Library PE

PE: Partial Evaluation



Equiv.v introduced constant folding as an example of a program transformation and proved that it preserves the meaning of the program. Constant folding operates on manifest constants such as ANum expressions. For example, it simplifies the command Y ::= APlus (ANum 3) (ANum 1) to the command Y ::= ANum 4. However, it does not propagate known constants along data flow. For example, it does not simplify the sequence X ::= ANum 3;; Y ::= APlus (AId X) (ANum 1) to X ::= ANum 3;; Y ::= ANum 4 because it forgets that X is 3 by the time it gets to Y.
We naturally want to enhance constant folding so that it propagates known constants and uses them to simplify programs. Doing so constitutes a rudimentary form of partial evaluation. As we will see, partial evaluation is so called because it is like running a program, except only part of the program can be evaluated because only part of the input to the program is known. For example, we can only simplify the program X ::= ANum 3;; Y ::= AMinus (APlus (AId X) (ANum 1)) (AId Y) to X ::= ANum 3;; Y ::= AMinus (ANum 4) (AId Y) without knowing the initial value of Y.

Require Export Imp.
Require Import FunctionalExtensionality.

Generalizing Constant Folding

The starting point of partial evaluation is to represent our partial knowledge about the state. For example, between the two assignments above, the partial evaluator may know only that X is 3 and nothing about any other variable.

Partial States

Conceptually speaking, we can think of such partial states as the type id → option nat (as opposed to the type id → nat of concrete, full states). However, in addition to looking up and updating the values of individual variables in a partial state, we may also want to compare two partial states to see if and where they differ, to handle conditional control flow. It is not possible to compare two arbitrary functions in this way, so we represent partial states in a more concrete format: as a list of id × nat pairs.

Definition pe_state := list (id × nat).

The idea is that a variable id appears in the list if and only if we know its current nat value. The pe_lookup function thus interprets this concrete representation. (If the same variable id appears multiple times in the list, the first occurrence wins, but we will define our partial evaluator to never construct such a pe_state.)

Fixpoint pe_lookup (pe_st : pe_state) (V:id) : option nat :=
  match pe_st with
  | [] ⇒ None
  | (V',n')::pe_st ⇒ if eq_id_dec V V' then Some n'
                      else pe_lookup pe_st V
  end.

For example, empty_pe_state represents complete ignorance about every variable -- the function that maps every id to None.

Definition empty_pe_state : pe_state := [].

More generally, if the list representing a pe_state does not contain some id, then that pe_state must map that id to None. Before we prove this fact, we first define a useful tactic for reasoning with id equality. The tactic compare V V' SCase means to reason by cases over eq_id_dec V V'. In the case where V = V', the tactic substitutes V for V' throughout.

Tactic Notation "compare" ident(i) ident(j) ident(c) :=
  let H := fresh "Heq" i j in
  destruct (eq_id_dec i j);
  [ Case_aux c "equal"; subst j
  | Case_aux c "not equal" ].

Theorem pe_domain: ∀ pe_st V n,
  pe_lookup pe_st V = Some n →
  In V (map (@fst _ _) pe_st).
Proof. intros pe_st V n H. induction pe_st as [| [V' n'] pe_st].
  Case "[]". inversion H.
  Case "::". simpl in H. simpl. compare V V' SCase; auto. Qed.

Aside on In.

We will make heavy use of the In predicate from the standard library. In is equivalent to the appears_in predicate introduced in Logic.v, but defined using a Fixpoint rather than an Inductive.

Print In.

In comes with various useful lemmas.

Check in_or_app.

Check filter_In.

Check in_dec.

Note that we can compute with in_dec, just as with eq_id_dec.

Arithmetic Expressions

Partial evaluation of aexp is straightforward -- it is basically the same as constant folding, fold_constants_aexp, except that sometimes the partial state tells us the current value of a variable and we can replace it by a constant expression.

Fixpoint pe_aexp (pe_st : pe_state) (a : aexp) : aexp :=
  match a with
  | ANum n ⇒ ANum n
  | AId i ⇒ match pe_lookup pe_st i with
             | Some n ⇒ ANum n
             | None ⇒ AId i
             end
  | APlus a1 a2 ⇒
      match (pe_aexp pe_st a1, pe_aexp pe_st a2) with
      | (ANum n1, ANum n2) ⇒ ANum (n1 + n2)
      | (a1', a2') ⇒ APlus a1' a2'
      end
  | AMinus a1 a2 ⇒
      match (pe_aexp pe_st a1, pe_aexp pe_st a2) with
      | (ANum n1, ANum n2) ⇒ ANum (n1 - n2)
      | (a1', a2') ⇒ AMinus a1' a2'
      end
  | AMult a1 a2 ⇒
      match (pe_aexp pe_st a1, pe_aexp pe_st a2) with
      | (ANum n1, ANum n2) ⇒ ANum (n1 × n2)
      | (a1', a2') ⇒ AMult a1' a2'
      end
  end.

This partial evaluator folds constants but does not apply the associativity of addition.

Example test_pe_aexp1:
  pe_aexp [(X,3)] (APlus (APlus (AId X) (ANum 1)) (AId Y))
  = APlus (ANum 4) (AId Y).
Proof. reflexivity. Qed.

Example text_pe_aexp2:
  pe_aexp [(Y,3)] (APlus (APlus (AId X) (ANum 1)) (AId Y))
  = APlus (APlus (AId X) (ANum 1)) (ANum 3).
Proof. reflexivity. Qed.

Now, in what sense is pe_aexp correct? It is reasonable to define the correctness of pe_aexp as follows: whenever a full state st:state is consistent with a partial state pe_st:pe_state (in other words, every variable to which pe_st assigns a value is assigned the same value by st), evaluating a and evaluating pe_aexp pe_st a in st yields the same result. This statement is indeed true.

Definition pe_consistent (st:state) (pe_st:pe_state) :=
  ∀ V n, Some n = pe_lookup pe_st V → st V = n.

Theorem pe_aexp_correct_weak: ∀ st pe_st, pe_consistent st pe_st →
  ∀ a, aeval st a = aeval st (pe_aexp pe_st a).
Proof. unfold pe_consistent. intros st pe_st H a.
  aexp_cases (induction a) Case; simpl;
    try reflexivity;
    try (destruct (pe_aexp pe_st a1);
         destruct (pe_aexp pe_st a2);
         rewrite IHa1; rewrite IHa2; reflexivity).
  Case "AId".
    remember (pe_lookup pe_st i) as l. destruct l.
    SCase "Some". rewrite H with (n:=n) by apply Heql. reflexivity.
    SCase "None". reflexivity.
Qed.

However, we will soon want our partial evaluator to remove assignments. For example, it will simplify X ::= ANum 3;; Y ::= AMinus (AId X) (AId Y);; X ::= ANum 4 to just Y ::= AMinus (ANum 3) (AId Y);; X ::= ANum 4 by delaying the assignment to X until the end. To accomplish this simplification, we need the result of partial evaluating pe_aexp (X,3) (AMinus (AId X) (AId Y)) to be equal to AMinus (ANum 3) (AId Y) and not the original expression AMinus (AId X) (AId Y). After all, it would be incorrect, not just inefficient, to transform X ::= ANum 3;; Y ::= AMinus (AId X) (AId Y);; X ::= ANum 4 to Y ::= AMinus (AId X) (AId Y);; X ::= ANum 4 even though the output expressions AMinus (ANum 3) (AId Y) and AMinus (AId X) (AId Y) both satisfy the correctness criterion that we just proved. Indeed, if we were to just define pe_aexp pe_st a = a then the theorem pe_aexp_correct' would already trivially hold.
Instead, we want to prove that the pe_aexp is correct in a stronger sense: evaluating the expression produced by partial evaluation (aeval st (pe_aexp pe_st a)) must not depend on those parts of the full state st that are already specified in the partial state pe_st. To be more precise, let us define a function pe_override, which updates st with the contents of pe_st. In other words, pe_override carries out the assignments listed in pe_st on top of st.

Fixpoint pe_override (st:state) (pe_st:pe_state) : state :=
  match pe_st with
  | [] ⇒ st
  | (V,n)::pe_st ⇒ update (pe_override st pe_st) V n
  end.

Example test_pe_override:
  pe_override (update empty_state Y 1) [(X,3);(Z,2)]
  = update (update (update empty_state Y 1) Z 2) X 3.
Proof. reflexivity. Qed.

Although pe_override operates on a concrete list representing a pe_state, its behavior is defined entirely by the pe_lookup interpretation of the pe_state.

Theorem pe_override_correct: ∀ st pe_st V0,
  pe_override st pe_st V0 =
  match pe_lookup pe_st V0 with
  | Some n ⇒ n
  | None ⇒ st V0
  end.
Proof. intros. induction pe_st as [| [V n] pe_st]. reflexivity.
  simpl in ×. unfold update.
  compare V0 V Case; auto. rewrite eq_id; auto. rewrite neq_id; auto. Qed.

We can relate pe_consistent to pe_override in two ways. First, overriding a state with a partial state always gives a state that is consistent with the partial state. Second, if a state is already consistent with a partial state, then overriding the state with the partial state gives the same state.

