Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Encoding.Body

namespace OAI

namespace DFVSGames.Foundations.PCP.AlphabetTable

open DFVSGames.Foundations.Complexity

theorem Driver.finiteAlphabet (q : Nat) :
    MachineFiniteAlphabet.FiniteAlphabet (Driver.machine q) := by
  intro k
  change Finite Bool
  infer_instance

theorem Runtime.finiteAlphabet (q : Nat) :
    MachineFiniteAlphabet.FiniteAlphabet (Runtime.tablePolynomialTime q).tm :=
  Driver.finiteAlphabet q

end DFVSGames.Foundations.PCP.AlphabetTable

namespace DFVSGames.Foundations.Complexity.MachineStateEquiv

open Turing.TM2

variable {K Λ σ τ : Type} {Γ : K → Type}

abbrev statement (e : σ ≃ τ) : Stmt Γ Λ σ → Stmt Γ Λ τ :=
  MachineControl.statement (id : Λ → Λ) e

@[simp] theorem statement_symm_statement (e : σ ≃ τ) (q : Stmt Γ Λ σ) :
    statement e.symm (statement e q) = q := by
  induction q <;>
    simp_all only [statement, MachineControl.statement, Equiv.symm_symm,
      Equiv.symm_apply_apply, id_eq]

def configuration (e : σ ≃ τ) (c : Cfg Γ Λ σ) : Cfg Γ Λ τ :=
  ⟨c.l, e c.var, c.stk⟩

@[simp] theorem configuration_label (e : σ ≃ τ) (c : Cfg Γ Λ σ) :
    (configuration e c).l = c.l := rfl

@[simp] theorem configuration_state (e : σ ≃ τ) (c : Cfg Γ Λ σ) :
    (configuration e c).var = e c.var := rfl

@[simp] theorem configuration_tapes (e : σ ≃ τ) (c : Cfg Γ Λ σ) :
    (configuration e c).stk = c.stk := rfl

@[simp] theorem configuration_symm_configuration (e : σ ≃ τ) (c : Cfg Γ Λ σ) :
    configuration e.symm (configuration e c) = c := by
  cases c
  simp only [configuration, Equiv.symm_apply_apply]

@[simp] theorem configuration_configuration_symm (e : σ ≃ τ) (c : Cfg Γ Λ τ) :
    configuration e (configuration e.symm c) = c := by
  cases c
  simp only [configuration, Equiv.apply_symm_apply]

def configurationEquiv (e : σ ≃ τ) : Cfg Γ Λ σ ≃ Cfg Γ Λ τ where
  toFun := configuration e
  invFun := configuration e.symm
  left_inv := configuration_symm_configuration e
  right_inv := configuration_configuration_symm e

@[simp] theorem configurationEquiv_apply (e : σ ≃ τ) (c : Cfg Γ Λ σ) :
    configurationEquiv e c = configuration e c := rfl

@[simp] theorem control_configuration (e : σ ≃ τ) (c : Cfg Γ Λ σ) :
    MachineControl.configuration (id : Λ → Λ) e c = configuration e c := by
  cases c
  simp [MachineControl.configuration, configuration]

def program (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ) : Λ → Stmt Γ Λ τ :=
  fun l => statement e (source l)

@[simp] theorem program_symm_program (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ) :
    program e.symm (program e source) = source := by
  funext l
  exact statement_symm_statement e (source l)

theorem statementPushBound (e : σ ≃ τ) (q : Stmt Γ Λ σ) :
    Runtime.statementPushBound (statement e q) = Runtime.statementPushBound q := by
  induction q <;>
    simp_all only [statement, MachineControl.statement, Runtime.statementPushBound]

theorem supportsStmt_iff (e : σ ≃ τ) (S : Finset Λ) (q : Stmt Γ Λ σ) :
    SupportsStmt S (statement e q) ↔ SupportsStmt S q := by
  induction q with
  | push k f next ih =>
      simpa only [statement, MachineControl.statement, SupportsStmt] using ih
  | peek k f next ih =>
      simpa only [statement, MachineControl.statement, SupportsStmt] using ih
  | pop k f next ih =>
      simpa only [statement, MachineControl.statement, SupportsStmt] using ih
  | load f next ih =>
      simpa only [statement, MachineControl.statement, SupportsStmt] using ih
  | branch f yes no ihYes ihNo =>
      simp only [statement, MachineControl.statement, SupportsStmt, ihYes, ihNo]
  | goto f =>
      change (∀ state : τ, f (e.symm state) ∈ S) ↔ ∀ state : σ, f state ∈ S
      constructor
      · intro h state
        simpa only [Equiv.symm_apply_apply] using h (e state)
      · intro h state
        exact h (e.symm state)
  | halt => rfl

theorem supports_iff [Inhabited Λ] (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ)
    (S : Finset Λ) : Supports (program e source) S ↔ Supports source S := by
  simp only [Supports, program, supportsStmt_iff]

variable [DecidableEq K]

