lean/OAI/Computability/DirectedFeedback/Machines/MachineDummyRows.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.MachineTableRows

namespace OAI

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namespace DFVSGames.Foundations.Complexity.MachineDummyRows

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

def trueRelation : RelationTable := Vector.replicate 4096 true

def trueBits : List Bool := encodeWords (relationWords trueRelation)

theorem trueBits_length : trueBits.length = 8192 := by
  simp only [trueBits, encodeWords_length, relationWords, trueRelation,
    Vector.toList_replicate, List.map_replicate, List.sum_replicate_nat,
    List.length_replicate] ; rfl

def rowBits (v e : Nat) : List Bool := encodeWord v ++ encodeWord e ++ trueBits

theorem rowBits_length (v e : Nat) : (rowBits v e).length = v + e + 8194 := by
  simp [rowBits, trueBits_length, encodeWord_length]
  omega

theorem rowBits_eq_graph_row {n m : Nat} (v : Fin n) (e : Fin m) :
    rowBits v.val e.val = encodeWords (rowWords (⟨v, e, trueRelation⟩ : DartRow n m)) := by
  rw [MachineTableRows.rowBits_eq]
  rfl

def rowsBits (v e : Nat) : Nat → List Bool
  | 0 => []
  | count + 1 => rowBits v e ++ rowsBits v (e + 1) count

abbrev Label (d : Nat) := Unit ⊕ (Fin d × (MachineTableRows.Label ⊕ Unit))

def start (d : Nat) : Label d := .inl ()
def rowLabel {d : Nat} (p : Fin d) (stage : MachineTableRows.Label) : Label d :=
  .inr (p, .inl stage)
def bumpLabel {d : Nat} (p : Fin d) : Label d := .inr (p, .inr ())

def fields : Fin 3 → Fin 6 := fun i => i.castLE (by decide)

def afterPort {d : Nat} (p : Fin d) : Option (Label d) :=
  if h : p.val + 1 < d then some (rowLabel ⟨p.val + 1, h⟩ .relationRead) else none

def entry (d : Nat) : Option (Label d) :=
  if h : 0 < d then some (rowLabel ⟨0, h⟩ .relationRead) else none

variable {σ : Type}

abbrev Alphabet (_ : Fin 6) := Bool

def continueAt {d : Nat} (exit : Option (Label d)) :
    TM2.Stmt Alphabet (Label d) (σ × Option Bool) :=
  match exit with
  | none => .halt
  | some label => .goto fun _ => label

theorem stepAux_continueAt {d : Nat} (exit : Option (Label d))
    (state : σ × Option Bool) (tapes : Fin 6 → List Bool) :
    TM2.stepAux (continueAt exit) state tapes = ⟨exit, state, tapes⟩ := by
  cases exit <;> rfl

def program (d : Nat) : Label d → TM2.Stmt Alphabet (Label d) (σ × Option Bool)
  | .inl () => pushWord 2 trueBits.reverse
      (.load (fun state => (state.1, none)) (continueAt (entry d)))
  | .inr (p, .inl stage) => MachineTableRows.routine fields 3 4 5
      (rowLabel p) (some (bumpLabel p)) stage
  | .inr (p, .inr ()) => .push 1 (fun _ => true) (continueAt (afterPort p))

def fieldTapes (v e : Nat) (output : List Bool) : Fin 6 → List Bool :=
  ![encodeWord v, encodeWord e, trueBits, [], output, []]

def initialTapes (v e : Nat) (output : List Bool) : Fin 6 → List Bool :=
  Function.update (fieldTapes v e output) 2 []

theorem initializeStep (d v e : Nat) (output : List Bool)
    (ambient : σ) (register : Option Bool) :
    TM2.step (program d) ⟨some (start d), (ambient, register), initialTapes v e output⟩ =
      some ⟨entry d, (ambient, none), fieldTapes v e output⟩ := by
  change some (TM2.stepAux (program d (start d)) (ambient, register)
    (initialTapes v e output)) = _
  simp only [start, program, stepAux_pushWord, List.reverse_reverse, TM2.stepAux,
    initialTapes, Function.update_self, List.append_nil, Function.update_idem]
  have ht : Function.update (fieldTapes v e output) 2 trueBits = fieldTapes v e output := by
    funext i
    fin_cases i <;> simp [fieldTapes]
  rw [ht, stepAux_continueAt]

theorem rowTrace {d : Nat} (p : Fin d) (v e : Nat) (output : List Bool)
    (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step (program d)))^[
        4 * (v + e + 8194) + 2 * output.length + 9]
      (some ⟨some (rowLabel p .relationRead), (ambient, register), fieldTapes v e output⟩) =
      some ⟨some (bumpLabel p), (ambient, none), fieldTapes v e (output ++ rowBits v e)⟩ := by
  have hfields (i : Fin 3) : fields i ≠ (3 : Fin 6) ∧ fields i ≠ (5 : Fin 6) := by
    fin_cases i <;> decide
  have h := MachineTableRows.appendTrace fields (3 : Fin 6) 4 5 hfields
    (by decide) (by decide) (by decide) (rowLabel p) (some (bumpLabel p))
    (program d) (fun _ => rfl) (fieldTapes v e output) rfl rfl ambient register
  have hbits : MachineTableRows.fieldBits fields (fieldTapes v e output) = rowBits v e := by
    rfl
  have hsize : MachineTableRows.fieldSize fields (fieldTapes v e output) = v + e + 8194 := by
    rw [← MachineTableRows.fieldBits_length, hbits, rowBits_length]
  have ht : Function.update (fieldTapes v e output) 4 (output ++ rowBits v e) =
      fieldTapes v e (output ++ rowBits v e) := by
    funext i
    fin_cases i <;> simp [fieldTapes]
  simpa only [hsize, hbits, show fieldTapes v e output 4 = output from rfl, ht] using h