Theorem pe_override_consistent: ∀ st pe_st,
  pe_consistent (pe_override st pe_st) pe_st.
Proof. intros st pe_st V n H. rewrite pe_override_correct.
  destruct (pe_lookup pe_st V); inversion H. reflexivity. Qed.

Theorem pe_consistent_override: ∀ st pe_st,
  pe_consistent st pe_st → ∀ V, st V = pe_override st pe_st V.
Proof. intros st pe_st H V. rewrite pe_override_correct.
  remember (pe_lookup pe_st V) as l. destruct l; auto. Qed.

Now we can state and prove that pe_aexp is correct in the stronger sense that will help us define the rest of the partial evaluator.
Intuitively, running a program using partial evaluation is a two-stage process. In the first, static stage, we partially evaluate the given program with respect to some partial state to get a residual program. In the second, dynamic stage, we evaluate the residual program with respect to the rest of the state. This dynamic state provides values for those variables that are unknown in the static (partial) state. Thus, the residual program should be equivalent to prepending the assignments listed in the partial state to the original program.

Theorem pe_aexp_correct: ∀ (pe_st:pe_state) (a:aexp) (st:state),
  aeval (pe_override st pe_st) a = aeval st (pe_aexp pe_st a).
Proof.
  intros pe_st a st.
  aexp_cases (induction a) Case; simpl;
    try reflexivity;
    try (destruct (pe_aexp pe_st a1);
         destruct (pe_aexp pe_st a2);
         rewrite IHa1; rewrite IHa2; reflexivity).
  rewrite pe_override_correct. destruct (pe_lookup pe_st i); reflexivity.
Qed.

Boolean Expressions

The partial evaluation of boolean expressions is similar. In fact, it is entirely analogous to the constant folding of boolean expressions, because our language has no boolean variables.

Fixpoint pe_bexp (pe_st : pe_state) (b : bexp) : bexp :=
  match b with
  | BTrue ⇒ BTrue
  | BFalse ⇒ BFalse
  | BEq a1 a2 ⇒
      match (pe_aexp pe_st a1, pe_aexp pe_st a2) with
      | (ANum n1, ANum n2) ⇒ if beq_nat n1 n2 then BTrue else BFalse
      | (a1', a2') ⇒ BEq a1' a2'
      end
  | BLe a1 a2 ⇒
      match (pe_aexp pe_st a1, pe_aexp pe_st a2) with
      | (ANum n1, ANum n2) ⇒ if ble_nat n1 n2 then BTrue else BFalse
      | (a1', a2') ⇒ BLe a1' a2'
      end
  | BNot b1 ⇒
      match (pe_bexp pe_st b1) with
      | BTrue ⇒ BFalse
      | BFalse ⇒ BTrue
      | b1' ⇒ BNot b1'
      end
  | BAnd b1 b2 ⇒
      match (pe_bexp pe_st b1, pe_bexp pe_st b2) with
      | (BTrue, BTrue) ⇒ BTrue
      | (BTrue, BFalse) ⇒ BFalse
      | (BFalse, BTrue) ⇒ BFalse
      | (BFalse, BFalse) ⇒ BFalse
      | (b1', b2') ⇒ BAnd b1' b2'
      end
  end.

Example test_pe_bexp1:
  pe_bexp [(X,3)] (BNot (BLe (AId X) (ANum 3)))
  = BFalse.
Proof. reflexivity. Qed.

Example test_pe_bexp2: ∀ b,
  b = BNot (BLe (AId X) (APlus (AId X) (ANum 1))) →
  pe_bexp [] b = b.
Proof. intros b H. rewrite → H. reflexivity. Qed.

The correctness of pe_bexp is analogous to the correctness of pe_aexp above.

Theorem pe_bexp_correct: ∀ (pe_st:pe_state) (b:bexp) (st:state),
  beval (pe_override st pe_st) b = beval st (pe_bexp pe_st b).
Proof.
  intros pe_st b st.
  bexp_cases (induction b) Case; simpl;
    try reflexivity;
    try (remember (pe_aexp pe_st a) as a';
         remember (pe_aexp pe_st a0) as a0';
         assert (Ha: aeval (pe_override st pe_st) a = aeval st a');
         assert (Ha0: aeval (pe_override st pe_st) a0 = aeval st a0');
           try (subst; apply pe_aexp_correct);
         destruct a'; destruct a0'; rewrite Ha; rewrite Ha0;
         simpl; try destruct (beq_nat n n0); try destruct (ble_nat n n0);
         reflexivity);
    try (destruct (pe_bexp pe_st b); rewrite IHb; reflexivity);
    try (destruct (pe_bexp pe_st b1);
         destruct (pe_bexp pe_st b2);
         rewrite IHb1; rewrite IHb2; reflexivity).
Qed.

Partial Evaluation of Commands, Without Loops

What about the partial evaluation of commands? The analogy between partial evaluation and full evaluation continues: Just as full evaluation of a command turns an initial state into a final state, partial evaluation of a command turns an initial partial state into a final partial state. The difference is that, because the state is partial, some parts of the command may not be executable at the static stage. Therefore, just as pe_aexp returns a residual aexp and pe_bexp returns a residual bexp above, partially evaluating a command yields a residual command.
Another way in which our partial evaluator is similar to a full evaluator is that it does not terminate on all commands. It is not hard to build a partial evaluator that terminates on all commands; what is hard is building a partial evaluator that terminates on all commands yet automatically performs desired optimizations such as unrolling loops. Often a partial evaluator can be coaxed into terminating more often and performing more optimizations by writing the source program differently so that the separation between static and dynamic information becomes more apparent. Such coaxing is the art of binding-time improvement. The binding time of a variable tells when its value is known -- either "static", or "dynamic."
Anyway, for now we will just live with the fact that our partial evaluator is not a total function from the source command and the initial partial state to the residual command and the final partial state. To model this non-termination, just as with the full evaluation of commands, we use an inductively defined relation. We write c1 / st || c1' / st' to mean that partially evaluating the source command c1 in the initial partial state st yields the residual command c1' and the final partial state st'. For example, we want something like (X ::= ANum 3 ;; Y ::= AMult (AId Z) (APlus (AId X) (AId X))) / ☐ || (Y ::= AMult (AId Z) (ANum 6)) / (X,3) to hold. The assignment to X appears in the final partial state, not the residual command.

Assignment

Let's start by considering how to partially evaluate an assignment. The two assignments in the source program above needs to be treated differently. The first assignment X ::= ANum 3, is static: its right-hand-side is a constant (more generally, simplifies to a constant), so we should update our partial state at X to 3 and produce no residual code. (Actually, we produce a residual SKIP.) The second assignment Y ::= AMult (AId Z) (APlus (AId X) (AId X)) is dynamic: its right-hand-side does not simplify to a constant, so we should leave it in the residual code and remove Y, if present, from our partial state. To implement these two cases, we define the functions pe_add and pe_remove. Like pe_override above, these functions operate on a concrete list representing a pe_state, but the theorems pe_add_correct and pe_remove_correct specify their behavior by the pe_lookup interpretation of the pe_state.

Fixpoint pe_remove (pe_st:pe_state) (V:id) : pe_state :=
  match pe_st with
  | [] ⇒ []
  | (V',n')::pe_st ⇒ if eq_id_dec V V' then pe_remove pe_st V
                      else (V',n') :: pe_remove pe_st V
  end.

Theorem pe_remove_correct: ∀ pe_st V V0,
  pe_lookup (pe_remove pe_st V) V0
  = if eq_id_dec V V0 then None else pe_lookup pe_st V0.
Proof. intros pe_st V V0. induction pe_st as [| [V' n'] pe_st].
  Case "[]". destruct (eq_id_dec V V0); reflexivity.
  Case "::". simpl. compare V V' SCase.
    SCase "equal". rewrite IHpe_st.
      destruct (eq_id_dec V V0). reflexivity. rewrite neq_id; auto.
    SCase "not equal". simpl. compare V0 V' SSCase.
      SSCase "equal". rewrite neq_id; auto.
      SSCase "not equal". rewrite IHpe_st. reflexivity.
Qed.

Definition pe_add (pe_st:pe_state) (V:id) (n:nat) : pe_state :=
  (V,n) :: pe_remove pe_st V.

Theorem pe_add_correct: ∀ pe_st V n V0,
  pe_lookup (pe_add pe_st V n) V0
  = if eq_id_dec V V0 then Some n else pe_lookup pe_st V0.
Proof. intros pe_st V n V0. unfold pe_add. simpl.
  compare V V0 Case.
  Case "equal". rewrite eq_id; auto.
  Case "not equal". rewrite pe_remove_correct. repeat rewrite neq_id; auto.
Qed.

We will use the two theorems below to show that our partial evaluator correctly deals with dynamic assignments and static assignments, respectively.

Theorem pe_override_update_remove: ∀ st pe_st V n,
  update (pe_override st pe_st) V n =
  pe_override (update st V n) (pe_remove pe_st V).
Proof. intros st pe_st V n. apply functional_extensionality. intros V0.
  unfold update. rewrite !pe_override_correct. rewrite pe_remove_correct.
  destruct (eq_id_dec V V0); reflexivity. Qed.

