lean/OAI/Computability/BinPacking/Computation/MachineRegularInternalRow.lean

import OAI.Computability.BinPacking.Computation.MachineCloudSelect
import OAI.Computability.BinPacking.Computation.MachineVertexPadding
import OAI.Computability.BinPacking.Computation.PoweringGlobalEnumeration

namespace OAI

namespace BinPackingGames.Foundations.Complexity.MachineRegularInternalRow

open Turing MachineComposition
open PCP
open Reduction.MachineSubstitution (pushWord stepAux_pushWord)

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

abbrev Alphabet (_ : K) := Bool

def equalityBits : List Bool :=
  encodeWords (GraphTables.relationWords PreprocessingInternalRows.equalityRelation)

theorem equalityBits_length_le : equalityBits.length ≤ 8192 :=
  GraphTables.relationBits_length_le _

def emittedBits (q x z r : Nat) : List Bool :=
  encodeWord x ++ encodeWord ((q + 1) * z + r) ++ equalityBits

theorem emittedBits_length (q x z r : Nat) :
    (emittedBits q x z r).length = x + ((q + 1) * z + r) + equalityBits.length + 2 := by
  simp only [emittedBits, List.length_append, encodeWord_length]
  omega

theorem emittedBits_eq_row (t : GraphTables.Table) (padding : Fin t.vertices → Nat)
    {q : Nat} (tables : ∀ v, ExpanderTables.Table
      (PreprocessingCloudIndex.cloudSize t v + padding v) q)
    (v : Fin t.vertices) (x : PreprocessingCloudIndex.PaddedCloud t padding v) (p : Fin q) :
    emittedBits q (PreprocessingRegularTables.vertexOrder t padding x.val).val
      (PreprocessingInternalRows.selectedIndex t padding v
        (PreprocessingInternalRows.localStep t padding tables v x p).1)
      (PreprocessingInternalRows.localStep t padding tables v x p).2.val =
        encodeWords (GraphTables.rowWords
          (PreprocessingInternalRows.row t padding tables v x p)) := by
  rw [MachineTableRows.rowBits_eq, PreprocessingInternalRows.row_tail,
    PreprocessingInternalRows.row_reverse_selectedIndex, PreprocessingInternalRows.row_relation]
  rfl

theorem selected_rotation {n q : Nat} (table : ExpanderTables.Table n q)
    (i : Fin n) (p : Fin q) :
    (ExpanderTableWords.rotationWords table)[q * i.val + p.val]? =
      some (q * (ExpanderTables.lookup table (i, p)).1.val +
        (ExpanderTables.lookup table (i, p)).2.val) := by
  simpa only [Nat.add_comm] using ExpanderTableWords.rotationWords_row_order table (i, p)

theorem selected_rotation_div {n q : Nat} (positive : 0 < q)
    (table : ExpanderTables.Table n q) (i : Fin n) (p : Fin q) :
    (q * (ExpanderTables.lookup table (i, p)).1.val +
      (ExpanderTables.lookup table (i, p)).2.val) / q =
        (ExpanderTables.lookup table (i, p)).1.val := by
  rw [Nat.add_comm, Nat.add_mul_div_left _ _ positive,
    Nat.div_eq_of_lt (ExpanderTables.lookup table (i, p)).2.isLt, Nat.zero_add]

theorem selected_rotation_mod {n q : Nat} (_positive : 0 < q)
    (table : ExpanderTables.Table n q) (i : Fin n) (p : Fin q) :
    (q * (ExpanderTables.lookup table (i, p)).1.val +
      (ExpanderTables.lookup table (i, p)).2.val) % q =
        (ExpanderTables.lookup table (i, p)).2.val := by
  rw [Nat.mul_add_mod_self_left,
    Nat.mod_eq_of_lt (ExpanderTables.lookup table (i, p)).2.isLt]

theorem rotationLookupTrace {n q : Nat} (table : ExpanderTables.Table n q)
    (i : Fin n) (p : Fin q) (source : K) (tape : Fin 5 → K)
    (distinct : Function.Injective tape) (outside : ∀ a, source ≠ tape a)
    (labels : MachineAffineLookup.Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (code : ∀ l, program (labels l) =
      MachineAffineLookup.instruction source tape q p.val labels exit l)
    (base : K → List Bool)
    (tableWord : base (tape 0) = encodeWords (ExpanderTableWords.rotationWords table))
    (scratchEmpty : base (tape 4) = []) (suffix : List Bool)
    (sourceWord : base source = encodeWord i.val ++ suffix)
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[MachineAffineLookup.steps
        (ExpanderTableWords.rotationWords table) i.val q p.val]
      (some ⟨some (labels .seed), (ambient, register), base⟩) =
      some ⟨exit, (ambient, none), MachineAffineLookup.finalTapes tape base
        (ExpanderTableWords.rotationWords table) (q * i.val + p.val)
        (q * (ExpanderTables.lookup table (i, p)).1.val +
          (ExpanderTables.lookup table (i, p)).2.val)⟩ :=
  MachineAffineLookup.affineLookupTrace source tape distinct outside q p.val labels exit
    program code base _ tableWord scratchEmpty i.val suffix sourceWord _
    (selected_rotation table i p) ambient register

inductive FinishLabel
  | seedReturn | copyReturn | restoreReturn | affine | restoreTarget | equality
  | row (stage : MachineTableRows.Label)
  | clearReverse | clearEquality
  deriving DecidableEq, Fintype

def finishFields (tape : Fin 8 → K) : Fin 3 → K := fun i =>
  if i = 0 then tape 0 else if i = 1 then tape 4 else tape 5

omit [DecidableEq K] in
@[simp] theorem finishFields_zero (tape : Fin 8 → K) : finishFields tape 0 = tape 0 := rfl
omit [DecidableEq K] in
@[simp] theorem finishFields_one (tape : Fin 8 → K) : finishFields tape 1 = tape 4 := rfl
omit [DecidableEq K] in
@[simp] theorem finishFields_two (tape : Fin 8 → K) : finishFields tape 2 = tape 5 := rfl