theorem bumpStep {d : Nat} (p : Fin d) (v e : Nat) (output : List Bool) (ambient : σ) :
    TM2.step (program d) ⟨some (bumpLabel p), (ambient, none), fieldTapes v e output⟩ =
      some ⟨afterPort p, (ambient, none), fieldTapes v (e + 1) output⟩ := by
  change some (TM2.stepAux (program d (bumpLabel p)) (ambient, none)
    (fieldTapes v e output)) = _
  simp only [bumpLabel, program, TM2.stepAux]
  rw [stepAux_continueAt]
  congr 2
  funext i
  fin_cases i <;> simp [fieldTapes, encodeWord, List.replicate_succ]

theorem rowAndBumpTrace {d : Nat} (p : Fin d) (v e : Nat) (output : List Bool)
    (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step (program d)))^[
        4 * (v + e + 8194) + 2 * output.length + 10]
      (some ⟨some (rowLabel p .relationRead), (ambient, register), fieldTapes v e output⟩) =
      some ⟨afterPort p, (ambient, none), fieldTapes v (e + 1) (output ++ rowBits v e)⟩ := by
  rw [show 4 * (v + e + 8194) + 2 * output.length + 10 =
      (4 * (v + e + 8194) + 2 * output.length + 9) + 1 by omega,
    Function.iterate_succ_apply', rowTrace, MachineComposition.advance_some]
  exact bumpStep p v e (output ++ rowBits v e) ambient

def steps (v e : Nat) (output : List Bool) : Nat → Nat
  | 0 => 0
  | count + 1 => (4 * (v + e + 8194) + 2 * output.length + 10) +
      steps v (e + 1) (output ++ rowBits v e) count

theorem suffixTrace {d : Nat} (count : Nat) (p : Fin d) (hp : p.val + count = d)
    (v e : Nat) (output : List Bool) (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step (program d)))^[steps v e output count]
      (some ⟨some (rowLabel p .relationRead), (ambient, register), fieldTapes v e output⟩) =
      some ⟨none, (ambient, none), fieldTapes v (e + count) (output ++ rowsBits v e count)⟩ := by
  induction count generalizing p v e output register with
  | zero =>
    have hlt := p.isLt
    omega
  | succ count ih =>
    rw [steps, Nat.add_comm (4 * (v + e + 8194) + 2 * output.length + 10),
      Function.iterate_add_apply, rowAndBumpTrace]
    by_cases hc : count = 0
    · subst count
      have hlast : ¬ p.val + 1 < d := by omega
      simp [afterPort, hlast, steps, rowsBits]
    · have hnext : p.val + 1 < d := by omega
      rw [afterPort, dite_eq_left hnext]
      have hh := ih ⟨p.val + 1, hnext⟩ (by change (p.val + 1) + count = d; omega)
        v (e + 1) (output ++ rowBits v e) none
      have he : (e + 1) + count = e + (count + 1) := by omega
      simpa only [rowsBits, List.append_assoc, he] using hh

theorem allTrace (d v e : Nat) (output : List Bool) (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step (program d)))^[steps v e output d + 1]
      (some ⟨some (start d), (ambient, register), initialTapes v e output⟩) =
      some ⟨none, (ambient, none), fieldTapes v (e + d) (output ++ rowsBits v e d)⟩ := by
  rw [Function.iterate_succ_apply]
  change (MachineComposition.advance (TM2.step (program d)))^[steps v e output d]
    (TM2.step (program d) ⟨some (start d), (ambient, register), initialTapes v e output⟩) = _
  rw [initializeStep]
  by_cases hd : 0 < d
  · rw [entry, dite_eq_left hd]
    exact suffixTrace d ⟨0, hd⟩ (by simp) v e output ambient none
  · have hz : d = 0 := by omega
    subst d
    simp [entry, steps, rowsBits]

def graphRows {n m : Nat} (v : Fin n) (e : Nat) :
    (count : Nat) → e + count ≤ m → List (DartRow n m)
  | 0, _ => []
  | count + 1, h =>
      ⟨v, ⟨e, by omega⟩, trueRelation⟩ ::
        graphRows v (e + 1) count (by omega)

theorem graphRows_length {n m : Nat} (v : Fin n) (e count : Nat) (h : e + count ≤ m) :
    (graphRows v e count h).length = count := by
  induction count generalizing e with
  | zero => rfl
  | succ count ih => simp [graphRows, ih]

theorem rowsBits_eq_graphRows {n m : Nat} (v : Fin n) (e count : Nat)
    (h : e + count ≤ m) :
    rowsBits v.val e count = encodeWords ((graphRows v e count h).flatMap rowWords) := by
  induction count generalizing e with
  | zero => rfl
  | succ count ih =>
    simp [rowsBits, graphRows, encodeWords_append, MachineTableRows.rowBits_eq,
      rowBits, trueBits]
    exact ih (e + 1) (by omega)

def dummyRows {n : Nat} (d : Nat) (v : Fin n) : List (DartRow n (n * d)) :=
  graphRows v (d * v.val) d (by
    have h := Nat.mul_le_mul_right d (Nat.succ_le_of_lt v.isLt)
    simpa only [Nat.succ_mul, Nat.mul_comm] using h)

theorem dummyRows_length {n : Nat} (d : Nat) (v : Fin n) :
    (dummyRows d v).length = d := graphRows_length _ _ _ _

