# EquivProgram Equivalence

(* \$Date: 2012-04-06 19:32:51 -0400 (Fri, 06 Apr 2012) \$ *)

Require Export Imp.

### Some general advice for working on problems

• We've tried to make sure that most of the Coq proofs we ask you to do are similar to proofs that we've provided. Before starting to work on the homework problems, take the time to work through our proofs (both informally, on paper, and in Coq) and make sure you understand them in detail. This will save you a lot of time.
• The Coq proofs we're doing now are sufficiently complicated that it is more or less impossible to complete them simply by "following your nose" or random hacking. You need to start with an idea about why the property is true and how the proof is going to go. The best way to do this is to write out at least a sketch of an informal proof on paper — one that intuitively convinces you of the truth of the theorem — before starting to work on the formal one.
• Use automation to save work! Some of the proofs in this chapter's exercises are pretty long if you try to write out all the cases explicitly.

# Behavioral Equivalence

In the last chapter, we investigated the correctness of a very simple program transformation: the optimize_0plus function. The programming language we were considering was the first version of the language of arithmetic expressions — with no variables — so in that setting it was very easy to define what it means for a program transformation to be correct: it should always yield a program that evaluates to the same number as the original.
To go further and talk about the correctness of program transformations in the full Imp language, we need to consider the role of variables and state.

## Definitions

For aexps and bexps with variables, the definition we want is clear. We say that two aexps or bexps are behaviorally equivalent if they evaluate to the same result in every state.

Definition aequiv (a1 a2 : aexp) : Prop :=
(st:state),
aeval st a1 = aeval st a2.

Definition bequiv (b1 b2 : bexp) : Prop :=
(st:state),
beval st b1 = beval st b2.

For commands, the situation is a little more subtle. We can't simply say "two commands are behaviorally equivalent if they evaluate to the same ending state whenever they are started in the same initial state," because some commands (in some starting states) don't terminate in any final state at all! What we need instead is this: two commands are behaviorally equivalent if, for any given starting state, they either both diverge or both terminate in the same final state. A compact way to express this is "if the first one terminates in a particular state then so does the second, and vice versa."

Definition cequiv (c1 c2 : com) : Prop :=
(st st' : state),
(c1 / st st') (c2 / st st').

#### Exercise: 2 stars, optional (pairs_equiv)

Which of the following pairs of programs are equivalent? Write "yes" or "no" for each one.
(a)
WHILE (BLe (ANum 1) (AId X)) DO
X ::= APlus (AId X) (ANum 1)
END
and
WHILE (BLe (ANum 2) (AId X)) DO
X ::= APlus (AId X) (ANum 1)
END
(* FILL IN HERE *)
(b)
WHILE BTrue DO
WHILE BFalse DO X ::= APlus (AId X) (ANum 1) END
END
and
WHILE BFalse DO
WHILE BTrue DO X ::= APlus (AId X) (ANum 1) END
END
(* FILL IN HERE *)

#### Exercise: 3 stars (equiv_classes)

Given the following programs, group together those that are equivalent in Imp. For example, if you think programs (a) through (h) are all equivalent to each other, but not to (i), your answer should look like this: {a,b,c,d,e,f,g,h} {i}.
(a)
WHILE X > 0 DO
X ::= X + 1
END
(b)
IFB X = 0 THEN
X ::= X + 1;
Y ::= 1
ELSE
Y ::= 0
FI;
X ::= X - Y;
Y ::= 0
(c)
SKIP
(d)
WHILE X <> 0 DO
X ::= X * Y + 1
END
(e)
Y ::= 0
(f)
Y ::= X + 1;
WHILE X <> Y DO
Y ::= X + 1
END
(g)
WHILE BTrue DO
SKIP
END
(h)
WHILE X <> X DO
X ::= X + 1
END
(i)
WHILE X <> Y DO
X ::= Y + 1
END
(* FILL IN HERE *)

## Examples

Here are some simple examples of equivalences of arithmetic and boolean expressions.

Theorem aequiv_example:
aequiv (AMinus (AId X) (AId X)) (ANum 0).
Proof.
intros st. simpl. apply minus_diag.
Qed.

Theorem bequiv_example:
bequiv (BEq (AMinus (AId X) (AId X)) (ANum 0)) BTrue.
Proof.
intros st. unfold beval.
rewrite aequiv_example. reflexivity.
Qed.