def finishInstruction (q : Nat) (tape : Fin 8 → K) (labels : FinishLabel → Λ)
    (exit : Option Λ) : FinishLabel → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)
  | .seedReturn => MachineUnaryAffineAt.seed (tape 4) 0 (labels .copyReturn)
  | .copyReturn => MachineUnaryAffineAt.scan (tape 2) (tape 3) (tape 4) 1
      (labels .copyReturn) (labels .restoreReturn)
  | .restoreReturn => Reduction.MachineTransfer.loopAt (tape 3) (tape 2) id false
      (labels .restoreReturn) (some (labels .affine))
  | .affine => MachineUnaryAffineAt.scan (tape 1) (tape 3) (tape 4) (q + 1)
      (labels .affine) (labels .restoreTarget)
  | .restoreTarget => Reduction.MachineTransfer.loopAt (tape 3) (tape 1) id false
      (labels .restoreTarget) (some (labels .equality))
  | .equality => pushWord (tape 5) equalityBits.reverse
      (.goto fun _ => labels (.row .relationRead))
  | .row stage => MachineTableRows.routine (finishFields tape) (tape 6) (tape 7) (tape 3)
      (fun l => labels (.row l)) (some (labels .clearReverse)) stage
  | .clearReverse => MachineDrain.drain (tape 4) (labels .clearReverse)
      (some (labels .clearEquality))
  | .clearEquality => MachineDrain.drain (tape 5) (labels .clearEquality) exit

def finishSteps (q x z r outputLength : Nat) : Nat :=
  (2 * (r + 1) + 1) + 2 * (z + 1) + 1 +
    (4 * (x + ((q + 1) * z + r) + equalityBits.length + 2) + 2 * outputLength + 9) +
    (((q + 1) * z + r) + 2) + (equalityBits.length + 1)

theorem finishSteps_le (q x z r outputLength : Nat) :
    finishSteps q x z r outputLength ≤
      (5 * q + 7) * z + 7 * r + 4 * x + 2 * outputLength + 40986 := by
  have h := equalityBits_length_le
  unfold finishSteps
  nlinarith

private theorem joinTrace {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 finishTrace (q : Nat) (tape : Fin 8 → K) (distinct : Function.Injective tape)
    (labels : FinishLabel → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (code : ∀ l, program (labels l) = finishInstruction q tape labels exit l)
    (base : K → List Bool) (x z r : Nat)
    (tailWord : base (tape 0) = encodeWord x)
    (targetWord : base (tape 1) = encodeWord z)
    (returnWord : base (tape 2) = encodeWord r)
    (scratchEmpty : base (tape 3) = []) (reverseEmpty : base (tape 4) = [])
    (equalityEmpty : base (tape 5) = []) (rowEmpty : base (tape 6) = [])
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[finishSteps q x z r (base (tape 7)).length]
      (some ⟨some (labels .seedReturn), (ambient, register), base⟩) =
      some ⟨exit, (ambient, none),
        Function.update base (tape 7) (base (tape 7) ++ emittedBits q x z r)⟩ := by
  have hd (i j : Fin 8) (h : i ≠ j) : tape i ≠ tape j := fun e => h (distinct e)
  let w := (q + 1) * z + r
  let copied := Function.update base (tape 4) (encodeWord r)
  let computed := Function.update base (tape 4) (encodeWord w)
  let initialized := Function.update computed (tape 5) equalityBits
  let appended := Function.update initialized (tape 7)
    (base (tape 7) ++ emittedBits q x z r)
  let drained := Function.update appended (tape 4) []
  have copyRun : (advance (TM2.step program))^[2 * (r + 1) + 1]
      (some ⟨some (labels .seedReturn), (ambient, register), base⟩) =
      some ⟨some (labels .affine), (ambient, none), copied⟩ := by
    simpa only [copied, Nat.one_mul, Nat.add_zero, reverseEmpty, List.append_nil] using
      MachineUnaryAffineAt.seededAffineTrace (tape 2) (tape 3) (tape 4)
        (hd 2 3 (by decide)) (hd 2 4 (by decide)) (hd 3 4 (by decide)) 1 0
        (labels .seedReturn) (labels .copyReturn) (labels .restoreReturn)
        (some (labels .affine)) program (code .seedReturn) (code .copyReturn)
        (code .restoreReturn) base r [] (by simpa using returnWord) scratchEmpty ambient register
  have affineFrame (u : Nat) :
      MachineUnaryAffineAt.tapes (tape 1) (tape 3) (tape 4) base
        (encodeWord z ++ []) [] (encodeWord u ++ []) =
        Function.update base (tape 4) (encodeWord u) := by
    simp only [List.append_nil]
    rw [← targetWord, ← scratchEmpty]
    simp only [MachineUnaryAffineAt.tapes, MachineCopy.forkTapes, Function.update_eq_self]
  have affineRun : (advance (TM2.step program))^[2 * (z + 1)]
      (some ⟨some (labels .affine), (ambient, none), copied⟩) =
      some ⟨some (labels .equality), (ambient, none), computed⟩ := by
    simpa only [affineFrame, copied, computed, w] using
      MachineUnaryAffineAt.affineTrace (tape 1) (tape 3) (tape 4)
        (hd 1 3 (by decide)) (hd 1 4 (by decide)) (hd 3 4 (by decide)) (q + 1)
        (labels .affine) (labels .restoreTarget) (some (labels .equality)) program
        (code .affine) (code .restoreTarget) base z r [] [] ambient none
  have initRun : (advance (TM2.step program))^[1]
      (some ⟨some (labels .equality), (ambient, none), computed⟩) =
      some ⟨some (labels (.row .relationRead)), (ambient, none), initialized⟩ := by
    change some (TM2.stepAux (program (labels .equality)) _ _) = _
    rw [code]
    simp [finishInstruction, stepAux_pushWord, TM2.stepAux, computed, initialized,
      hd 5 4 (by decide), equalityEmpty]
  have hfields : ∀ i, finishFields tape i ≠ tape 6 ∧ finishFields tape i ≠ tape 3 := by
    intro i
    fin_cases i
    all_goals exact ⟨hd _ _ (by decide), hd _ _ (by decide)⟩
  have hbits : MachineTableRows.fieldBits (finishFields tape) initialized =
      emittedBits q x z r := by
    simp [MachineTableRows.fieldBits, initialized, computed, finishFields, tailWord,
      hd 0 4 (by decide), hd 0 5 (by decide), hd 4 5 (by decide), emittedBits, w]
  have hsize : MachineTableRows.fieldSize (finishFields tape) initialized =
      x + w + equalityBits.length + 2 := by
    rw [← MachineTableRows.fieldBits_length, hbits, emittedBits_length]
  have hout : initialized (tape 7) = base (tape 7) := by
    simp [initialized, computed, hd 7 4 (by decide), hd 7 5 (by decide)]
  have rowRun : (advance (TM2.step program))^[
      4 * (x + w + equalityBits.length + 2) + 2 * (base (tape 7)).length + 9]
      (some ⟨some (labels (.row .relationRead)), (ambient, none), initialized⟩) =
      some ⟨some (labels .clearReverse), (ambient, none), appended⟩ := by
    simpa only [hbits, hsize, hout, appended] using
      MachineTableRows.appendTrace (finishFields tape) (tape 6) (tape 7) (tape 3)
        hfields (hd 6 7 (by decide)) (hd 6 3 (by decide)) (hd 7 3 (by decide))
        (fun l => labels (.row l)) (some (labels .clearReverse)) program
        (fun l => code (.row l)) initialized
        (by simp [initialized, computed, hd 6 4 (by decide), hd 6 5 (by decide), rowEmpty])
        (by simp [initialized, computed, hd 3 4 (by decide), hd 3 5 (by decide), scratchEmpty])
        ambient none
  have hreverse : appended (tape 4) = encodeWord w := by
    simp [appended, initialized, computed, hd 4 7 (by decide), hd 4 5 (by decide)]
  have reverseRun : (advance (TM2.step program))^[w + 2]
      (some ⟨some (labels .clearReverse), (ambient, none), appended⟩) =
      some ⟨some (labels .clearEquality), (ambient, none), drained⟩ := by
    have h := MachineDrain.drainTrace (tape 4) (labels .clearReverse)
      (some (labels .clearEquality)) program (code .clearReverse)
      appended (appended (tape 4)) ambient none
    rw [Function.update_eq_self, hreverse, encodeWord_length] at h
    exact h
  have hequality : drained (tape 5) = equalityBits := by
    simp [drained, appended, initialized, hd 5 4 (by decide), hd 5 7 (by decide)]
  have finalFrame : Function.update drained (tape 5) [] =
      Function.update base (tape 7) (base (tape 7) ++ emittedBits q x z r) := by
    funext k
    by_cases h4 : k = tape 4
    · subst k
      simp [drained, hd 4 5 (by decide), hd 4 7 (by decide), reverseEmpty]
    · by_cases h5 : k = tape 5
      · subst k
        simp [hd 5 7 (by decide), equalityEmpty]
      · by_cases h7 : k = tape 7
        · subst k
          simp [drained, appended, h4, h5]
        · simp [drained, appended, initialized, computed, h4, h5, h7]
  have equalityRun : (advance (TM2.step program))^[equalityBits.length + 1]
      (some ⟨some (labels .clearEquality), (ambient, none), drained⟩) =
      some ⟨exit, (ambient, none),
        Function.update base (tape 7) (base (tape 7) ++ emittedBits q x z r)⟩ := by
    have h := MachineDrain.drainTrace (tape 5) (labels .clearEquality) exit program
      (code .clearEquality) drained (drained (tape 5)) ambient none
    rw [Function.update_eq_self, hequality, finalFrame] at h
    exact h
  exact joinTrace (joinTrace (joinTrace (joinTrace (joinTrace
    copyRun affineRun) initRun) rowRun) reverseRun) equalityRun

