Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Encoding.PreprocessingStageMaps
import OAI.Computability.DirectedFeedback.Machines.VertexLabel

namespace OAI

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namespace DFVSGames.Foundations.Complexity.MachineOverlayTable

open Turing MachineComposition
open PCP PCP.GraphTables PCP.PreprocessingOverlayWords
open MachineCloudPadding

variable {A : Type}

abbrev Tape := Fin 5 ⊕ Fin 12
abbrev Ambient (A : Type) (d e : Nat) := ((A × Fin e) × MachineLazyRows.Buffer) × Fin d
abbrev State (A : Type) (d e : Nat) := Ambient A d e × Option Bool

def graphArchive : Tape := .inl 0
def graphStream : Tape := .inl 1
def scratch : Tape := .inl 2
def vertexCount : Tape := .inl 3
def dartCount : Tape := .inl 4
def expanderArchive : Tape := .inr 0
def expanderStream : Tape := .inr 1
def fuel : Tape := .inr 2
def vertex : Tape := .inr 3
def tail : Tape := .inr 4
def oldReverse : Tape := .inr 5
def relation : Tape := .inr 6
def dividedCopy : Tape := .inr 7
def quotient : Tape := .inr 8
def newReverse : Tape := .inr 9
def rowBuffer : Tape := .inr 10
def output : Tape := .inr 11

def oldTapes : Fin 10 → Tape :=
  ![tail, oldReverse, relation, scratch, dividedCopy, quotient, newReverse,
    output, rowBuffer, graphStream]

def expanderTapes : Fin 10 → Tape :=
  ![tail, oldReverse, relation, scratch, dividedCopy, quotient, newReverse,
    output, rowBuffer, expanderStream]

theorem oldTapes_injective : Function.Injective oldTapes := by
  intro i j h
  fin_cases i <;> fin_cases j <;>
    simp_all [oldTapes, tail, oldReverse, relation, scratch, dividedCopy, quotient,
      newReverse, output, rowBuffer, graphStream]

theorem expanderTapes_injective : Function.Injective expanderTapes := by
  intro i j h
  fin_cases i <;> fin_cases j <;>
    simp_all [expanderTapes, tail, oldReverse, relation, scratch, dividedCopy, quotient,
      newReverse, output, rowBuffer, expanderStream]

def clean (d e : Nat) (hd : 0 < d) (he : 0 < e) (a : A) : State A d e :=
  MachineLazyRows.streamClean d hd (a, MachineFixedDivMod.residue e he 0)

def expanderState (A : Type) (d e : Nat) :
    MachineOverlayRows.State ((A × MachineLazyRows.Buffer) × Fin d) e ≃ State A d e where
  toFun
    | ((((a, buffer), rd), re), bit) => ((((a, re), buffer), rd), bit)
  invFun
    | ((((a, re), buffer), rd), bit) => ((((a, buffer), rd), re), bit)
  left_inv := by rintro ⟨⟨⟨⟨a, buffer⟩, rd⟩, re⟩, bit⟩; rfl
  right_inv := by rintro ⟨⟨⟨⟨a, re⟩, buffer⟩, rd⟩, bit⟩; rfl

theorem expanderState_clean (d e : Nat) (hd : 0 < d) (he : 0 < e) (a : A) :
    expanderState A d e (MachinePortReindex.clean e he
      ((a, MachineRegularOriginalRow.zeroBuffer), MachineFixedDivMod.residue d hd 0)) =
      clean d e hd he a := rfl

inductive MainLabel (d e : Nat)
  | copyHFirst | copyHSecond
  | countSeed | countScan | countRestore
  | headerSeed | headerScan | headerRestore
  | fuelSeed | fuelScan | fuelRestore | vertexSeed | guard
  | old (stage : MachineLazyRows.VertexLabel d)
  | tailSeed | tailScan | tailRestore
  | expander (stage : MachineOverlayRows.VertexLabel e)
  | drainTail | increment
  | drainFuel | drainVertex | drainVertexCount | drainDartCount
  deriving DecidableEq, Fintype

abbrev Label (d e : Nat) := MachineTableSplit.Label ⊕ MainLabel d e

def next {d e : Nat} (l : MainLabel d e) : Option (Label d e) := some (.inr l)

def mainInstruction (d e : Nat) (hd : 0 < d) (he : 0 < e) :
    MainLabel d e → TM2.Stmt (fun _ : Tape => Bool) (Label d e) (State A d e)
  | .copyHFirst => Reduction.MachineTransfer.loopAt expanderArchive scratch id false
      (.inr .copyHFirst) (next .copyHSecond)
  | .copyHSecond => MachineCopy.forkLoop scratch expanderArchive expanderStream false
      (.inr .copyHSecond) (next .countSeed)
  | .countSeed => MachineUnaryAffineAt.seed output 0 (.inr .countScan)
  | .countScan => MachineUnaryAffineAt.scan vertexCount scratch output (d + e)
      (.inr .countScan) (.inr .countRestore)
  | .countRestore => Reduction.MachineTransfer.loopAt scratch vertexCount id false
      (.inr .countRestore) (next .headerSeed)
  | .headerSeed => MachineUnaryAffineAt.seed output 0 (.inr .headerScan)
  | .headerScan => MachineUnaryAffineAt.scan vertexCount scratch output 1
      (.inr .headerScan) (.inr .headerRestore)
  | .headerRestore => Reduction.MachineTransfer.loopAt scratch vertexCount id false
      (.inr .headerRestore) (next .fuelSeed)
  | .fuelSeed => MachineUnaryAffineAt.seed fuel 0 (.inr .fuelScan)
  | .fuelScan => MachineUnaryAffineAt.scan vertexCount scratch fuel 1
      (.inr .fuelScan) (.inr .fuelRestore)
  | .fuelRestore => Reduction.MachineTransfer.loopAt scratch vertexCount id false
      (.inr .fuelRestore) (next .vertexSeed)
  | .vertexSeed => .push vertex (fun _ => false) (.goto fun _ => .inr .guard)
  | .guard => MachineUnaryCounter.guard fuel (.inr (.old (0, none))) (.inr .drainFuel)
  | .old stage => MachineLazyRows.vertexInstruction d hd (d + e) 0 oldTapes
      (fun stage => .inr (.old stage)) (next .tailSeed) stage
  | .tailSeed => MachineUnaryAffineAt.seed tail 0 (.inr .tailScan)
  | .tailScan => MachineUnaryAffineAt.scan vertex scratch tail 1
      (.inr .tailScan) (.inr .tailRestore)
  | .tailRestore => Reduction.MachineTransfer.loopAt scratch vertex id false
      (.inr .tailRestore) (next (.expander (0, none)))
  | .expander stage => MachineStateEquiv.statement (expanderState A d e)
      (MachineOverlayRows.vertexInstruction d e he expanderTapes
        (fun stage => .inr (.expander stage)) (next .drainTail) stage)
  | .drainTail => MachineDrain.drain tail (.inr .drainTail) (next .increment)
  | .increment => .push vertex (fun _ => true) (.goto fun _ => .inr .guard)
  | .drainFuel => MachineDrain.drain fuel (.inr .drainFuel) (next .drainVertex)
  | .drainVertex => MachineDrain.drain vertex (.inr .drainVertex) (next .drainVertexCount)
  | .drainVertexCount => MachineDrain.drain vertexCount (.inr .drainVertexCount) (next .drainDartCount)
  | .drainDartCount => MachineDrain.drain dartCount (.inr .drainDartCount) none