theorem stepAux_transport (e : σ ≃ τ) (q : Stmt Γ Λ σ) (state : σ)
    (tapes : ∀ k, List (Γ k)) :
    stepAux (statement e q) (e state) tapes =
      configuration e (stepAux q state tapes) := by
  simpa only [statement, control_configuration] using
    MachineControl.stepAux_simulation (id : Λ → Λ) e q state tapes

theorem stepAux_transport_symm (e : σ ≃ τ) (q : Stmt Γ Λ σ) (state : τ)
    (tapes : ∀ k, List (Γ k)) :
    stepAux (statement e q) state tapes =
      configuration e (stepAux q (e.symm state) tapes) := by
  simpa only [Equiv.apply_symm_apply] using
    stepAux_transport e q (e.symm state) tapes

theorem step_transport (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ) (c : Cfg Γ Λ σ) :
    step (program e source) (configuration e c) =
      (step source c).map (configuration e) := by
  cases c with
  | mk l state tapes =>
      cases l with
      | none => rfl
      | some l =>
          change some (stepAux (statement e (source l)) (e state) tapes) = _
          rw [stepAux_transport]
          rfl

theorem step_iff (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ) (a b : Cfg Γ Λ σ) :
    step (program e source) (configuration e a) = some (configuration e b) ↔
      step source a = some b := by
  rw [step_transport]
  change (step source a).map (configuration e) = (some b).map (configuration e) ↔ _
  constructor
  · intro h
    apply Option.map_injective (configurationEquiv e).injective
    change (step source a).map (configuration e) = (some b).map (configuration e)
    exact h
  · exact congrArg (Option.map (configuration e))

theorem advance_transport (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ)
    (c : Option (Cfg Γ Λ σ)) :
    MachineComposition.advance (step (program e source)) (c.map (configuration e)) =
      (MachineComposition.advance (step source) c).map (configuration e) := by
  cases c with
  | none => rfl
  | some c => exact step_transport e source c

theorem iterate_transport (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ) (n : Nat)
    (c : Option (Cfg Γ Λ σ)) :
    (MachineComposition.advance (step (program e source)))^[n]
        (c.map (configuration e)) =
      ((MachineComposition.advance (step source))^[n] c).map (configuration e) := by
  induction n with
  | zero => rfl
  | succ n ih =>
      rw [Function.iterate_succ_apply', ih, advance_transport,
        Function.iterate_succ_apply']

theorem trace_iff (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ) (n : Nat)
    (a b : Cfg Γ Λ σ) :
    (MachineComposition.advance (step (program e source)))^[n]
        (some (configuration e a)) = some (configuration e b) ↔
      (MachineComposition.advance (step source))^[n] (some a) = some b := by
  change (MachineComposition.advance (step (program e source)))^[n]
      ((some a).map (configuration e)) = (some b).map (configuration e) ↔ _
  rw [iterate_transport]
  constructor
  · intro h
    apply Option.map_injective (configurationEquiv e).injective
    change ((MachineComposition.advance (step source))^[n] (some a)).map (configuration e) =
      (some b).map (configuration e)
    exact h
  · exact congrArg (Option.map (configuration e))

theorem trace (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ) (n : Nat)
    (a b : Cfg Γ Λ σ)
    (run : (MachineComposition.advance (step source))^[n] (some a) = some b) :
    (MachineComposition.advance (step (program e source)))^[n]
      (some (configuration e a)) = some (configuration e b) :=
  (trace_iff e source n a b).2 run

def execution (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ)
    {start : Cfg Γ Λ σ} {finish : Option (Cfg Γ Λ σ)} {budget : Nat}
    (run : StateTransition.EvalsToInTime (step source) start finish budget) :
    StateTransition.EvalsToInTime (step (program e source))
      (configuration e start) (finish.map (configuration e)) budget where
  steps := run.steps
  evals_in_steps := by
    have h := run.evals_in_steps
    change (MachineComposition.advance (step source))^[run.steps] (some start) = finish at h
    change (MachineComposition.advance (step (program e source)))^[run.steps]
      ((some start).map (configuration e)) = finish.map (configuration e)
    rw [iterate_transport, h]
  steps_le_m := run.steps_le_m

@[simp] theorem execution_steps (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ)
    {start : Cfg Γ Λ σ} {finish : Option (Cfg Γ Λ σ)} {budget : Nat}
    (run : StateTransition.EvalsToInTime (step source) start finish budget) :
    (execution e source run).steps = run.steps := rfl

def execution_reflect (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ)
    {start : Cfg Γ Λ σ} {finish : Option (Cfg Γ Λ σ)} {budget : Nat}
    (run : StateTransition.EvalsToInTime (step (program e source))
      (configuration e start) (finish.map (configuration e)) budget) :
    StateTransition.EvalsToInTime (step source) start finish budget where
  steps := run.steps
  evals_in_steps := by
    change (MachineComposition.advance (step source))^[run.steps] (some start) = finish
    apply Option.map_injective (configurationEquiv e).injective
    change ((MachineComposition.advance (step source))^[run.steps] (some start)).map
      (configuration e) = finish.map (configuration e)
    rw [← iterate_transport]
    exact run.evals_in_steps
  steps_le_m := run.steps_le_m

