frap/CompilerCorrectness_template.v

(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
  * Chapter 9: Compiler Correctness
  * Author: Adam Chlipala
  * License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)

Require Import Frap.

Set Implicit Arguments.


Inductive arith : Set :=
| Const (n : nat)
| Var (x : var)
| Plus (e1 e2 : arith)
| Minus (e1 e2 : arith)
| Times (e1 e2 : arith).

Inductive cmd :=
| Skip
| Assign (x : var) (e : arith)
| Sequence (c1 c2 : cmd)
| If (e : arith) (then_ else_ : cmd)
| While (e : arith) (body : cmd)
| Output (e : arith).
(* ^-- new constructor *)

Coercion Const : nat >-> arith.
Coercion Var : var >-> arith.
Infix "+" := Plus : arith_scope.
Infix "-" := Minus : arith_scope.
Infix "*" := Times : arith_scope.
Delimit Scope arith_scope with arith.
Notation "x <- e" := (Assign x e%arith) (at level 75).
Infix ";;" := Sequence (at level 76). (* This one changed slightly, to avoid parsing clashes. *)
Notation "'when' e 'then' then_ 'else' else_ 'done'" := (If e%arith then_ else_) (at level 75, e at level 0).
Notation "'while' e 'loop' body 'done'" := (While e%arith body) (at level 75).

Definition valuation := fmap var nat.
Fixpoint interp (e : arith) (v : valuation) : nat :=
  match e with
  | Const n => n
  | Var x =>
    match v $? x with
    | None => 0
    | Some n => n
    end
  | Plus e1 e2 => interp e1 v + interp e2 v
  | Minus e1 e2 => interp e1 v - interp e2 v
  | Times e1 e2 => interp e1 v * interp e2 v
  end.

Inductive context :=
| Hole
| CSeq (C : context) (c : cmd).

Inductive plug : context -> cmd -> cmd -> Prop :=
| PlugHole : forall c, plug Hole c c
| PlugSeq : forall c C c' c2,
  plug C c c'
  -> plug (CSeq C c2) c (Sequence c' c2).

Inductive step0 : valuation * cmd -> option nat -> valuation * cmd -> Prop :=
| Step0Assign : forall v x e,
  step0 (v, Assign x e) None (v $+ (x, interp e v), Skip)
| Step0Seq : forall v c2,
  step0 (v, Sequence Skip c2) None (v, c2)
| Step0IfTrue : forall v e then_ else_,
  interp e v <> 0
  -> step0 (v, If e then_ else_) None (v, then_)
| Step0IfFalse : forall v e then_ else_,
  interp e v = 0
  -> step0 (v, If e then_ else_) None (v, else_)
| Step0WhileTrue : forall v e body,
  interp e v <> 0
  -> step0 (v, While e body) None (v, Sequence body (While e body))
| Step0WhileFalse : forall v e body,
  interp e v = 0
  -> step0 (v, While e body) None (v, Skip)
| Step0Output : forall v e,
  step0 (v, Output e) (Some (interp e v)) (v, Skip).

Inductive cstep : valuation * cmd -> option nat -> valuation * cmd -> Prop :=
| CStep : forall C v c l v' c' c1 c2,
  plug C c c1
  -> step0 (v, c) l (v', c')
  -> plug C c' c2
  -> cstep (v, c1) l (v', c2).

Inductive generate : valuation * cmd -> list (option nat) -> Prop :=
| GenDone : forall vc,
  generate vc []
| GenSkip : forall v,
  generate (v, Skip) [None]
| GenSilent : forall vc vc' ns,
  cstep vc None vc'
  -> generate vc' ns
  -> generate vc ns
| GenOutput : forall vc n vc' ns,
  cstep vc (Some n) vc'
  -> generate vc' ns
  -> generate vc (Some n :: ns).

Hint Constructors plug step0 cstep generate.

Definition traceInclusion (vc1 vc2 : valuation * cmd) :=
  forall ns, generate vc1 ns -> generate vc2 ns.
Infix "<|" := traceInclusion (at level 70).

Definition traceEquivalence (vc1 vc2 : valuation * cmd) :=
  vc1 <| vc2 /\ vc2 <| vc1.
Infix "=|" := traceEquivalence (at level 70).

Definition daysPerWeek := 7.
Definition weeksPerMonth := 4.
Definition daysPerMonth := (daysPerWeek * weeksPerMonth)%arith.
(* We are purposely building an expression with arithmetic that can be resolved
 * at compile time, to give our optimizations something to do. *)

Example month_boundaries_in_days :=
  "acc" <- 0;;
  while 1 loop
    when daysPerMonth then
      "acc" <- "acc" + daysPerMonth;;
      Output "acc"
    else
      (* Oh no!  We must have calculated it wrong, since we got zero! *)
      (* And, yes, we know this spot can never be reached.  Some of our
       * optimizations will prove it for us! *)
      Skip
    done
  done.

Hint Extern 1 (interp _ _ = _) => simplify; equality.
Hint Extern 1 (interp _ _ <> _) => simplify; equality.