def program (d e : Nat) (hd : 0 < d) (he : 0 < e) :
    Label d e → TM2.Stmt (fun _ : Tape => Bool) (Label d e) (State A d e)
  | .inl l => Placement.statement Sum.inl Sum.inl (next .copyHFirst)
      (MachineTableSplit.program l)
  | .inr l => mainInstruction d e hd he l

theorem program_main (d e : Nat) (hd : 0 < d) (he : 0 < e) (l : MainLabel d e) :
    program (A := A) d e hd he (.inr l) = mainInstruction d e hd he l := rfl

def extraTapes (rawH : List Bool) : Fin 12 → List Bool :=
  fun i => if i = 0 then rawH else []

def splitView : Tape → Option (Fin 5)
  | .inl i => some i
  | .inr _ => none

def combine (left : Fin 5 → List Bool) (rawH : List Bool) : Tape → List Bool
  | .inl i => left i
  | .inr i => extraTapes rawH i

def initialTapes (graph rawH : List Bool) : Tape → List Bool :=
  combine (MachineTableSplit.initialTapes graph) rawH

def splitTapes (n m : Nat) (rows rawH : List Bool) : Tape → List Bool :=
  combine (MachineTableSplit.resultTapes n m rows) rawH

theorem placedTapes (left : Fin 5 → List Bool) (rawH : List Bool) :
    Placement.tapes splitView left (initialTapes [] rawH) = combine left rawH := by
  funext k
  cases k <;> rfl

theorem splitTrace (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (table : GraphTables.Table) (rawH : List Bool) (a : A) :
    (advance (TM2.step (program d e hd he)))^[MachineTableSplit.exactSteps
      table.vertices table.darts (MachineTableSplit.rowsBits table)]
      (some ⟨some (.inl .copyFirst), clean d e hd he a,
        initialTapes (GraphTables.tableBits table) rawH⟩) =
      some ⟨next .copyHFirst, clean d e hd he a,
        splitTapes table.vertices table.darts (MachineTableSplit.rowsBits table) rawH⟩ := by
  let ambient := (clean d e hd he a).1
  have source := MachineTableSplit.tableTrace table ambient none
  have placed := Placement.trace (Sum.inl : Fin 5 → Tape) splitView (fun _ => rfl)
    (by intro j k h; cases j <;> simp_all [splitView])
    (Sum.inl : MachineTableSplit.Label → Label d e) (next .copyHFirst)
    (initialTapes [] rawH) MachineTableSplit.program (program d e hd he) (fun _ => rfl)
    _ _ _ source
  simp only [Placement.configuration, Placement.label] at placed
  rw [placedTapes, placedTapes] at placed
  simpa only [initialTapes,
    splitTapes, ambient, clean, MachineLazyRows.streamClean, MachinePortReindex.clean] using placed

def memory (g h gs hs n m f v t out : List Bool) : Tape → List Bool
  | .inl i => ![g, gs, [], n, m] i
  | .inr i => ![h, hs, f, v, t, [], [], [], [], [], [], out] i

@[simp] theorem memory_graphArchive (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out graphArchive = g := rfl
@[simp] theorem memory_graphStream (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out graphStream = gs := rfl
@[simp] theorem memory_scratch (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out scratch = [] := rfl
@[simp] theorem memory_vertexCount (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out vertexCount = n := rfl
@[simp] theorem memory_dartCount (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out dartCount = m := rfl
@[simp] theorem memory_expanderArchive (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out expanderArchive = h := rfl
@[simp] theorem memory_expanderStream (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out expanderStream = hs := rfl
@[simp] theorem memory_fuel (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out fuel = f := rfl
@[simp] theorem memory_vertex (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out vertex = v := rfl
@[simp] theorem memory_tail (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out tail = t := rfl
@[simp] theorem memory_oldReverse (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out oldReverse = [] := rfl
@[simp] theorem memory_relation (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out relation = [] := rfl
@[simp] theorem memory_dividedCopy (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out dividedCopy = [] := rfl
@[simp] theorem memory_quotient (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out quotient = [] := rfl
@[simp] theorem memory_newReverse (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out newReverse = [] := rfl
@[simp] theorem memory_rowBuffer (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out rowBuffer = [] := rfl
@[simp] theorem memory_output (g h gs hs n m f v t out : List Bool) :
    memory g h gs hs n m f v t out output = out := rfl

@[simp] theorem update_output (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) output x = memory g h gs hs n m f v t x := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl
@[simp] theorem update_expanderStream (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) expanderStream x =
      memory g h gs x n m f v t out := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl
@[simp] theorem update_graphStream (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) graphStream x =
      memory g h x hs n m f v t out := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl
@[simp] theorem update_fuel (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) fuel x = memory g h gs hs n m x v t out := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl
@[simp] theorem update_vertex (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) vertex x = memory g h gs hs n m f x t out := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl
@[simp] theorem update_tail (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) tail x = memory g h gs hs n m f v x out := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl
@[simp] theorem update_vertexCount (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) vertexCount x = memory g h gs hs x m f v t out := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl
@[simp] theorem update_dartCount (g h gs hs n m f v t out x : List Bool) :
    Function.update (memory g h gs hs n m f v t out) dartCount x = memory g h gs hs n x f v t out := by
  funext k; rcases k with i | i <;> fin_cases i <;> rfl

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

theorem initialTapes_memory (g h : List Bool) :
    initialTapes g h = memory g h [] [] [] [] [] [] [] [] := by
  funext k
  rcases k with i | i <;> fin_cases i <;>
    simp [initialTapes, combine, extraTapes, MachineTableSplit.initialTapes,
      MachineTableSplit.tapes, memory]

theorem splitTapes_memory (n m : Nat) (rows rawH : List Bool) :
    splitTapes n m rows rawH = memory (encodeWords [n, m] ++ rows) rawH rows []
      (encodeWord n) (encodeWord m) [] [] [] [] := by
  funext k
  rcases k with i | i <;> fin_cases i <;>
    simp [splitTapes, combine, extraTapes, MachineTableSplit.resultTapes,
      MachineTableSplit.tapes, memory]

def preparationSteps (n : Nat) (h : List Bool) : Nat := 2 * (h.length + 1) + 6 * (n + 1) + 4