theorem dummyRowsTrace {n : Nat} (d : Nat) (v : Fin n) (output : List Bool)
    (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step (program d)))^[steps v.val (d * v.val) output d + 1]
      (some ⟨some (start d), (ambient, register), initialTapes v.val (d * v.val) output⟩) =
      some ⟨none, (ambient, none), fieldTapes v.val (d * v.val + d)
        (output ++ encodeWords ((dummyRows d v).flatMap rowWords))⟩ := by
  have h := allTrace d v.val (d * v.val) output ambient register
  rw [rowsBits_eq_graphRows v (d * v.val) d (by
    have h := Nat.mul_le_mul_right d (Nat.succ_le_of_lt v.isLt)
    simpa only [Nat.succ_mul, Nat.mul_comm] using h)] at h
  exact h

theorem steps_le_bound (v e : Nat) (output : List Bool) (count : Nat) :
    steps v e output count ≤ count *
      (4 * (v + e + count + 8194) +
        2 * (output.length + count * (v + e + count + 8194)) + 10) := by
  induction count generalizing e output with
  | zero => simp [steps]
  | succ count ih =>
    let B := v + e + (count + 1) + 8194
    let C := 4 * B + 2 * (output.length + (count + 1) * B) + 10
    have hb : v + (e + 1) + count + 8194 = B := by dsimp [B]; omega
    have hi := ih (e + 1) (output ++ rowBits v e)
    rw [List.length_append, rowBits_length, hb] at hi
    have hrow : v + e + 8194 ≤ B := by dsimp [B]; omega
    have hout : output.length + (v + e + 8194) + count * B ≤
        output.length + (count + 1) * B := by
      calc
        _ ≤ output.length + B + count * B :=
          Nat.add_le_add_right (Nat.add_le_add_left hrow output.length) _
        _ = _ := by simp only [Nat.add_mul, Nat.one_mul]; omega
    have hconstant : 4 * B + 2 * (output.length + (v + e + 8194) + count * B) + 10 ≤ C := by
      exact Nat.add_le_add_right
        (Nat.add_le_add_left (Nat.mul_le_mul_left 2 hout) (4 * B)) 10
    have hremaining := hi.trans (Nat.mul_le_mul_left count hconstant)
    have hbody : 4 * (v + e + 8194) + 2 * output.length + 10 ≤ C := by
      exact Nat.add_le_add_right (Nat.add_le_add (Nat.mul_le_mul_left 4 hrow)
        (Nat.mul_le_mul_left 2 (Nat.le_add_right output.length ((count + 1) * B)))) 10
    change (4 * (v + e + 8194) + 2 * output.length + 10) +
      steps v (e + 1) (output ++ rowBits v e) count ≤ (count + 1) * C
    calc
      _ ≤ C + count * C := Nat.add_le_add hbody hremaining
      _ = (count + 1) * C := by rw [Nat.add_mul, Nat.one_mul]; omega

def inputSize (v e : Nat) (output : List Bool) : Nat :=
  (encodeWord v).length + (encodeWord e).length + output.length

noncomputable def timePolynomial (d : Nat) : Polynomial Nat :=
  Polynomial.C d *
    (Polynomial.C 4 * (Polynomial.X + Polynomial.C (d + 8192)) +
      Polynomial.C 2 * (Polynomial.X + Polynomial.C d *
        (Polynomial.X + Polynomial.C (d + 8192))) + Polynomial.C 10) + 1

theorem timePolynomial_bounds (d v e : Nat) (output : List Bool) :
    steps v e output d + 1 ≤ (timePolynomial d).eval (inputSize v e output) := by
  have h := steps_le_bound v e output d
  have hB : v + e + d + 8194 ≤ inputSize v e output + d + 8192 := by
    simp only [inputSize, encodeWord_length]
    omega
  have hO : output.length ≤ inputSize v e output := by
    simp only [inputSize, encodeWord_length]
    omega
  have hm := Nat.mul_le_mul_left d hB
  have hinner :
      4 * (v + e + d + 8194) + 2 * (output.length + d * (v + e + d + 8194)) + 10 ≤
      4 * (inputSize v e output + d + 8192) +
        2 * (inputSize v e output + d * (inputSize v e output + d + 8192)) + 10 := by
    exact Nat.add_le_add_right (Nat.add_le_add (Nat.mul_le_mul_left 4 hB)
      (Nat.mul_le_mul_left 2 (Nat.add_le_add hO hm))) 10
  have htotal := h.trans (Nat.mul_le_mul_left d hinner)
  simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_one]
  simpa only [Nat.add_assoc] using Nat.add_le_add_right htotal 1

def machine (d : Nat) : FinTM2 where
  K := Fin 6
  k₀ := 0
  k₁ := 4
  Γ _ := Bool
  Λ := Label d
  main := start d
  σ := Unit × Option Bool
  initialState := ((), none)
  m := program d

def machineInTime (d v e : Nat) (output : List Bool) (register : Option Bool) :
    StateTransition.EvalsToInTime (machine d).step
      ⟨some (start d), ((), register), initialTapes v e output⟩
      (some ⟨none, ((), none), fieldTapes v (e + d) (output ++ rowsBits v e d)⟩)
      ((timePolynomial d).eval (inputSize v e output)) where
  steps := steps v e output d + 1
  evals_in_steps := by
    convert allTrace d v e output () register using 1 ; rfl
  steps_le_m := timePolynomial_bounds d v e output