For examples of command equivalence, let's start by looking at some trivial program transformations involving SKIP:

Theorem skip_left: c,
cequiv
(SKIP; c)
c.
Proof.
(* WORKED IN CLASS *)
intros c st st'.
split; intros H.
Case "".
inversion H. subst.
inversion H2. subst.
assumption.
Case "".
apply E_Seq with st.
apply E_Skip.
assumption.
Qed.

#### Exercise: 2 stars (skip_right)

Theorem skip_right: c,
cequiv
(c; SKIP)
c.
Proof.
(* FILL IN HERE *) Admitted.
Similarly, here is a simple transformations that simplifies IFB commands:

Theorem IFB_true_simple: c1 c2,
cequiv
(IFB BTrue THEN c1 ELSE c2 FI)
c1.
Proof.
intros c1 c2.
split; intros H.
Case "".
inversion H; subst. assumption. inversion H5.
Case "".
apply E_IfTrue. reflexivity. assumption. Qed.

Of course, few programmers would be tempted to write a conditional whose guard is literally BTrue. A more interesting case is when the guard is equivalent to true:
Theorem: If b is equivalent to BTrue, then IFB b THEN c1 ELSE c2 FI is equivalent to c1.
Proof:
• () We must show, for all st and st', that if IFB b THEN c1 ELSE c2 FI / st st' then c1 / st st'.
Proceed by cases on the rules that could possibly have been used to show IFB b THEN c1 ELSE c2 FI / st st', namely E_IfTrue and E_IfFalse.
• Suppose the final rule rule in the derivation of IFB b THEN c1 ELSE c2 FI / st st' was E_IfTrue. We then have, by the premises of E_IfTrue, that c1 / st st'. This is exactly what we set out to prove.
• On the other hand, suppose the final rule in the derivation of IFB b THEN c1 ELSE c2 FI / st st' was E_IfFalse. We then know that beval st b = false and c2 / st st'.
Recall that b is equivalent to BTrue, i.e. forall st, beval st b = beval st BTrue. In particular, this means that beval st b = true, since beval st BTrue = true. But this is a contradiction, since E_IfFalse requires that beval st b = false. Thus, the final rule could not have been E_IfFalse.
• () We must show, for all st and st', that if c1 / st st' then IFB b THEN c1 ELSE c2 FI / st st'.
Since b is equivalent to BTrue, we know that beval st b = beval st BTrue = true. Together with the assumption that c1 / st st', we can apply E_IfTrue to derive IFB b THEN c1 ELSE c2 FI / st st'.
Here is the formal version of this proof:

Theorem IFB_true: b c1 c2,
bequiv b BTrue
cequiv
(IFB b THEN c1 ELSE c2 FI)
c1.
Proof.
intros b c1 c2 Hb.
split; intros H.
Case "".
inversion H; subst.
SCase "b evaluates to true".
assumption.
SCase "b evaluates to false (contradiction)".
rewrite Hb in H5.
inversion H5.
Case "".
apply E_IfTrue; try assumption.
rewrite Hb. reflexivity. Qed.

#### Exercise: 2 stars, recommended (IFB_false)

Theorem IFB_false: b c1 c2,
bequiv b BFalse
cequiv
(IFB b THEN c1 ELSE c2 FI)
c2.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars (swap_if_branches)

Theorem swap_if_branches: b e1 e2,
cequiv
(IFB b THEN e1 ELSE e2 FI)
(IFB BNot b THEN e2 ELSE e1 FI).
Proof.
(* FILL IN HERE *) Admitted.
For WHILE loops, we can give a similar pair of theorems. A loop whose guard is equivalent to BFalse is equivalent to SKIP, while a loop whose guard is equivalent to BTrue is equivalent to WHILE BTrue DO SKIP END (or any other non-terminating program). The first of these facts is easy.

Theorem WHILE_false : b c,
bequiv b BFalse
cequiv
(WHILE b DO c END)
SKIP.
Proof.
intros b c Hb. split; intros H.
Case "".
inversion H; subst.
SCase "E_WhileEnd".
apply E_Skip.
SCase "E_WhileLoop".
rewrite Hb in H2. inversion H2.
Case "".
inversion H; subst.
apply E_WhileEnd.
rewrite Hb.
reflexivity. Qed.