theorem preparationTrace (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (g h gs : List Bool) (n m : Nat) (a : A) :
    (advance (TM2.step (program d e hd he)))^[preparationSteps n h]
      (some ⟨next .copyHFirst, clean d e hd he a,
        memory g h gs [] (encodeWord n) (encodeWord m) [] [] [] []⟩) =
      some ⟨next .guard, clean d e hd he a,
        memory g h gs h (encodeWord n) (encodeWord m) (encodeWord n) (encodeWord 0) []
          (encodeWords [n, n * (d + e)])⟩ := by
  let b0 := memory g h gs [] (encodeWord n) (encodeWord m) [] [] [] []
  let b1 := memory g h gs h (encodeWord n) (encodeWord m) [] [] [] []
  let b2 := memory g h gs h (encodeWord n) (encodeWord m) [] [] [] (encodeWord ((d + e) * n))
  let b3 := memory g h gs h (encodeWord n) (encodeWord m) [] [] []
    (encodeWords [n, n * (d + e)])
  let b4 := memory g h gs h (encodeWord n) (encodeWord m) (encodeWord n) [] []
    (encodeWords [n, n * (d + e)])
  let ambient := (clean d e hd he a).1
  have hcopy := MachineCopy.copyTrace expanderArchive expanderStream scratch
    (by decide) (by decide) (by decide) false (.inr .copyHFirst) (.inr .copyHSecond)
    (next .countSeed) (program d e hd he) rfl rfl b0 rfl ambient none
  have copyRun : (advance (TM2.step (program d e hd he)))^[2 * (h.length + 1)]
      (some ⟨next .copyHFirst, clean d e hd he a, b0⟩) =
      some ⟨next .countSeed, clean d e hd he a, b1⟩ := by
    simpa only [b0, b1, memory_expanderArchive, memory_expanderStream, next, List.append_nil,
      update_expanderStream, ambient, clean, MachineLazyRows.streamClean,
      MachinePortReindex.clean] using hcopy
  have hcount := MachineUnaryAffineAt.seededAffineTrace vertexCount scratch output
    (by decide) (by decide) (by decide) (d + e) 0
    (.inr .countSeed) (.inr .countScan) (.inr .countRestore) (next .headerSeed)
    (program d e hd he) rfl rfl rfl b1 n [] (by simp [b1, memory, vertexCount]) rfl ambient none
  have countRun : (advance (TM2.step (program d e hd he)))^[2 * (n + 1) + 1]
      (some ⟨next .countSeed, clean d e hd he a, b1⟩) =
      some ⟨next .headerSeed, clean d e hd he a, b2⟩ := by
    simpa only [b1, b2, memory_output, next, List.append_nil, Nat.add_zero, update_output,
      ambient, clean, MachineLazyRows.streamClean, MachinePortReindex.clean] using hcount
  have hheader := MachineUnaryAffineAt.seededAffineTrace vertexCount scratch output
    (by decide) (by decide) (by decide) 1 0
    (.inr .headerSeed) (.inr .headerScan) (.inr .headerRestore) (next .fuelSeed)
    (program d e hd he) rfl rfl rfl b2 n [] (by simp [b2, memory, vertexCount]) rfl ambient none
  have headerRun : (advance (TM2.step (program d e hd he)))^[2 * (n + 1) + 1]
      (some ⟨next .headerSeed, clean d e hd he a, b2⟩) =
      some ⟨next .fuelSeed, clean d e hd he a, b3⟩ := by
    simpa only [b2, b3, memory_output, next, Nat.one_mul, Nat.add_zero, update_output,
      Nat.mul_comm (d + e) n, encodeWords, List.append_nil,
      ambient, clean, MachineLazyRows.streamClean, MachinePortReindex.clean] using hheader
  have hfuel := MachineUnaryAffineAt.seededAffineTrace vertexCount scratch fuel
    (by decide) (by decide) (by decide) 1 0
    (.inr .fuelSeed) (.inr .fuelScan) (.inr .fuelRestore) (next .vertexSeed)
    (program d e hd he) rfl rfl rfl b3 n [] (by simp [b3, memory, vertexCount]) rfl ambient none
  have fuelRun : (advance (TM2.step (program d e hd he)))^[2 * (n + 1) + 1]
      (some ⟨next .fuelSeed, clean d e hd he a, b3⟩) =
      some ⟨next .vertexSeed, clean d e hd he a, b4⟩ := by
    simpa only [b3, b4, memory_fuel, next, List.append_nil, Nat.one_mul, Nat.add_zero,
      update_fuel, ambient, clean, MachineLazyRows.streamClean, MachinePortReindex.clean] using hfuel
  have seedRun : (advance (TM2.step (program d e hd he)))^[1]
      (some ⟨next .vertexSeed, clean d e hd he a, b4⟩) =
      some ⟨next .guard, clean d e hd he a,
        memory g h gs h (encodeWord n) (encodeWord m) (encodeWord n) (encodeWord 0) []
          (encodeWords [n, n * (d + e)])⟩ := by
    change some (TM2.stepAux (program d e hd he (.inr .vertexSeed)) _ _) = _
    simp only [program, mainInstruction, TM2.stepAux, b4, memory_vertex,
      update_vertex, next, encodeWord, List.replicate_zero, List.nil_append]
  have all := joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable copyRun countRun) headerRun) fuelRun) seedRun
  have count : preparationSteps n h =
      (((2 * (h.length + 1) + (2 * (n + 1) + 1)) + (2 * (n + 1) + 1)) +
        (2 * (n + 1) + 1)) + 1 := by unfold preparationSteps; omega
  rw [count]
  exact all

private theorem encodeWords_flatMap_inline_MachineOverlayTable {α : Type*} (items : List α) (words : α → List Nat) :
    encodeWords (items.flatMap words) = items.flatMap (fun x => encodeWords (words x)) := by
  induction items with
  | nil => rfl
  | cons item items ih => simp only [List.flatMap_cons, encodeWords_append, ih]

private theorem encodeWords_flatten_inline_MachineOverlayTable (items : List (List Nat)) :
    encodeWords items.flatten = (items.map encodeWords).flatten := by
  induction items with
  | nil => rfl
  | cons item items ih => simp only [List.flatten_cons, List.map_cons, encodeWords_append, ih]

private theorem encodeWords_flatten_flatMap_inline_MachineOverlayTable {α : Type*} (items : List (List α))
    (words : α → List Nat) :
    encodeWords (items.flatten.flatMap words) =
      (items.map fun xs => encodeWords (xs.flatMap words)).flatten := by
  induction items with
  | nil => rfl
  | cons item items ih =>
      simp only [List.flatten_cons, List.flatMap_append, encodeWords_append, List.map_cons, ih]

def oldRows {n d : Nat} (G : PortTables.Table n d) (v : Fin n) := List.ofFn (row G v)
def expanderValues {n e : Nat} (H : ExpanderTables.Table n e) (v : Fin n) (p : Fin e) : Nat :=
  (ExpanderTables.reverseIndex H (ExpanderTables.rowIndex n e (v, p))).val

def oldBlock {n d : Nat} (G : PortTables.Table n d) (v : Fin n) : List Bool :=
  MachineLazyRows.recordsInput (oldRows G v)
def expanderBlock {n e : Nat} (H : ExpanderTables.Table n e) (v : Fin n) : List Bool :=
  encodeWords (List.ofFn (expanderValues H v))

def oldOutput {n d : Nat} (G : PortTables.Table n d) (e : Nat) (v : Fin n) : List Bool :=
  MachineLazyRows.recordsOutput d (d + e) 0 (oldRows G v)
def expanderOutput {n e : Nat} (d : Nat) (H : ExpanderTables.Table n e) (v : Fin n) : List Bool :=
  MachineOverlayRows.streamOutput d e v.val (List.ofFn (expanderValues H v))
def outputBlock {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e)
    (v : Fin n) : List Bool := oldOutput G e v ++ expanderOutput d H v

theorem oldOutput_eq {n d : Nat} (G : PortTables.Table n d) (e : Nat) (v : Fin n) :
    oldOutput G e v = encodeWords ((List.ofFn (oldRow G e v)).flatMap rowWords) := by
  rw [encodeWords_flatMap_inline_MachineOverlayTable]
  simp only [oldOutput, oldRows, MachineLazyRows.recordsOutput, List.flatMap_def, List.map_ofFn]
  apply congrArg List.flatten
  apply congrArg List.ofFn
  funext p
  change MachineLazyRows.reindexRowBits d (d + e) 0 (row G v p).tail.val
    (row G v p).reverseIndex.val (row G v p).relation = encodeWords (rowWords (oldRow G e v p))
  rw [oldRow_words]
  simp only [MachineLazyRows.reindexRowBits, MachinePortReindex.value, Nat.add_zero,
    oldReverseMap, row, PreprocessingLazyWords.row, List.cons_append, List.nil_append,
    encodeWords, List.append_assoc]