inductive RotorLabel (q : Nat)
  | lookup (stage : MachineAffineLookup.Label)
  | initialize | divide | emit (r : Fin q)
  deriving DecidableEq, Fintype

def rotorLookupTapes (tape : Fin 8 → K) (j : Fin 5) : K :=
  tape ⟨j.val + 1, by omega⟩

omit [DecidableEq K] in
theorem rotorLookupTapes_injective (tape : Fin 8 → K) (h : Function.Injective tape) :
    Function.Injective (rotorLookupTapes tape) := by
  intro i j hij
  have hv := congrArg Fin.val (h hij)
  apply Fin.ext
  simpa only [Fin.val_mk, Nat.add_left_inj] using hv

omit [DecidableEq K] in
theorem rotorLookupTapes_outside (tape : Fin 8 → K) (h : Function.Injective tape)
    (i : Fin 5) : tape 0 ≠ rotorLookupTapes tape i := by
  intro he
  have hv := congrArg Fin.val (h he)
  change 0 = i.val + 1 at hv
  omega

def rotorInstruction (q : Nat) (positive : 0 < q) (p : Fin q)
    (tape : Fin 8 → K) (labels : RotorLabel q → Λ) (exit : Option Λ) :
    RotorLabel q → TM2.Stmt (Alphabet (K := K)) Λ (MachineFixedDivMod.State σ q)
  | .lookup stage => MachineAffineLookup.instruction (tape 0) (rotorLookupTapes tape)
      q p.val (fun l => labels (.lookup l)) (some (labels .initialize)) stage
  | .initialize => .push (tape 6) (fun _ => false)
      (.push (tape 7) (fun _ => false) (.goto fun _ => labels .divide))
  | .divide => MachineFixedDivMod.scanLoop q positive (tape 4) (tape 6)
      (labels .divide) (fun r => labels (.emit r))
  | .emit r => MachineFixedDivMod.emitter q positive (tape 7) exit r

def rotorFinalTapes {n q : Nat} (table : ExpanderTables.Table n q) (i : Fin n) (p : Fin q)
    (tape : Fin 8 → K) (base : K → List Bool) : K → List Bool :=
  MachineFixedDivMod.unaryTapes (tape 4) (tape 6) (tape 7)
    (MachineAffineLookup.finalTapes (rotorLookupTapes tape) base
      (ExpanderTableWords.rotationWords table) (q * i.val + p.val)
      (q * (ExpanderTables.lookup table (i, p)).1.val +
        (ExpanderTables.lookup table (i, p)).2.val))
    0 (ExpanderTables.lookup table (i, p)).1.val
      (ExpanderTables.lookup table (i, p)).2.val [] [] []

def rotorSteps {n q : Nat} (table : ExpanderTables.Table n q) (i : Fin n) (p : Fin q) : Nat :=
  MachineAffineLookup.steps (ExpanderTableWords.rotationWords table) i.val q p.val +
    1 + (q * (ExpanderTables.lookup table (i, p)).1.val +
      (ExpanderTables.lookup table (i, p)).2.val + 2)