Theorem first_few_values :
  generate ($0, month_boundaries_in_days) [Some 28; Some 56].
Proof.
  unfold month_boundaries_in_days.
  eapply GenSilent.
  eapply CStep with (C := CSeq Hole _); eauto.
  eapply GenSilent.
  eapply CStep with (C := Hole); eauto.
  eapply GenSilent.
  eapply CStep with (C := Hole); eauto.
  eapply GenSilent.
  eapply CStep with (C := CSeq Hole _); eauto.
  eapply GenSilent.
  eapply CStep with (C := CSeq (CSeq Hole _) _); eauto.
  eapply GenSilent.
  eapply CStep with (C := CSeq Hole _); eauto.
  eapply GenOutput.
  eapply CStep with (C := CSeq Hole _); eauto.
  replace 28 with (interp "acc"
    ($0 $+ ("acc", interp 0 $0)
      $+ ("acc", interp ("acc" + daysPerMonth)%arith ($0 $+ ("acc", interp 0 $0))))); eauto.
  eapply GenSilent.
  eapply CStep with (C := Hole); eauto.
  eapply GenSilent.
  eapply CStep with (C := Hole); eauto.
  eapply GenSilent.
  eapply CStep with (C := CSeq Hole _); eauto.
  eapply GenSilent.
  eapply CStep with (C := CSeq (CSeq Hole _) _); eauto.
  eapply GenSilent.
  eapply CStep with (C := CSeq Hole _); eauto.
  eapply GenOutput.
  eapply CStep with (C := CSeq Hole _); eauto.
  replace 56 with (interp "acc"
    ($0 $+ ("acc", interp 0 $0) $+ ("acc",
     interp ("acc" + daysPerMonth)%arith ($0 $+ ("acc", interp 0 $0))) $+ ("acc",
     interp ("acc" + daysPerMonth)%arith
       ($0 $+ ("acc", interp 0 $0) $+ ("acc",
        interp ("acc" + daysPerMonth)%arith ($0 $+ ("acc", interp 0 $0))))))); eauto.
  constructor.
Qed.


Fixpoint cfoldArith (e : arith) : arith :=
  match e with
  | Const _ => e
  | Var _ => e
  | Plus e1 e2 =>
    let e1' := cfoldArith e1 in
    let e2' := cfoldArith e2 in
    match e1', e2' with
    | Const n1, Const n2 => Const (n1 + n2)
    | _, _ => Plus e1' e2'
    end
  | Minus e1 e2 =>
    let e1' := cfoldArith e1 in
    let e2' := cfoldArith e2 in
    match e1', e2' with
    | Const n1, Const n2 => Const (n1 - n2)
    | _, _ => Minus e1' e2'
    end
  | Times e1 e2 =>
    let e1' := cfoldArith e1 in
    let e2' := cfoldArith e2 in
    match e1', e2' with
    | Const n1, Const n2 => Const (n1 * n2)
    | _, _ => Times e1' e2'
    end
  end.

Theorem cfoldArith_ok : forall v e,
    interp (cfoldArith e) v = interp e v.
Proof.
Admitted.

Fixpoint cfoldExprs (c : cmd) : cmd :=
  match c with
  | Skip => c
  | Assign x e => Assign x (cfoldArith e)
  | Sequence c1 c2 => Sequence (cfoldExprs c1) (cfoldExprs c2)
  | If e then_ else_ => If (cfoldArith e) (cfoldExprs then_) (cfoldExprs else_)
  | While e body => While (cfoldArith e) (cfoldExprs body)
  | Output e => Output (cfoldArith e)
  end.

Compute cfoldExprs month_boundaries_in_days.

Theorem skip_or_step : forall v c,
    c = Skip
    \/ exists v' l c', cstep (v, c) l (v', c').
Proof.
  induct c; simplify; first_order; subst;
    try match goal with
        | [ H : cstep _ _ _ |- _ ] => invert H
        end;
    try match goal with
        | [ |- context[cstep (?v, If ?e _ _)] ] => cases (interp e v ==n 0)
        | [ |- context[cstep (?v, While ?e _)] ] => cases (interp e v ==n 0)
        end; eauto 10.
Qed.

Theorem plug_function : forall C c1 c2, plug C c1 c2
  -> forall c2', plug C c1 c2'
  -> c2 = c2'.
Proof.
  induct 1; invert 1; eauto.
  apply IHplug in H5.
  equality.
Qed.

Lemma peel_cseq : forall C1 C2 c (c1 c2 : cmd),
    C1 = C2 /\ c1 = c2
    -> CSeq C1 c = CSeq C2 c /\ c1 = c2.
Proof.
  equality.
Qed.

Hint Resolve peel_cseq.

Lemma plug_deterministic : forall v C c1 c2, plug C c1 c2
  -> forall l vc1, step0 (v, c1) l vc1
  -> forall C' c1', plug C' c1' c2
  -> forall l' vc1', step0 (v, c1') l' vc1'
  -> C' = C /\ c1' = c1.
Proof.
  induct 1; invert 1; invert 1; invert 1; auto;
    try match goal with
        | [ H : plug _ _ _ |- _ ] => invert1 H
        end; eauto.
Qed.

Lemma deterministic0 : forall vc l vc',
  step0 vc l vc'
  -> forall l' vc'', step0 vc l' vc''
                     -> l = l' /\ vc'' = vc'.
Proof.
  invert 1; invert 1; simplify; propositional.
Qed.