theorem expanderOutput_eq {n e : Nat} (d : Nat) (H : ExpanderTables.Table n e) (v : Fin n) :
    expanderOutput d H v = encodeWords ((List.ofFn (expanderRow d H v)).flatMap rowWords) :=
  MachineOverlayRows.streamOutput_expanderRows d H v

theorem outputBlock_eq {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) :
    outputBlock G H v = encodeWords ((vertexRows G H v).flatMap rowWords) := by
  rw [outputBlock, oldOutput_eq, expanderOutput_eq]
  simp only [vertexRows, List.flatMap_append, encodeWords_append]

theorem graphRows_blocks {n d : Nat} (G : PortTables.Table n d) :
    MachineTableSplit.rowsBits (PortTables.graphTable G) = (List.ofFn (oldBlock G)).flatten := by
  change encodeWords ((PortTables.flatRows G).toList.flatMap rowWords) = _
  rw [PreprocessingLazyWords.flatRows_list, encodeWords_flatten_flatMap_inline_MachineOverlayTable, List.map_ofFn]
  apply congrArg List.flatten
  apply congrArg List.ofFn
  funext v
  exact (MachineLazyRows.recordsInput_eq_codec (oldRows G v)).symm

theorem expanderRows_blocks {n e : Nat} (H : ExpanderTables.Table n e) :
    encodeWords (ExpanderTableWords.rotationWords H) = (List.ofFn (expanderBlock H)).flatten := by
  rw [rotationWords_vertices, encodeWords_flatten_inline_MachineOverlayTable, List.map_ofFn]
  rfl

theorem tableBits_blocks {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) :
    PortTables.tableBits (PreprocessingOverlayTables.overlay G H) =
      encodeWords [n, n * (d + e)] ++ (List.ofFn (outputBlock G H)).flatten := by
  rw [tableBits_overlay, encodeWords_append, encodeWords_flatten_flatMap_inline_MachineOverlayTable, List.map_ofFn]
  congr 1
  apply congrArg List.flatten
  apply congrArg List.ofFn
  funext v
  exact (outputBlock_eq G H v).symm

def remaining {n : Nat} (block : Fin n → List Bool) (k : Nat) : List Bool :=
  ((List.ofFn block).drop k).flatten
def emittedPrefix {n : Nat} (block : Fin n → List Bool) (k : Nat) : List Bool :=
  ((List.ofFn block).take k).flatten

theorem remaining_step {n : Nat} (block : Fin n → List Bool) (v : Fin n) :
    remaining block v.val = block v ++ remaining block (v.val + 1) := by
  have hs := List.getElem_cons_drop (as := List.ofFn block) (i := v.val)
    (by simpa only [List.length_ofFn] using v.isLt)
  unfold remaining
  rw [← hs]
  simp only [List.flatten_cons, List.getElem_ofFn, Fin.eta]

theorem prefix_step {n : Nat} (block : Fin n → List Bool) (v : Fin n) :
    emittedPrefix block (v.val + 1) = emittedPrefix block v.val ++ block v := by
  unfold emittedPrefix
  rw [List.take_succ_eq_append_getElem (by simpa only [List.length_ofFn] using v.isLt)]
  simp only [List.flatten_append, List.flatten_cons, List.flatten_nil, List.append_nil,
    List.getElem_ofFn, Fin.eta]

@[simp] theorem remaining_zero {n : Nat} (block : Fin n → List Bool) :
    remaining block 0 = (List.ofFn block).flatten := by simp [remaining]
@[simp] theorem remaining_end {n : Nat} (block : Fin n → List Bool) :
    remaining block n = [] := by
  unfold remaining
  rw [List.drop_eq_nil_iff.mpr (by simpa only [List.length_ofFn] using (Nat.le_refl n)), List.flatten_nil]
@[simp] theorem prefix_zero {n : Nat} (block : Fin n → List Bool) : emittedPrefix block 0 = [] := by
  simp [emittedPrefix]
@[simp] theorem prefix_end {n : Nat} (block : Fin n → List Bool) :
    emittedPrefix block n = (List.ofFn block).flatten := by
  unfold emittedPrefix
  have ht : (List.ofFn block).take n = List.ofFn block :=
    List.take_of_length_le (by simpa only [List.length_ofFn] using (Nat.le_refl n))
  rw [ht]

theorem oldFrame {n m : Nat} (g h gs hs nv mv f v out : List Bool)
    (rows : List (DartRow n m)) :
    MachineLazyRows.streamFrame oldTapes (memory g h gs hs nv mv f v [] []) rows out =
      memory g h (MachineLazyRows.recordsInput rows ++ gs) hs nv mv f v [] out := by
  funext k
  rcases k with i | i <;> fin_cases i <;>
    simp [MachineLazyRows.streamFrame, oldTapes, memory, tail, oldReverse, relation,
      scratch, dividedCopy, quotient, newReverse, output, rowBuffer, graphStream]

theorem expanderFrame (g h gs hs nv mv f v t out : List Bool) (words : List Nat) :
    MachineOverlayRows.streamFrame expanderTapes (memory g h gs hs nv mv f v t []) words out =
      memory g h gs (encodeWords words ++ hs) nv mv f v t out := by
  funext k
  rcases k with i | i <;> fin_cases i <;>
    simp [MachineOverlayRows.streamFrame, expanderTapes, memory, tail, oldReverse,
      relation, scratch, dividedCopy, quotient, newReverse, output, rowBuffer, expanderStream]

theorem oldVertexTrace {n : Nat} (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (G : PortTables.Table n d) (v : Fin n) (g h gs hs nv mv f current out : List Bool) (a : A) :
    (advance (TM2.step (program d e hd he)))^[MachineLazyRows.vertexSteps d (d + e) 0 (oldRows G v) out]
      (some ⟨next (.old (0, none)), clean d e hd he a,
        memory g h (oldBlock G v ++ gs) hs nv mv f current [] out⟩) =
      some ⟨next .tailSeed, clean d e hd he a,
        memory g h gs hs nv mv f current [] (out ++ oldOutput G e v)⟩ := by
  have hrun := MachineLazyRows.vertexTrace d hd (d + e) 0 oldTapes oldTapes_injective
    (fun stage => .inr (.old stage)) (next .tailSeed) (program d e hd he) (fun _ => rfl)
    (memory g h gs hs nv mv f current [] []) (row G v)
    (by
      intro i hi7 hi9
      fin_cases i <;> simp_all [oldTapes, memory, tail, oldReverse, relation, scratch,
        dividedCopy, quotient, newReverse, output, rowBuffer, graphStream])
    (a, MachineFixedDivMod.residue e he 0) out
  simpa only [next, oldFrame, MachineLazyRows.recordsInput, List.flatMap_nil, List.nil_append,
    clean, oldBlock, oldRows, oldOutput] using hrun