#### Exercise: 2 stars (WHILE_false_informal)

Write an informal proof of WHILE_false.
(* FILL IN HERE *)
To prove the second fact, we need an auxiliary lemma stating that WHILE loops whose guards are equivalent to BTrue never terminate:
Lemma: If b is equivalent to BTrue, then it cannot be the case that (WHILE b DO c END) / st st'.
Proof: Suppose that (WHILE b DO c END) / st st'. We show, by induction on a derivation of (WHILE b DO c END) / st st', that this assumption leads to a contradiction.
• Suppose (WHILE b DO c END) / st st' is proved using rule E_WhileEnd. Then by assumption beval st b = false. But this contradicts the assumption that b is equivalent to BTrue.
• Suppose (WHILE b DO c END) / st st' is proved using rule E_WhileLoop. Then we are given the induction hypothesis that (WHILE b DO c END) / st st' is contradictory, which is exactly what we are trying to prove!
• Since these are the only rules that could have been used to prove (WHILE b DO c END) / st st', the other cases of the induction are immediately contradictory.

Lemma WHILE_true_nonterm : b c st st',
bequiv b BTrue
~( (WHILE b DO c END) / st st' ).
Proof.
(* WORKED IN CLASS *)
intros b c st st' Hb.
intros H.
remember (WHILE b DO c END) as cw.
ceval_cases (induction H) Case;
(* Most rules don't apply, and we can rule them out
by inversion *)

inversion Heqcw; subst; clear Heqcw.
(* The two interesting cases are the ones for WHILE loops: *)
Case "E_WhileEnd". (* contradictory -- b is always true! *)
unfold bequiv in Hb.
rewrite Hb in H. inversion H.
Case "E_WhileLoop". (* immediate from the IH *)
apply IHceval2. reflexivity. Qed.

#### Exercise: 2 stars, optional (WHILE_true_nonterm_informal)

Explain what the lemma WHILE_true_nonterm means in English.
(* FILL IN HERE *)

#### Exercise: 2 stars, recommended (WHILE_true)

Hint: You'll want to use WHILE_true_nonterm here.

Theorem WHILE_true: b c,
bequiv b BTrue
cequiv
(WHILE b DO c END)
(WHILE BTrue DO SKIP END).
Proof.
(* FILL IN HERE *) Admitted.

Theorem loop_unrolling: b c,
cequiv
(WHILE b DO c END)
(IFB b THEN (c; WHILE b DO c END) ELSE SKIP FI).
Proof.
(* WORKED IN CLASS *)
intros b c st st'.
split; intros Hce.
Case "".
inversion Hce; subst.
SCase "loop doesn't run".
apply E_IfFalse. assumption. apply E_Skip.
SCase "loop runs".
apply E_IfTrue. assumption.
apply E_Seq with (st' := st'0). assumption. assumption.
Case "".
inversion Hce; subst.
SCase "loop runs".
inversion H5; subst.
apply E_WhileLoop with (st' := st'0).
assumption. assumption. assumption.
SCase "loop doesn't run".
inversion H5; subst. apply E_WhileEnd. assumption. Qed.

#### Exercise: 2 stars, optional (seq_assoc)

Theorem seq_assoc : c1 c2 c3,
cequiv ((c1;c2);c3) (c1;(c2;c3)).
Proof.
(* FILL IN HERE *) Admitted.

## The Functional Equivalence Axiom

Finally, let's look at simple equivalences involving assignments. For example, we might expect to be able to show that X ::= AId X is equivalent to SKIP. However, when we try to show it, we get stuck in an interesting way.

Theorem identity_assignment_first_try : (X:id),
cequiv (X ::= AId X) SKIP.
Proof.
intros. split; intro H.
Case "".
inversion H; subst. simpl.
replace (update st X (st X)) with st.
constructor.
(* Here we're stuck. The goal looks reasonable,
but in fact it is not provable!  If we look back
at the set of lemmas we proved about update in
the last chapter, we can see that lemma
update_same almost does the job, but not quite:
it says that the original and updated states
agree at all values, but this is not the same
thing as saying that they are = in Coq's
sense! *)

What is going on here? Recall that our states are just functions from identifiers to values. For Coq, functions are only equal when their definitions are syntactically the same, modulo simplification. (This is the only way we can legally apply the refl_equal constructor of the inductively defined proposition eq!) In practice, for functions built up by repeated uses of the update operation, this means that two functions can be proven equal only if they were constructed using the same update operations, applied in the same order. In the theorem above, the sequence of updates on the first parameter cequiv is one longer than for the second parameter, so it is no wonder that the equality doesn't hold.
This problem is actually quite general. If we try to prove other simple facts, such as
cequiv (X ::= APlus (AId X ANum 1) ;
X ::= APlus (AId X ANum 1))
(X ::= APlus (AId X ANum 2))
or
cequiv (X ::= ANum 1; Y ::= ANum 2)
(y ::= ANum 2; X ::= ANum 1)

we'll get stuck in the same way: we'll have two functions that behave the same way on all inputs, but cannot be proven to be eq to each other.
The reasoning principle we would like to use in these situations is called functional extensionality:
 ∀ x, f x = g x f = g
Although this principle is not derivable in Coq's built-in logic, it is safe to add it as an additional axiom.