def machineDummyInTime {n : Nat} (d : Nat) (v : Fin n)
    (output : List Bool) (register : Option Bool) :
    StateTransition.EvalsToInTime (machine d).step
      ⟨some (start d), ((), register), initialTapes v.val (d * v.val) output⟩
      (some ⟨none, ((), none), fieldTapes v.val (d * v.val + d)
        (output ++ encodeWords ((dummyRows d v).flatMap rowWords))⟩)
      ((timePolynomial d).eval (inputSize v.val (d * v.val) output)) where
  steps := steps v.val (d * v.val) output d + 1
  evals_in_steps := by
    convert dummyRowsTrace d v output () register using 1 ; rfl
  steps_le_m := timePolynomial_bounds d v.val (d * v.val) output

theorem finalFrame (v e : Nat) (output : List Bool) :
    fieldTapes v e output 0 = encodeWord v ∧
    fieldTapes v e output 2 = trueBits ∧
    fieldTapes v e output 3 = [] ∧ fieldTapes v e output 5 = [] :=
  ⟨rfl, rfl, rfl, rfl⟩

end DFVSGames.Foundations.Complexity.MachineDummyRows

end

section

namespace DFVSGames.Foundations.Complexity.MachineRegularDummyRow

open Turing MachineComposition PCP.GraphTables

abbrev Tape := Fin 6
abbrev Alphabet (_ : Tape) := Bool
abbrev State (σ : Type) := σ × Option Bool

inductive Label
  | affine (stage : MachineUnaryAffineAt.Label)
  | row (stage : MachineDummyRows.Label 1)
  | reverse | relation
  deriving DecidableEq, Fintype

def reverseIndex (q x : Nat) : Nat := (q + 1) * x + q

def emittedBits (q x : Nat) : List Bool :=
  MachineDummyRows.rowBits x (reverseIndex q x)

def instruction {σ Λ : Type} (q : Nat) (labels : Label → Λ) (exit : Option Λ) :
    Label → TM2.Stmt Alphabet Λ (State σ)
  | .affine .seed => MachineUnaryAffineAt.seed 1 q (labels (.affine .scan))
  | .affine .scan => MachineUnaryAffineAt.scan 0 3 1 (q + 1)
      (labels (.affine .scan)) (labels (.affine .restore))
  | .affine .restore => Reduction.MachineTransfer.loopAt 3 0 id false
      (labels (.affine .restore)) (some (labels (.row (MachineDummyRows.start 1))))
  | .row stage => MachineCloudPadding.Placement.statement (id : Tape → Tape)
      (fun l => labels (.row l)) (some (labels .reverse))
      (MachineDummyRows.program 1 stage)
  | .reverse => MachineDrain.drain 1 (labels .reverse) (some (labels .relation))
  | .relation => MachineDrain.drain 2 (labels .relation) exit

structure Input (x : Nat) (base : Tape → List Bool) : Prop where
  source : base 0 = encodeWord x
  reverseEmpty : base 1 = []
  relationEmpty : base 2 = []
  scratchEmpty : base 3 = []
  rowEmpty : base 5 = []

def memory (x : Nat) (output : List Bool) : Tape → List Bool :=
  ![encodeWord x, [], [], [], output, []]

theorem memory_input (x : Nat) (output : List Bool) : Input x (memory x output) :=
  ⟨rfl, rfl, rfl, rfl, rfl⟩

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

theorem affine_frame (x e : Nat) (base : Tape → List Bool) (input : Input x base) :
    Function.update base 1 (encodeWord e) =
      MachineDummyRows.initialTapes x e (base 4) := by
  funext k
  fin_cases k <;>
    simp [MachineDummyRows.initialTapes, MachineDummyRows.fieldTapes,
      input.source, input.relationEmpty, input.scratchEmpty, input.rowEmpty]

theorem cleanup_frame (x e : Nat) (output : List Bool)
    (base : Tape → List Bool) (input : Input x base) :
    Function.update (Function.update (MachineDummyRows.fieldTapes x e output) 1 []) 2 [] =
      Function.update base 4 output := by
  funext k
  fin_cases k <;>
    simp [MachineDummyRows.fieldTapes, input.source, input.reverseEmpty,
      input.relationEmpty, input.scratchEmpty, input.rowEmpty]

theorem rowTraceAt {σ Λ : Type} (q : Nat) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Alphabet Λ (State σ))
    (code : ∀ l, program (labels l) = instruction q labels exit l)
    (x e : Nat) (output : List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[MachineDummyRows.steps x e output 1 + 1]
      (some ⟨some (labels (.row (MachineDummyRows.start 1))), (ambient, register),
        MachineDummyRows.initialTapes x e output⟩) =
      some ⟨some (labels .reverse), (ambient, none),
        MachineDummyRows.fieldTapes x (e + 1) (output ++ MachineDummyRows.rowBits x e)⟩ := by
  have run := MachineDummyRows.allTrace 1 x e output ambient register
  have placed := MachineCloudPadding.Placement.trace (id : Tape → Tape) (fun k => some k)
    (fun _ => rfl) (fun _ _ h => (Option.some.inj h).symm)
    (fun l => labels (.row l)) (some (labels .reverse)) (fun _ => [])
    (MachineDummyRows.program 1) program
    (fun l => code (.row l)) _ _ _ run
  have identityTapes (source : Tape → List Bool) :
      MachineCloudPadding.Placement.tapes (fun k : Tape => some k) source
        (fun _ : Tape => []) = source := rfl
  simpa only [MachineCloudPadding.Placement.configuration,
    MachineCloudPadding.Placement.label, identityTapes,
    MachineDummyRows.rowsBits, List.append_nil] using placed