@[simp] theorem execution_reflect_steps (e : σ ≃ τ) (source : Λ → Stmt Γ Λ σ)
    {start : Cfg Γ Λ σ} {finish : Option (Cfg Γ Λ σ)} {budget : Nat}
    (run : StateTransition.EvalsToInTime (step (program e source))
      (configuration e start) (finish.map (configuration e)) budget) :
    (execution_reflect e source run).steps = run.steps := rfl

end DFVSGames.Foundations.Complexity.MachineStateEquiv

namespace DFVSGames.Foundations.Complexity.MachineUnaryEqualityBit

open Turing
open MachineComposition

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

abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := (A × Bool × Option Bool) × Option Bool

def clean (ambient : A) : State A := ((ambient, false, none), none)

def compareStateEquiv (A : Type) : MachineCompare.State A ≃ State A where
  toFun s := ((s.1, s.2.1, s.2.2.1), s.2.2.2)
  invFun s := (s.1.1, s.1.2.1, s.1.2.2, s.2)
  left_inv := by rintro ⟨a, b, c, d⟩; rfl
  right_inv := by rintro ⟨⟨a, b, c⟩, d⟩; rfl

inductive Label
  | seedLeft | scanLeft | restoreLeft
  | seedRight | scanRight | restoreRight
  | compare | equal | different
  deriving DecidableEq

instance : Fintype Label where
  elems := {.seedLeft, .scanLeft, .restoreLeft, .seedRight, .scanRight,
    .restoreRight, .compare, .equal, .different}
  complete label := by cases label <;> simp

def exitAt (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  match exit with
  | none => .halt
  | some label => .goto fun _ => label

def bitEncoding (bit : Bool) : List Bool := encodeWord (if bit then 1 else 0)

def emit (destination : K) (bit : Bool) (exit : Option Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  .push destination (fun _ => false)
    (if bit then .push destination (fun _ => true) (exitAt exit) else exitAt exit)

def instruction (tape : Fin 6 → K) (labels : Label → Λ) (exit : Option Λ) :
    Label → TM2.Stmt (Alphabet (K := K)) Λ (State A)
  | .seedLeft => MachineUnaryAffineAt.seed (tape 2) 0 (labels .scanLeft)
  | .scanLeft => MachineUnaryAffineAt.scan (tape 0) (tape 4) (tape 2) 1
      (labels .scanLeft) (labels .restoreLeft)
  | .restoreLeft => Reduction.MachineTransfer.loopAt (tape 4) (tape 0) id false
      (labels .restoreLeft) (some (labels .seedRight))
  | .seedRight => MachineUnaryAffineAt.seed (tape 3) 0 (labels .scanRight)
  | .scanRight => MachineUnaryAffineAt.scan (tape 1) (tape 4) (tape 3) 1
      (labels .scanRight) (labels .restoreRight)
  | .restoreRight => Reduction.MachineTransfer.loopAt (tape 4) (tape 1) id false
      (labels .restoreRight) (some (labels .compare))
  | .compare => MachineStateEquiv.statement (compareStateEquiv A)
      (MachineCompare.loop (tape 2) (tape 3) (labels .compare)
        (some (labels .equal)) (some (labels .different)))
  | .equal => emit (tape 5) true exit
  | .different => emit (tape 5) false exit

def copySteps (n : Nat) : Nat := 2 * (n + 1) + 1
def compareSteps (a b : Nat) : Nat := max (a + 1) (b + 1) + 1
def steps (a b : Nat) : Nat := copySteps a + copySteps b + compareSteps a b + 1

theorem steps_le (a b : Nat) : steps a b ≤ 3 * (a + b) + 9 := by
  unfold steps copySteps compareSteps
  omega

theorem encodeWord_eq_iff (a b : Nat) : encodeWord a = encodeWord b ↔ a = b := by
  constructor
  · intro h
    have hlength := congrArg List.length h
    simp only [encodeWord_length] at hlength
    omega
  · rintro rfl
    rfl

theorem emit_step (destination : K) (bit : Bool) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
    (label : Λ) (atEmit : program label = emit destination bit exit)
    (state : State A) (base : K → List Bool) :
    TM2.step program ⟨some label, state, base⟩ =
      some ⟨exit, state,
        Function.update base destination (bitEncoding bit ++ base destination)⟩ := by
  change some (TM2.stepAux (program label) state base) = _
  rw [atEmit]
  cases bit <;> cases exit <;>
    simp [emit, exitAt, TM2.stepAux, bitEncoding, encodeWord, Function.update_idem]