theorem rotorTrace {n q : Nat} (positive : 0 < q) (table : ExpanderTables.Table n q)
    (i : Fin n) (p : Fin q) (tape : Fin 8 → K) (distinct : Function.Injective tape)
    (labels : RotorLabel q → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (MachineFixedDivMod.State σ q))
    (code : ∀ l, program (labels l) = rotorInstruction q positive p tape labels exit l)
    (base : K → List Bool)
    (sourceWord : base (tape 0) = encodeWord i.val)
    (tableWord : base (tape 1) = encodeWords (ExpanderTableWords.rotationWords table))
    (flatEmpty : base (tape 4) = []) (scratchEmpty : base (tape 5) = [])
    (localEmpty : base (tape 6) = []) (returnEmpty : base (tape 7) = [])
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[rotorSteps table i p]
      (some ⟨some (labels (.lookup .seed)),
        ((ambient, MachineFixedDivMod.residue q positive 0), register), base⟩) =
      some ⟨exit, ((ambient, MachineFixedDivMod.residue q positive 0), none),
        rotorFinalTapes table i p tape base⟩ := by
  have hd (a b : Fin 8) (h : a ≠ b) : tape a ≠ tape b := fun e => h (distinct e)
  let a := q * (ExpanderTables.lookup table (i, p)).1.val +
    (ExpanderTables.lookup table (i, p)).2.val
  let looked := MachineAffineLookup.finalTapes (rotorLookupTapes tape) base
    (ExpanderTableWords.rotationWords table) (q * i.val + p.val) a
  let initialized := Function.update (Function.update looked (tape 6) (encodeWord 0))
    (tape 7) (encodeWord 0)
  have lookupRun : (advance (TM2.step program))^[MachineAffineLookup.steps
      (ExpanderTableWords.rotationWords table) i.val q p.val]
      (some ⟨some (labels (.lookup .seed)),
        ((ambient, MachineFixedDivMod.residue q positive 0), register), base⟩) =
      some ⟨some (labels .initialize),
        ((ambient, MachineFixedDivMod.residue q positive 0), none), looked⟩ := by
    exact rotationLookupTrace table i p (tape 0) (rotorLookupTapes tape)
      (rotorLookupTapes_injective tape distinct) (rotorLookupTapes_outside tape distinct)
      (fun l => labels (.lookup l)) (some (labels .initialize)) program
      (fun l => code (.lookup l)) base tableWord scratchEmpty []
      (by simpa using sourceWord) (ambient, MachineFixedDivMod.residue q positive 0) register
  have lookedOther (k : Fin 8) (h2 : k ≠ 2) (h3 : k ≠ 3) (h4 : k ≠ 4) :
      looked (tape k) = base (tape k) := by
    exact MachineAffineLookup.finalTapes_other (rotorLookupTapes tape) base
      (ExpanderTableWords.rotationWords table) (q * i.val + p.val) a (tape k)
      (hd k 2 h2) (hd k 3 h3) (hd k 4 h4)
  have lookedFlat : looked (tape 4) = encodeWord a := by
    change MachineAffineLookup.finalTapes (rotorLookupTapes tape) base
      (ExpanderTableWords.rotationWords table) (q * i.val + p.val) a
      (rotorLookupTapes tape 3) = _
    rw [MachineAffineLookup.finalTapes_output]
    change encodeWord a ++ base (tape 4) = _
    rw [flatEmpty, List.append_nil]
  have initRun : (advance (TM2.step program))^[1]
      (some ⟨some (labels .initialize),
        ((ambient, MachineFixedDivMod.residue q positive 0), none), looked⟩) =
      some ⟨some (labels .divide),
        ((ambient, MachineFixedDivMod.residue q positive 0), none), initialized⟩ := by
    change some (TM2.stepAux (program (labels .initialize)) _ _) = _
    rw [code]
    simp [rotorInstruction, TM2.stepAux, initialized, encodeWord,
      lookedOther 6 (by decide) (by decide) (by decide),
      lookedOther 7 (by decide) (by decide) (by decide),
      hd 7 6 (by decide), localEmpty, returnEmpty]
  have initialFrame : MachineFixedDivMod.unaryTapes (tape 4) (tape 6) (tape 7)
      looked a 0 0 [] [] [] = initialized := by
    simp only [MachineFixedDivMod.unaryTapes, MachineFixedDivMod.tapes,
      MachineCopy.forkTapes, List.append_nil, ← lookedFlat, Function.update_eq_self,
      initialized]
  have finalFrame : MachineFixedDivMod.unaryTapes (tape 4) (tape 6) (tape 7)
      looked 0 (a / q) (a % q) [] [] [] = rotorFinalTapes table i p tape base := by
    dsimp only [a]
    rw [selected_rotation_div positive, selected_rotation_mod positive]
    rfl
  have divideRun : (advance (TM2.step program))^[a + 2]
      (some ⟨some (labels .divide),
        ((ambient, MachineFixedDivMod.residue q positive 0), none), initialized⟩) =
      some ⟨exit, ((ambient, MachineFixedDivMod.residue q positive 0), none),
        rotorFinalTapes table i p tape base⟩ := by
    simpa only [initialFrame, finalFrame] using
      MachineFixedDivMod.divModTrace q positive (tape 4) (tape 6) (tape 7)
        (hd 4 6 (by decide)) (hd 4 7 (by decide)) (hd 6 7 (by decide))
        (labels .divide) (fun r => labels (.emit r)) exit program (code .divide)
        (fun r => code (.emit r)) looked a [] [] [] ambient none
  exact joinTrace (joinTrace lookupRun initRun) divideRun

theorem rotorSteps_le {n q : Nat} (table : ExpanderTables.Table n q) (i : Fin n) (p : Fin q) :
    rotorSteps table i p ≤ 2 * i.val +
      5 * (encodeWords (ExpanderTableWords.rotationWords table)).length + n * q + 8 := by
  have hl := MachineAffineLookup.steps_le (ExpanderTableWords.rotationWords table)
    i.val q p.val _ (selected_rotation table i p)
  have hb := (ExpanderTables.rowIndex n q (ExpanderTables.lookup table (i, p))).isLt
  rw [ExpanderTables.rowIndex_val] at hb
  unfold rotorSteps
  omega

inductive SelectExtraTape
  | localIndex | owner | cloudSize | prefixOffset | darts
  | savedLeft | savedRight | localWork | sizeWork
  deriving DecidableEq

protected abbrev SelectExtraTape.enumList : List SelectExtraTape := [.localIndex, .owner,
  .cloudSize, .prefixOffset, .darts, .savedLeft, .savedRight, .localWork, .sizeWork]

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

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

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

abbrev SelectTape := MachineCloudSelect.Tape ⊕ SelectExtraTape
abbrev RawSelectAlphabet : SelectTape → Type :=
  MachineEmbedding.Alphabet MachineCloudSelect.Alphabet (fun _ : SelectExtraTape => Bool)
abbrev SelectAlphabet (_ : SelectTape) := Bool

theorem selectAlphabet_eq : RawSelectAlphabet = SelectAlphabet := by
  funext k
  cases k with
  | inl k => cases k <;> rfl
  | inr k => rfl

def selectToBoolWord : (k : SelectTape) → List (RawSelectAlphabet k) → List Bool
  | .inl (.inl _), word => word
  | .inl (.inr _), word => word
  | .inr _, word => word

def selectToBoolTapes (base : ∀ k, List (RawSelectAlphabet k)) : SelectTape → List Bool :=
  fun k => selectToBoolWord k (base k)

private theorem alphabet_tapes_apply {K : Type} {Γ Δ : K → Type}
    (h : Γ = Δ) (base : ∀ k, List (Γ k)) (k : K) :
    MachineAlphabetTransport.tapes h base k =
      Eq.mp (congrArg (fun alphabet => List (alphabet k)) h) (base k) := by
  cases h
  rfl