theorem cleanupTraceAt {σ Λ : Type} (q : Nat) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Alphabet Λ (State σ))
    (code : ∀ l, program (labels l) = instruction q labels exit l)
    (x e : Nat) (output : List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[
        ((encodeWord e).length + 1) + (MachineDummyRows.trueBits.length + 1)]
      (some ⟨some (labels .reverse), (ambient, register),
        MachineDummyRows.fieldTapes x e output⟩) =
      some ⟨exit, (ambient, none),
        Function.update (Function.update (MachineDummyRows.fieldTapes x e output) 1 []) 2 []⟩ := by
  let base := MachineDummyRows.fieldTapes x e output
  let mid := Function.update base (1 : Tape) []
  have first := MachineDrain.drainTrace (1 : Tape) (labels .reverse)
    (some (labels .relation)) program (code .reverse) base (base 1) ambient register
  have second := MachineDrain.drainTrace (2 : Tape) (labels .relation)
    exit program (code .relation) mid (mid 2) ambient none
  simp only [Function.update_eq_self] at first second
  have hreverse : base 1 = encodeWord e := rfl
  have hrelation : mid 2 = MachineDummyRows.trueBits := by
    simp only [mid, Function.update_of_ne (by decide : (2 : Tape) ≠ 1)]
    rfl
  rw [hreverse] at first
  rw [hrelation] at second
  exact join_trace_inline_MachineRegularDummyRow first second

def steps (q x : Nat) (output : List Bool) : Nat :=
  ((2 * (x + 1) + 1) + (MachineDummyRows.steps x (reverseIndex q x) output 1 + 1)) +
    (((encodeWord (reverseIndex q x + 1)).length + 1) +
      (MachineDummyRows.trueBits.length + 1))

def timeBound (q x outputLength : Nat) : Nat :=
  (5 * q + 11) * x + 2 * outputLength + 5 * q + 40986

theorem steps_eq (q x : Nat) (output : List Bool) :
    steps q x output = timeBound q x output.length := by
  simp only [steps, MachineDummyRows.steps, encodeWord_length, MachineDummyRows.trueBits_length,
    reverseIndex, timeBound]
  ring

theorem traceAt {σ Λ : Type} (q : Nat) (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt Alphabet Λ (State σ))
    (code : ∀ l, program (labels l) = instruction q labels exit l)
    (x : Nat) (base : Tape → List Bool) (input : Input x base)
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[steps q x (base 4)]
      (some ⟨some (labels (.affine .seed)), (ambient, register), base⟩) =
      some ⟨exit, (ambient, none), Function.update base 4 (base 4 ++ emittedBits q x)⟩ := by
  have affine := MachineUnaryAffineAt.seededAffineTrace (0 : Tape) 3 1
    (by decide) (by decide) (by decide) (q + 1) q
    (labels (.affine .seed)) (labels (.affine .scan)) (labels (.affine .restore))
    (some (labels (.row (MachineDummyRows.start 1)))) program
    (code (.affine .seed)) (code (.affine .scan)) (code (.affine .restore))
    base x [] (by simpa only [List.append_nil] using input.source)
    input.scratchEmpty ambient register
  simp only [input.reverseEmpty, List.append_nil] at affine
  change (advance (TM2.step program))^[2 * (x + 1) + 1]
    (some ⟨some (labels (.affine .seed)), (ambient, register), base⟩) =
      some ⟨some (labels (.row (MachineDummyRows.start 1))), (ambient, none),
        Function.update base 1 (encodeWord (reverseIndex q x))⟩ at affine
  rw [affine_frame x (reverseIndex q x) base input] at affine
  have row := rowTraceAt q labels exit program code x (reverseIndex q x) (base 4) ambient none
  have clean := cleanupTraceAt q labels exit program code x (reverseIndex q x + 1)
    (base 4 ++ MachineDummyRows.rowBits x (reverseIndex q x)) ambient none
  rw [cleanup_frame x (reverseIndex q x + 1) _ base input] at clean
  exact join_trace_inline_MachineRegularDummyRow (join_trace_inline_MachineRegularDummyRow affine row) clean

def machine (q : Nat) : FinTM2 where
  K := Tape
  k₀ := 0
  k₁ := 4
  Γ := Alphabet
  Λ := Label
  main := .affine .seed
  σ := State Unit
  initialState := ((), none)
  m := instruction q id none

def machineInTime (q x : Nat) (base : Tape → List Bool) (input : Input x base)
    (register : Option Bool) :
    StateTransition.EvalsToInTime (machine q).step
      ⟨some (.affine .seed), ((), register), base⟩
      (some ⟨none, ((), none), Function.update base (4 : Tape)
        (base 4 ++ emittedBits q x)⟩)
      (timeBound q x (base 4).length) where
  steps := steps q x (base 4)
  evals_in_steps := traceAt q id none (instruction q id none)
    (fun _ => rfl) x base input () register
  steps_le_m := (steps_eq q x (base 4)).le

end DFVSGames.Foundations.Complexity.MachineRegularDummyRow

end

section

namespace DFVSGames.Foundations.Complexity.MachinePaddingRows

open Turing MachineComposition

inductive Control | initialize | guard | bump
  deriving DecidableEq

instance : Fintype Control := derive_fintype% _

abbrev Tape := Fin 6 ⊕ Unit
abbrev Label (d : Nat) := MachineDummyRows.Label d ⊕ Control
abbrev Alphabet : Tape → Type := MachineEmbedding.Alphabet
  (fun _ : Fin 6 => Bool) (fun _ : Unit => Bool)
abbrev State := (Unit × Option Bool) × Unit

def bodyEntry (d : Nat) (hd : 0 < d) : Label d :=
  .inl (MachineDummyRows.rowLabel ⟨0, hd⟩ .relationRead)