private theorem joinTrace_inline_MachineUnaryEqualityBit {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 equalityTrace (tape : Fin 6 → K) (distinct : Function.Injective tape)
    (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
    (atLabels : ∀ l, program (labels l) = instruction tape labels exit l)
    (base : K → List Bool) (a b : Nat) (leftSuffix rightSuffix : List Bool)
    (leftWord : base (tape 0) = encodeWord a ++ leftSuffix)
    (rightWord : base (tape 1) = encodeWord b ++ rightSuffix)
    (leftEmpty : base (tape 2) = []) (rightEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) (ambient : A) :
    (advance (TM2.step program))^[steps a b]
      (some ⟨some (labels .seedLeft), clean ambient, base⟩) =
      some ⟨exit, clean ambient,
        Function.update base (tape 5) (bitEncoding (decide (a = b)) ++ base (tape 5))⟩ := by
  have hd (i j : Fin 6) (hne : i ≠ j) : tape i ≠ tape j :=
    fun h => hne (distinct h)
  let midLeft := Function.update base (tape 2) (encodeWord a)
  let midBoth := Function.update midLeft (tape 3) (encodeWord b)
  have leftRun : (advance (TM2.step program))^[copySteps a]
      (some ⟨some (labels .seedLeft), clean ambient, base⟩) =
      some ⟨some (labels .seedRight), clean ambient, midLeft⟩ := by
    simpa only [copySteps, clean, midLeft, Nat.one_mul, Nat.add_zero,
      leftEmpty, List.append_nil] using
      MachineUnaryAffineAt.seededAffineTrace (tape 0) (tape 4) (tape 2)
        (hd 0 4 (by decide)) (hd 0 2 (by decide)) (hd 4 2 (by decide)) 1 0
        (labels .seedLeft) (labels .scanLeft) (labels .restoreLeft)
        (some (labels .seedRight)) program (atLabels .seedLeft)
        (atLabels .scanLeft) (atLabels .restoreLeft)
        base a leftSuffix leftWord scratchEmpty (ambient, false, none) none
  have rightSource : midLeft (tape 1) = encodeWord b ++ rightSuffix := by
    simpa [midLeft, hd 1 2 (by decide)] using rightWord
  have scratchAfterLeft : midLeft (tape 4) = [] := by
    simpa [midLeft, hd 4 2 (by decide)] using scratchEmpty
  have rightCopyAfterLeft : midLeft (tape 3) = [] := by
    simpa [midLeft, hd 3 2 (by decide)] using rightEmpty
  have rightRun : (advance (TM2.step program))^[copySteps b]
      (some ⟨some (labels .seedRight), clean ambient, midLeft⟩) =
      some ⟨some (labels .compare), clean ambient, midBoth⟩ := by
    simpa only [copySteps, clean, midBoth, Nat.one_mul, Nat.add_zero,
      rightCopyAfterLeft, List.append_nil] using
      MachineUnaryAffineAt.seededAffineTrace (tape 1) (tape 4) (tape 3)
        (hd 1 4 (by decide)) (hd 1 3 (by decide)) (hd 4 3 (by decide)) 1 0
        (labels .seedRight) (labels .scanRight) (labels .restoreRight)
        (some (labels .compare)) program (atLabels .seedRight)
        (atLabels .scanRight) (atLabels .restoreRight)
        midLeft b rightSuffix rightSource scratchAfterLeft (ambient, false, none) none
  have copyLeftWord : midBoth (tape 2) = encodeWord a := by
    simp [midBoth, midLeft, hd 2 3 (by decide)]
  have copyRightWord : midBoth (tape 3) = encodeWord b := by simp [midBoth]
  have restored : Reduction.MachineTransfer.tapesAt (tape 2) (tape 3) midBoth [] [] = base := by
    funext k
    by_cases hl : k = tape 2
    · subst k
      simp [Reduction.MachineTransfer.tapesAt, midBoth, midLeft,
        hd 2 3 (by decide), leftEmpty]
    · by_cases hr : k = tape 3
      · subst k
        simp [Reduction.MachineTransfer.tapesAt, midBoth, midLeft, rightEmpty]
      · simp [Reduction.MachineTransfer.tapesAt, midBoth, midLeft, hl, hr]
  let back := MachineStateEquiv.program (compareStateEquiv A).symm program
  have atCompare : back (labels .compare) =
      MachineCompare.loop (tape 2) (tape 3) (labels .compare)
        (some (labels .equal)) (some (labels .different)) := by
    change MachineStateEquiv.statement (compareStateEquiv A).symm
      (program (labels .compare)) = _
    rw [atLabels .compare]
    exact MachineStateEquiv.statement_symm_statement (compareStateEquiv A) _
  have backForward : MachineStateEquiv.program (compareStateEquiv A) back = program := by
    funext l
    change MachineStateEquiv.statement (compareStateEquiv A)
      (MachineStateEquiv.statement (compareStateEquiv A).symm (program l)) = program l
    simpa only [Equiv.symm_symm] using
      MachineStateEquiv.statement_symm_statement (compareStateEquiv A).symm (program l)
  have compareRun : (advance (TM2.step program))^[compareSteps a b]
      (some ⟨some (labels .compare), clean ambient, midBoth⟩) =
      some ⟨if a = b then some (labels .equal) else some (labels .different),
        clean ambient, base⟩ := by
    have native := MachineCompare.compareTrace_fromTapes (tape 2) (tape 3)
      (hd 2 3 (by decide)) (labels .compare) (some (labels .equal))
      (some (labels .different)) back atCompare midBoth ambient none none
    rw [copyLeftWord, copyRightWord, restored] at native
    simp only [encodeWord_eq_iff, encodeWord_length] at native
    have transported := MachineStateEquiv.trace (compareStateEquiv A) back _ _ _ native
    rw [backForward] at transported
    simpa [MachineStateEquiv.configuration, compareStateEquiv, clean, compareSteps] using
      transported
  have emitRun : TM2.step program
      ⟨if a = b then some (labels .equal) else some (labels .different),
        clean ambient, base⟩ =
      some ⟨exit, clean ambient,
        Function.update base (tape 5) (bitEncoding (decide (a = b)) ++ base (tape 5))⟩ := by
    by_cases heq : a = b
    · simpa only [heq, ite_true, decide_true] using
        emit_step (tape 5) true exit program (labels .equal) (atLabels .equal)
          (clean ambient) base
    · simpa only [heq, ite_false, decide_false] using
        emit_step (tape 5) false exit program (labels .different) (atLabels .different)
          (clean ambient) base
  have copied := joinTrace_inline_MachineUnaryEqualityBit leftRun rightRun
  have compared := joinTrace_inline_MachineUnaryEqualityBit copied compareRun
  rw [steps, Function.iterate_succ_apply', compared, advance_some]
  exact emitRun