Theorem pe_override_update_add: ∀ st pe_st V n,
  update (pe_override st pe_st) V n =
  pe_override st (pe_add pe_st V n).
Proof. intros st pe_st V n. apply functional_extensionality. intros V0.
  unfold update. rewrite !pe_override_correct. rewrite pe_add_correct.
  destruct (eq_id_dec V V0); reflexivity. Qed.

Conditional

Trickier than assignments to partially evaluate is the conditional, IFB b1 THEN c1 ELSE c2 FI. If b1 simplifies to BTrue or BFalse then it's easy: we know which branch will be taken, so just take that branch. If b1 does not simplify to a constant, then we need to take both branches, and the final partial state may differ between the two branches!
The following program illustrates the difficulty: X ::= ANum 3;; IFB BLe (AId Y) (ANum 4) THEN Y ::= ANum 4;; IFB BEq (AId X) (AId Y) THEN Y ::= ANum 999 ELSE SKIP FI ELSE SKIP FI Suppose the initial partial state is empty. We don't know statically how Y compares to 4, so we must partially evaluate both branches of the (outer) conditional. On the THEN branch, we know that Y is set to 4 and can even use that knowledge to simplify the code somewhat. On the ELSE branch, we still don't know the exact value of Y at the end. What should the final partial state and residual program be?
One way to handle such a dynamic conditional is to take the intersection of the final partial states of the two branches. In this example, we take the intersection of (Y,4),(X,3) and (X,3), so the overall final partial state is (X,3). To compensate for forgetting that Y is 4, we need to add an assignment Y ::= ANum 4 to the end of the THEN branch. So, the residual program will be something like SKIP;; IFB BLe (AId Y) (ANum 4) THEN SKIP;; SKIP;; Y ::= ANum 4 ELSE SKIP FI
Programming this case in Coq calls for several auxiliary functions: we need to compute the intersection of two pe_states and turn their difference into sequences of assignments.
First, we show how to compute whether two pe_states to disagree at a given variable. In the theorem pe_disagree_domain, we prove that two pe_states can only disagree at variables that appear in at least one of them.

Definition pe_disagree_at (pe_st1 pe_st2 : pe_state) (V:id) : bool :=
  match pe_lookup pe_st1 V, pe_lookup pe_st2 V with
  | Some x, Some y ⇒ negb (beq_nat x y)
  | None, None ⇒ false
  | _, _ ⇒ true
  end.

Theorem pe_disagree_domain: ∀ (pe_st1 pe_st2 : pe_state) (V:id),
  true = pe_disagree_at pe_st1 pe_st2 V →
  In V (map (@fst _ _) pe_st1 ++ map (@fst _ _) pe_st2).
Proof. unfold pe_disagree_at. intros pe_st1 pe_st2 V H.
  apply in_or_app.
  remember (pe_lookup pe_st1 V) as lookup1.
  destruct lookup1 as [n1|]. left. apply pe_domain with n1. auto.
  remember (pe_lookup pe_st2 V) as lookup2.
  destruct lookup2 as [n2|]. right. apply pe_domain with n2. auto.
  inversion H. Qed.

We define the pe_compare function to list the variables where two given pe_states disagree. This list is exact, according to the theorem pe_compare_correct: a variable appears on the list if and only if the two given pe_states disagree at that variable. Furthermore, we use the pe_unique function to eliminate duplicates from the list.

Fixpoint pe_unique (l : list id) : list id :=
  match l with
  | [] ⇒ []
  | x::l ⇒ x :: filter (fun y ⇒ if eq_id_dec x y then false else true) (pe_unique l)
  end.

Theorem pe_unique_correct: ∀ l x,
  In x l ↔ In x (pe_unique l).
Proof. intros l x. induction l as [| h t]. reflexivity.
  simpl in ×. split.
  Case "->".
    intros. inversion H; clear H.
      left. assumption.
      destruct (eq_id_dec h x).
         left. assumption.
         right. apply filter_In. split.
           apply IHt. assumption.
           rewrite neq_id; auto.
  Case "<-".
    intros. inversion H; clear H.
       left. assumption.
       apply filter_In in H0. inversion H0. right. apply IHt. assumption.
Qed.

Definition pe_compare (pe_st1 pe_st2 : pe_state) : list id :=
  pe_unique (filter (pe_disagree_at pe_st1 pe_st2)
    (map (@fst _ _) pe_st1 ++ map (@fst _ _) pe_st2)).

Theorem pe_compare_correct: ∀ pe_st1 pe_st2 V,
  pe_lookup pe_st1 V = pe_lookup pe_st2 V ↔
  ¬ In V (pe_compare pe_st1 pe_st2).
Proof. intros pe_st1 pe_st2 V.
  unfold pe_compare. rewrite <- pe_unique_correct. rewrite filter_In.
  split; intros Heq.
  Case "->".
    intro. destruct H. unfold pe_disagree_at in H0. rewrite Heq in H0.
    destruct (pe_lookup pe_st2 V).
    rewrite <- beq_nat_refl in H0. inversion H0.
    inversion H0.
  Case "<-".
    assert (Hagree: pe_disagree_at pe_st1 pe_st2 V = false).
      SCase "Proof of assertion".
      remember (pe_disagree_at pe_st1 pe_st2 V) as disagree.
      destruct disagree; [| reflexivity].
      apply pe_disagree_domain in Heqdisagree.
      apply ex_falso_quodlibet. apply Heq. split. assumption. reflexivity.
    unfold pe_disagree_at in Hagree.
    destruct (pe_lookup pe_st1 V) as [n1|];
    destruct (pe_lookup pe_st2 V) as [n2|];
      try reflexivity; try solve by inversion.
    rewrite negb_false_iff in Hagree.
    apply beq_nat_true in Hagree. subst. reflexivity. Qed.

The intersection of two partial states is the result of removing from one of them all the variables where the two disagree. We define the function pe_removes, in terms of pe_remove above, to perform such a removal of a whole list of variables at once.
The theorem pe_compare_removes testifies that the pe_lookup interpretation of the result of this intersection operation is the same no matter which of the two partial states we remove the variables from. Because pe_override only depends on the pe_lookup interpretation of partial states, pe_override also does not care which of the two partial states we remove the variables from; that theorem pe_compare_override is used in the correctness proof shortly.

Fixpoint pe_removes (pe_st:pe_state) (ids : list id) : pe_state :=
  match ids with
  | [] ⇒ pe_st
  | V::ids ⇒ pe_remove (pe_removes pe_st ids) V
  end.

Theorem pe_removes_correct: ∀ pe_st ids V,
  pe_lookup (pe_removes pe_st ids) V =
  if in_dec eq_id_dec V ids then None else pe_lookup pe_st V.
Proof. intros pe_st ids V. induction ids as [| V' ids]. reflexivity.
  simpl. rewrite pe_remove_correct. rewrite IHids.
  compare V' V Case.
    reflexivity.
    destruct (in_dec eq_id_dec V ids);
      reflexivity.
Qed.

Theorem pe_compare_removes: ∀ pe_st1 pe_st2 V,
  pe_lookup (pe_removes pe_st1 (pe_compare pe_st1 pe_st2)) V =
  pe_lookup (pe_removes pe_st2 (pe_compare pe_st1 pe_st2)) V.
Proof. intros pe_st1 pe_st2 V. rewrite !pe_removes_correct.
  destruct (in_dec eq_id_dec V (pe_compare pe_st1 pe_st2)).
    reflexivity.
    apply pe_compare_correct. auto. Qed.

Theorem pe_compare_override: ∀ pe_st1 pe_st2 st,
  pe_override st (pe_removes pe_st1 (pe_compare pe_st1 pe_st2)) =
  pe_override st (pe_removes pe_st2 (pe_compare pe_st1 pe_st2)).
Proof. intros. apply functional_extensionality. intros V.
  rewrite !pe_override_correct. rewrite pe_compare_removes. reflexivity.
Qed.

Finally, we define an assign function to turn the difference between two partial states into a sequence of assignment commands. More precisely, assign pe_st ids generates an assignment command for each variable listed in ids.

Fixpoint assign (pe_st : pe_state) (ids : list id) : com :=
  match ids with
  | [] ⇒ SKIP
  | V::ids ⇒ match pe_lookup pe_st V with
              | Some n ⇒ (assign pe_st ids;; V ::= ANum n)
              | None ⇒ assign pe_st ids
              end
  end.

The command generated by assign always terminates, because it is just a sequence of assignments. The (total) function assigned below computes the effect of the command on the (dynamic state). The theorem assign_removes then confirms that the generated assignments perfectly compensate for removing the variables from the partial state.

Definition assigned (pe_st:pe_state) (ids : list id) (st:state) : state :=
  fun V ⇒ if in_dec eq_id_dec V ids then
                match pe_lookup pe_st V with
                | Some n ⇒ n
                | None ⇒ st V
                end
           else st V.

Theorem assign_removes: ∀ pe_st ids st,
  pe_override st pe_st =
  pe_override (assigned pe_st ids st) (pe_removes pe_st ids).
Proof. intros pe_st ids st. apply functional_extensionality. intros V.
  rewrite !pe_override_correct. rewrite pe_removes_correct. unfold assigned.
  destruct (in_dec eq_id_dec V ids); destruct (pe_lookup pe_st V); reflexivity.
Qed.