Theorem deterministic : forall vc l vc',
  cstep vc l vc'
  -> forall l' vc'', cstep vc l' vc''
                     -> l = l' /\ vc' = vc''.
Proof.
  invert 1; invert 1; simplify.
  eapply plug_deterministic in H0; eauto.
  invert H0.
  match goal with
  | [ H : step0 _ _ _, H' : step0 _ _ _ |- _ ] => eapply deterministic0 in H; [ | apply H' ]
  end.
  propositional; subst; auto.
  invert H0.
  auto.
  eapply plug_function in H2; eauto.
  equality.
Qed.





















Section simulation.
  Variable R : valuation * cmd -> valuation * cmd -> Prop.

  Hypothesis one_step : forall vc1 vc2, R vc1 vc2
    -> forall vc1' l, cstep vc1 l vc1'
      -> exists vc2', cstep vc2 l vc2' /\ R vc1' vc2'.

  Hypothesis agree_on_termination : forall v1 v2 c2, R (v1, Skip) (v2, c2)
    -> c2 = Skip.

  Lemma simulation_fwd' : forall vc1 ns, generate vc1 ns
    -> forall vc2, R vc1 vc2
      -> generate vc2 ns.
  Proof.
  Admitted.

  Theorem simulation_fwd : forall vc1 vc2, R vc1 vc2
    -> vc1 <| vc2.
  Proof.
    unfold traceInclusion; eauto using simulation_fwd'.
  Qed.

  Lemma simulation_bwd' : forall vc2 ns, generate vc2 ns
    -> forall vc1, R vc1 vc2
      -> generate vc1 ns.
  Proof.
  Admitted.

  Theorem simulation_bwd : forall vc1 vc2, R vc1 vc2
    -> vc2 <| vc1.
  Proof.
    unfold traceInclusion; eauto using simulation_bwd'.
  Qed.

  Theorem simulation : forall vc1 vc2, R vc1 vc2
    -> vc1 =| vc2.
  Proof.
    simplify; split; auto using simulation_fwd, simulation_bwd.
  Qed.
End simulation.

Theorem cfoldExprs_ok : forall v c,
    (v, c) =| (v, cfoldExprs c).
Proof.
Admitted.


Fixpoint cfold (c : cmd) : cmd :=
  match c with
  | Skip => c
  | Assign x e => Assign x (cfoldArith e)
  | Sequence c1 c2 => Sequence (cfold c1) (cfold c2)
  | If e then_ else_ =>
    let e' := cfoldArith e in
    match e' with
    | Const n => if n ==n 0 then cfold else_ else cfold then_
    | _ => If e' (cfold then_) (cfold else_)
    end
  | While e body => While (cfoldArith e) (cfold body)
  | Output e => Output (cfoldArith e)
  end.

Compute cfold month_boundaries_in_days.




















Notation silent_cstep := (fun a b => cstep a None b).

Lemma silent_generate_fwd : forall ns vc vc',
    silent_cstep^* vc vc'
    -> generate vc ns
    -> generate vc' ns.
Proof.
  induct 1; simplify; eauto.
  invert H1; auto.

  invert H.
  invert H3.
  invert H4.

  eapply deterministic in H; eauto.
  propositional; subst.
  auto.

  eapply deterministic in H; eauto.
  equality.
Qed.

Lemma silent_generate_bwd : forall ns vc vc',
    silent_cstep^* vc vc'
    -> generate vc' ns
    -> generate vc ns.
Proof.
  induct 1; eauto.
Qed.

Lemma generate_Skip : forall v a ns,
    generate (v, Skip) (Some a :: ns)
    -> False.
Proof.
  induct 1; simplify.

  invert H.
  invert H3.
  invert H4.

  invert H.
  invert H3.
  invert H4.
Qed.

Hint Resolve silent_generate_fwd silent_generate_bwd generate_Skip.

Section simulation_skipping.
  Variable R : nat -> valuation * cmd -> valuation * cmd -> Prop.

  Hypothesis one_step : forall n vc1 vc2, R n vc1 vc2
    -> forall vc1' l, cstep vc1 l vc1'
                      -> (exists n', n = S n' /\ l = None /\ R n' vc1' vc2)
                         \/ exists n' vc2', cstep vc2 l vc2' /\ R n' vc1' vc2'.

  Hypothesis agree_on_termination : forall n v1 v2 c2, R n (v1, Skip) (v2, c2)
    -> c2 = Skip.

  Lemma simulation_skipping_fwd' : forall vc1 ns, generate vc1 ns
    -> forall n vc2, R n vc1 vc2
      -> generate vc2 ns.
  Proof.
    induct 1; simplify; eauto.

    cases vc2.
    apply agree_on_termination in H.
    subst.
    auto.

    eapply one_step in H; eauto.
    first_order.
    eauto.

    eapply one_step in H1; eauto.
    first_order.
    equality.
    eauto.
  Qed.

  Theorem simulation_skipping_fwd : forall n vc1 vc2, R n vc1 vc2
    -> vc1 <| vc2.
  Proof.
    unfold traceInclusion; eauto using simulation_skipping_fwd'.
  Qed.

