d) (suffix : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  position_zero : finalState.2 = zeroPosition positive
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H))
    ⟨some (.inr .initialize), state, initialTapes v (oldTableWord G) suffix⟩
    (some ⟨none, finalState,
      finalTapes v (oldTableWord G) (encodeWords (rotationWords (step G H))) suffix⟩)
    ((timePolynomial d).eval (oldTableWord G).length)

private def initializeInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (state : State ρ d) (suffix : List Bool) :
    StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inr .initialize), state, initialTapes v (oldTableWord G) suffix⟩
      (some ⟨some (.inr .vertexGuard), initialState positive H (caller state),
        vertexLoopTapes G H 0 v suffix⟩) 1 where
  steps := 1
  evals_in_steps := by
    change TM2.step (program (ρ := ρ) positive H)
      ⟨some (.inr .initialize), state, initialTapes v (oldTableWord G) suffix⟩ =
      some ⟨some (.inr .vertexGuard), initialState positive H (caller state),
        vertexLoopTapes G H 0 v suffix⟩
    simpa only [vertexLoopTapes, accumulatedVertices_zero] using
      initialize_boundaryStep positive H v (oldTableWord G) suffix state
  steps_le_m := Nat.le_refl _

def tableInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (state : State ρ d) (suffix : List Bool) : TableRun positive G H state suffix := by
  let loop := vertexLoopInTime positive G H 0 v
    (initialState positive H (caller state)) suffix (by omega) rfl
  let initialRun := initializeInTime positive G H state suffix
  have finish : StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inr .reverseOutput), loop.finalState, vertexLoopTapes G H v 0 suffix⟩
      (some ⟨none, clearRegister loop.finalState,
        finalTapes v (oldTableWord G) (encodeWords (rotationWords (step G H))) suffix⟩)
      ((encodeWords (rotationWords (step G H))).length + 2) := by
    simpa only [vertexLoopTapes, finalTapes, accumulatedVertices_full, List.reverse_reverse,
      List.append_nil, List.length_reverse] using
      reverseOutputHaltInTime positive H v 0 (oldTableWord G)
        (accumulatedVertices G H v) suffix [] loop.finalState
  let first := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
    _ _ _ _ _ initialRun loop.execution
  let all := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
    _ _ _ _ _ first finish
  refine ⟨clearRegister loop.finalState, ?_, ?_, ?_⟩
  · exact (caller_clearRegister loop.finalState).trans loop.caller_preserved
  · simpa only [clearRegister] using loop.position_zero
  · refine { toEvalsTo := all.toEvalsTo, steps_le_m := ?_ }
    have bound := all.steps_le_m
    have polynomialBound := completeBudget_le positive G H
    unfold completeBudget at polynomialBound
    omega

end BinPackingGames.Foundations.Complexity.MachineExpanderTable

end OAI