def extra (d : Nat) (hd : 0 < d) : Control → TM2.Stmt Alphabet (Label d) State
  | .initialize => Reduction.MachineSubstitution.pushWord (.inl 2)
      MachineDummyRows.trueBits.reverse
      (.load (fun _ => (((), none), ())) (.goto (fun _ => .inr .guard)))
  | .guard => .pop (.inr ()) (fun _ bit => (((), bit), ()))
      (.branch (fun s => s.1.2.getD false)
        (.load (fun _ => (((), none), ())) (.goto (fun _ => bodyEntry d hd)))
        (.push (.inr ()) (fun _ => false)
          (.load (fun _ => (((), none), ())) .halt)))
  | .bump => .push (.inl 0) (fun _ => true) (.goto (fun _ => .inr .guard))

def program (d : Nat) (hd : 0 < d) : Label d → TM2.Stmt Alphabet (Label d) State :=
  MachineEmbedding.program (some (.inr .bump)) (MachineDummyRows.program d) (extra d hd)

def tapes (v e fuel : Nat) (output : List Bool) : ∀ k, List (Alphabet k) :=
  MachineEmbedding.tapes (MachineDummyRows.fieldTapes v e output)
    (fun _ : Unit => encodeWord fuel)

def cfg (d : Nat) (label : Option (Label d)) (v e fuel : Nat)
    (output : List Bool) : TM2.Cfg Alphabet (Label d) State :=
  ⟨label, (((), none), ()), tapes v e fuel output⟩

theorem bodyTrace (d : Nat) (hd : 0 < d) (v e fuel : Nat) (output : List Bool) :
    (advance (TM2.step (program d hd)))^[MachineDummyRows.steps v e output d]
      (some (cfg d (some (bodyEntry d hd)) v e fuel output)) =
      some (cfg d (some (.inr .bump)) v (e + d) fuel
        (output ++ MachineDummyRows.rowsBits v e d)) := by
  have run := MachineDummyRows.suffixTrace d ⟨0, hd⟩ (by simp)
    v e output () none
  have lifted := liftSuccessfulTrace
    (TM2.step (MachineDummyRows.program d)) (TM2.step (program d hd))
    (MachineEmbedding.configuration (some (.inr .bump)) ()
      (fun _ : Unit => encodeWord fuel))
    (by
      intro a b h
      exact MachineEmbedding.step_simulation (some (.inr Control.bump)) ()
        (fun _ : Unit => encodeWord fuel) (MachineDummyRows.program d) (extra d hd) a b h)
    (MachineDummyRows.steps v e output d) _ _ run
  exact lifted

theorem guard_succ (d : Nat) (hd : 0 < d) (v e fuel : Nat) (output : List Bool) :
    TM2.step (program d hd) (cfg d (some (.inr .guard)) v e (fuel + 1) output) =
      some (cfg d (some (bodyEntry d hd)) v e fuel output) := by
  change some (TM2.stepAux (extra d hd .guard) _ _) = _
  simp only [extra, TM2.stepAux, tapes, MachineEmbedding.tapes_inr,
    encodeWord, List.replicate_succ, List.cons_append, List.head?_cons,
    List.tail_cons, Option.getD_some, Bool.cond_true]
  congr 2
  funext k
  cases k <;> simp [tapes, MachineEmbedding.tapes, Function.update, encodeWord]

theorem guard_zero (d : Nat) (hd : 0 < d) (v e : Nat) (output : List Bool) :
    TM2.step (program d hd) (cfg d (some (.inr .guard)) v e 0 output) =
      some (cfg d none v e 0 output) := by
  change some (TM2.stepAux (extra d hd .guard) _ _) = _
  simp only [extra, TM2.stepAux, tapes, MachineEmbedding.tapes_inr,
    encodeWord, List.replicate_zero, List.nil_append, List.head?_cons,
    List.tail_cons, Option.getD_some, Bool.cond_false]
  congr 2
  funext k
  cases k <;> simp [tapes, MachineEmbedding.tapes, Function.update, encodeWord]

theorem bump_step (d : Nat) (hd : 0 < d) (v e fuel : Nat) (output : List Bool) :
    TM2.step (program d hd) (cfg d (some (.inr .bump)) v e fuel output) =
      some (cfg d (some (.inr .guard)) (v + 1) e fuel output) := by
  change some (TM2.stepAux (extra d hd .bump) _ _) = _
  simp only [extra, TM2.stepAux]
  congr 2
  funext k
  cases k with
  | inl k => fin_cases k <;> simp [tapes, MachineEmbedding.tapes,
      MachineDummyRows.fieldTapes, encodeWord, List.replicate_succ]
  | inr k => simp [tapes, MachineEmbedding.tapes]

def paddingBits (d v e : Nat) : Nat → List Bool
  | 0 => []
  | count + 1 => MachineDummyRows.rowsBits v e d ++ paddingBits d (v + 1) (e + d) count

def steps (d v e : Nat) (output : List Bool) : Nat → Nat
  | 0 => 1
  | count + 1 => (MachineDummyRows.steps v e output d + 2) +
      steps d (v + 1) (e + d) (output ++ MachineDummyRows.rowsBits v e d) count