theorem selectToBoolTapes_eq (base : ∀ k, List (RawSelectAlphabet k)) :
    selectToBoolTapes base = MachineAlphabetTransport.tapes selectAlphabet_eq base := by
  funext k
  rw [alphabet_tapes_apply]
  cases k with
  | inl k => cases k <;> rfl
  | inr k => rfl

abbrev SelectState (σ : Type) := MachineUnaryLessAt.State (σ × MachineCloudSelect.Phase)

def selectStateEquiv (σ : Type) : (MachineCloudSelect.State σ × Unit) ≃ SelectState σ where
  toFun s := (((s.1.1.1.1, s.1.2), s.1.1.1.2), s.1.1.2)
  invFun s := ((((s.1.1.1, s.1.2), s.2), s.1.1.2), ())
  left_inv s := by rcases s with ⟨⟨⟨⟨a, b⟩, c⟩, d⟩, ⟨⟩⟩; rfl
  right_inv s := by rcases s with ⟨⟨⟨a, b⟩, c⟩, d⟩; rfl

def selectScratch : SelectTape := .inl (.inl .scratch)
def selectQuery : SelectTape := .inl (.inl .target)
def selectOutput : SelectTape := .inl (.inr ())

def compareSlots : Fin 4 ↪ SelectTape where
  toFun i := ![.inr .localIndex, .inr .cloudSize, .inr .savedLeft, .inr .savedRight] i
  inj' := by intro i j h; fin_cases i <;> fin_cases j <;> simp_all

def subtractSlots : Fin 4 ↪ SelectTape where
  toFun i := ![.inr .localWork, .inr .sizeWork, .inr .savedLeft, .inr .savedRight] i
  inj' := by intro i j h; fin_cases i <;> fin_cases j <;> simp_all

inductive SelectCopy
  | oldLocal | oldOwner | dummyLocal | dummySize | dummyDifference
  deriving DecidableEq

protected abbrev SelectCopy.enumList : List SelectCopy := [.oldLocal, .oldOwner, .dummyLocal,
  .dummySize, .dummyDifference]

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

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

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

inductive SelectAdd
  | prefixOffset | darts
  deriving DecidableEq

protected abbrev SelectAdd.enumList : List SelectAdd := [.prefixOffset, .darts]

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

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

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

inductive SelectAddStage
  | scan | restore
  deriving DecidableEq

protected abbrev SelectAddStage.enumList : List SelectAddStage := [.scan, .restore]

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

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

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

inductive SelectControl
  | compare (stage : MachineUnaryLessAt.Label)
  | choose
  | copy (kind : SelectCopy) (stage : MachineUnaryAffineAt.Label)
  | subtract
  | add (kind : SelectAdd) (stage : SelectAddStage)
  | cleanup (i : Fin 5)
  deriving DecidableEq, Fintype

abbrev SelectLabel := MachineCloudSelect.Label ⊕ SelectControl

def copySource : SelectCopy → SelectTape
  | .oldLocal | .dummyLocal => .inr .localIndex
  | .oldOwner => .inr .owner
  | .dummySize => .inr .cloudSize
  | .dummyDifference => .inr .localWork

def copyDestination : SelectCopy → SelectTape
  | .oldLocal | .oldOwner => selectQuery
  | .dummyLocal => .inr .localWork
  | .dummySize => .inr .sizeWork
  | .dummyDifference => selectOutput

def afterCopy : SelectCopy → SelectLabel
  | .oldLocal => .inr (.copy .oldOwner .seed)
  | .oldOwner => .inl (.inr .initialize)
  | .dummyLocal => .inr (.copy .dummySize .seed)
  | .dummySize => .inr .subtract
  | .dummyDifference => .inr (.add .prefixOffset .scan)

def addSource : SelectAdd → SelectTape
  | .prefixOffset => .inr .prefixOffset
  | .darts => .inr .darts

def afterAdd : SelectAdd → SelectLabel
  | .prefixOffset => .inr (.add .darts .scan)
  | .darts => .inr (.cleanup 0)

def cleanupTape (i : Fin 5) : SelectTape :=
  ![selectQuery, .inr .localWork, .inr .sizeWork, .inr .savedLeft, .inr .savedRight] i

def afterCleanup (i : Fin 5) : Option SelectLabel :=
  if h : i.val + 1 < 5 then some (.inr (.cleanup ⟨i.val + 1, h⟩)) else none

def selectControl : SelectControl → TM2.Stmt SelectAlphabet SelectLabel (SelectState σ)
  | .compare stage => MachineUnaryLessAt.statement compareSlots
      (fun s => .inr (.compare s)) (some (.inr .choose)) stage
  | .choose => .branch (fun s => s.1.2)
      (.load (fun s => ((s.1.1, false), none))
        (.goto fun _ => .inr (.copy .oldLocal .seed)))
      (.load (fun s => ((s.1.1, false), none))
        (.goto fun _ => .inr (.copy .dummyLocal .seed)))
  | .copy kind .seed => MachineUnaryAffineAt.seed (copyDestination kind) 0
      (.inr (.copy kind .scan))
  | .copy kind .scan => MachineUnaryAffineAt.scan (copySource kind) selectScratch
      (copyDestination kind) 1 (.inr (.copy kind .scan)) (.inr (.copy kind .restore))
  | .copy kind .restore => Reduction.MachineTransfer.loopAt selectScratch (copySource kind)
      id false (.inr (.copy kind .restore)) (some (afterCopy kind))
  | .subtract => MachineUnaryLessAt.scan subtractSlots (.inr .subtract)
      (.inr (.copy .dummyDifference .seed))
  | .add kind .scan => MachineUnaryAffineAt.scan (addSource kind) selectScratch selectOutput 1
      (.inr (.add kind .scan)) (.inr (.add kind .restore))
  | .add kind .restore => Reduction.MachineTransfer.loopAt selectScratch (addSource kind)
      id false (.inr (.add kind .restore)) (some (afterAdd kind))
  | .cleanup i => MachineDrain.drain (cleanupTape i) (.inr (.cleanup i)) (afterCleanup i)

def rawSelectProgram : SelectLabel → TM2.Stmt RawSelectAlphabet SelectLabel (SelectState σ) :=
  MachineStateEquiv.program (selectStateEquiv σ)
    (MachineEmbedding.program (some (.inr (.cleanup 0))) MachineCloudSelect.program
      (fun control => MachineStateEquiv.statement (selectStateEquiv σ).symm
        (MachineAlphabetTransport.statement selectAlphabet_eq.symm (selectControl control))))

def selectProgram : SelectLabel → TM2.Stmt SelectAlphabet SelectLabel (SelectState σ) :=
  MachineAlphabetTransport.program selectAlphabet_eq rawSelectProgram