Lemma ceval_extensionality: ∀ c st st1 st2,
  c / st || st1 → (∀ V, st1 V = st2 V) → c / st || st2.
Proof. intros c st st1 st2 H Heq.
  apply functional_extensionality in Heq. rewrite <- Heq. apply H. Qed.

Theorem eval_assign: ∀ pe_st ids st,
  assign pe_st ids / st || assigned pe_st ids st.
Proof. intros pe_st ids st. induction ids as [| V ids]; simpl.
  Case "[]". eapply ceval_extensionality. apply E_Skip. reflexivity.
  Case "V::ids".
    remember (pe_lookup pe_st V) as lookup. destruct lookup.
    SCase "Some". eapply E_Seq. apply IHids. unfold assigned. simpl.
      eapply ceval_extensionality. apply E_Ass. simpl. reflexivity.
      intros V0. unfold update. compare V V0 SSCase.
      SSCase "equal". rewrite <- Heqlookup. reflexivity.
      SSCase "not equal". destruct (in_dec eq_id_dec V0 ids); auto.
    SCase "None". eapply ceval_extensionality. apply IHids.
      unfold assigned. intros V0. simpl. compare V V0 SSCase.
      SSCase "equal". rewrite <- Heqlookup.
        destruct (in_dec eq_id_dec V ids); reflexivity.
      SSCase "not equal". destruct (in_dec eq_id_dec V0 ids); reflexivity. Qed.

The Partial Evaluation Relation

At long last, we can define a partial evaluator for commands without loops, as an inductive relation! The inequality conditions in PE_AssDynamic and PE_If are just to keep the partial evaluator deterministic; they are not required for correctness.

Reserved Notation "c1 '/' st '||' c1' '/' st'"
  (at level 40, st at level 39, c1' at level 39).

Inductive pe_com : com → pe_state → com → pe_state → Prop :=
  | PE_Skip : ∀ pe_st,
      SKIP / pe_st || SKIP / pe_st
  | PE_AssStatic : ∀ pe_st a1 n1 l,
      pe_aexp pe_st a1 = ANum n1 →
      (l ::= a1) / pe_st || SKIP / pe_add pe_st l n1
  | PE_AssDynamic : ∀ pe_st a1 a1' l,
      pe_aexp pe_st a1 = a1' →
      (∀ n, a1' ≠ ANum n) →
      (l ::= a1) / pe_st || (l ::= a1') / pe_remove pe_st l
  | PE_Seq : ∀ pe_st pe_st' pe_st'' c1 c2 c1' c2',
      c1 / pe_st || c1' / pe_st' →
      c2 / pe_st' || c2' / pe_st'' →
      (c1 ;; c2) / pe_st || (c1' ;; c2') / pe_st''
  | PE_IfTrue : ∀ pe_st pe_st' b1 c1 c2 c1',
      pe_bexp pe_st b1 = BTrue →
      c1 / pe_st || c1' / pe_st' →
      (IFB b1 THEN c1 ELSE c2 FI) / pe_st || c1' / pe_st'
  | PE_IfFalse : ∀ pe_st pe_st' b1 c1 c2 c2',
      pe_bexp pe_st b1 = BFalse →
      c2 / pe_st || c2' / pe_st' →
      (IFB b1 THEN c1 ELSE c2 FI) / pe_st || c2' / pe_st'
  | PE_If : ∀ pe_st pe_st1 pe_st2 b1 c1 c2 c1' c2',
      pe_bexp pe_st b1 ≠ BTrue →
      pe_bexp pe_st b1 ≠ BFalse →
      c1 / pe_st || c1' / pe_st1 →
      c2 / pe_st || c2' / pe_st2 →
      (IFB b1 THEN c1 ELSE c2 FI) / pe_st
        || (IFB pe_bexp pe_st b1
             THEN c1' ;; assign pe_st1 (pe_compare pe_st1 pe_st2)
             ELSE c2' ;; assign pe_st2 (pe_compare pe_st1 pe_st2) FI)
            / pe_removes pe_st1 (pe_compare pe_st1 pe_st2)

  where "c1 '/' st '||' c1' '/' st'" := (pe_com c1 st c1' st').

Tactic Notation "pe_com_cases" tactic(first) ident(c) :=
  first;
  [ Case_aux c "PE_Skip"
  | Case_aux c "PE_AssStatic" | Case_aux c "PE_AssDynamic"
  | Case_aux c "PE_Seq"
  | Case_aux c "PE_IfTrue" | Case_aux c "PE_IfFalse" | Case_aux c "PE_If" ].

Hint Constructors pe_com.
Hint Constructors ceval.

Examples

Below are some examples of using the partial evaluator. To make the pe_com relation actually usable for automatic partial evaluation, we would need to define more automation tactics in Coq. That is not hard to do, but it is not needed here.

Example pe_example1:
  (X ::= ANum 3 ;; Y ::= AMult (AId Z) (APlus (AId X) (AId X)))
  / [] || (SKIP;; Y ::= AMult (AId Z) (ANum 6)) / [(X,3)].
Proof. eapply PE_Seq. eapply PE_AssStatic. reflexivity.
  eapply PE_AssDynamic. reflexivity. intros n H. inversion H. Qed.

Example pe_example2:
  (X ::= ANum 3 ;; IFB BLe (AId X) (ANum 4) THEN X ::= ANum 4 ELSE SKIP FI)
  / [] || (SKIP;; SKIP) / [(X,4)].
Proof. eapply PE_Seq. eapply PE_AssStatic. reflexivity.
  eapply PE_IfTrue. reflexivity.
  eapply PE_AssStatic. reflexivity. Qed.

Example pe_example3:
  (X ::= ANum 3;;
   IFB BLe (AId Y) (ANum 4) THEN
     Y ::= ANum 4;;
     IFB BEq (AId X) (AId Y) THEN Y ::= ANum 999 ELSE SKIP FI
   ELSE SKIP FI) / []
  || (SKIP;;
       IFB BLe (AId Y) (ANum 4) THEN
         (SKIP;; SKIP);; (SKIP;; Y ::= ANum 4)
       ELSE SKIP;; SKIP FI)
      / [(X,3)].
Proof. erewrite f_equal2 with (f := fun c st ⇒ _ / _ || c / st).
  eapply PE_Seq. eapply PE_AssStatic. reflexivity.
  eapply PE_If; intuition eauto; try solve by inversion.
  econstructor. eapply PE_AssStatic. reflexivity.
  eapply PE_IfFalse. reflexivity. econstructor.
  reflexivity. reflexivity. Qed.

Correctness of Partial Evaluation

Finally let's prove that this partial evaluator is correct!

Reserved Notation "c' '/' pe_st' '/' st '||' st''"
  (at level 40, pe_st' at level 39, st at level 39).

Inductive pe_ceval
  (c':com) (pe_st':pe_state) (st:state) (st'':state) : Prop :=
  | pe_ceval_intro : ∀ st',
    c' / st || st' →
    pe_override st' pe_st' = st'' →
    c' / pe_st' / st || st''
  where "c' '/' pe_st' '/' st '||' st''" := (pe_ceval c' pe_st' st st'').

Hint Constructors pe_ceval.

Theorem pe_com_complete:
  ∀ c pe_st pe_st' c', c / pe_st || c' / pe_st' →
  ∀ st st'',
  (c / pe_override st pe_st || st'') →
  (c' / pe_st' / st || st'').
Proof. intros c pe_st pe_st' c' Hpe.
  pe_com_cases (induction Hpe) Case; intros st st'' Heval;
  try (inversion Heval; subst;
       try (rewrite → pe_bexp_correct, → H in *; solve by inversion);
       []);
  eauto.
  Case "PE_AssStatic". econstructor. econstructor.
    rewrite → pe_aexp_correct. rewrite <- pe_override_update_add.
    rewrite → H. reflexivity.
  Case "PE_AssDynamic". econstructor. econstructor. reflexivity.
    rewrite → pe_aexp_correct. rewrite <- pe_override_update_remove.
    reflexivity.
  Case "PE_Seq".
    edestruct IHHpe1. eassumption. subst.
    edestruct IHHpe2. eassumption.
    eauto.
  Case "PE_If". inversion Heval; subst.
    SCase "E'IfTrue". edestruct IHHpe1. eassumption.
      econstructor. apply E_IfTrue. rewrite <- pe_bexp_correct. assumption.
      eapply E_Seq. eassumption. apply eval_assign.
      rewrite <- assign_removes. eassumption.
    SCase "E_IfFalse". edestruct IHHpe2. eassumption.
      econstructor. apply E_IfFalse. rewrite <- pe_bexp_correct. assumption.
      eapply E_Seq. eassumption. apply eval_assign.
      rewrite → pe_compare_override.
      rewrite <- assign_removes. eassumption.
Qed.