theorem loopTrace (d : Nat) (hd : 0 < d) (count v e : Nat) (output : List Bool) :
    (advance (TM2.step (program d hd)))^[steps d v e output count]
      (some (cfg d (some (.inr .guard)) v e count output)) =
      some (cfg d none (v + count) (e + count * d) 0
        (output ++ paddingBits d v e count)) := by
  induction count generalizing v e output with
  | zero => simpa [steps, paddingBits] using guard_zero d hd v e output
  | succ count ih =>
      rw [steps, Nat.add_comm (MachineDummyRows.steps v e output d + 2),
        Function.iterate_add_apply]
      have hb : (advance (TM2.step (program d hd)))^[MachineDummyRows.steps v e output d + 2]
          (some (cfg d (some (.inr .guard)) v e (count + 1) output)) =
          some (cfg d (some (.inr .guard)) (v + 1) (e + d) count
            (output ++ MachineDummyRows.rowsBits v e d)) := by
        rw [show MachineDummyRows.steps v e output d + 2 =
          (MachineDummyRows.steps v e output d + 1) + 1 by omega,
          Function.iterate_succ_apply, advance_some, guard_succ,
          Function.iterate_succ_apply', bodyTrace, advance_some, bump_step]
      rw [hb, ih]
      simp only [paddingBits, List.append_assoc, Nat.succ_mul]
      congr 2 <;> omega

def initialTapes (v e fuel : Nat) (output : List Bool) : ∀ k, List (Alphabet k) :=
  Function.update (tapes v e fuel output) (.inl 2) []

theorem initialize_step (d : Nat) (hd : 0 < d) (v e fuel : Nat) (output : List Bool) :
    TM2.step (program d hd)
      ⟨some (.inr .initialize), (((), none), ()), initialTapes v e fuel output⟩ =
      some (cfg d (some (.inr .guard)) v e fuel output) := by
  change some (TM2.stepAux (extra d hd .initialize) _ _) = _
  simp only [extra, Reduction.MachineSubstitution.stepAux_pushWord,
    List.reverse_reverse, initialTapes, Function.update_self, List.append_nil,
    Function.update_idem, TM2.stepAux]
  congr 2
  funext k
  cases k with
  | inl k => fin_cases k <;> simp [tapes, MachineEmbedding.tapes,      MachineDummyRows.fieldTapes]
  | inr k => simp [tapes, MachineEmbedding.tapes]

theorem paddingTrace (d : Nat) (hd : 0 < d) (count v e : Nat) (output : List Bool) :
    (advance (TM2.step (program d hd)))^[steps d v e output count + 1]
      (some ⟨some (.inr .initialize), (((), none), ()), initialTapes v e count output⟩) =
      some (cfg d none (v + count) (e + count * d) 0
        (output ++ paddingBits d v e count)) := by
  rw [Function.iterate_succ_apply, advance_some, initialize_step, loopTrace]

def machine (d : Nat) (hd : 0 < d) : FinTM2 where
  K := Tape
  k₀ := .inl 0
  k₁ := .inl 4
  Γ := Alphabet
  Λ := Label d
  main := .inr .initialize
  σ := State
  initialState := (((), none), ())
  m := program d hd

def execution (d : Nat) (hd : 0 < d) (count v e : Nat) (output : List Bool) :
    StateTransition.EvalsToInTime (machine d hd).step
      ⟨some (.inr .initialize), (((), none), ()), initialTapes v e count output⟩
      (some (cfg d none (v + count) (e + count * d) 0
        (output ++ paddingBits d v e count)))
      (steps d v e output count + 1) where
  steps := steps d v e output count + 1
  evals_in_steps := by convert paddingTrace d hd count v e output using 1 ; rfl
  steps_le_m := le_rfl

end DFVSGames.Foundations.Complexity.MachinePaddingRows
end

section

namespace DFVSGames.Foundations.PCP.PreprocessingPaddingWords

open Complexity PortTables

variable {n m d : Nat}

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

theorem ofFn_split {α : Type*} (h : n ≤ m) (f : Fin m → α) :
    List.ofFn f =
      List.ofFn (fun v : Fin n => f (v.castLE h)) ++
        List.ofFn (fun v : Fin (m - n) => f (PreprocessingPaddingTables.vertexEquiv h (.inr v))) := by
  calc
    List.ofFn f = List.ofFn
        (fun v : Fin (n + (m - n)) => f (Fin.cast (Nat.add_sub_of_le h) v)) :=
      List.ofFn_congr (Nat.add_sub_of_le h).symm f
    _ = _ := by rw [List.ofFn_add]; rfl

theorem flatten_ofFn_rowIndex {α : Type*} (f : Fin (n * d) → List α) :
    (List.ofFn f).flatten =
      (List.ofFn (fun v : Fin n =>
        (List.ofFn (fun p : Fin d => f (rowIndex n d (v, p)))).flatten)).flatten := by
  rw [List.ofFn_mul, List.flatten_flatten, List.map_ofFn]
  apply congrArg List.flatten
  apply congrArg List.ofFn
  funext v
  apply congrArg List.flatten
  apply congrArg List.ofFn
  funext p
  apply congrArg f
  apply Fin.ext
  simp only [rowIndex_val, Nat.mul_comm, Nat.add_comm]

def rowBits (table : Table n d) (v : Fin n) (p : Fin d) : List Bool :=
  encodeWord v.val ++ encodeWord (table.reverseIndex[rowIndex n d (v, p)]).val ++
    encodeWords (GraphTables.relationWords table.relations[rowIndex n d (v, p)])

def vertexBits (table : Table n d) (v : Fin n) : List Bool :=
  (List.ofFn (fun p : Fin d => rowBits table v p)).flatten

def rowsBits (table : Table n d) : List Bool :=
  encodeWords ((flatRows table).toList.flatMap GraphTables.rowWords)

theorem rowBits_flat (table : Table n d) (i : Fin (n * d)) :
    encodeWords (GraphTables.rowWords ((flatRows table)[i])) =
      rowBits table ((rowIndex n d).symm i).1 ((rowIndex n d).symm i).2 := by
  simp only [flatRows, Vector.getElem_ofFn, Fin.getElem_fin,
    MachineTableRows.rowBits_eq, rowBits, Prod.eta, Equiv.apply_symm_apply]