def equalityInTime (tape : Fin 6 → K) (distinct : Function.Injective tape)
    (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
    (atLabels : ∀ l, program (labels l) = instruction tape labels exit l)
    (base : K → List Bool) (a b : Nat) (leftSuffix rightSuffix : List Bool)
    (leftWord : base (tape 0) = encodeWord a ++ leftSuffix)
    (rightWord : base (tape 1) = encodeWord b ++ rightSuffix)
    (leftEmpty : base (tape 2) = []) (rightEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) (ambient : A) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .seedLeft), clean ambient, base⟩
      (some ⟨exit, clean ambient,
        Function.update base (tape 5) (bitEncoding (decide (a = b)) ++ base (tape 5))⟩)
      (3 * (a + b) + 9) where
  steps := steps a b
  evals_in_steps := equalityTrace tape distinct labels exit program atLabels base a b
    leftSuffix rightSuffix leftWord rightWord leftEmpty rightEmpty scratchEmpty ambient
  steps_le_m := steps_le a b

def machine : FinTM2 where
  K := Fin 6
  k₀ := 0
  k₁ := 5
  Γ _ := Bool
  Λ := Label
  main := .seedLeft
  σ := State Unit
  initialState := clean ()
  m := instruction id id none

def machineInTime (base : Fin 6 → List Bool) (a b : Nat)
    (leftSuffix rightSuffix : List Bool)
    (leftWord : base 0 = encodeWord a ++ leftSuffix)
    (rightWord : base 1 = encodeWord b ++ rightSuffix)
    (leftEmpty : base 2 = []) (rightEmpty : base 3 = []) (scratchEmpty : base 4 = []) :
    StateTransition.EvalsToInTime machine.step
      ⟨some .seedLeft, clean (), base⟩
      (some ⟨none, clean (),
        Function.update base (5 : Fin 6) (bitEncoding (decide (a = b)) ++ base (5 : Fin 6))⟩)
      (3 * (a + b) + 9) :=
  equalityInTime id (fun _ _ h => h) id none (instruction id id none) (fun _ => rfl)
    base a b leftSuffix rightSuffix leftWord rightWord leftEmpty rightEmpty scratchEmpty ()

end DFVSGames.Foundations.Complexity.MachineUnaryEqualityBit

namespace DFVSGames.Foundations.Complexity.PoweringMachineRow

open Turing
open MachineFixedBlockMap
open DFVSGames.Foundations.PCP

abbrev Field (S : Nat) := (GraphTables.Label × GraphTables.Label) ⊕ (Fin S ⊕ Fin S)

def inputSize (t S : Nat) : Nat := t * (4096 + (S + S))

def fieldEquiv (S : Nat) : Field S ≃ Fin (4096 + (S + S)) :=
  (Equiv.sumCongr GraphTables.relationIndex
    (finSumFinEquiv : (Fin S ⊕ Fin S) ≃ Fin (S + S))).trans finSumFinEquiv

def inputEquiv (t S : Nat) : (Fin t × Field S) ≃ Fin (inputSize t S) :=
  (Equiv.prodCongr (Equiv.refl (Fin t)) (fieldEquiv S)).trans finProdFinEquiv

def packData {t S : Nat} (data : Fin t → Field S → Bool) : Buffer (inputSize t S) :=
  fun i => data ((inputEquiv t S).symm i).1 ((inputEquiv t S).symm i).2