  Lemma match_step : forall n vc2 l vc2' vc1,
      cstep vc2 l vc2'
      -> R n vc1 vc2
      -> exists vc1' vc1'' n', silent_cstep^* vc1 vc1'
                              /\ cstep vc1' l vc1''
                              /\ R n' vc1'' vc2'.
  Proof.
    induct n; simplify.

    cases vc1; cases vc2.
    assert (c = Skip \/ exists v' l' c', cstep (v, c) l' (v', c')) by apply skip_or_step.
    first_order; subst.
    apply agree_on_termination in H0; subst.
    invert H.
    invert H2.
    invert H3.
    eapply one_step in H0; eauto.
    first_order; subst.
    equality.
    eapply deterministic in H; eauto.
    first_order; subst.
    eauto 6.
    
    cases vc1; cases vc2.
    assert (c = Skip \/ exists v' l' c', cstep (v, c) l' (v', c')) by apply skip_or_step.
    first_order; subst.
    apply agree_on_termination in H0; subst.
    invert H.
    invert H2.
    invert H3.
    eapply one_step in H0; eauto.
    first_order; subst.
    invert H0.
    eapply IHn in H3; eauto.
    first_order.
    eauto 8.
    eapply deterministic in H; eauto.
    first_order; subst.
    eauto 6.
  Qed.

  Lemma step_to_termination : forall vc v,
      silent_cstep^* vc (v, Skip)
      -> generate vc [None].
  Proof.
    clear; induct 1; eauto.
  Qed.

  Hint Resolve step_to_termination.

  Lemma R_Skip : forall n vc1 v,
      R n vc1 (v, Skip)
      -> exists v', silent_cstep^* vc1 (v', Skip).
  Proof.
    induct n; simplify.

    cases vc1.
    assert (c = Skip \/ exists v' l c', cstep (v0, c) l (v', c')) by apply skip_or_step.
    first_order; subst.
    eauto.
    eapply one_step in H; eauto.
    first_order.
    equality.
    invert H.
    invert H4.
    invert H5.

    cases vc1.
    assert (c = Skip \/ exists v' l c', cstep (v0, c) l (v', c')) by apply skip_or_step.
    first_order; subst.
    eauto.
    eapply one_step in H; eauto.
    first_order; subst.
    invert H.
    apply IHn in H2.
    first_order.
    eauto.
    invert H.
    invert H4.
    invert H5.
  Qed.

  Lemma simulation_skipping_bwd' : forall ns vc2, generate vc2 ns
    -> forall n vc1, R n vc1 vc2
      -> generate vc1 ns.
  Proof.
    induct 1; simplify; eauto.

    cases vc1.
    apply R_Skip in H; first_order.
    eauto.

    eapply match_step in H1; eauto.
    first_order.
    eauto.

    eapply match_step in H1; eauto.
    first_order.
    eauto.
  Qed.

  Theorem simulation_skipping_bwd : forall n vc1 vc2, R n vc1 vc2
    -> vc2 <| vc1.
  Proof.
    unfold traceInclusion; eauto using simulation_skipping_bwd'.
  Qed.

  Theorem simulation_skipping : forall n vc1 vc2, R n vc1 vc2
    -> vc1 =| vc2.
  Proof.
    simplify; split; eauto using simulation_skipping_fwd, simulation_skipping_bwd.
  Qed.
End simulation_skipping.

Hint Extern 1 (_ < _) => linear_arithmetic.
Hint Extern 1 (_ >= _) => linear_arithmetic.
Hint Extern 1 (_ <> _) => linear_arithmetic.

Lemma cfold_ok : forall v c,
    (v, c) =| (v, cfold c).
Proof.
Admitted.


Fixpoint tempVar (n : nat) : string :=
  match n with
  | O => "_tmp"
  | S n' => tempVar n' ++ "'"
  end%string.

Compute tempVar 0.
Compute tempVar 1.
Compute tempVar 2.

Fixpoint noUnderscoreVar (x : var) : bool :=
  match x with
  | String "_" _ => false
  | _ => true
  end.

Lemma append_assoc : forall a b c,
    (a ++ (b ++ c) = (a ++ b) ++ c)%string.
Proof.
  induct a; simplify; equality.
Qed.

Lemma append_assoc_String : forall a b,
    (String a b = String a "" ++ b)%string.
Proof.
  induct b; simplify; equality.
Qed.

Lemma noUnderscoreVar_tempVar' : forall n,
    exists s, tempVar n = ("_tmp" ++ s)%string.
Proof.
  induct n; simplify; first_order.

  exists ""; auto.

  rewrite H.
  exists (x ++ "'")%string.
  repeat match goal with
         | [ |- context[String ?c ?x] ] =>
           match x with
           | "" => fail 1
           | _ => rewrite (append_assoc_String c x)
           end
         end.
  repeat rewrite append_assoc.
  reflexivity.
Qed.  

Theorem noUnderscoreVar_tempVar : forall x,
    noUnderscoreVar x = true
    -> forall n, x <> tempVar n.
Proof.
  unfold not; simplify.
  subst.
  pose proof (noUnderscoreVar_tempVar' n).
  first_order.
  rewrite H0 in H.
  simplify.
  equality.
Qed.