theorem expanderVertexTrace {n : Nat} (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (H : ExpanderTables.Table n e) (v : Fin n) (g h gs hs nv mv f current out : List Bool) (a : A) :
    (advance (TM2.step (program d e hd he)))^[MachineOverlayRows.vertexSteps d e v.val
      (List.ofFn (expanderValues H v)) out]
      (some ⟨next (.expander (0, none)), clean d e hd he a,
        memory g h gs (expanderBlock H v ++ hs) nv mv f current (encodeWord v.val) out⟩) =
      some ⟨next .drainTail, clean d e hd he a,
        memory g h gs hs nv mv f current (encodeWord v.val) (out ++ expanderOutput d H v)⟩ := by
  let source := MachineStateEquiv.program (expanderState A d e).symm (program d e hd he)
  have code (stage : MachineOverlayRows.VertexLabel e) : source (.inr (.expander stage)) =
      MachineOverlayRows.vertexInstruction d e he expanderTapes
        (fun stage => .inr (.expander stage)) (next .drainTail) stage := by
    change MachineStateEquiv.statement (expanderState A d e).symm
      (MachineStateEquiv.statement (expanderState A d e) _) = _
    exact MachineStateEquiv.statement_symm_statement _ _
  have native := MachineOverlayRows.vertexTrace d e he expanderTapes expanderTapes_injective
    (fun stage => .inr (.expander stage)) (next .drainTail) source code
    (memory g h gs hs nv mv f current (encodeWord v.val) []) v.val (expanderValues H v)
    rfl
    (by
      intro i hi0 hi7 hi9
      fin_cases i <;> simp_all [expanderTapes, memory, tail, oldReverse, relation, scratch,
        dividedCopy, quotient, newReverse, output, rowBuffer, expanderStream])
    ((a, MachineRegularOriginalRow.zeroBuffer), MachineFixedDivMod.residue d hd 0) out
  have transported := MachineStateEquiv.trace (expanderState A d e) source _ _ _ native
  have back : MachineStateEquiv.program (expanderState A d e) source = program d e hd he :=
    MachineStateEquiv.program_symm_program (expanderState A d e).symm _
  rw [back] at transported
  simpa only [next, MachineStateEquiv.configuration, expanderState_clean, expanderFrame,
    encodeWords, List.nil_append, expanderBlock, expanderOutput] using transported

theorem tailCopyTrace (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (g h gs hs nv mv f out : List Bool) (k : Nat) (a : A) :
    (advance (TM2.step (program d e hd he)))^[2 * (k + 1) + 1]
      (some ⟨next .tailSeed, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord k) [] out⟩) =
      some ⟨next (.expander (0, none)), clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord k) (encodeWord k) out⟩ := by
  let ambient := (clean d e hd he a).1
  have hrun := MachineUnaryAffineAt.seededAffineTrace vertex scratch tail
    (by decide) (by decide) (by decide) 1 0
    (.inr .tailSeed) (.inr .tailScan) (.inr .tailRestore) (next (.expander (0, none)))
    (program d e hd he) rfl rfl rfl
    (memory g h gs hs nv mv f (encodeWord k) [] out) k []
    (by simp [memory, vertex]) rfl ambient none
  simpa only [memory_tail, next, List.append_nil, Nat.one_mul, Nat.add_zero, update_tail,
    ambient, clean, MachineLazyRows.streamClean, MachinePortReindex.clean] using hrun

theorem tailFinishTrace (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (g h gs hs nv mv f out : List Bool) (k : Nat) (a : A) :
    (advance (TM2.step (program d e hd he)))^[k + 3]
      (some ⟨next .drainTail, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord k) (encodeWord k) out⟩) =
      some ⟨next .guard, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord (k + 1)) [] out⟩ := by
  let ambient := (clean d e hd he a).1
  have hd₀ := MachineDrain.drainTrace tail (.inr .drainTail) (next .increment)
    (program d e hd he) rfl (memory g h gs hs nv mv f (encodeWord k) [] out)
    (encodeWord k) ambient none
  have drain : (advance (TM2.step (program d e hd he)))^[k + 2]
      (some ⟨next .drainTail, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord k) (encodeWord k) out⟩) =
      some ⟨next .increment, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord k) [] out⟩ := by
    simpa only [next, encodeWord_length, Nat.add_assoc, update_tail, ambient, clean,
      MachineLazyRows.streamClean, MachinePortReindex.clean] using hd₀
  have increment : (advance (TM2.step (program d e hd he)))^[1]
      (some ⟨next .increment, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord k) [] out⟩) =
      some ⟨next .guard, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord (k + 1)) [] out⟩ := by
    change some (TM2.stepAux (program d e hd he (.inr .increment)) _ _) = _
    simp only [program, mainInstruction, TM2.stepAux, memory_vertex, next,
      update_vertex, encodeWord, List.replicate_succ, List.cons_append]
  have all := joinTrace_inline_MachineOverlayTable drain increment
  simpa only [Nat.add_assoc] using all

def pairSteps {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e)
    (v : Fin n) (out : List Bool) : Nat :=
  MachineLazyRows.vertexSteps d (d + e) 0 (oldRows G v) out + (2 * (v.val + 1) + 1) +
    MachineOverlayRows.vertexSteps d e v.val (List.ofFn (expanderValues H v))
      (out ++ oldOutput G e v) + (v.val + 3)

theorem pairTrace {n : Nat} (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (v : Fin n)
    (g h gs hs nv mv f out : List Bool) (a : A) :
    (advance (TM2.step (program d e hd he)))^[pairSteps G H v out]
      (some ⟨next (.old (0, none)), clean d e hd he a,
        memory g h (oldBlock G v ++ gs) (expanderBlock H v ++ hs)
          nv mv f (encodeWord v.val) [] out⟩) =
      some ⟨next .guard, clean d e hd he a,
        memory g h gs hs nv mv f (encodeWord (v.val + 1)) [] (out ++ outputBlock G H v)⟩ := by
  have h₁ := oldVertexTrace d e hd he G v g h gs (expanderBlock H v ++ hs)
    nv mv f (encodeWord v.val) out a
  have h₂ := tailCopyTrace d e hd he g h gs (expanderBlock H v ++ hs)
    nv mv f (out ++ oldOutput G e v) v.val a
  have h₃ := expanderVertexTrace d e hd he H v g h gs hs nv mv f
    (encodeWord v.val) (out ++ oldOutput G e v) a
  have h₄ := tailFinishTrace d e hd he g h gs hs nv mv f
    ((out ++ oldOutput G e v) ++ expanderOutput d H v) v.val a
  simpa only [pairSteps, outputBlock, List.append_assoc] using
    joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable h₁ h₂) h₃) h₄