@[simp] theorem packData_inputEquiv {t S : Nat}
    (data : Fin t → Field S → Bool) (k : Fin t) (f : Field S) :
    packData data (inputEquiv t S (k, f)) = data k f := by
  simp [packData]

def basePredicate {t S : Nat} (bits : Buffer (inputSize t S))
    (k : Fin t) (a b : GraphTables.Label) : Bool :=
  bits (inputEquiv t S (k, .inl (a, b)))

def leftMatches {t S : Nat} (bits : Buffer (inputSize t S))
    (k : Fin t) (i : Fin S) : Bool :=
  bits (inputEquiv t S (k, .inr (.inl i)))

def rightMatches {t S : Nat} (bits : Buffer (inputSize t S))
    (k : Fin t) (i : Fin S) : Bool :=
  bits (inputEquiv t S (k, .inr (.inr i)))

def firstMatch {S : Nat} (mask : Fin S → Bool) : Option (Fin S) :=
  (List.ofFn id).find? mask

theorem firstMatch_spec {S : Nat} (mask : Fin S → Bool) (i : Fin S) :
    firstMatch mask = some i ↔
      mask i = true ∧ ∃ before after,
        List.ofFn id = before ++ i :: after ∧ ∀ j ∈ before, mask j = false := by
  simpa only [firstMatch, Bool.not_eq_eq_eq_not, Bool.not_true] using
    (List.find?_eq_some_iff_append (xs := List.ofFn id) (p := mask) (b := i))

theorem firstMatch_none {S : Nat} (mask : Fin S → Bool) :
    firstMatch mask = none ↔ ∀ i, mask i = false := by
  simp [firstMatch, List.find?_eq_none, List.mem_ofFn]

def edgeAccept {t S q : Nat} (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (k : Fin t) (a b : Fin q) : Bool :=
  match firstMatch (leftMatches bits k), firstMatch (rightMatches bits k) with
  | some i, some j => basePredicate bits k (labelAt a i) (labelAt b j)
  | _, _ => false

theorem edgeAccept_of_matches {t S q : Nat}
    (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (k : Fin t) (a b : Fin q) (i j : Fin S)
    (hl : firstMatch (leftMatches bits k) = some i)
    (hr : firstMatch (rightMatches bits k) = some j) :
    edgeAccept labelAt bits k a b = basePredicate bits k (labelAt a i) (labelAt b j) := by
  simp [edgeAccept, hl, hr]

def rowAccept {t S q : Nat} (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (a b : Fin q) : Bool :=
  (List.ofFn id).all (fun k : Fin t => edgeAccept labelAt bits k a b)

theorem rowAccept_eq_true {t S q : Nat}
    (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (a b : Fin q) :
    rowAccept labelAt bits a b = true ↔ ∀ k, edgeAccept labelAt bits k a b = true := by
  simp [rowAccept, List.all_eq_true, List.mem_ofFn]

def rowBlock {t S q : Nat} (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) : Buffer (q * q) :=
  fun i => rowAccept labelAt bits
    ((GenericGraphTables.relationIndex q).symm i).1
    ((GenericGraphTables.relationIndex q).symm i).2

@[simp] theorem rowBlock_at {t S q : Nat}
    (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (a b : Fin q) :
    rowBlock labelAt bits (GenericGraphTables.relationIndex q (a, b)) =
      rowAccept labelAt bits a b := by
  simp [rowBlock]

def encodeBit (b : Bool) : List Bool := if b then [true, false] else [false]

def encodeBits : List Bool → List Bool
  | [] => []
  | b :: bits => encodeBit b ++ encodeBits bits

@[simp] theorem encodeBits_append (xs ys : List Bool) :
    encodeBits (xs ++ ys) = encodeBits xs ++ encodeBits ys := by
  induction xs with
  | nil => rfl
  | cons b xs ih => simp [encodeBits, ih, List.append_assoc]

theorem encodeBits_graphWords (bits : List Bool) :
    encodeBits bits = encodeWords (bits.map GraphTables.bitWord) := by
  induction bits with
  | nil => rfl
  | cons b bits ih => cases b <;> simp [encodeBits, encodeBit, encodeWords,
      GraphTables.bitWord, encodeWord, ih]

theorem encodeBits_length_le (bits : List Bool) :
    (encodeBits bits).length ≤ 2 * bits.length := by
  induction bits with
  | nil => simp [encodeBits]
  | cons b bits ih =>
      cases b <;> simp only [encodeBits, encodeBit, Bool.false_eq_true,
        ite_false, ite_true, List.length_append, List.length_cons, List.length_nil] <;> omega

section Chains

variable {K Λ σ : Type} {N : Nat}

def readEncodedSlots (src : K) : List (Fin N) →
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N) →
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N)
  | [], next => next
  | i :: slots, next =>
      .pop src (fun state head =>
        (state.1, Function.update state.2 i (head.getD false)))
        (.branch (fun state => state.2 i)
          (.pop src (fun state _ => state) (readEncodedSlots src slots next))
          (readEncodedSlots src slots next))

variable [DecidableEq K]