@[simp] theorem selectProgram_control (l : SelectControl) :
    selectProgram (σ := σ) (.inr l) = selectControl l := by
  change MachineAlphabetTransport.statement selectAlphabet_eq
    (MachineStateEquiv.statement (selectStateEquiv σ)
      (MachineStateEquiv.statement (selectStateEquiv σ).symm
        (MachineAlphabetTransport.statement selectAlphabet_eq.symm (selectControl l)))) = _
  have h := MachineStateEquiv.statement_symm_statement (selectStateEquiv σ).symm
    (MachineAlphabetTransport.statement selectAlphabet_eq.symm (selectControl (σ := σ) l))
  simp only [Equiv.symm_symm] at h
  rw [h]
  exact MachineAlphabetTransport.statement_roundtrip selectAlphabet_eq _

def selectCleanState (ambient : σ) (register : Option Bool) : SelectState σ :=
  (((ambient, .checking), false), register)

theorem selectComparisonTrace (base : SelectTape → List Bool) (j k : Nat)
    (hj : base (.inr .localIndex) = encodeWord j)
    (hk : base (.inr .cloudSize) = encodeWord k)
    (hl : base (.inr .savedLeft) = []) (hr : base (.inr .savedRight) = [])
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step selectProgram))^[MachineUnaryLessAt.steps j k + 1]
      (some ⟨some (.inr (.compare .scan)), selectCleanState ambient register, base⟩) =
      some ⟨some (.inr (.copy (if j < k then .oldLocal else .dummyLocal) .seed)),
        selectCleanState ambient none, base⟩ := by
  have h := MachineUnaryLessAt.lessThanTrace compareSlots
    (fun l => .inr (.compare l)) (some (.inr .choose)) selectProgram
    (fun l => by rw [selectProgram_control]; rfl)
    base j k [] [] (by change base (.inr .localIndex) = _; simpa only [List.append_nil] using hj)
    (by change base (.inr .cloudSize) = _; simpa only [List.append_nil] using hk) hl hr
    (ambient, MachineCloudSelect.Phase.checking) false register
  simp only [selectCleanState]
  rw [Function.iterate_succ_apply', h, advance_some]
  change some (TM2.stepAux (selectProgram (.inr .choose)) _ _) = _
  rw [selectProgram_control]
  by_cases hlt : j < k <;> simp [selectControl, TM2.stepAux, hlt]

theorem selectCopyTrace (kind : SelectCopy) (base : SelectTape → List Bool)
    (a : Nat) (hword : base (copySource kind) = encodeWord a)
    (hscratch : base selectScratch = []) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step selectProgram))^[2 * (a + 1) + 1]
      (some ⟨some (.inr (.copy kind .seed)), selectCleanState ambient register, base⟩) =
      some ⟨some (afterCopy kind), selectCleanState ambient none,
        Function.update base (copyDestination kind)
          (encodeWord a ++ base (copyDestination kind))⟩ := by
  have h₁ : copySource kind ≠ selectScratch := by cases kind <;> decide
  have h₂ : copySource kind ≠ copyDestination kind := by cases kind <;> decide
  have h₃ : selectScratch ≠ copyDestination kind := by cases kind <;> decide
  simpa only [Nat.one_mul, Nat.add_zero, selectCleanState] using
    MachineUnaryAffineAt.seededAffineTrace (copySource kind) selectScratch (copyDestination kind)
      h₁ h₂ h₃ 1 0 (.inr (.copy kind .seed)) (.inr (.copy kind .scan))
      (.inr (.copy kind .restore)) (some (afterCopy kind)) selectProgram
      (by rw [selectProgram_control]; rfl) (by rw [selectProgram_control]; rfl)
      (by rw [selectProgram_control]; rfl) base a [] (by simpa using hword) hscratch
      ((ambient, MachineCloudSelect.Phase.checking), false) register

theorem selectAddTrace (kind : SelectAdd) (base : SelectTape → List Bool)
    (a b : Nat) (hword : base (addSource kind) = encodeWord a)
    (houtput : base selectOutput = encodeWord b) (hscratch : base selectScratch = [])
    (ambient : σ) :
    (advance (TM2.step selectProgram))^[2 * (a + 1)]
      (some ⟨some (.inr (.add kind .scan)), selectCleanState ambient none, base⟩) =
      some ⟨some (afterAdd kind), selectCleanState ambient none,
        Function.update base selectOutput (encodeWord (a + b))⟩ := by
  have h₁ : addSource kind ≠ selectScratch := by cases kind <;> decide
  have h₂ : addSource kind ≠ selectOutput := by cases kind <;> decide
  have h₃ : selectScratch ≠ selectOutput := by decide
  have frame (n : Nat) : MachineUnaryAffineAt.tapes (addSource kind) selectScratch selectOutput
      base (encodeWord a ++ []) [] (encodeWord n ++ []) =
        Function.update base selectOutput (encodeWord n) := by
    simp only [List.append_nil]
    rw [← hword, ← hscratch]
    simp only [MachineUnaryAffineAt.tapes, MachineCopy.forkTapes, Function.update_eq_self]
  have h := MachineUnaryAffineAt.affineTrace (addSource kind) selectScratch selectOutput
    h₁ h₂ h₃ 1 (.inr (.add kind .scan)) (.inr (.add kind .restore))
    (some (afterAdd kind)) selectProgram
    (by rw [selectProgram_control]; rfl) (by rw [selectProgram_control]; rfl)
    base a b [] [] ((ambient, MachineCloudSelect.Phase.checking), false) none
  rw [frame, frame] at h
  rw [← houtput, Function.update_eq_self] at h
  simpa only [Nat.one_mul, selectCleanState] using h

abbrev CoreTape := Fin 27
abbrev CoreState (σ : Type) (q : Nat) :=
  (σ × (MachineFixedBlockMap.Buffer 4096 ×
    (Fin q × (Bool × MachineCloudSelect.Phase)))) × Option Bool

inductive CoreLabel (q : Nat)
  | rotor (stage : RotorLabel q)
  | select (stage : SelectLabel)
  | finish (stage : FinishLabel)
  | cleanup (i : Fin 10)
  deriving DecidableEq, Fintype

def coreEntry (q : Nat) : CoreLabel q := .rotor (.lookup .seed)