def cleanupSteps (n m : Nat) : Nat := 2 * n + m + 8

theorem cleanupTrace (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (g h out : List Bool) (n m : Nat) (a : A) :
    (advance (TM2.step (program d e hd he)))^[cleanupSteps n m]
      (some ⟨next .drainFuel, clean d e hd he a,
        memory g h [] [] (encodeWord n) (encodeWord m) (encodeWord 0) (encodeWord n) [] out⟩) =
      some ⟨none, clean d e hd he a, memory g h [] [] [] [] [] [] [] out⟩ := by
  let ambient := (clean d e hd he a).1
  have h₁ := MachineDrain.drainTrace fuel (.inr .drainFuel) (next .drainVertex)
    (program d e hd he) rfl
    (memory g h [] [] (encodeWord n) (encodeWord m) [] (encodeWord n) [] out)
    (encodeWord 0) ambient none
  have h₂ := MachineDrain.drainTrace vertex (.inr .drainVertex) (next .drainVertexCount)
    (program d e hd he) rfl
    (memory g h [] [] (encodeWord n) (encodeWord m) [] [] [] out)
    (encodeWord n) ambient none
  have h₃ := MachineDrain.drainTrace vertexCount (.inr .drainVertexCount) (next .drainDartCount)
    (program d e hd he) rfl (memory g h [] [] [] (encodeWord m) [] [] [] out)
    (encodeWord n) ambient none
  have h₄ := MachineDrain.drainTrace dartCount (.inr .drainDartCount) none
    (program d e hd he) rfl (memory g h [] [] [] [] [] [] [] out)
    (encodeWord m) ambient none
  simp only [encodeWord_length, update_fuel, update_vertex, update_vertexCount, update_dartCount] at h₁ h₂ h₃ h₄
  have all := joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable h₁ h₂) h₃) h₄
  have hsteps : cleanupSteps n m = ((0 + 1 + 1 + (n + 1 + 1)) + (n + 1 + 1)) + (m + 1 + 1) := by
    unfold cleanupSteps
    omega
  rw [hsteps]
  exact all

def rawExpander {n e : Nat} (H : ExpanderTables.Table n e) : List Bool :=
  encodeWords (ExpanderTableWords.rotationWords H)

def outputPrefix {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e)
    (k : Nat) : List Bool := encodeWords [n, n * (d + e)] ++ emittedPrefix (outputBlock G H) k

theorem outputPrefix_step {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) :
    outputPrefix G H (v.val + 1) = outputPrefix G H v.val ++ outputBlock G H v := by
  simp only [outputPrefix, prefix_step, List.append_assoc]

def loopBase {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e)
    (r : Nat) : Tape → List Bool :=
  memory (PortTables.tableBits G) (rawExpander H)
    (remaining (oldBlock G) (n - r)) (remaining (expanderBlock H) (n - r))
    (encodeWord n) (encodeWord (n * d)) [] (encodeWord (n - r)) [] (outputPrefix G H (n - r))

def loopCost {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e)
    (r : Nat) : Nat :=
  if hr : r < n then
    pairSteps G H ⟨n - (r + 1), by omega⟩ (outputPrefix G H (n - (r + 1)))
  else 0

theorem loopBodyTraces {n : Nat} (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (a : A) :
    MachineCountedLoop.BodyTraces fuel (.inr .guard) (.inr (.old (0, none)))
      (program d e hd he) [] (fun _ => (clean d e hd he a).1) (fun _ => none)
      (loopBase G H) (loopCost G H) n := by
  intro r hr
  let v : Fin n := ⟨n - (r + 1), by omega⟩
  have hv : v.val + 1 = n - r := by dsimp [v]; omega
  have hinput : MachineUnaryCounter.counterTapes fuel (loopBase G H (r + 1)) r [] =
      memory (PortTables.tableBits G) (rawExpander H)
        (oldBlock G v ++ remaining (oldBlock G) (v.val + 1))
        (expanderBlock H v ++ remaining (expanderBlock H) (v.val + 1))
        (encodeWord n) (encodeWord (n * d)) (encodeWord r) (encodeWord v.val) []
        (outputPrefix G H v.val) := by
    simp only [MachineUnaryCounter.counterTapes, loopBase, List.append_nil, update_fuel]
    change memory _ _ (remaining (oldBlock G) v.val) (remaining (expanderBlock H) v.val)
      _ _ _ _ _ _ = _
    rw [remaining_step (oldBlock G) v, remaining_step (expanderBlock H) v]
  have houtput : MachineUnaryCounter.counterTapes fuel (loopBase G H r) r [] =
      memory (PortTables.tableBits G) (rawExpander H)
        (remaining (oldBlock G) (v.val + 1)) (remaining (expanderBlock H) (v.val + 1))
        (encodeWord n) (encodeWord (n * d)) (encodeWord r) (encodeWord (v.val + 1)) []
        (outputPrefix G H v.val ++ outputBlock G H v) := by
    simp only [MachineUnaryCounter.counterTapes, loopBase, List.append_nil, update_fuel]
    rw [← hv, outputPrefix_step]
  have run := pairTrace d e hd he G H v (PortTables.tableBits G) (rawExpander H)
    (remaining (oldBlock G) (v.val + 1)) (remaining (expanderBlock H) (v.val + 1))
    (encodeWord n) (encodeWord (n * d)) (encodeWord r) (outputPrefix G H v.val) a
  change (advance (TM2.step (program d e hd he)))^[loopCost G H r]
    (some ⟨next (.old (0, none)), clean d e hd he a,
      MachineUnaryCounter.counterTapes fuel (loopBase G H (r + 1)) r []⟩) =
    some ⟨next .guard, clean d e hd he a,
      MachineUnaryCounter.counterTapes fuel (loopBase G H r) r []⟩
  rw [hinput, houtput]
  simpa only [loopCost, dite_eq_left hr, v] using run

theorem loopTrace {n : Nat} (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (a : A) :
    (advance (TM2.step (program d e hd he)))^[MachineCountedLoop.totalSteps (loopCost G H) n]
      (some ⟨next .guard, clean d e hd he a,
        memory (PortTables.tableBits G) (rawExpander H)
          (MachineTableSplit.rowsBits (PortTables.graphTable G)) (rawExpander H)
          (encodeWord n) (encodeWord (n * d)) (encodeWord n) (encodeWord 0) []
          (encodeWords [n, n * (d + e)])⟩) =
      some ⟨next .drainFuel, clean d e hd he a,
        memory (PortTables.tableBits G) (rawExpander H) [] []
          (encodeWord n) (encodeWord (n * d)) (encodeWord 0) (encodeWord n) []
          (PortTables.tableBits (PreprocessingOverlayTables.overlay G H))⟩ := by
  have run := MachineCountedLoop.loopTrace fuel (.inr .guard) (.inr (.old (0, none)))
    (.inr .drainFuel) (program d e hd he) rfl []
    (fun _ => (clean d e hd he a).1) (fun _ => none) (loopBase G H) (loopCost G H) n
    (loopBodyTraces d e hd he G H a)
  have startFrame : MachineUnaryCounter.counterTapes fuel (loopBase G H n) n [] =
      memory (PortTables.tableBits G) (rawExpander H)
        (MachineTableSplit.rowsBits (PortTables.graphTable G)) (rawExpander H)
        (encodeWord n) (encodeWord (n * d)) (encodeWord n) (encodeWord 0) []
        (encodeWords [n, n * (d + e)]) := by
    simp only [MachineUnaryCounter.counterTapes, loopBase, List.append_nil, update_fuel,
      Nat.sub_self, remaining_zero, outputPrefix, prefix_zero]
    simp only [graphRows_blocks, rawExpander, expanderRows_blocks]
  have endFrame : MachineUnaryCounter.counterTapes fuel (loopBase G H 0) 0 [] =
      memory (PortTables.tableBits G) (rawExpander H) [] []
        (encodeWord n) (encodeWord (n * d)) (encodeWord 0) (encodeWord n) []
        (PortTables.tableBits (PreprocessingOverlayTables.overlay G H)) := by
    simp only [MachineUnaryCounter.counterTapes, loopBase, List.append_nil, update_fuel,
      Nat.sub_zero, remaining_end, outputPrefix, prefix_end, tableBits_blocks]
  simp only [MachineCountedLoop.guardConfiguration, MachineCountedLoop.exitConfiguration] at run
  rw [startFrame, endFrame] at run
  simpa only [next, clean, MachineLazyRows.streamClean, MachinePortReindex.clean] using run