Lemma tempVar_inj' : forall s1 s2,
    (s1 ++ "'" = s2 ++ "'")%string
    -> s1 = s2.
Proof.
  induct s1; simplify.

  cases s2; simplify; try equality.
  invert H.
  cases s2; simplify; equality.

  cases s2; simplify.
  invert H.
  cases s1; simplify; equality.
  invert H.
  f_equal; auto.
Qed.

Theorem tempVar_inj : forall n1 n2,
    tempVar n1 = tempVar n2
    -> n1 = n2.
Proof.
  induct n1; simplify; cases n2; simplify; try equality.

  repeat match goal with
         | [ _ : context[(?s ++ "'")%string] |- _ ] => cases s; simplify; try equality
         end.

  repeat match goal with
         | [ _ : context[(?s ++ "'")%string] |- _ ] => cases s; simplify; try equality
         end.

  auto using tempVar_inj'.
Qed.

Fixpoint noUnderscoreArith (e : arith) : bool :=
  match e with
  | Const _ => true
  | Var x => noUnderscoreVar x
  | Plus e1 e2 => noUnderscoreArith e1 && noUnderscoreArith e2
  | Minus e1 e2 => noUnderscoreArith e1 && noUnderscoreArith e2
  | Times e1 e2 => noUnderscoreArith e1 && noUnderscoreArith e2
  end.

Fixpoint noUnderscore (c : cmd) : bool :=
  match c with
  | Skip => true
  | Assign x e => noUnderscoreVar x && noUnderscoreArith e
  | Sequence c1 c2 => noUnderscore c1 && noUnderscore c2
  | If e then_ else_ => noUnderscoreArith e && noUnderscore then_ && noUnderscore else_
  | While e body => noUnderscoreArith e && noUnderscore body
  | Output e => noUnderscoreArith e
  end.

Compute noUnderscore month_boundaries_in_days.

Fixpoint flattenArith (tempCount : nat) (dst : var) (e : arith) : cmd :=
  match e with
  | Const _
  | Var _ => Assign dst e
  | Plus e1 e2 =>
    let x1 := tempVar tempCount in
    let c1 := flattenArith (S tempCount) x1 e1 in
    let x2 := tempVar (S tempCount) in
    let c2 := flattenArith (S (S tempCount)) x2 e2 in
    Sequence c1 (Sequence c2 (Assign dst (Plus x1 x2)))
  | Minus e1 e2 =>
    let x1 := tempVar tempCount in
    let c1 := flattenArith (S tempCount) x1 e1 in
    let x2 := tempVar (S tempCount) in
    let c2 := flattenArith (S (S tempCount)) x2 e2 in
    Sequence c1 (Sequence c2 (Assign dst (Minus x1 x2)))
  | Times e1 e2 =>
    let x1 := tempVar tempCount in
    let c1 := flattenArith (S tempCount) x1 e1 in
    let x2 := tempVar (S tempCount) in
    let c2 := flattenArith (S (S tempCount)) x2 e2 in
    Sequence c1 (Sequence c2 (Assign dst (Times x1 x2)))
  end.

Fixpoint flatten (c : cmd) : cmd :=
  match c with
  | Skip => c
  | Assign x e => flattenArith 0 x e
  | Sequence c1 c2 => Sequence (flatten c1) (flatten c2)
  | If e then_ else_ => If e (flatten then_) (flatten else_)
  | While e body => While e (flatten body)
  | Output _ => c
  end.

Compute flatten month_boundaries_in_days.



























Section simulation_multiple.
  Variable R : valuation * cmd -> valuation * cmd -> Prop.

  Hypothesis one_step : forall vc1 vc2, R vc1 vc2
    -> forall vc1' l, cstep vc1 l vc1'
                      -> exists vc2' vc2'',
                        silent_cstep^* vc2 vc2'
                        /\ cstep vc2' l vc2''
                        /\ R vc1' vc2''.

  Hypothesis agree_on_termination : forall v1 v2 c2, R (v1, Skip) (v2, c2)
    -> c2 = Skip.

  Lemma simulation_multiple_fwd' : forall vc1 ns, generate vc1 ns
    -> forall vc2, R vc1 vc2
      -> generate vc2 ns.
  Proof.
    induct 1; simplify; eauto.

    cases vc2.
    apply agree_on_termination in H; subst.
    auto.

    eapply one_step in H; eauto.
    first_order.
    eauto.

    eapply one_step in H1; eauto.
    first_order.
    eauto.
  Qed.

  Theorem simulation_multiple_fwd : forall vc1 vc2, R vc1 vc2
    -> vc1 <| vc2.
  Proof.
    unfold traceInclusion; eauto using simulation_multiple_fwd'.
  Qed.

  Inductive generateN : nat -> valuation * cmd -> list (option nat) -> Prop :=
  | GenDoneN : forall vc,
      generateN 0 vc []
  | GenSkupN : forall v,
      generateN 0 (v, Skip) [None]
  | GenSilentN : forall sc vc vc' ns,
      cstep vc None vc'
      -> generateN sc vc' ns
      -> generateN (S sc) vc ns
  | GenOutputN : forall sc vc n vc' ns,
      cstep vc (Some n) vc'
      -> generateN sc vc' ns
      -> generateN (S sc) vc (Some n :: ns).

  (* We won't comment on the other proof details, though they could be
   * interesting reading. *)

  Hint Constructors generateN.