theorem selectCloudTrace (t : GraphTables.Table) (v : Fin t.vertices)
    (i : Fin (PreprocessingCloudIndex.cloudSize t v))
    (extra : SelectExtraTape → List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step selectProgram))^[MachineCloudSelect.cloudSelectSteps t v i []]
      (some ⟨some (.inl (.inr .initialize)), selectCleanState ambient register,
        selectToBoolTapes (MachineEmbedding.tapes
          (MachineCloudSelect.memory (GraphTables.tableBits t) []
            (MachineCloudSelect.queryWord v.val i.val []) [] [] [] []) extra)⟩) =
      some ⟨some (.inr (.cleanup 0)), selectCleanState ambient none,
        selectToBoolTapes (MachineEmbedding.tapes
          (MachineCloudSelect.memory (GraphTables.tableBits t) []
            (MachineCloudSelect.queryWord v.val i.val []) [] [] []
            (encodeWord (PreprocessingCloudIndex.cloudSelect t v i).val.val)) extra)⟩ := by
  let source := MachineCloudSelect.program (σ := σ)
  let more := fun c => MachineStateEquiv.statement (selectStateEquiv σ).symm
    (MachineAlphabetTransport.statement selectAlphabet_eq.symm (selectControl (σ := σ) c))
  let caller := MachineEmbedding.program (some (.inr (.cleanup 0))) source more
  let embed := MachineEmbedding.configuration (K := MachineCloudSelect.Tape)
    (Γ := MachineCloudSelect.Alphabet) (σ := MachineCloudSelect.State σ)
    (some (Sum.inr (SelectControl.cleanup 0)) : Option SelectLabel) () extra
  have sourceRun := MachineCloudSelect.cloudSelectTrace t v i [] [] ambient register
  have lifted := liftSuccessfulTrace (TM2.step source) (TM2.step caller) embed
    (MachineEmbedding.step_simulation (some (.inr (.cleanup 0))) () extra source more)
    _ _ _ sourceRun
  have transported := MachineStateEquiv.trace (selectStateEquiv σ) caller _ _ _ lifted
  have boolean := MachineAlphabetTransport.successfulTrace selectAlphabet_eq
    (MachineStateEquiv.program (selectStateEquiv σ) caller) _ _ _ transported
  simp only [embed, MachineCloudSelect.cfg, MachineStateEquiv.configuration, MachineEmbedding.configuration,
    MachineEmbedding.label, selectStateEquiv, List.append_nil,
    MachineAlphabetTransport.configuration_mk, ← selectToBoolTapes_eq] at boolean
  convert boolean using 1 <;> rfl

def selectExtraMemory (j v k offset m : Nat)
    (savedLeft savedRight localWork sizeWork : List Bool) :
    SelectExtraTape → List Bool
  | .localIndex => encodeWord j
  | .owner => encodeWord v
  | .cloudSize => encodeWord k
  | .prefixOffset => encodeWord offset
  | .darts => encodeWord m
  | .savedLeft => savedLeft
  | .savedRight => savedRight
  | .localWork => localWork
  | .sizeWork => sizeWork

def selectMemory (graph : List Bool) (j v k offset m : Nat)
    (query output savedLeft savedRight localWork sizeWork : List Bool) :
    SelectTape → List Bool :=
  selectToBoolTapes (MachineEmbedding.tapes
    (MachineCloudSelect.memory graph [] query [] [] [] output)
    (selectExtraMemory j v k offset m savedLeft savedRight localWork sizeWork))

@[simp] theorem selectMemory_query (graph : List Bool) (j v k o m : Nat)
    (query output a b c d : List Bool) :
    selectMemory graph j v k o m query output a b c d selectQuery = query := rfl

@[simp] theorem selectMemory_output (graph : List Bool) (j v k o m : Nat)
    (query output a b c d : List Bool) :
    selectMemory graph j v k o m query output a b c d selectOutput = output := rfl

@[simp] theorem selectMemory_scratch (graph : List Bool) (j v k o m : Nat)
    (query output a b c d : List Bool) :
    selectMemory graph j v k o m query output a b c d selectScratch = [] := rfl

@[simp] theorem selectMemory_extra (graph : List Bool) (j v k o m : Nat)
    (query output a b c d : List Bool) (z : SelectExtraTape) :
    selectMemory graph j v k o m query output a b c d (.inr z) =
      selectExtraMemory j v k o m a b c d z := rfl

@[simp] theorem update_selectMemory_query (graph : List Bool) (j v k o m : Nat)
    (query output a b c d word : List Bool) :
    Function.update (selectMemory graph j v k o m query output a b c d) selectQuery word =
      selectMemory graph j v k o m word output a b c d := by
  funext z
  cases z with
  | inl z =>
    cases z with
    | inl z => cases z <;> rfl
    | inr z => cases z; rfl
  | inr z => cases z <;> rfl

@[simp] theorem update_selectMemory_output (graph : List Bool) (j v k o m : Nat)
    (query output a b c d word : List Bool) :
    Function.update (selectMemory graph j v k o m query output a b c d) selectOutput word =
      selectMemory graph j v k o m query word a b c d := by
  funext z
  cases z with
  | inl z =>
    cases z with
    | inl z => cases z <;> rfl
    | inr z => cases z; rfl
  | inr z => cases z <;> rfl

@[simp] theorem update_selectMemory_local (graph : List Bool) (j v k o m : Nat)
    (query output a b c d word : List Bool) :
    Function.update (selectMemory graph j v k o m query output a b c d) (.inr .localWork) word =
      selectMemory graph j v k o m query output a b word d := by
  funext z
  cases z with
  | inl z =>
    cases z with
    | inl z => cases z <;> rfl
    | inr z => cases z; rfl
  | inr z => cases z <;> rfl

@[simp] theorem update_selectMemory_size (graph : List Bool) (j v k o m : Nat)
    (query output a b c d word : List Bool) :
    Function.update (selectMemory graph j v k o m query output a b c d) (.inr .sizeWork) word =
      selectMemory graph j v k o m query output a b c word := by
  funext z
  cases z with
  | inl z =>
    cases z with
    | inl z => cases z <;> rfl
    | inr z => cases z; rfl
  | inr z => cases z <;> rfl

@[simp] theorem update_selectMemory_left (graph : List Bool) (j v k o m : Nat)
    (query output a b c d word : List Bool) :
    Function.update (selectMemory graph j v k o m query output a b c d) (.inr .savedLeft) word =
      selectMemory graph j v k o m query output word b c d := by
  funext z
  cases z with
  | inl z =>
    cases z with
    | inl z => cases z <;> rfl
    | inr z => cases z; rfl
  | inr z => cases z <;> rfl