theorem stepAux_readEncodedSlots (src : K) (slots : List (Fin N))
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (ambient : σ) (bits buffer : Buffer N) (tapes : K → List Bool)
    (suffix : List Bool) :
    TM2.stepAux (readEncodedSlots src slots next) (ambient, buffer)
        (Function.update tapes src (encodeBits (slots.map bits) ++ suffix)) =
      TM2.stepAux next (ambient, fill slots bits buffer)
        (Function.update tapes src suffix) := by
  induction slots generalizing buffer tapes with
  | nil => simp [readEncodedSlots, encodeBits, fill]
  | cons i slots ih =>
      cases hb : bits i <;>
        simpa only [readEncodedSlots, List.map_cons, encodeBits, encodeBit, hb,
          Bool.false_eq_true, ite_false, ite_true, List.cons_append,
          List.nil_append, TM2.stepAux, Function.update_self, List.head?_cons,
          Option.getD_some, List.tail_cons, Function.update_idem, Bool.cond_false,
          Bool.cond_true, fill, List.foldl_cons] using
          ih (Function.update buffer i (bits i)) tapes

theorem stepAux_readEncodedAll (src : K)
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (state : σ × Buffer N) (bits : Buffer N) (tapes : K → List Bool)
    (suffix : List Bool) (hinput : tapes src = encodeBits (List.ofFn bits) ++ suffix) :
    TM2.stepAux (readEncodedSlots src (List.ofFn id) next) state tapes =
      TM2.stepAux next (state.1, bits) (Function.update tapes src suffix) := by
  have h := stepAux_readEncodedSlots src (List.ofFn id) next
    state.1 bits state.2 tapes suffix
  have hin : Function.update tapes src
      (encodeBits ((List.ofFn id).map bits) ++ suffix) = tapes := by
    simpa only [List.map_ofFn, Function.comp_id, ← hinput] using
      Function.update_eq_self src tapes
  rw [hin, fill_all] at h
  exact h

omit [DecidableEq K] in
theorem statementPushBound_readEncodedSlots (src : K) (slots : List (Fin N))
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N)) :
    Runtime.statementPushBound (readEncodedSlots src slots next) =
      Runtime.statementPushBound next := by
  induction slots with
  | nil => rfl
  | cons i slots ih => simp [readEncodedSlots, Runtime.statementPushBound, ih]

def writeEncodedSlots {M : Nat} (dst : K) (emit : σ → Buffer M) : List (Fin M) →
    TM2.Stmt (fun _ : K => Bool) Λ σ → TM2.Stmt (fun _ : K => Bool) Λ σ
  | [], next => next
  | i :: slots, next => .push dst (fun _ => false)
      (.branch (fun state => emit state i)
        (.push dst (fun _ => true) (writeEncodedSlots dst emit slots next))
        (writeEncodedSlots dst emit slots next))

theorem stepAux_writeEncodedSlots {M : Nat} (dst : K) (emit : σ → Buffer M)
    (slots : List (Fin M)) (next : TM2.Stmt (fun _ : K => Bool) Λ σ)
    (state : σ) (tapes : K → List Bool) :
    TM2.stepAux (writeEncodedSlots dst emit slots next) state tapes =
      TM2.stepAux next state
        (Function.update tapes dst
          (encodeBits ((slots.map (emit state)).reverse) ++ tapes dst)) := by
  induction slots generalizing tapes with
  | nil => simp [writeEncodedSlots, encodeBits]
  | cons i slots ih =>
      cases hb : emit state i <;>
        simp only [writeEncodedSlots, TM2.stepAux, hb, Bool.cond_false, Bool.cond_true]
      all_goals rw [ih]
      all_goals simp [Function.update_idem, List.map_cons, List.reverse_cons,
        encodeBits_append, encodeBits, encodeBit, hb, List.append_assoc]

omit [DecidableEq K] in
theorem statementPushBound_writeEncodedSlots {M : Nat}
    (dst : K) (emit : σ → Buffer M) (slots : List (Fin M))
    (next : TM2.Stmt (fun _ : K => Bool) Λ σ) :
    Runtime.statementPushBound (writeEncodedSlots dst emit slots next) =
      2 * slots.length + Runtime.statementPushBound next := by
  induction slots with
  | nil => simp [writeEncodedSlots]
  | cons i slots ih =>
      simp only [writeEncodedSlots, Runtime.statementPushBound, ih, List.length_cons]
      omega

end Chains

section Block

variable {K Λ σ : Type} {N M : Nat}

def encodedBlockStmt (src dst : K) (F : Buffer N → Buffer M)
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N)) :
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N) :=
  readEncodedSlots src (List.ofFn id)
    (writeEncodedSlots dst (fun state => F state.2) (List.ofFn id).reverse
      (.load (fun state => (state.1, emptyBuffer N)) next))

def encodedBlockAt (src dst : K) (F : Buffer N → Buffer M) (exit : Option Λ) :
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N) :=
  encodedBlockStmt src dst F (finishAt exit)

theorem statementPushBound_encodedBlockAt (src dst : K)
    (F : Buffer N → Buffer M) (exit : Option Λ) :
    Runtime.statementPushBound (encodedBlockAt (σ := σ) src dst F exit) = 2 * M := by
  cases exit <;> simp [encodedBlockAt, encodedBlockStmt,
    statementPushBound_readEncodedSlots, statementPushBound_writeEncodedSlots,
    finishAt, Runtime.statementPushBound]