  Lemma generateN_fwd : forall sc vc ns,
      generateN sc vc ns
      -> generate vc ns.
  Proof.
    induct 1; eauto.
  Qed.

  Hint Resolve generateN_fwd.

  Lemma generateN_bwd : forall vc ns,
      generate vc ns
      -> exists sc, generateN sc vc ns.
  Proof.
    induct 1; first_order; eauto.
  Qed.

  Lemma generateN_silent_cstep : forall sc vc ns,
      generateN sc vc ns
      -> forall vc', silent_cstep^* vc vc'
      -> exists sc', sc' <= sc /\ generateN sc' vc' ns.
  Proof.
    clear; induct 1; simplify; eauto.

    invert H; eauto.
    invert H0.
    invert H3.
    invert H4.

    invert H1; eauto.
    eapply deterministic in H; eauto.
    propositional; subst.
    apply IHgenerateN in H3.
    first_order.
    eauto.

    invert H1; eauto.
    eapply deterministic in H; eauto.
    equality.
  Qed.

  Lemma termination_is_last : forall sc vc ns,
      generateN sc vc (None :: ns)
      -> ns = [].
  Proof.
    induct 1; eauto.
  Qed.

  Lemma simulation_multiple_bwd' : forall sc sc', sc' < sc
    -> forall vc2 ns, generateN sc' vc2 ns
    -> forall vc1, R vc1 vc2
    -> generate vc1 ns.
  Proof.
    induct sc; simplify.

    linear_arithmetic.

    cases sc'.
    invert H0.
    auto.

    cases vc1.
    assert (c = Skip \/ exists v' l c', cstep (v0, c) l (v', c')) by apply skip_or_step.
    first_order; subst.
    auto.
    eapply one_step in H1; eauto.
    first_order.
    invert H1.
    invert H2.
    invert H5.
    invert H6.
    invert H4.
    invert H7.
    invert H8.

    cases vc1; cases vc2.
    assert (c = Skip \/ exists v' l c', cstep (v, c) l (v', c')) by apply skip_or_step.
    first_order; subst.
    apply agree_on_termination in H1; subst.
    cases ns; auto.
    cases o.
    exfalso; eauto.
    eapply termination_is_last in H0; subst.
    auto.

    eapply one_step in H1; eauto.
    first_order.
    eapply generateN_silent_cstep in H0; eauto.
    first_order.
    invert H5; auto.
    invert H3.
    invert H7.
    invert H8.
    eapply deterministic in H3; eauto.
    propositional; subst.
    econstructor.
    eauto.
    eapply IHsc; try eassumption.
    linear_arithmetic.

    eapply deterministic in H3; eauto.
    propositional; subst.
    eapply GenOutput.
    eauto.
    eapply IHsc; try eassumption.
    linear_arithmetic.
  Qed.

  Theorem simulation_multiple_bwd : forall vc1 vc2, R vc1 vc2
    -> vc2 <| vc1.
  Proof.
    unfold traceInclusion; simplify.
    apply generateN_bwd in H0.
    first_order.
    eauto using simulation_multiple_bwd'.
  Qed.

  Theorem simulation_multiple : forall vc1 vc2, R vc1 vc2
    -> vc1 =| vc2.
  Proof.
    simplify; split; auto using simulation_multiple_fwd, simulation_multiple_bwd.
  Qed.
End simulation_multiple.


































Definition agree (v v' : valuation) :=
  forall x,
    noUnderscoreVar x = true
    -> v $? x = v' $? x.

Ltac bool :=
  simplify;
  repeat match goal with
         | [ H : _ && _ = true |- _ ] => apply andb_true_iff in H; propositional
         end.

Lemma interp_agree : forall v v', agree v v'
  -> forall e, noUnderscoreArith e = true
  -> interp e v = interp e v'.
Proof.
  induct e; bool; try equality.

  unfold agree in H.
  specialize (H _ H0).
  rewrite H.
  equality.
Qed.

Lemma agree_add : forall v v' x n,
    agree v v'
    -> agree (v $+ (x, n)) (v' $+ (x, n)).
Proof.
  unfold agree; simplify.
  apply H in H0.
  cases (x ==v x0); simplify; auto.
Qed.

Lemma agree_add_tempVar_fwd : forall v v' n nv,
    agree v v'
    -> agree (v $+ (tempVar n, nv)) v'.
Proof.
  unfold agree; simplify.
  cases (x ==v tempVar n); simplify; subst; auto.
  eapply noUnderscoreVar_tempVar in H0.
  propositional.
Qed.

Lemma agree_add_tempVar_bwd : forall v v' n nv,
    agree (v $+ (tempVar n, nv)) v'
    -> agree v v'.
Proof.
  unfold agree; simplify.
  specialize (H _ H0).
  cases (x ==v tempVar n); simplify; subst; auto.
  eapply noUnderscoreVar_tempVar in H0.
  propositional.
Qed.

Lemma agree_add_tempVar_bwd_prime : forall v v' n nv,
    agree (v $+ (tempVar n ++ "'", nv)%string) v'
    -> agree v v'.
Proof.
  simplify.
  change (tempVar n ++ "'")%string with (tempVar (S n)) in *.
  eauto using agree_add_tempVar_bwd.
Qed.