Axiom functional_extensionality : {X Y: Type} {f g : X Y},
((x: X), f x = g x) f = g.

It can be shown that adding this axiom doesn't introduce any inconsistencies into Coq. (In this way, it is similar to adding one of the classical logic axioms, such as excluded_middle.)
With the benefit of this axiom we can prove our theorem.

Theorem identity_assignment : (X:id),
cequiv
(X ::= AId X)
SKIP.
Proof.
intros. split; intro H.
Case "".
inversion H; subst. simpl.
replace (update st X (st X)) with st.
constructor.
apply functional_extensionality. intro.
rewrite update_same; reflexivity.
Case "".
inversion H; subst.
assert (st' = (update st' X (st' X))).
apply functional_extensionality. intro.
rewrite update_same; reflexivity.
rewrite H0 at 2.
constructor. reflexivity.
Qed.

#### Exercise: 2 stars, recommended (assign_aequiv)

Theorem assign_aequiv : X e,
aequiv (AId X) e
cequiv SKIP (X ::= e).
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 2 stars, optional (functional_extensionality_failed_false)

As we just mentioned, functional extensionality doesn't add any inconsistencies into Coq. However, consider the following theorem. Inspired by it, we present an attempt to prove False. Briefly explain why it will fail.

Theorem feff_1 : true = false False.
Proof.
intros. inversion H.
Qed.

Lemma feff_2 : empty_state = update empty_state X 0.
Proof.
apply functional_extensionality. intros.
destruct x as [n]. destruct n as [| n'].
Case "x = AId 0". reflexivity.
Case "x = AId (S n')". reflexivity.
Qed.

(* Lemma feff_3 : empty_state = update empty_state X 0 -> False.
Proof.
intros. inversion H. (* And this makes no progress! *)
Qed. *)

If we were able to prove feff_3, then together with feff_2, a proof of False is immediate. Explain why the inversion H fails, though it works in feff_1.

(* FILL IN HERE *)

# Properties of Behavioral Equivalence

We now turn to developing some of the properties of the program equivalences we have defined.

## Behavioral Equivalence is an Equivalence

First, we verify that the equivalences on aexps, bexps, and coms really are equivalences — i.e., that they are reflexive, symmetric, and transitive:

Lemma refl_aequiv : (a : aexp), aequiv a a.
Proof.
intros a st. reflexivity. Qed.

Lemma sym_aequiv : (a1 a2 : aexp),
aequiv a1 a2 aequiv a2 a1.
Proof.
intros a1 a2 H. intros st. symmetry. apply H. Qed.

Lemma trans_aequiv : (a1 a2 a3 : aexp),
aequiv a1 a2 aequiv a2 a3 aequiv a1 a3.
Proof.
unfold aequiv. intros a1 a2 a3 H12 H23 st.
rewrite (H12 st). rewrite (H23 st). reflexivity. Qed.

Lemma refl_bequiv : (b : bexp), bequiv b b.
Proof.
unfold bequiv. intros b st. reflexivity. Qed.

Lemma sym_bequiv : (b1 b2 : bexp),
bequiv b1 b2 bequiv b2 b1.
Proof.
unfold bequiv. intros b1 b2 H. intros st. symmetry. apply H. Qed.

Lemma trans_bequiv : (b1 b2 b3 : bexp),
bequiv b1 b2 bequiv b2 b3 bequiv b1 b3.
Proof.
unfold bequiv. intros b1 b2 b3 H12 H23 st.
rewrite (H12 st). rewrite (H23 st). reflexivity. Qed.

Lemma refl_cequiv : (c : com), cequiv c c.
Proof.
unfold cequiv. intros c st st'. apply iff_refl. Qed.

Lemma sym_cequiv : (c1 c2 : com),
cequiv c1 c2 cequiv c2 c1.
Proof.
unfold cequiv. intros c1 c2 H st st'.
assert (c1 / st st' c2 / st st') as H'.
SCase "Proof of assertion". apply H.
apply iff_sym. assumption.
Qed.