def tableSteps {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e) : Nat :=
  MachineTableSplit.exactSteps n (n * d) (MachineTableSplit.rowsBits (PortTables.graphTable G)) +
    preparationSteps n (rawExpander H) + MachineCountedLoop.totalSteps (loopCost G H) n +
    cleanupSteps n (n * d)

theorem tableTrace {n : Nat} (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (a : A) :
    (advance (TM2.step (program d e hd he)))^[tableSteps G H]
      (some ⟨some (.inl .copyFirst), clean d e hd he a,
        initialTapes (PortTables.tableBits G) (rawExpander H)⟩) =
      some ⟨none, clean d e hd he a,
        memory (PortTables.tableBits G) (rawExpander H) [] [] [] [] [] [] []
          (PortTables.tableBits (PreprocessingOverlayTables.overlay G H))⟩ := by
  have h₁ := splitTrace d e hd he (PortTables.graphTable G) (rawExpander H) a
  rw [splitTapes_memory] at h₁
  have header : encodeWords [n, n * d] ++ MachineTableSplit.rowsBits (PortTables.graphTable G) =
      PortTables.tableBits G := (MachineTableSplit.tableBits_header_rows (PortTables.graphTable G)).symm
  change (advance (TM2.step (program d e hd he)))^[MachineTableSplit.exactSteps n (n * d)
      (MachineTableSplit.rowsBits (PortTables.graphTable G))]
    (some ⟨some (.inl .copyFirst), clean d e hd he a,
      initialTapes (PortTables.tableBits G) (rawExpander H)⟩) =
    some ⟨next .copyHFirst, clean d e hd he a,
      memory (encodeWords [n, n * d] ++ MachineTableSplit.rowsBits (PortTables.graphTable G))
        (rawExpander H) (MachineTableSplit.rowsBits (PortTables.graphTable G)) []
        (encodeWord n) (encodeWord (n * d)) [] [] [] []⟩ at h₁
  rw [header] at h₁
  have h₂ := preparationTrace d e hd he (PortTables.tableBits G) (rawExpander H)
    (MachineTableSplit.rowsBits (PortTables.graphTable G)) n (n * d) a
  have h₃ := loopTrace d e hd he G H a
  have h₄ := cleanupTrace d e hd he (PortTables.tableBits G) (rawExpander H)
    (PortTables.tableBits (PreprocessingOverlayTables.overlay G H)) n (n * d) a
  exact joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable (joinTrace_inline_MachineOverlayTable h₁ h₂) h₃) h₄

private theorem flatMap_length_le_inline_MachineOverlayTable {α : Type*} (items : List α) (f : α → List Bool)
    (B : Nat) (bounded : ∀ x ∈ items, (f x).length ≤ B) :
    (items.flatMap f).length ≤ items.length * B := by
  induction items with
  | nil => simp
  | cons x xs ih =>
      have hx := bounded x (by simp)
      have ht := ih (fun y hy => bounded y (by simp [hy]))
      simp only [List.flatMap_cons, List.length_append, List.length_cons]
      rw [Nat.add_mul, Nat.one_mul]
      omega

private theorem flatten_length_le_inline_MachineOverlayTable (items : List (List Bool)) (B : Nat)
    (bounded : ∀ x ∈ items, x.length ≤ B) : items.flatten.length ≤ items.length * B := by
  simpa only [List.flatMap_id] using flatMap_length_le_inline_MachineOverlayTable items id B bounded

def oldSize (d e n : Nat) : Nat := MachineLazyRows.rowSizeBound (d + e) 0 n (n * d)
def expanderSize (d e n : Nat) : Nat := MachineOverlayRows.rowSizeBound d e n (n * e)
def blockSize (d e n : Nat) : Nat := d * oldSize d e n + e * expanderSize d e n

theorem oldOutput_length_le {n d : Nat} (G : PortTables.Table n d) (e : Nat) (v : Fin n) :
    (oldOutput G e v).length ≤ d * oldSize d e n := by
  have bound := flatMap_length_le_inline_MachineOverlayTable (oldRows G v)
    (fun r => MachineLazyRows.reindexRowBits d (d + e) 0 r.tail.val r.reverseIndex.val r.relation)
    (oldSize d e n) (fun r _ => MachineLazyRows.reindexRowBits_length_le d (d + e) 0 r)
  simpa only [oldOutput, MachineLazyRows.recordsOutput, oldRows, List.length_ofFn] using bound

theorem expanderOutput_length_le {n e : Nat} (d : Nat) (H : ExpanderTables.Table n e)
    (v : Fin n) : (expanderOutput d H v).length ≤ e * expanderSize d e n := by
  have bound := flatMap_length_le_inline_MachineOverlayTable (List.ofFn (expanderValues H v))
    (MachineOverlayRows.rowBits d e v.val) (expanderSize d e n) (by
      intro j hj
      obtain ⟨p, rfl⟩ := List.mem_ofFn.mp hj
      have hrow := MachineOverlayRows.rowBits_length_le d e v.val (expanderValues H v p)
        (n * e) (MachineOverlayRows.expanderValues_bounded H v p)
      have hv := v.isLt.le
      unfold expanderSize MachineOverlayRows.rowSizeBound at *
      omega)
  simpa only [expanderOutput, MachineOverlayRows.streamOutput, List.length_ofFn] using bound

theorem outputBlock_length_le {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) :
    (outputBlock G H v).length ≤ blockSize d e n := by
  have hg := oldOutput_length_le G e v
  have hh := expanderOutput_length_le d H v
  simp only [outputBlock, List.length_append, blockSize]
  omega

def outputCapacity (d e n : Nat) : Nat := n + n * (d + e) + 2 + n * blockSize d e n

theorem outputPrefix_length_le {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (k : Nat) :
    (outputPrefix G H k).length ≤ outputCapacity d e n := by
  have hblocks := flatten_length_le_inline_MachineOverlayTable ((List.ofFn (outputBlock G H)).take k)
    (blockSize d e n) (by
      intro xs hx
      have hm := List.mem_of_mem_take hx
      obtain ⟨v, rfl⟩ := List.mem_ofFn.mp hm
      exact outputBlock_length_le G H v)
  have hlen : ((List.ofFn (outputBlock G H)).take k).length ≤ n := by
    simp only [List.length_take, List.length_ofFn]
    exact Nat.min_le_right _ _
  have hcap := Nat.mul_le_mul_right (blockSize d e n) hlen
  simp only [outputPrefix, emittedPrefix, List.length_append, encodeWords_length,
    List.sum_cons, List.sum_nil, List.length_cons, List.length_nil, Nat.add_zero, outputCapacity]
  omega

def pairBudget (d e n : Nat) : Nat :=
  (d * (MachineLazyRows.rowCostBound (d + e) 0 n (n * d) +
    2 * (outputCapacity d e n + d * oldSize d e n) + 1) + 1) +
  (e * (MachineOverlayRows.rowCostBound d e n (n * e) +
    2 * (outputCapacity d e n + d * oldSize d e n + e * expanderSize d e n) + 1) + 1) +
  (3 * n + 6)