Lemma agree_refl : forall v,
    agree v v.
Proof.
  first_order.
Qed.

Hint Resolve agree_add agree_add_tempVar_fwd agree_add_tempVar_bwd agree_add_tempVar_bwd_prime agree_refl.

Lemma silent_csteps_front : forall c v1 v2 c1 c2,
    silent_cstep^* (v1, c1) (v2, c2)
    -> silent_cstep^* (v1, c1;; c) (v2, c2;; c).
Proof.
  induct 1; eauto.
  invert H.
  eauto 6.
Qed.

Hint Resolve silent_csteps_front.

Lemma tempVar_contra : forall n1 n2,
    tempVar n1 = tempVar n2
    -> n1 <> n2
    -> False.
Proof.
  pose proof tempVar_inj.
  first_order.
Qed.

Hint Resolve tempVar_contra.

Lemma self_prime_contra : forall s,
    (s ++ "'")%string = s -> False.
Proof.
  induct s; simplify; equality.
Qed.

Hint Resolve self_prime_contra.

Opaque tempVar.

(*Lemma flatten_Assign : forall e dst tempCount,
  noUnderscoreArith e = true
  -> (forall n, n >= tempCount -> dst <> tempVar n)
  -> forall v1 v2, agree v1 v2
  -> exists v c v2',
    silent_cstep^* (v2, flattenArith tempCount dst e) (v, c)
    /\ cstep (v, c) None (v2', Skip)
    /\ agree (v1 $+ (dst, interp e v1)) v2'
    /\ v2' $? dst = Some (interp e v1)
    /\ (forall n, n < tempCount -> dst <> tempVar n -> v2' $? tempVar n = v2 $? tempVar n).
Proof.
  induct e; bool.

  do 3 eexists.
  split.
  auto.
  split.
  eauto.
  split.
  eauto.
  propositional; auto.
  simplify; auto.
  simplify.
  cases (dst ==v tempVar n0); simplify; subst; auto.

  do 3 eexists.
  split.
  auto.
  split.
  eauto.
  split.
  eauto.
  propositional; auto.
  simplify.
  unfold agree in H1.
  apply H1 in H.
  rewrite H.
  eauto.
  simplify.
  unfold agree in H1.
  apply H1 in H.
  rewrite H.
  split.
  equality.
  simplify.
  equality.

  eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
  first_order.
  eapply IHe2 with (dst := tempVar (S tempCount)) (tempCount := S (S tempCount)) in H4; eauto; clear IHe2.
  first_order.
  eexists; exists (dst <- tempVar tempCount + tempVar (S tempCount)); eexists.
  split.
  apply trc_trans with (y := (x2, x3;; dst <- tempVar tempCount + tempVar (S tempCount))).
  apply trc_trans with (y := (x1, flattenArith (S (S tempCount)) (tempVar (S tempCount)) e2;; dst <- tempVar tempCount + tempVar (S tempCount))).
  eauto 7 using trc_trans.
  eauto 7 using trc_trans.
  eauto 7 using trc_trans.
  split.
  eauto.
  split.
  simplify.
  rewrite H9.
  rewrite H10 by eauto.
  rewrite H5.
  erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
  eauto.
  simplify.
  propositional.
  rewrite H9.
  rewrite H10 by eauto.
  rewrite H5.
  erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
  auto.
  simplify.
  rewrite H10 by eauto.
  eauto.

  (* Apologies for the copy-and-paste between the binary-operator cases! *)
  eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
  first_order.
  eapply IHe2 with (dst := tempVar (S tempCount)) (tempCount := S (S tempCount)) in H4; eauto; clear IHe2.
  first_order.
  eexists; exists (dst <- tempVar tempCount - tempVar (S tempCount)); eexists.
  split.
  apply trc_trans with (y := (x2, x3;; dst <- tempVar tempCount - tempVar (S tempCount))).
  apply trc_trans with (y := (x1, flattenArith (S (S tempCount)) (tempVar (S tempCount)) e2;; dst <- tempVar tempCount - tempVar (S tempCount))).
  eauto 7 using trc_trans.
  eauto 7 using trc_trans.
  eauto 7 using trc_trans.
  split.
  eauto.
  split.
  simplify.
  rewrite H9.
  rewrite H10 by eauto.
  rewrite H5.
  erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
  eauto.
  simplify.
  propositional.
  rewrite H9.
  rewrite H10 by eauto.
  rewrite H5.
  erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
  auto.
  simplify.
  rewrite H10 by eauto.
  eauto.

  eapply IHe1 with (dst := tempVar tempCount) (tempCount := S tempCount) in H1; eauto; clear IHe1.
  first_order.
  eapply IHe2 with (dst := tempVar (S tempCount)) (tempCount := S (S tempCount)) in H4; eauto; clear IHe2.
  first_order.
  eexists; exists (dst <- tempVar tempCount * tempVar (S tempCount)); eexists.
  split.
  apply trc_trans with (y := (x2, x3;; dst <- tempVar tempCount * tempVar (S tempCount))).
  apply trc_trans with (y := (x1, flattenArith (S (S tempCount)) (tempVar (S tempCount)) e2;; dst <- tempVar tempCount * tempVar (S tempCount))).
  eauto 7 using trc_trans.
  eauto 7 using trc_trans.
  eauto 7 using trc_trans.
  split.
  eauto.
  split.
  simplify.
  rewrite H9.
  rewrite H10 by eauto.
  rewrite H5.
  erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
  eauto.
  simplify.
  propositional.
  rewrite H9.
  rewrite H10 by eauto.
  rewrite H5.
  erewrite interp_agree with (v := v1 $+ (tempVar tempCount, interp e1 v1)) (v' := v1) by eauto.
  auto.
  simplify.
  rewrite H10 by eauto.
  eauto.
Qed.