Lemma iff_trans : (P1 P2 P3 : Prop),
(P1 P2) (P2 P3) (P1 P3).
Proof.
intros P1 P2 P3 H12 H23.
inversion H12. inversion H23.
split; intros A.
apply H1. apply H. apply A.
apply H0. apply H2. apply A. Qed.

Lemma trans_cequiv : (c1 c2 c3 : com),
cequiv c1 c2 cequiv c2 c3 cequiv c1 c3.
Proof.
unfold cequiv. intros c1 c2 c3 H12 H23 st st'.
apply iff_trans with (c2 / st st'). apply H12. apply H23. Qed.

(* ########################################################*)

## Behavioral Equivalence is a Congruence

Less obviously, behavioral equivalence is also a congruence. That is, the equivalence of two subprograms implies the equivalence of the larger programs in which they are embedded:
 aequiv a1 a1' cequiv (i ::= a1) (i ::= a1')
 cequiv c1 c1' cequiv c2 c2' cequiv (c1;c2) (c1';c2')
...and so on. (Note that we are using the inference rule notation here not as part of a definition, but simply to write down some valid implications in a readable format. We prove these implications below.)
We will see a concrete example of why these congruence properties are important in the following section (in the proof of fold_constants_com_sound), but the main idea is that they allow us to replace a small part of a large program with an equivalent small part and know that the whole large programs are equivalent without doing an explicit proof about the non-varying parts — i.e., the "proof burden" of a small change to a large program is proportional to the size of the change, not the program.

Theorem CAss_congruence : i a1 a1',
aequiv a1 a1'
cequiv (CAss i a1) (CAss i a1').
Proof.
intros i a1 a2 Heqv st st'.
split; intros Hceval.
Case "".
inversion Hceval. subst. apply E_Ass.
rewrite Heqv. reflexivity.
Case "".
inversion Hceval. subst. apply E_Ass.
rewrite Heqv. reflexivity. Qed.

The congruence property for loops is a little more interesting, since it requires induction.
Theorem: Equivalence is a congruence for WHILE — that is, if b1 is equivalent to b1' and c1 is equivalent to c1', then WHILE b1 DO c1 END is equivalent to WHILE b1' DO c1' END.
Proof: Suppose b1 is equivalent to b1' and c1 is equivalent to c1'. We must show, for every st and st', that WHILE b1 DO c1 END / st st' iff WHILE b1' DO c1' END / st st'. We consider the two directions separately.
• () We show that WHILE b1 DO c1 END / st st' implies WHILE b1' DO c1' END / st st', by induction on a derivation of WHILE b1 DO c1 END / st st'. The only nontrivial cases are when the final rule in the derivation is E_WhileEnd or E_WhileLoop.
• E_WhileEnd: In this case, the form of the rule gives us beval st b1 = false and st = st'. But then, since b1 and b1' are equivalent, we have beval st b1' = false, and E-WhileEnd applies, giving us WHILE b1' DO c1' END / st st', as required.
• E_WhileLoop: The form of the rule now gives us beval st b1 = true, with c1 / st st'0 and WHILE b1 DO c1 END / st'0 st' for some state st'0, with the induction hypothesis WHILE b1' DO c1' END / st'0 st'.
Since c1 and c1' are equivalent, we know that c1' / st st'0. And since b1 and b1' are equivalent, we have beval st b1' = true. Now E-WhileLoop applies, giving us WHILE b1' DO c1' END / st st', as required.
• () Similar.

Theorem CWhile_congruence : b1 b1' c1 c1',
bequiv b1 b1' cequiv c1 c1'
cequiv (WHILE b1 DO c1 END) (WHILE b1' DO c1' END).
Proof.
(* WORKED IN CLASS *)
unfold bequiv,cequiv.
intros b1 b1' c1 c1' Hb1e Hc1e st st'.
split; intros Hce.
Case "".
remember (WHILE b1 DO c1 END) as cwhile.
induction Hce; inversion Heqcwhile; subst.
SCase "E_WhileEnd".
apply E_WhileEnd. rewrite Hb1e. apply H.
SCase "E_WhileLoop".
apply E_WhileLoop with (st' := st').
SSCase "show loop runs". rewrite Hb1e. apply H.
SSCase "body execution".
apply (Hc1e st st'). apply Hce1.
SSCase "subsequent loop execution".
apply IHHce2. reflexivity.
Case "".
remember (WHILE b1' DO c1' END) as c'while.
induction Hce; inversion Heqc'while; subst.
SCase "E_WhileEnd".
apply E_WhileEnd. rewrite Hb1e. apply H.
SCase "E_WhileLoop".
apply E_WhileLoop with (st' := st').
SSCase "show loop runs". rewrite Hb1e. apply H.
SSCase "body execution".
apply (Hc1e st st'). apply Hce1.
SSCase "subsequent loop execution".
apply IHHce2. reflexivity. Qed.