Theorem pe_com_sound:
  ∀ c pe_st pe_st' c', c / pe_st || c' / pe_st' →
  ∀ st st'',
  (c' / pe_st' / st || st'') →
  (c / pe_override st pe_st || st'').
Proof. intros c pe_st pe_st' c' Hpe.
  pe_com_cases (induction Hpe) Case;
    intros st st'' [st' Heval Heq];
    try (inversion Heval; []; subst); auto.
  Case "PE_AssStatic". rewrite <- pe_override_update_add. apply E_Ass.
    rewrite → pe_aexp_correct. rewrite → H. reflexivity.
  Case "PE_AssDynamic". rewrite <- pe_override_update_remove. apply E_Ass.
    rewrite <- pe_aexp_correct. reflexivity.
  Case "PE_Seq". eapply E_Seq; eauto.
  Case "PE_IfTrue". apply E_IfTrue.
    rewrite → pe_bexp_correct. rewrite → H. reflexivity. eauto.
  Case "PE_IfFalse". apply E_IfFalse.
    rewrite → pe_bexp_correct. rewrite → H. reflexivity. eauto.
  Case "PE_If".
    inversion Heval; subst; inversion H7;
      (eapply ceval_deterministic in H8; [| apply eval_assign]); subst.
    SCase "E_IfTrue".
      apply E_IfTrue. rewrite → pe_bexp_correct. assumption.
      rewrite <- assign_removes. eauto.
    SCase "E_IfFalse".
      rewrite → pe_compare_override.
      apply E_IfFalse. rewrite → pe_bexp_correct. assumption.
      rewrite <- assign_removes. eauto.
Qed.

The main theorem. Thanks to David Menendez for this formulation!

Corollary pe_com_correct:
  ∀ c pe_st pe_st' c', c / pe_st || c' / pe_st' →
  ∀ st st'',
  (c / pe_override st pe_st || st'') ↔
  (c' / pe_st' / st || st'').
Proof. intros c pe_st pe_st' c' H st st''. split.
  Case "->". apply pe_com_complete. apply H.
  Case "<-". apply pe_com_sound. apply H.
Qed.

Partial Evaluation of Loops

It may seem straightforward at first glance to extend the partial evaluation relation pe_com above to loops. Indeed, many loops are easy to deal with. Considered this repeated-squaring loop, for example: WHILE BLe (ANum 1) (AId X) DO Y ::= AMult (AId Y) (AId Y);; X ::= AMinus (AId X) (ANum 1) END If we know neither X nor Y statically, then the entire loop is dynamic and the residual command should be the same. If we know X but not Y, then the loop can be unrolled all the way and the residual command should be Y ::= AMult (AId Y) (AId Y);; Y ::= AMult (AId Y) (AId Y);; Y ::= AMult (AId Y) (AId Y) if X is initially 3 (and finally 0). In general, a loop is easy to partially evaluate if the final partial state of the loop body is equal to the initial state, or if its guard condition is static.
But there are other loops for which it is hard to express the residual program we want in Imp. For example, take this program for checking if Y is even or odd: X ::= ANum 0;; WHILE BLe (ANum 1) (AId Y) DO Y ::= AMinus (AId Y) (ANum 1);; X ::= AMinus (ANum 1) (AId X) END The value of X alternates between 0 and 1 during the loop. Ideally, we would like to unroll this loop, not all the way but two-fold, into something like WHILE BLe (ANum 1) (AId Y) DO Y ::= AMinus (AId Y) (ANum 1);; IF BLe (ANum 1) (AId Y) THEN Y ::= AMinus (AId Y) (ANum 1) ELSE X ::= ANum 1;; EXIT FI END;; X ::= ANum 0 Unfortunately, there is no EXIT command in Imp. Without extending the range of control structures available in our language, the best we can do is to repeat loop-guard tests or add flag variables. Neither option is terribly attractive.
Still, as a digression, below is an attempt at performing partial evaluation on Imp commands. We add one more command argument c'' to the pe_com relation, which keeps track of a loop to roll up.

Module Loop.

Reserved Notation "c1 '/' st '||' c1' '/' st' '/' c''"
  (at level 40, st at level 39, c1' at level 39, st' at level 39).

Inductive pe_com : com → pe_state → com → pe_state → com → Prop :=
  | PE_Skip : ∀ pe_st,
      SKIP / pe_st || SKIP / pe_st / SKIP
  | PE_AssStatic : ∀ pe_st a1 n1 l,
      pe_aexp pe_st a1 = ANum n1 →
      (l ::= a1) / pe_st || SKIP / pe_add pe_st l n1 / SKIP
  | PE_AssDynamic : ∀ pe_st a1 a1' l,
      pe_aexp pe_st a1 = a1' →
      (∀ n, a1' ≠ ANum n) →
      (l ::= a1) / pe_st || (l ::= a1') / pe_remove pe_st l / SKIP
  | PE_Seq : ∀ pe_st pe_st' pe_st'' c1 c2 c1' c2' c'',
      c1 / pe_st || c1' / pe_st' / SKIP →
      c2 / pe_st' || c2' / pe_st'' / c'' →
      (c1 ;; c2) / pe_st || (c1' ;; c2') / pe_st'' / c''
  | PE_IfTrue : ∀ pe_st pe_st' b1 c1 c2 c1' c'',
      pe_bexp pe_st b1 = BTrue →
      c1 / pe_st || c1' / pe_st' / c'' →
      (IFB b1 THEN c1 ELSE c2 FI) / pe_st || c1' / pe_st' / c''
  | PE_IfFalse : ∀ pe_st pe_st' b1 c1 c2 c2' c'',
      pe_bexp pe_st b1 = BFalse →
      c2 / pe_st || c2' / pe_st' / c'' →
      (IFB b1 THEN c1 ELSE c2 FI) / pe_st || c2' / pe_st' / c''
  | PE_If : ∀ pe_st pe_st1 pe_st2 b1 c1 c2 c1' c2' c'',
      pe_bexp pe_st b1 ≠ BTrue →
      pe_bexp pe_st b1 ≠ BFalse →
      c1 / pe_st || c1' / pe_st1 / c'' →
      c2 / pe_st || c2' / pe_st2 / c'' →
      (IFB b1 THEN c1 ELSE c2 FI) / pe_st
        || (IFB pe_bexp pe_st b1
             THEN c1' ;; assign pe_st1 (pe_compare pe_st1 pe_st2)
             ELSE c2' ;; assign pe_st2 (pe_compare pe_st1 pe_st2) FI)
            / pe_removes pe_st1 (pe_compare pe_st1 pe_st2)
            / c''
  | PE_WhileEnd : ∀ pe_st b1 c1,
      pe_bexp pe_st b1 = BFalse →
      (WHILE b1 DO c1 END) / pe_st || SKIP / pe_st / SKIP
  | PE_WhileLoop : ∀ pe_st pe_st' pe_st'' b1 c1 c1' c2' c2'',
      pe_bexp pe_st b1 = BTrue →
      c1 / pe_st || c1' / pe_st' / SKIP →
      (WHILE b1 DO c1 END) / pe_st' || c2' / pe_st'' / c2'' →
      pe_compare pe_st pe_st'' ≠ [] →
      (WHILE b1 DO c1 END) / pe_st || (c1';;c2') / pe_st'' / c2''
  | PE_While : ∀ pe_st pe_st' pe_st'' b1 c1 c1' c2' c2'',
      pe_bexp pe_st b1 ≠ BFalse →
      pe_bexp pe_st b1 ≠ BTrue →
      c1 / pe_st || c1' / pe_st' / SKIP →
      (WHILE b1 DO c1 END) / pe_st' || c2' / pe_st'' / c2'' →
      pe_compare pe_st pe_st'' ≠ [] →
      (c2'' = SKIP ∨ c2'' = WHILE b1 DO c1 END) →
      (WHILE b1 DO c1 END) / pe_st
        || (IFB pe_bexp pe_st b1
             THEN c1';; c2';; assign pe_st'' (pe_compare pe_st pe_st'')
             ELSE assign pe_st (pe_compare pe_st pe_st'') FI)
            / pe_removes pe_st (pe_compare pe_st pe_st'')
            / c2''
  | PE_WhileFixedEnd : ∀ pe_st b1 c1,
      pe_bexp pe_st b1 ≠ BFalse →
      (WHILE b1 DO c1 END) / pe_st || SKIP / pe_st / (WHILE b1 DO c1 END)
  | PE_WhileFixedLoop : ∀ pe_st pe_st' pe_st'' b1 c1 c1' c2',
      pe_bexp pe_st b1 = BTrue →
      c1 / pe_st || c1' / pe_st' / SKIP →
      (WHILE b1 DO c1 END) / pe_st'
        || c2' / pe_st'' / (WHILE b1 DO c1 END) →
      pe_compare pe_st pe_st'' = [] →
      (WHILE b1 DO c1 END) / pe_st
        || (WHILE BTrue DO SKIP END) / pe_st / SKIP
      
  | PE_WhileFixed : ∀ pe_st pe_st' pe_st'' b1 c1 c1' c2',
      pe_bexp pe_st b1 ≠ BFalse →
      pe_bexp pe_st b1 ≠ BTrue →
      c1 / pe_st || c1' / pe_st' / SKIP →
      (WHILE b1 DO c1 END) / pe_st'
        || c2' / pe_st'' / (WHILE b1 DO c1 END) →
      pe_compare pe_st pe_st'' = [] →
      (WHILE b1 DO c1 END) / pe_st
        || (WHILE pe_bexp pe_st b1 DO c1';; c2' END) / pe_st / SKIP

  where "c1 '/' st '||' c1' '/' st' '/' c''" := (pe_com c1 st c1' st' c'').