Lemma flatten_ok' : forall v1 c1 l v2 c2,
    step0 (v1, c1) l (v2, c2)
    -> noUnderscore c1 = true
    -> forall v1', agree v1 v1'
    -> exists v c v2', silent_cstep^* (v1', flatten c1) (v, c)
                       /\ cstep (v, c) l (v2', flatten c2)
                       /\ agree v2 v2'.
Proof.
  invert 1; simplify; bool;
    repeat erewrite interp_agree in * by eassumption; eauto 10.

  assert (Hnu : noUnderscoreArith e = true) by assumption.
  eapply flatten_Assign with (tempCount := 0) (dst := x) in Hnu; eauto.
  first_order.
  do 3 eexists.
  split.
  eassumption.
  split.
  eassumption.
  erewrite <- interp_agree; eauto.
  simplify.
  eauto using noUnderscoreVar_tempVar.
Qed.

Lemma noUnderscore_plug : forall C c0 c1,
    plug C c0 c1
    -> noUnderscore c1 = true
    -> noUnderscore c0 = true.
Proof.
  induct 1; bool; auto.
Qed.

Hint Immediate noUnderscore_plug.

Lemma silent_csteps_plug : forall C c1 c1',
    plug C c1 c1'
    -> forall v1 v2 c2 c2', plug C c2 c2'
    -> silent_cstep^* (v1, c1) (v2, c2)
    -> silent_cstep^* (v1, c1') (v2, c2').
Proof.
  induct 1; invert 1; eauto.
Qed.

Hint Resolve silent_csteps_plug.

Fixpoint flattenContext (C : context) : context :=
  match C with
  | Hole => Hole
  | CSeq C c => CSeq (flattenContext C) (flatten c)
  end.

Lemma plug_flatten : forall C c1 c2, plug C c1 c2
  -> plug (flattenContext C) (flatten c1) (flatten c2).
Proof.
  induct 1; simplify; eauto.
Qed.

Hint Resolve plug_flatten.

Lemma plug_total : forall c C, exists c', plug C c c'.
Proof.
  induct C; first_order; eauto.
Qed.

Lemma plug_cstep : forall C c1 c1', plug C c1 c1'
  -> forall c2 c2', plug C c2 c2'
  -> forall v l v', cstep (v, c1) l (v', c2)
                    -> cstep (v, c1') l (v', c2').
Proof.
  induct 1; invert 1; first_order; eauto.
  eapply IHplug in H0; eauto.
  first_order.
  invert H0.
  eauto.
Qed.

Hint Resolve plug_cstep.

Lemma step0_noUnderscore : forall v c l v' c',
    step0 (v, c) l (v', c')
    -> noUnderscore c = true
    -> noUnderscore c' = true.
Proof.
  invert 1; bool; auto.
  rewrite H0, H1.
  reflexivity.
Qed.

Hint Resolve step0_noUnderscore.

Fixpoint noUnderscoreContext (C : context) : bool :=
  match C with
  | Hole => true
  | CSeq C' c => noUnderscoreContext C' && noUnderscore c
  end.

Lemma noUnderscore_plug_context : forall C c0 c1,
    plug C c0 c1
    -> noUnderscore c1 = true
    -> noUnderscoreContext C = true.
Proof.
  induct 1; bool; auto.
  rewrite H0, H2; reflexivity.
Qed.

Lemma noUnderscore_plug_fwd : forall C c0 c1,
    plug C c0 c1
    -> noUnderscoreContext C = true
    -> noUnderscore c0 = true
    -> noUnderscore c1 = true.
Proof.
  induct 1; bool; auto.
  rewrite H4, H3; reflexivity.
Qed.

Hint Resolve noUnderscore_plug_context noUnderscore_plug_fwd.

Lemma flatten_ok : forall v c,
    noUnderscore c = true
    -> (v, c) =| (v, flatten c).
Proof.
  simplify.
  (* Note that our simulation relation remembers lack of underscores, and it
   * enforces mere agreement between valuations, rather than full equality. *)
  apply simulation_multiple with (R := fun vc1 vc2 => noUnderscore (snd vc1) = true
                                                      /\ agree (fst vc1) (fst vc2)
                                                      /\ snd vc2 = flatten (snd vc1));
    simplify; propositional; eauto.

  invert H1; simplify; subst.
  assert (noUnderscore c2 = true) by eauto.
  eapply flatten_ok' in H5; eauto.
  first_order.
  cases vc2; simplify; subst.
  pose proof (plug_total x0 (flattenContext C)).
  first_order.
  do 2 eexists.
  split.
  eapply silent_csteps_plug; try apply H4; eauto.
  eauto 6.
Qed.*)