#### Exercise: 3 stars, optional (CSeq_congruence)

Theorem CSeq_congruence : c1 c1' c2 c2',
cequiv c1 c1' cequiv c2 c2'
cequiv (c1;c2) (c1';c2').
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars (CIf_congruence)

Theorem CIf_congruence : b b' c1 c1' c2 c2',
bequiv b b' cequiv c1 c1' cequiv c2 c2'
cequiv (IFB b THEN c1 ELSE c2 FI) (IFB b' THEN c1' ELSE c2' FI).
Proof.
(* FILL IN HERE *) Admitted.
For example, here are two equivalent programs and a proof of their equivalence...

Example congruence_example:
cequiv
(X ::= ANum 0;
IFB (BEq (AId X) (ANum 0))
THEN
Y ::= ANum 0
ELSE
Y ::= ANum 42
FI)
(X ::= ANum 0;
IFB (BEq (AId X) (ANum 0))
THEN
Y ::= AMinus (AId X) (AId X) (* <--- changed here *)
ELSE
Y ::= ANum 42
FI).
Proof.
apply CSeq_congruence.
apply refl_cequiv.
apply CIf_congruence.
apply refl_bequiv.
apply CAss_congruence. unfold aequiv. simpl.
symmetry. apply minus_diag.
apply refl_cequiv.
Qed.

# Case Study: Constant Folding

A program transformation is a function that takes a program as input and produces some variant of the program as its output. Compiler optimizations such as constant folding are a canonical example, but there are many others.

## Soundness of Program Transformations

A program transformation is sound if it preserves the behavior of the original program.
We can define a notion of soundness for translations of aexps, bexps, and coms.

Definition atrans_sound (atrans : aexp aexp) : Prop :=
(a : aexp),
aequiv a (atrans a).

Definition btrans_sound (btrans : bexp bexp) : Prop :=
(b : bexp),
bequiv b (btrans b).

Definition ctrans_sound (ctrans : com com) : Prop :=
(c : com),
cequiv c (ctrans c).

## The Constant-Folding Transformation

An expression is constant when it contains no variable references.
Constant folding is an optimization that finds constant expressions and replaces them by their values.

Fixpoint fold_constants_aexp (a : aexp) : aexp :=
match a with
| ANum n => ANum n
| AId i => AId i
| APlus a1 a2 =>
match (fold_constants_aexp a1, fold_constants_aexp a2) with
| (ANum n1, ANum n2) => ANum (n1 + n2)
| (a1', a2') => APlus a1' a2'
end
| AMinus a1 a2 =>
match (fold_constants_aexp a1, fold_constants_aexp a2) with
| (ANum n1, ANum n2) => ANum (n1 - n2)
| (a1', a2') => AMinus a1' a2'
end
| AMult a1 a2 =>
match (fold_constants_aexp a1, fold_constants_aexp a2) with
| (ANum n1, ANum n2) => ANum (n1 * n2)
| (a1', a2') => AMult a1' a2'
end
end.

Example fold_aexp_ex1 :
fold_constants_aexp
(AMult (APlus (ANum 1) (ANum 2)) (AId X))
= AMult (ANum 3) (AId X).
Proof. reflexivity. Qed.

Note that this version of constant folding doesn't eliminate trivial additions, etc. — we are focusing attention on a single optimization for the sake of simplicity. It is not hard to incorporate other ways of simplifying expressions; the definitions and proofs just get longer.

Example fold_aexp_ex2 :
fold_constants_aexp
(AMinus (AId X) (APlus (AMult (ANum 0) (ANum 6)) (AId Y)))
= AMinus (AId X) (APlus (ANum 0) (AId Y)).
Proof. reflexivity. Qed.

Not only can we lift fold_constants_aexp to bexps (in the BEq and BLe cases), we can also find constant boolean expressions and reduce them in-place.