theorem rowsBits_eq_vertices (table : Table n d) :
    rowsBits table = (List.ofFn (fun v : Fin n => vertexBits table v)).flatten := by
  have hflat : (flatRows table).toList =
      List.ofFn (fun i : Fin (n * d) => (flatRows table)[i]) := by
    simp only [flatRows, Vector.toList_ofFn, Vector.getElem_ofFn, Fin.getElem_fin]
  unfold rowsBits
  rw [hflat, encodeWords_flatMap_inline_PreprocessingPaddingWords, List.flatMap_def, List.map_ofFn]
  change (List.ofFn (fun i : Fin (n * d) =>
    encodeWords (GraphTables.rowWords ((flatRows table)[i])))).flatten = _
  simp_rw [rowBits_flat]
  simpa only [Equiv.symm_apply_apply, vertexBits] using
    flatten_ofFn_rowIndex (fun i : Fin (n * d) =>
      rowBits table ((rowIndex n d).symm i).1 ((rowIndex n d).symm i).2)

theorem rowBits_pad_old (table : Table n d) (h : n ≤ m) (v : Fin n) (p : Fin d) :
    rowBits (PreprocessingPaddingTables.pad table h) (v.castLE h) p = rowBits table v p := by
  unfold rowBits
  rw [PreprocessingPaddingTables.reverseIndex_pad_old,
    PreprocessingPaddingTables.relations_pad_old]
  have he : (rowIndex m d ((rotation table (v, p)).1.castLE h,
      (rotation table (v, p)).2)).val = (table.reverseIndex[rowIndex n d (v, p)]).val := by
    rw [← rowIndex_rotation]
    rfl
  rw [he]
  rfl

theorem rowBits_pad_new (table : Table n d) (h : n ≤ m)
    (v : Fin (m - n)) (p : Fin d) :
    rowBits (PreprocessingPaddingTables.pad table h)
        (PreprocessingPaddingTables.vertexEquiv h (.inr v)) p =
      MachineDummyRows.rowBits (n + v.val) (p.val + d * (n + v.val)) := by
  unfold rowBits
  rw [PreprocessingPaddingTables.reverseIndex_pad_new,
    PreprocessingPaddingTables.relations_pad_new]
  rfl

theorem dummy_rowsBits_eq_ofFn (v e count : Nat) :
    MachineDummyRows.rowsBits v e count =
      (List.ofFn (fun p : Fin count => MachineDummyRows.rowBits v (e + p.val))).flatten := by
  induction count generalizing e with
  | zero => rfl
  | succ count ih =>
    rw [MachineDummyRows.rowsBits, List.ofFn_succ, List.flatten_cons, ih]
    simp only [Fin.val_zero, Nat.add_zero, Fin.val_succ]
    congr 1
    apply congrArg (fun f : Fin count → List Bool => (List.ofFn f).flatten)
    funext p
    congr 1
    omega

theorem paddingBits_eq_ofFn (d v e count : Nat) :
    MachinePaddingRows.paddingBits d v e count =
      (List.ofFn (fun k : Fin count =>
        MachineDummyRows.rowsBits (v + k.val) (e + k.val * d) d)).flatten := by
  induction count generalizing v e with
  | zero => rfl
  | succ count ih =>
    rw [MachinePaddingRows.paddingBits, List.ofFn_succ, List.flatten_cons, ih]
    simp only [Fin.val_zero, Nat.zero_mul, Nat.add_zero, Fin.val_succ, Nat.add_mul, Nat.one_mul]
    congr 1
    apply congrArg (fun f : Fin count → List Bool => (List.ofFn f).flatten)
    funext k
    congr 1 <;> omega

@[simp] theorem vertexBits_pad_old (table : Table n d) (h : n ≤ m) (v : Fin n) :
    vertexBits (PreprocessingPaddingTables.pad table h) (v.castLE h) = vertexBits table v := by
  simp only [vertexBits, rowBits_pad_old]

theorem vertexBits_pad_new (table : Table n d) (h : n ≤ m) (v : Fin (m - n)) :
    vertexBits (PreprocessingPaddingTables.pad table h)
        (PreprocessingPaddingTables.vertexEquiv h (.inr v)) =
      MachineDummyRows.rowsBits (n + v.val) (n * d + v.val * d) d := by
  unfold vertexBits
  simp_rw [rowBits_pad_new]
  rw [dummy_rowsBits_eq_ofFn]
  apply congrArg (fun f : Fin d → List Bool => (List.ofFn f).flatten)
  funext p
  congr 1
  ring

theorem rowsBits_pad (table : Table n d) (h : n ≤ m) :
    rowsBits (PreprocessingPaddingTables.pad table h) =
      rowsBits table ++ MachinePaddingRows.paddingBits d n (n * d) (m - n) := by
  rw [rowsBits_eq_vertices, ofFn_split h, List.flatten_append]
  simp_rw [vertexBits_pad_old, vertexBits_pad_new]
  rw [← rowsBits_eq_vertices, ← paddingBits_eq_ofFn]

theorem tableBits_pad (table : Table n d) (h : n ≤ m) :
    tableBits (PreprocessingPaddingTables.pad table h) =
      encodeWords [m, m * d] ++ rowsBits table ++
        MachinePaddingRows.paddingBits d n (n * d) (m - n) := by
  change encodeWords (PortTables.tableWords (PreprocessingPaddingTables.pad table h)) = _
  rw [tableWords_eq, encodeWords_append]
  change encodeWords [m, m * d] ++ rowsBits (PreprocessingPaddingTables.pad table h) = _
  rw [rowsBits_pad, List.append_assoc]

end DFVSGames.Foundations.PCP.PreprocessingPaddingWords

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