variable [DecidableEq K]

theorem stepAux_encodedBlockStmt (src dst : K) (F : Buffer N → Buffer M)
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (hne : src ≠ dst) (bits : Buffer N) (suffix : List Bool)
    (state : σ × Buffer N) (tapes : K → List Bool)
    (hinput : tapes src = encodeBits (List.ofFn bits) ++ suffix) :
    TM2.stepAux (encodedBlockStmt src dst F next) state tapes =
      TM2.stepAux next (state.1, emptyBuffer N)
        (Function.update (Function.update tapes src suffix) dst
          (encodeBits (List.ofFn (F bits)) ++ tapes dst)) := by
  unfold encodedBlockStmt
  rw [stepAux_readEncodedAll src _ state bits tapes suffix hinput,
    stepAux_writeEncodedSlots]
  simp only [List.map_reverse, List.map_ofFn, Function.comp_id, List.reverse_reverse,
    Function.update_of_ne (Ne.symm hne), TM2.stepAux]

theorem stepAux_encodedBlockAt (src dst : K) (F : Buffer N → Buffer M)
    (exit : Option Λ) (hne : src ≠ dst) (bits : Buffer N) (suffix : List Bool)
    (state : σ × Buffer N) (tapes : K → List Bool)
    (hinput : tapes src = encodeBits (List.ofFn bits) ++ suffix) :
    TM2.stepAux (encodedBlockAt src dst F exit) state tapes =
      { l := exit, var := (state.1, emptyBuffer N),
        stk := Function.update (Function.update tapes src suffix) dst
          (encodeBits (List.ofFn (F bits)) ++ tapes dst) } := by
  rw [encodedBlockAt, stepAux_encodedBlockStmt src dst F _ hne bits suffix state tapes hinput]
  cases exit <;> rfl

theorem step_encodedBlockAt (src dst : K) (F : Buffer N → Buffer M)
    (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (label : Λ) (hprogram : program label = encodedBlockAt src dst F exit)
    (hne : src ≠ dst) (bits : Buffer N) (suffix : List Bool)
    (state : σ × Buffer N) (tapes : K → List Bool)
    (hinput : tapes src = encodeBits (List.ofFn bits) ++ suffix) :
    TM2.step program { l := some label, var := state, stk := tapes } =
      some { l := exit
             var := (state.1, emptyBuffer N)
             stk := Function.update (Function.update tapes src suffix) dst
               (encodeBits (List.ofFn (F bits)) ++ tapes dst) } := by
  simp only [TM2.step, hprogram,
    stepAux_encodedBlockAt src dst F exit hne bits suffix state tapes hinput]

end Block

def rowAt {K Λ σ : Type} {t S q : Nat} (src dst : K)
    (labelAt : Fin q → Fin S → GraphTables.Label) (exit : Option Λ) :
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer (inputSize t S)) :=
  encodedBlockAt src dst (rowBlock labelAt) exit

def machine {t S q : Nat} (labelAt : Fin q → Fin S → GraphTables.Label) : FinTM2 where
  K := Bool
  k₀ := false
  k₁ := true
  Γ _ := Bool
  Λ := Unit
  main := ()
  σ := Unit × Buffer (inputSize t S)
  initialState := ((), emptyBuffer (inputSize t S))
  m _ := rowAt false true labelAt none

theorem machine_statementPushBound {t S q : Nat}
    (labelAt : Fin q → Fin S → GraphTables.Label) :
    Runtime.statementPushBound ((machine (t := t) labelAt).m ()) = 2 * (q * q) :=
  statementPushBound_encodedBlockAt false true (rowBlock labelAt) none

def machineInTime {t S q : Nat} (labelAt : Fin q → Fin S → GraphTables.Label)
    (bits : Buffer (inputSize t S)) (suffix : List Bool)
    (register : Buffer (inputSize t S)) (tapes : Bool → List Bool)
    (hinput : tapes false = encodeBits (List.ofFn bits) ++ suffix) :
    StateTransition.EvalsToInTime (machine (t := t) labelAt).step
      ⟨some (), ((), register), tapes⟩
      (some ⟨none, ((), emptyBuffer (inputSize t S)),
        Function.update (Function.update tapes false suffix) true
          (encodeBits (List.ofFn (rowBlock labelAt bits)) ++ tapes true)⟩) 1 where
  steps := 1
  evals_in_steps := by
    change TM2.step (fun _ : Unit => rowAt false true labelAt none)
        { l := some (), var := ((), register), stk := tapes } =
      some { l := none
             var := ((), emptyBuffer (inputSize t S))
             stk := Function.update (Function.update tapes false suffix) true
               (encodeBits (List.ofFn (rowBlock labelAt bits)) ++ tapes true) }
    exact step_encodedBlockAt false true (rowBlock labelAt) (none : Option Unit)
      (fun _ => rowAt false true labelAt none) () rfl (by decide)
      bits suffix ((), register) tapes hinput
  steps_le_m := le_rfl

end DFVSGames.Foundations.Complexity.PoweringMachineRow

end OAI