Fixpoint fold_constants_bexp (b : bexp) : bexp :=
match b with
| BTrue => BTrue
| BFalse => BFalse
| BEq a1 a2 =>
match (fold_constants_aexp a1, fold_constants_aexp 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 (fold_constants_aexp a1, fold_constants_aexp a2) with
| (ANum n1, ANum n2) => if ble_nat n1 n2 then BTrue else BFalse
| (a1', a2') => BLe a1' a2'
end
| BNot b1 =>
match (fold_constants_bexp b1) with
| BTrue => BFalse
| BFalse => BTrue
| b1' => BNot b1'
end
| BAnd b1 b2 =>
match (fold_constants_bexp b1, fold_constants_bexp b2) with
| (BTrue, BTrue) => BTrue
| (BTrue, BFalse) => BFalse
| (BFalse, BTrue) => BFalse
| (BFalse, BFalse) => BFalse
| (b1', b2') => BAnd b1' b2'
end
end.

Example fold_bexp_ex1 :
fold_constants_bexp (BAnd BTrue (BNot (BAnd BFalse BTrue)))
= BTrue.
Proof. reflexivity. Qed.

Example fold_bexp_ex2 :
fold_constants_bexp
(BAnd (BEq (AId X) (AId Y))
(BEq (ANum 0)
(AMinus (ANum 2) (APlus (ANum 1) (ANum 1)))))
= BAnd (BEq (AId X) (AId Y)) BTrue.
Proof. reflexivity. Qed.

To fold constants in a command, we apply the appropriate folding functions on all embedded expressions.

Fixpoint fold_constants_com (c : com) : com :=
match c with
| SKIP =>
SKIP
| i ::= a =>
CAss i (fold_constants_aexp a)
| c1 ; c2 =>
(fold_constants_com c1) ; (fold_constants_com c2)
| IFB b THEN c1 ELSE c2 FI =>
match fold_constants_bexp b with
| BTrue => fold_constants_com c1
| BFalse => fold_constants_com c2
| b' => IFB b' THEN fold_constants_com c1
ELSE fold_constants_com c2 FI
end
| WHILE b DO c END =>
match fold_constants_bexp b with
| BTrue => WHILE BTrue DO SKIP END
| BFalse => SKIP
| b' => WHILE b' DO (fold_constants_com c) END
end
end.

Example fold_com_ex1 :
fold_constants_com
(X ::= APlus (ANum 4) (ANum 5);
Y ::= AMinus (AId X) (ANum 3);
IFB BEq (AMinus (AId X) (AId Y)) (APlus (ANum 2) (ANum 4)) THEN
SKIP
ELSE
Y ::= ANum 0
FI;
IFB BLe (ANum 0) (AMinus (ANum 4) (APlus (ANum 2) (ANum 1))) THEN
Y ::= ANum 0
ELSE
SKIP
FI;
WHILE BEq (AId Y) (ANum 0) DO
X ::= APlus (AId X) (ANum 1)
END) =
(X ::= ANum 9;
Y ::= AMinus (AId X) (ANum 3);
IFB BEq (AMinus (AId X) (AId Y)) (ANum 6) THEN
SKIP
ELSE
(Y ::= ANum 0)
FI;
Y ::= ANum 0;
WHILE BEq (AId Y) (ANum 0) DO
X ::= APlus (AId X) (ANum 1)
END).
Proof. reflexivity. Qed.

## Soundness of Constant Folding

Now we need to show that what we've done is correct. Here's the proof for arithmetic expressions:

Theorem fold_constants_aexp_sound :
atrans_sound fold_constants_aexp.
Proof.
unfold atrans_sound. intros a. unfold aequiv. intros st.
aexp_cases (induction a) Case; simpl;
(* ANum and AId follow immediately *)
try reflexivity;
(* APlus, AMinus, and AMult follow from the IH
and the observation that
aeval st (APlus a1 a2)
= ANum ((aeval st a1) + (aeval st a2))
= aeval st (ANum ((aeval st a1) + (aeval st a2)))
(and similarly for AMinus/minus and AMult/mult) *)

try (destruct (fold_constants_aexp a1);
destruct (fold_constants_aexp a2);
rewrite IHa1; rewrite IHa2; reflexivity). Qed.