Tactic Notation "pe_com_cases" tactic(first) ident(c) :=
  first;
  [ Case_aux c "PE_Skip"
  | Case_aux c "PE_AssStatic" | Case_aux c "PE_AssDynamic"
  | Case_aux c "PE_Seq"
  | Case_aux c "PE_IfTrue" | Case_aux c "PE_IfFalse" | Case_aux c "PE_If"
  | Case_aux c "PE_WhileEnd" | Case_aux c "PE_WhileLoop"
  | Case_aux c "PE_While" | Case_aux c "PE_WhileFixedEnd"
  | Case_aux c "PE_WhileFixedLoop" | Case_aux c "PE_WhileFixed" ].

Hint Constructors pe_com.

Examples


Ltac step i :=
  (eapply i; intuition eauto; try solve by inversion);
  repeat (try eapply PE_Seq;
          try (eapply PE_AssStatic; simpl; reflexivity);
          try (eapply PE_AssDynamic;
               [ simpl; reflexivity
               | intuition eauto; solve by inversion ])).

Definition square_loop: com :=
  WHILE BLe (ANum 1) (AId X) DO
    Y ::= AMult (AId Y) (AId Y);;
    X ::= AMinus (AId X) (ANum 1)
  END.

Example pe_loop_example1:
  square_loop / []
  || (WHILE BLe (ANum 1) (AId X) DO
         (Y ::= AMult (AId Y) (AId Y);;
          X ::= AMinus (AId X) (ANum 1));; SKIP
       END) / [] / SKIP.
Proof. erewrite f_equal2 with (f := fun c st ⇒ _ / _ || c / st / SKIP).
  step PE_WhileFixed. step PE_WhileFixedEnd. reflexivity.
  reflexivity. reflexivity. Qed.

Example pe_loop_example2:
  (X ::= ANum 3;; square_loop) / []
  || (SKIP;;
       (Y ::= AMult (AId Y) (AId Y);; SKIP);;
       (Y ::= AMult (AId Y) (AId Y);; SKIP);;
       (Y ::= AMult (AId Y) (AId Y);; SKIP);;
       SKIP) / [(X,0)] / SKIP.
Proof. erewrite f_equal2 with (f := fun c st ⇒ _ / _ || c / st / SKIP).
  eapply PE_Seq. eapply PE_AssStatic. reflexivity.
  step PE_WhileLoop.
  step PE_WhileLoop.
  step PE_WhileLoop.
  step PE_WhileEnd.
  inversion H. inversion H. inversion H.
  reflexivity. reflexivity. Qed.

Example pe_loop_example3:
  (Z ::= ANum 3;; subtract_slowly) / []
  || (SKIP;;
       IFB BNot (BEq (AId X) (ANum 0)) THEN
         (SKIP;; X ::= AMinus (AId X) (ANum 1));;
         IFB BNot (BEq (AId X) (ANum 0)) THEN
           (SKIP;; X ::= AMinus (AId X) (ANum 1));;
           IFB BNot (BEq (AId X) (ANum 0)) THEN
             (SKIP;; X ::= AMinus (AId X) (ANum 1));;
             WHILE BNot (BEq (AId X) (ANum 0)) DO
               (SKIP;; X ::= AMinus (AId X) (ANum 1));; SKIP
             END;;
             SKIP;; Z ::= ANum 0
           ELSE SKIP;; Z ::= ANum 1 FI;; SKIP
         ELSE SKIP;; Z ::= ANum 2 FI;; SKIP
       ELSE SKIP;; Z ::= ANum 3 FI) / [] / SKIP.
Proof. erewrite f_equal2 with (f := fun c st ⇒ _ / _ || c / st / SKIP).
  eapply PE_Seq. eapply PE_AssStatic. reflexivity.
  step PE_While.
  step PE_While.
  step PE_While.
  step PE_WhileFixed.
  step PE_WhileFixedEnd.
  reflexivity. inversion H. inversion H. inversion H.
  reflexivity. reflexivity. Qed.

Example pe_loop_example4:
  (X ::= ANum 0;;
   WHILE BLe (AId X) (ANum 2) DO
     X ::= AMinus (ANum 1) (AId X)
   END) / [] || (SKIP;; WHILE BTrue DO SKIP END) / [(X,0)] / SKIP.
Proof. erewrite f_equal2 with (f := fun c st ⇒ _ / _ || c / st / SKIP).
  eapply PE_Seq. eapply PE_AssStatic. reflexivity.
  step PE_WhileFixedLoop.
  step PE_WhileLoop.
  step PE_WhileFixedEnd.
  inversion H. reflexivity. reflexivity. reflexivity. Qed.

Correctness

Because this partial evaluator can unroll a loop n-fold where n is a (finite) integer greater than one, in order to show it correct we need to perform induction not structurally on dynamic evaluation but on the number of times dynamic evaluation enters a loop body.

Reserved Notation "c1 '/' st '||' st' '#' n"
  (at level 40, st at level 39, st' at level 39).

Inductive ceval_count : com → state → state → nat → Prop :=
  | E'Skip : ∀ st,
      SKIP / st || st # 0
  | E'Ass : ∀ st a1 n l,
      aeval st a1 = n →
      (l ::= a1) / st || (update st l n) # 0
  | E'Seq : ∀ c1 c2 st st' st'' n1 n2,
      c1 / st || st' # n1 →
      c2 / st' || st'' # n2 →
      (c1 ;; c2) / st || st'' # (n1 + n2)
  | E'IfTrue : ∀ st st' b1 c1 c2 n,
      beval st b1 = true →
      c1 / st || st' # n →
      (IFB b1 THEN c1 ELSE c2 FI) / st || st' # n
  | E'IfFalse : ∀ st st' b1 c1 c2 n,
      beval st b1 = false →
      c2 / st || st' # n →
      (IFB b1 THEN c1 ELSE c2 FI) / st || st' # n
  | E'WhileEnd : ∀ b1 st c1,
      beval st b1 = false →
      (WHILE b1 DO c1 END) / st || st # 0
  | E'WhileLoop : ∀ st st' st'' b1 c1 n1 n2,
      beval st b1 = true →
      c1 / st || st' # n1 →
      (WHILE b1 DO c1 END) / st' || st'' # n2 →
      (WHILE b1 DO c1 END) / st || st'' # S (n1 + n2)

  where "c1 '/' st '||' st' # n" := (ceval_count c1 st st' n).

Tactic Notation "ceval_count_cases" tactic(first) ident(c) :=
  first;
  [ Case_aux c "E'Skip" | Case_aux c "E'Ass" | Case_aux c "E'Seq"
  | Case_aux c "E'IfTrue" | Case_aux c "E'IfFalse"
  | Case_aux c "E'WhileEnd" | Case_aux c "E'WhileLoop" ].

Hint Constructors ceval_count.

Theorem ceval_count_complete: ∀ c st st',
  c / st || st' → ∃ n, c / st || st' # n.
Proof. intros c st st' Heval.
  induction Heval;
    try inversion IHHeval1;
    try inversion IHHeval2;
    try inversion IHHeval;
    eauto. Qed.

Theorem ceval_count_sound: ∀ c st st' n,
  c / st || st' # n → c / st || st'.
Proof. intros c st st' n Heval. induction Heval; eauto. Qed.

Theorem pe_compare_nil_lookup: ∀ pe_st1 pe_st2,
  pe_compare pe_st1 pe_st2 = [] →
  ∀ V, pe_lookup pe_st1 V = pe_lookup pe_st2 V.
Proof. intros pe_st1 pe_st2 H V.
  apply (pe_compare_correct pe_st1 pe_st2 V).
  rewrite H. intro. inversion H0. Qed.

Theorem pe_compare_nil_override: ∀ pe_st1 pe_st2,
  pe_compare pe_st1 pe_st2 = [] →
  ∀ st, pe_override st pe_st1 = pe_override st pe_st2.
Proof. intros pe_st1 pe_st2 H st.
  apply functional_extensionality. intros V.
  rewrite !pe_override_correct.
  apply pe_compare_nil_lookup with (V:=V) in H.
  rewrite H. reflexivity. Qed.

Reserved Notation "c' '/' pe_st' '/' c'' '/' st '||' st'' '#' n"
  (at level 40, pe_st' at level 39, c'' at level 39,
   st at level 39, st'' at level 39).

Inductive pe_ceval_count (c':com) (pe_st':pe_state) (c'':com)
                         (st:state) (st'':state) (n:nat) : Prop :=
  | pe_ceval_count_intro : ∀ st' n',
    c' / st || st' →
    c'' / pe_override st' pe_st' || st'' # n' →
    n' ≤ n →
    c' / pe_st' / c'' / st || st'' # n
  where "c' '/' pe_st' '/' c'' '/' st '||' st'' '#' n" :=
        (pe_ceval_count c' pe_st' c'' st st'' n).

Hint Constructors pe_ceval_count.

Lemma pe_ceval_count_le: ∀ c' pe_st' c'' st st'' n n',
  n' ≤ n →
  c' / pe_st' / c'' / st || st'' # n' →
  c' / pe_st' / c'' / st || st'' # n.
Proof. intros c' pe_st' c'' st st'' n n' Hle H. inversion H.
  econstructor; try eassumption. omega. Qed.