theorem pairSteps_le {n d e : Nat} (G : PortTables.Table n d)
    (H : ExpanderTables.Table n e) (v : Fin n) (out : List Bool)
    (hout : out.length ≤ outputCapacity d e n) : pairSteps G H v out ≤ pairBudget d e n := by
  have hold := MachineLazyRows.vertexSteps_le d (d + e) 0 (oldRows G v) out
  simp only [oldRows, List.length_ofFn] at hold
  have hgsize := oldOutput_length_le G e v
  have hnew := MachineOverlayRows.vertexSteps_le d e v.val (n * e)
    (List.ofFn (expanderValues H v)) (by
      intro j hj
      obtain ⟨p, rfl⟩ := List.mem_ofFn.mp hj
      exact MachineOverlayRows.expanderValues_bounded H v p) (out ++ oldOutput G e v)
  simp only [List.length_ofFn, List.length_append] at hnew
  have hv := v.isLt.le
  have hsize : MachineOverlayRows.rowSizeBound d e v.val (n * e) ≤ expanderSize d e n := by
    unfold expanderSize MachineOverlayRows.rowSizeBound
    omega
  have hcost : MachineOverlayRows.rowCostBound d e v.val (n * e) ≤
      MachineOverlayRows.rowCostBound d e n (n * e) := by
    unfold MachineOverlayRows.rowCostBound
    change _ ≤ 8 * (n * e) + 4 * expanderSize d e n + _ + _ + 28
    omega
  have hsizeMul := Nat.mul_le_mul_left e hsize
  have holdBudget := Nat.mul_le_mul_left d
    (show MachineLazyRows.rowCostBound (d + e) 0 n (n * d) +
        2 * (out.length + d * MachineLazyRows.rowSizeBound (d + e) 0 n (n * d)) + 1 ≤
      MachineLazyRows.rowCostBound (d + e) 0 n (n * d) +
        2 * (outputCapacity d e n + d * oldSize d e n) + 1 by

      unfold oldSize
      omega)
  have hnewBudget := Nat.mul_le_mul_left e
    (show MachineOverlayRows.rowCostBound d e v.val (n * e) +
        2 * (out.length + (oldOutput G e v).length + e * MachineOverlayRows.rowSizeBound d e v.val (n * e)) + 1 ≤
      MachineOverlayRows.rowCostBound d e n (n * e) +
        2 * (outputCapacity d e n + d * oldSize d e n + e * expanderSize d e n) + 1 by omega)
  unfold pairSteps pairBudget oldRows
  omega

theorem loopSteps_le {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e) :
    MachineCountedLoop.totalSteps (loopCost G H) n ≤ n * (pairBudget d e n + 1) + 1 := by
  apply MachineCountedLoop.totalSteps_le
  intro r hr
  rw [loopCost, dite_eq_left hr]
  exact pairSteps_le G H _ _ (outputPrefix_length_le G H _)

noncomputable def phasePolynomial (d e : Nat) : Polynomial Nat :=
  let x := Polynomial.X
  let C := Polynomial.C (R := Nat)
  let oldValue := C (d + e) * (x * C d) + x * C d + C 0
  let os := x + oldValue + C 8194
  let newValue := C (d + e) * (x * C e) + x * C e + C d
  let es := x + newValue + C MachineOverlayRows.trueBits.length + C 2
  let block := C d * os + C e * es
  let cap := x + x * C (d + e) + C 2 + x * block
  let oldCost := C 2 * x + C 8 * (x * C d) + oldValue + C 4 * os + C 8224
  let newCost := C 8 * (x * C e) + C 4 * es + newValue + C MachineOverlayRows.trueBits.length + C 28
  let pair := (C d * (oldCost + C 2 * (cap + C d * os) + C 1) + C 1) +
    (C e * (newCost + C 2 * (cap + C d * os + C e * es) + C 1) + C 1) +
    (C 3 * x + C 6)
  x * (pair + C 1) + C 1 + (C 2 * x + x * C d + C 8)

@[simp] theorem phasePolynomial_eval (d e n : Nat) :
    (phasePolynomial d e).eval n = n * (pairBudget d e n + 1) + 1 + cleanupSteps n (n * d) := by
  simp only [phasePolynomial, Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C,
    Polynomial.eval_X, pairBudget, outputCapacity, blockSize, oldSize, expanderSize,
    MachineLazyRows.rowSizeBound, MachineLazyRows.rowCostBound, MachineLazyRows.valueBound,
    MachineOverlayRows.rowSizeBound, MachineOverlayRows.rowCostBound,
    MachineOverlayRows.valueBound, cleanupSteps]

noncomputable def timePolynomial (d e : Nat) : Polynomial Nat :=
  Polynomial.C 12 * Polynomial.X + Polynomial.C 18 + phasePolynomial d e

theorem tableSteps_le {n d e : Nat} (G : PortTables.Table n d) (H : ExpanderTables.Table n e) :
    tableSteps G H ≤ (timePolynomial d e).eval ((PortTables.tableBits G).length + (rawExpander H).length) := by
  have hsplit := MachineTableSplit.exactSteps_le n (n * d)
    (MachineTableSplit.rowsBits (PortTables.graphTable G))
  have header : encodeWords [n, n * d] ++ MachineTableSplit.rowsBits (PortTables.graphTable G) =
      PortTables.tableBits G := (MachineTableSplit.tableBits_header_rows (PortTables.graphTable G)).symm
  rw [header, MachineTableSplit.timePolynomial_eval] at hsplit
  have hn := PortTables.vertices_le_tableBits_length G
  have hloop := loopSteps_le G H
  have hmono := natPolynomial_eval_mono (phasePolynomial d e)
    (show n ≤ (PortTables.tableBits G).length + (rawExpander H).length by omega)
  rw [phasePolynomial_eval] at hmono
  simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C,
    Polynomial.eval_X]
  unfold tableSteps preparationSteps
  omega

def tableInTime {n : Nat} (d e : Nat) (hd : 0 < d) (he : 0 < e)
    (G : PortTables.Table n d) (H : ExpanderTables.Table n e) (a : A) :
    StateTransition.EvalsToInTime (TM2.step (program d e hd he))
      ⟨some (.inl .copyFirst), clean d e hd he a,
        initialTapes (PortTables.tableBits G) (rawExpander H)⟩
      (some ⟨none, clean d e hd he a,
        memory (PortTables.tableBits G) (rawExpander H) [] [] [] [] [] [] []
          (PortTables.tableBits (PreprocessingOverlayTables.overlay G H))⟩)
      ((timePolynomial d e).eval ((PortTables.tableBits G).length + (rawExpander H).length)) where
  steps := tableSteps G H
  evals_in_steps := tableTrace d e hd he G H a
  steps_le_m := tableSteps_le G H

def machine (d e : Nat) (hd : 0 < d) (he : 0 < e) : FinTM2 where
  K := Tape
  k₀ := graphArchive
  k₁ := output
  Γ _ := Bool
  Λ := Label d e
  main := .inl .copyFirst
  σ := State Unit d e
  initialState := clean d e hd he ()
  m := program d e hd he

theorem machine_alphabet_finite (d e : Nat) (hd : 0 < d) (he : 0 < e) :
    ∀ k, Finite ((machine d e hd he).Γ k) := fun _ => inferInstanceAs (Finite Bool)

end DFVSGames.Foundations.Complexity.MachineOverlayTable
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