@[simp] theorem update_selectMemory_right (graph : List Bool) (j v k o m : Nat)
    (query output a b c d word : List Bool) :
    Function.update (selectMemory graph j v k o m query output a b c d) (.inr .savedRight) word =
      selectMemory graph j v k o m query output a word c d := by
  funext z
  cases z with
  | inl z =>
    cases z with
    | inl z => cases z <;> rfl
    | inr z => cases z; rfl
  | inr z => cases z <;> rfl

theorem selectCleanupTrace (graph : List Bool) (j v k o m : Nat)
    (query output a b c d : List Bool) (ambient : σ) :
    (advance (TM2.step selectProgram))^[query.length + c.length + d.length + a.length + b.length + 5]
      (some ⟨some (.inr (.cleanup 0)), selectCleanState ambient none,
        selectMemory graph j v k o m query output a b c d⟩) =
      some ⟨none, selectCleanState ambient none,
        selectMemory graph j v k o m [] output [] [] [] []⟩ := by
  have h0 := MachineDrain.drainTrace selectQuery (.inr (.cleanup 0))
    (some (.inr (.cleanup 1))) selectProgram
    (by rw [selectProgram_control]; rfl)
    (selectMemory graph j v k o m query output a b c d) query
    ((ambient, MachineCloudSelect.Phase.checking), false) none
  simp only [update_selectMemory_query] at h0
  have h1 := MachineDrain.drainTrace (.inr .localWork) (.inr (.cleanup 1))
    (some (.inr (.cleanup 2))) selectProgram
    (by rw [selectProgram_control]; rfl)
    (selectMemory graph j v k o m [] output a b c d) c
    ((ambient, MachineCloudSelect.Phase.checking), false) none
  simp only [update_selectMemory_local] at h1
  have h2 := MachineDrain.drainTrace (.inr .sizeWork) (.inr (.cleanup 2))
    (some (.inr (.cleanup 3))) selectProgram
    (by rw [selectProgram_control]; rfl)
    (selectMemory graph j v k o m [] output a b [] d) d
    ((ambient, MachineCloudSelect.Phase.checking), false) none
  simp only [update_selectMemory_size] at h2
  have h3 := MachineDrain.drainTrace (.inr .savedLeft) (.inr (.cleanup 3))
    (some (.inr (.cleanup 4))) selectProgram
    (by rw [selectProgram_control]; rfl)
    (selectMemory graph j v k o m [] output a b [] []) a
    ((ambient, MachineCloudSelect.Phase.checking), false) none
  simp only [update_selectMemory_left] at h3
  have h4 := MachineDrain.drainTrace (.inr .savedRight) (.inr (.cleanup 4))
    none selectProgram (by rw [selectProgram_control]; rfl)
    (selectMemory graph j v k o m [] output [] b [] []) b
    ((ambient, MachineCloudSelect.Phase.checking), false) none
  simp only [update_selectMemory_right] at h4
  have total := joinTrace (joinTrace (joinTrace (joinTrace h0 h1) h2) h3) h4
  have hn : query.length + c.length + d.length + a.length + b.length + 5 =
      ((((query.length + 1) + (c.length + 1)) + (d.length + 1)) + (a.length + 1)) +
        (b.length + 1) := by omega
  rw [hn]
  exact total

def oldSelectSteps (t : GraphTables.Table) (v : Fin t.vertices)
    (i : Fin (PreprocessingCloudIndex.cloudSize t v)) : Nat :=
  (2 * (i.val + 1) + 1) + (2 * (v.val + 1) + 1) +
    MachineCloudSelect.cloudSelectSteps t v i [] + (v.val + i.val + 7)

theorem selectOldTrace (t : GraphTables.Table) (v : Fin t.vertices)
    (i : Fin (PreprocessingCloudIndex.cloudSize t v)) (o : Nat) (ambient : σ) :
    (advance (TM2.step selectProgram))^[oldSelectSteps t v i]
      (some ⟨some (.inr (.copy .oldLocal .seed)), selectCleanState ambient none,
        selectMemory (GraphTables.tableBits t) i.val v.val
          (PreprocessingCloudIndex.cloudSize t v) o t.darts [] [] [] [] [] []⟩) =
      some ⟨none, selectCleanState ambient none,
        selectMemory (GraphTables.tableBits t) i.val v.val
          (PreprocessingCloudIndex.cloudSize t v) o t.darts []
          (encodeWord (PreprocessingCloudIndex.cloudSelect t v i).val.val) [] [] [] []⟩ := by
  let graph := GraphTables.tableBits t
  let k := PreprocessingCloudIndex.cloudSize t v
  have hi := selectCopyTrace .oldLocal
    (selectMemory graph i.val v.val k o t.darts [] [] [] [] [] [])
    i.val rfl rfl ambient none
  simp only [copyDestination, afterCopy, selectMemory_query, update_selectMemory_query, List.append_nil] at hi
  have hv := selectCopyTrace .oldOwner
    (selectMemory graph i.val v.val k o t.darts (encodeWord i.val) [] [] [] [] [])
    v.val rfl rfl ambient none
  simp only [copyDestination, afterCopy, selectMemory_query, update_selectMemory_query] at hv
  have hc := selectCloudTrace t v i (selectExtraMemory i.val v.val k o t.darts [] [] [] [])
    ambient none
  have hq : MachineCloudSelect.queryWord v.val i.val [] =
      encodeWord v.val ++ encodeWord i.val := by
    simp only [MachineCloudSelect.queryWord, MachineCloudRank.queryWord, List.append_nil]
  change (advance (TM2.step selectProgram))^[MachineCloudSelect.cloudSelectSteps t v i []]
    (some ⟨some (.inl (.inr .initialize)), selectCleanState ambient none,
      selectMemory graph i.val v.val k o t.darts
        (MachineCloudSelect.queryWord v.val i.val []) [] [] [] [] []⟩) =
    some ⟨some (.inr (.cleanup 0)), selectCleanState ambient none,
      selectMemory graph i.val v.val k o t.darts
        (MachineCloudSelect.queryWord v.val i.val [])
        (encodeWord (PreprocessingCloudIndex.cloudSelect t v i).val.val) [] [] [] []⟩ at hc
  rw [hq] at hc
  have hd := selectCleanupTrace graph i.val v.val k o t.darts
    (encodeWord v.val ++ encodeWord i.val)
    (encodeWord (PreprocessingCloudIndex.cloudSelect t v i).val.val) [] [] [] [] ambient
  have hn : (encodeWord v.val ++ encodeWord i.val).length + ([] : List Bool).length + ([] : List Bool).length +
      ([] : List Bool).length + ([] : List Bool).length + 5 = v.val + i.val + 7 := by
    simp only [List.length_append, encodeWord_length, List.length_nil]
    omega
  rw [hn] at hd
  exact joinTrace (joinTrace (joinTrace hi hv) hc) hd

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

end BinPackingGames.Foundations.Complexity.MachineRegularInternalRow

end OAI