Theorem pe_com_complete:
  ∀ c pe_st pe_st' c' c'', c / pe_st || c' / pe_st' / c'' →
  ∀ st st'' n,
  (c / pe_override st pe_st || st'' # n) →
  (c' / pe_st' / c'' / st || st'' # n).
Proof. intros c pe_st pe_st' c' c'' Hpe.
  pe_com_cases (induction Hpe) Case; intros st st'' n Heval;
  try (inversion Heval; subst;
       try (rewrite → pe_bexp_correct, → H in *; solve by inversion);
       []);
  eauto.
  Case "PE_AssStatic". econstructor. econstructor.
    rewrite → pe_aexp_correct. rewrite <- pe_override_update_add.
    rewrite → H. apply E'Skip. auto.
  Case "PE_AssDynamic". econstructor. econstructor. reflexivity.
    rewrite → pe_aexp_correct. rewrite <- pe_override_update_remove.
    apply E'Skip. auto.
  Case "PE_Seq".
    edestruct IHHpe1 as [? ? ? Hskip ?]. eassumption.
    inversion Hskip. subst.
    edestruct IHHpe2. eassumption.
    econstructor; eauto. omega.
  Case "PE_If". inversion Heval; subst.
    SCase "E'IfTrue". edestruct IHHpe1. eassumption.
      econstructor. apply E_IfTrue. rewrite <- pe_bexp_correct. assumption.
      eapply E_Seq. eassumption. apply eval_assign.
      rewrite <- assign_removes. eassumption. eassumption.
    SCase "E_IfFalse". edestruct IHHpe2. eassumption.
      econstructor. apply E_IfFalse. rewrite <- pe_bexp_correct. assumption.
      eapply E_Seq. eassumption. apply eval_assign.
      rewrite → pe_compare_override.
      rewrite <- assign_removes. eassumption. eassumption.
  Case "PE_WhileLoop".
    edestruct IHHpe1 as [? ? ? Hskip ?]. eassumption.
    inversion Hskip. subst.
    edestruct IHHpe2. eassumption.
    econstructor; eauto. omega.
  Case "PE_While". inversion Heval; subst.
    SCase "E_WhileEnd". econstructor. apply E_IfFalse.
      rewrite <- pe_bexp_correct. assumption.
      apply eval_assign.
      rewrite <- assign_removes. inversion H2; subst; auto.
      auto.
    SCase "E_WhileLoop".
      edestruct IHHpe1 as [? ? ? Hskip ?]. eassumption.
      inversion Hskip. subst.
      edestruct IHHpe2. eassumption.
      econstructor. apply E_IfTrue.
      rewrite <- pe_bexp_correct. assumption.
      repeat eapply E_Seq; eauto. apply eval_assign.
      rewrite → pe_compare_override, <- assign_removes. eassumption.
      omega.
  Case "PE_WhileFixedLoop". apply ex_falso_quodlibet.
    generalize dependent (S (n1 + n2)). intros n.
    clear - Case H H0 IHHpe1 IHHpe2. generalize dependent st.
    induction n using lt_wf_ind; intros st Heval. inversion Heval; subst.
    SCase "E'WhileEnd". rewrite pe_bexp_correct, H in H7. inversion H7.
    SCase "E'WhileLoop".
      edestruct IHHpe1 as [? ? ? Hskip ?]. eassumption.
      inversion Hskip. subst.
      edestruct IHHpe2. eassumption.
      rewrite <- (pe_compare_nil_override _ _ H0) in H7.
      apply H1 in H7; [| omega]. inversion H7.
  Case "PE_WhileFixed". generalize dependent st.
    induction n using lt_wf_ind; intros st Heval. inversion Heval; subst.
    SCase "E'WhileEnd". rewrite pe_bexp_correct in H8. eauto.
    SCase "E'WhileLoop". rewrite pe_bexp_correct in H5.
      edestruct IHHpe1 as [? ? ? Hskip ?]. eassumption.
      inversion Hskip. subst.
      edestruct IHHpe2. eassumption.
      rewrite <- (pe_compare_nil_override _ _ H1) in H8.
      apply H2 in H8; [| omega]. inversion H8.
      econstructor; [ eapply E_WhileLoop; eauto | eassumption | omega].
Qed.

Theorem pe_com_sound:
  ∀ c pe_st pe_st' c' c'', c / pe_st || c' / pe_st' / c'' →
  ∀ st st'' n,
  (c' / pe_st' / c'' / st || st'' # n) →
  (c / pe_override st pe_st || st'').
Proof. intros c pe_st pe_st' c' c'' Hpe.
  pe_com_cases (induction Hpe) Case;
    intros st st'' n [st' n' Heval Heval' Hle];
    try (inversion Heval; []; subst);
    try (inversion Heval'; []; subst); eauto.
  Case "PE_AssStatic". rewrite <- pe_override_update_add. apply E_Ass.
    rewrite → pe_aexp_correct. rewrite → H. reflexivity.
  Case "PE_AssDynamic". rewrite <- pe_override_update_remove. apply E_Ass.
    rewrite <- pe_aexp_correct. reflexivity.
  Case "PE_Seq". eapply E_Seq; eauto.
  Case "PE_IfTrue". apply E_IfTrue.
    rewrite → pe_bexp_correct. rewrite → H. reflexivity.
    eapply IHHpe. eauto.
  Case "PE_IfFalse". apply E_IfFalse.
    rewrite → pe_bexp_correct. rewrite → H. reflexivity.
    eapply IHHpe. eauto.
  Case "PE_If". inversion Heval; subst; inversion H7; subst; clear H7.
    SCase "E_IfTrue".
      eapply ceval_deterministic in H8; [| apply eval_assign]. subst.
      rewrite <- assign_removes in Heval'.
      apply E_IfTrue. rewrite → pe_bexp_correct. assumption.
      eapply IHHpe1. eauto.
    SCase "E_IfFalse".
      eapply ceval_deterministic in H8; [| apply eval_assign]. subst.
      rewrite → pe_compare_override in Heval'.
      rewrite <- assign_removes in Heval'.
      apply E_IfFalse. rewrite → pe_bexp_correct. assumption.
      eapply IHHpe2. eauto.
  Case "PE_WhileEnd". apply E_WhileEnd.
    rewrite → pe_bexp_correct. rewrite → H. reflexivity.
  Case "PE_WhileLoop". eapply E_WhileLoop.
    rewrite → pe_bexp_correct. rewrite → H. reflexivity.
    eapply IHHpe1. eauto. eapply IHHpe2. eauto.
  Case "PE_While". inversion Heval; subst.
    SCase "E_IfTrue".
      inversion H9. subst. clear H9.
      inversion H10. subst. clear H10.
      eapply ceval_deterministic in H11; [| apply eval_assign]. subst.
      rewrite → pe_compare_override in Heval'.
      rewrite <- assign_removes in Heval'.
      eapply E_WhileLoop. rewrite → pe_bexp_correct. assumption.
      eapply IHHpe1. eauto.
      eapply IHHpe2. eauto.
    SCase "E_IfFalse". apply ceval_count_sound in Heval'.
      eapply ceval_deterministic in H9; [| apply eval_assign]. subst.
      rewrite <- assign_removes in Heval'.
      inversion H2; subst.
      SSCase "c2'' = SKIP". inversion Heval'. subst. apply E_WhileEnd.
        rewrite → pe_bexp_correct. assumption.
      SSCase "c2'' = WHILE b1 DO c1 END". assumption.
  Case "PE_WhileFixedEnd". eapply ceval_count_sound. apply Heval'.
  Case "PE_WhileFixedLoop".
    apply loop_never_stops in Heval. inversion Heval.
  Case "PE_WhileFixed".
    clear - Case H1 IHHpe1 IHHpe2 Heval.
    remember (WHILE pe_bexp pe_st b1 DO c1';; c2' END) as c'.
    ceval_cases (induction Heval) SCase;
      inversion Heqc'; subst; clear Heqc'.
    SCase "E_WhileEnd". apply E_WhileEnd.
      rewrite pe_bexp_correct. assumption.
    SCase "E_WhileLoop".
      assert (IHHeval2' := IHHeval2 (refl_equal _)).
      apply ceval_count_complete in IHHeval2'. inversion IHHeval2'.
      clear IHHeval1 IHHeval2 IHHeval2'.
      inversion Heval1. subst.
      eapply E_WhileLoop. rewrite pe_bexp_correct. assumption. eauto.
      eapply IHHpe2. econstructor. eassumption.
      rewrite <- (pe_compare_nil_override _ _ H1). eassumption. apply le_n.
Qed.

Corollary pe_com_correct:
  ∀ c pe_st pe_st' c', c / pe_st || c' / pe_st' / SKIP →
  ∀ st st'',
  (c / pe_override st pe_st || st'') ↔
  (∃ st', c' / st || st' ∧ pe_override st' pe_st' = st'').
Proof. intros c pe_st pe_st' c' H st st''. split.
  Case "->". intros Heval.
    apply ceval_count_complete in Heval. inversion Heval as [n Heval'].
    apply pe_com_complete with (st:=st) (st'':=st'') (n:=n) in H.
    inversion H as [? ? ? Hskip ?]. inversion Hskip. subst. eauto.
    assumption.
  Case "<-". intros [st' [Heval Heq]]. subst st''.
    eapply pe_com_sound in H. apply H.
    econstructor. apply Heval. apply E'Skip. apply le_n.
Qed.

End Loop.

Partial Evaluation of Flowchart Programs

Instead of partially evaluating WHILE loops directly, the standard approach to partially evaluating imperative programs is to convert them into flowcharts. In other words, it turns out that adding labels and jumps to our language makes it much easier to partially evaluate. The result of partially evaluating a flowchart is a residual flowchart. If we are lucky, the jumps in the residual flowchart can be converted back to WHILE loops, but that is not possible in general; we do not pursue it here.

Basic blocks

A flowchart is made of basic blocks, which we represent with the inductive type block. A basic block is a sequence of assignments (the constructor Assign), concluding with a conditional jump (the constructor If) or an unconditional jump (the constructor Goto). The destinations of the jumps are specified by labels, which can be of any type. Therefore, we parameterize the block type by the type of labels.

Inductive block (Label:Type) : Type :=
  | Goto : Label → block Label
  | If : bexp → Label → Label → block Label
  | Assign : id → aexp → block Label → block Label.

Tactic Notation "block_cases" tactic(first) ident(c) :=
  first;
  [ Case_aux c "Goto" | Case_aux c "If" | Case_aux c "Assign" ].

Arguments Goto {Label} _.
Arguments If {Label} _ _ _.
Arguments Assign {Label} _ _ _.

We use the "even or odd" program, expressed above in Imp, as our running example. Converting this program into a flowchart turns out to require 4 labels, so we define the following type.

Inductive parity_label : Type :=
  | entry : parity_label
  | loop : parity_label
  | body : parity_label
  | done : parity_label.

The following block is the basic block found at the body label of the example program.

Definition parity_body : block parity_label :=
  Assign Y (AMinus (AId Y) (ANum 1))
   (Assign X (AMinus (ANum 1) (AId X))
     (Goto loop)).

To evaluate a basic block, given an initial state, is to compute the final state and the label to jump to next. Because basic blocks do not contain loops or other control structures, evaluation of basic blocks is a total function -- we don't need to worry about non-termination.

Fixpoint keval {L:Type} (st:state) (k : block L) : state × L :=
  match k with
  | Goto l ⇒ (st, l)
  | If b l1 l2 ⇒ (st, if beval st b then l1 else l2)
  | Assign i a k ⇒ keval (update st i (aeval st a)) k
  end.

Example keval_example:
  keval empty_state parity_body
  = (update (update empty_state Y 0) X 1, loop).
Proof. reflexivity. Qed.

Flowchart programs

A flowchart program is simply a lookup function that maps labels to basic blocks. Actually, some labels are halting states and do not map to any basic block. So, more precisely, a flowchart program whose labels are of type L is a function from L to option (block L).

Definition program (L:Type) : Type := L → option (block L).

Definition parity : program parity_label := fun l ⇒
  match l with
  | entry ⇒ Some (Assign X (ANum 0) (Goto loop))
  | loop ⇒ Some (If (BLe (ANum 1) (AId Y)) body done)
  | body ⇒ Some parity_body
  | done ⇒ None
  end.

Unlike a basic block, a program may not terminate, so we model the evaluation of programs by an inductive relation peval rather than a recursive function.

Inductive peval {L:Type} (p : program L)
  : state → L → state → L → Prop :=
  | E_None: ∀ st l,
    p l = None →
    peval p st l st l
  | E_Some: ∀ st l k st' l' st'' l'',
    p l = Some k →
    keval st k = (st', l') →
    peval p st' l' st'' l'' →
    peval p st l st'' l''.

Example parity_eval: peval parity empty_state entry empty_state done.
Proof. erewrite f_equal with (f := fun st ⇒ peval _ _ _ st _).
  eapply E_Some. reflexivity. reflexivity.
  eapply E_Some. reflexivity. reflexivity.
  apply E_None. reflexivity.
  apply functional_extensionality. intros i. rewrite update_same; auto.
Qed.

Tactic Notation "peval_cases" tactic(first) ident(c) :=
  first;
  [ Case_aux c "E_None" | Case_aux c "E_Some" ].

Partial evaluation of basic blocks and flowchart programs

Partial evaluation changes the label type in a systematic way: if the label type used to be L, it becomes pe_state × L. So the same label in the original program may be unfolded, or blown up, into multiple labels by being paired with different partial states. For example, the label loop in the parity program will become two labels: ([(X,0)], loop) and ([(X,1)], loop). This change of label type is reflected in the types of pe_block and pe_program defined presently.

Fixpoint pe_block {L:Type} (pe_st:pe_state) (k : block L)
  : block (pe_state × L) :=
  match k with
  | Goto l ⇒ Goto (pe_st, l)
  | If b l1 l2 ⇒
    match pe_bexp pe_st b with
    | BTrue ⇒ Goto (pe_st, l1)
    | BFalse ⇒ Goto (pe_st, l2)
    | b' ⇒ If b' (pe_st, l1) (pe_st, l2)
    end
  | Assign i a k ⇒
    match pe_aexp pe_st a with
    | ANum n ⇒ pe_block (pe_add pe_st i n) k
    | a' ⇒ Assign i a' (pe_block (pe_remove pe_st i) k)
    end
  end.

Example pe_block_example:
  pe_block [(X,0)] parity_body
  = Assign Y (AMinus (AId Y) (ANum 1)) (Goto ([(X,1)], loop)).
Proof. reflexivity. Qed.

Theorem pe_block_correct: ∀ (L:Type) st pe_st k st' pe_st' (l':L),
  keval st (pe_block pe_st k) = (st', (pe_st', l')) →
  keval (pe_override st pe_st) k = (pe_override st' pe_st', l').
Proof. intros. generalize dependent pe_st. generalize dependent st.
  block_cases (induction k as [l | b l1 l2 | i a k]) Case;
    intros st pe_st H.
  Case "Goto". inversion H; reflexivity.
  Case "If".
    replace (keval st (pe_block pe_st (If b l1 l2)))
       with (keval st (If (pe_bexp pe_st b) (pe_st, l1) (pe_st, l2)))
       in H by (simpl; destruct (pe_bexp pe_st b); reflexivity).
    simpl in ×. rewrite pe_bexp_correct.
    destruct (beval st (pe_bexp pe_st b)); inversion H; reflexivity.
  Case "Assign".
    simpl in ×. rewrite pe_aexp_correct.
    destruct (pe_aexp pe_st a); simpl;
      try solve [rewrite pe_override_update_add; apply IHk; apply H];
      solve [rewrite pe_override_update_remove; apply IHk; apply H].
Qed.

Definition pe_program {L:Type} (p : program L)
  : program (pe_state × L) :=
  fun pe_l ⇒ match pe_l with (pe_st, l) ⇒
                option_map (pe_block pe_st) (p l)
              end.

Inductive pe_peval {L:Type} (p : program L)
  (st:state) (pe_st:pe_state) (l:L) (st'o:state) (l':L) : Prop :=
  | pe_peval_intro : ∀ st' pe_st',
    peval (pe_program p) st (pe_st, l) st' (pe_st', l') →
    pe_override st' pe_st' = st'o →
    pe_peval p st pe_st l st'o l'.

Theorem pe_program_correct:
  ∀ (L:Type) (p : program L) st pe_st l st'o l',
  peval p (pe_override st pe_st) l st'o l' ↔
  pe_peval p st pe_st l st'o l'.
Proof. intros.
  split; [Case "->" | Case "<-"].
  Case "->". intros Heval.
    remember (pe_override st pe_st) as sto.
    generalize dependent pe_st. generalize dependent st.
    peval_cases (induction Heval as
      [ sto l Hlookup | sto l k st'o l' st''o l'' Hlookup Hkeval Heval ])
      SCase; intros st pe_st Heqsto; subst sto.
    SCase "E_None". eapply pe_peval_intro. apply E_None.
      simpl. rewrite Hlookup. reflexivity. reflexivity.
    SCase "E_Some".
      remember (keval st (pe_block pe_st k)) as x.
      destruct x as [st' [pe_st' l'_]].
      symmetry in Heqx. erewrite pe_block_correct in Hkeval by apply Heqx.
      inversion Hkeval. subst st'o l'_. clear Hkeval.
      edestruct IHHeval. reflexivity. subst st''o. clear IHHeval.
      eapply pe_peval_intro; [| reflexivity]. eapply E_Some; eauto.
      simpl. rewrite Hlookup. reflexivity.
  Case "<-". intros [st' pe_st' Heval Heqst'o].
    remember (pe_st, l) as pe_st_l.
    remember (pe_st', l') as pe_st'_l'.
    generalize dependent pe_st. generalize dependent l.
    peval_cases (induction Heval as
      [ st [pe_st_ l_] Hlookup
      | st [pe_st_ l_] pe_k st' [pe_st'_ l'_] st'' [pe_st'' l'']
        Hlookup Hkeval Heval ])
      SCase; intros l pe_st Heqpe_st_l;
      inversion Heqpe_st_l; inversion Heqpe_st'_l'; repeat subst.
    SCase "E_None". apply E_None. simpl in Hlookup.
      destruct (p l'); [ solve [ inversion Hlookup ] | reflexivity ].
    SCase "E_Some".
      simpl in Hlookup. remember (p l) as k.
      destruct k as [k|]; inversion Hlookup; subst.
      eapply E_Some; eauto. apply pe_block_correct. apply Hkeval.
Qed.