#### Exercise: 3 stars, optional (fold_bexp_BEq_informal)

Here is an informal proof of the BEq case of the soundness argument for boolean expression constant folding. Read it carefully and compare it to the formal proof that follows. Then fill in the BLe case of the formal proof (without looking at the BEq case, if possible).
Theorem: The constant folding function for booleans, fold_constants_bexp, is sound.
Proof: We must show that b is equivalent to fold_constants_bexp, for all boolean expressions b. Proceed by induction on b. We show just the case where b has the form BEq a1 a2.
In this case, we must show
beval st (BEq a1 a2
= beval st (fold_constants_bexp (BEq a1 a2)).
There are two cases to consider:
• First, suppose fold_constants_aexp a1 = ANum n1 and fold_constants_aexp a2 = ANum n2 for some n1 and n2.
In this case, we have
fold_constants_bexp (BEq a1 a2
= if beq_nat n1 n2 then BTrue else BFalse
and
beval st (BEq a1 a2
= beq_nat (aeval st a1) (aeval st a2).
By the soundness of constant folding for arithmetic expressions (Lemma fold_constants_aexp_sound), we know
aeval st a1
= aeval st (fold_constants_aexp a1
= aeval st (ANum n1
= n1
and
aeval st a2
= aeval st (fold_constants_aexp a2
= aeval st (ANum n2
= n2,
so
beval st (BEq a1 a2
= beq_nat (aeval a1) (aeval a2)
= beq_nat n1 n2.
Also, it is easy to see (by considering the cases n1 = n2 and n1 <> n2 separately) that
beval st (if beq_nat n1 n2 then BTrue else BFalse)
= if beq_nat n1 n2 then beval st BTrue else beval st BFalse
= if beq_nat n1 n2 then true else false
= beq_nat n1 n2.
So
beval st (BEq a1 a2
= beq_nat n1 n2.
= beval st (if beq_nat n1 n2 then BTrue else BFalse),
as required.
• Otherwise, one of fold_constants_aexp a1 and fold_constants_aexp a2 is not a constant. In this case, we must show
beval st (BEq a1 a2
= beval st (BEq (fold_constants_aexp a1)
(fold_constants_aexp a2)),
which, by the definition of beval, is the same as showing
beq_nat (aeval st a1) (aeval st a2
= beq_nat (aeval st (fold_constants_aexp a1))
(aeval st (fold_constants_aexp a2)).
But the soundness of constant folding for arithmetic expressions (fold_constants_aexp_sound) gives us
aeval st a1 = aeval st (fold_constants_aexp a1)
aeval st a2 = aeval st (fold_constants_aexp a2),
completing the case.

Theorem fold_constants_bexp_sound:
btrans_sound fold_constants_bexp.
Proof.
unfold btrans_sound. intros b. unfold bequiv. intros st.
bexp_cases (induction b) Case;
(* BTrue and BFalse are immediate *)
try reflexivity.
Case "BEq".
(* Doing induction when there are a lot of constructors makes
specifying variable names a chore, but Coq doesn't always
choose nice variable names.  We can rename entries in the
context with the rename tactic: rename a into a1 will
change a to a1 in the current goal and context. *)

rename a into a1. rename a0 into a2. simpl.
remember (fold_constants_aexp a1) as a1'.
remember (fold_constants_aexp a2) as a2'.
replace (aeval st a1) with (aeval st a1') by
(subst a1'; rewrite fold_constants_aexp_sound; reflexivity).
replace (aeval st a2) with (aeval st a2') by
(subst a2'; rewrite fold_constants_aexp_sound; reflexivity).
destruct a1'; destruct a2'; try reflexivity.
(* The only interesting case is when both a1 and a2
become constants after folding *)

simpl. destruct (beq_nat n n0); reflexivity.
Case "BLe".
(* FILL IN HERE *) admit.
Case "BNot".
simpl. remember (fold_constants_bexp b) as b'.
rewrite IHb.
destruct b'; reflexivity.
Case "BAnd".
simpl.
remember (fold_constants_bexp b1) as b1'.
remember (fold_constants_bexp b2) as b2'.
rewrite IHb1. rewrite IHb2.
destruct b1'; destruct b2'; reflexivity. Qed.

#### Exercise: 3 stars (fold_constants_com_sound)

Complete the WHILE case of the following proof.

Theorem fold_constants_com_sound :
ctrans_sound fold_constants_com.
Proof.
unfold ctrans_sound. intros c.
com_cases (induction c) Case; simpl.
Case "SKIP". apply refl_cequiv.
Case "::=". apply CAss_congruence. apply fold_constants_aexp_sound.
Case ";". apply CSeq_congruence; assumption.
Case "IFB".
assert (bequiv b (fold_constants_bexp b)).
SCase "Pf of assertion". apply fold_constants_bexp_sound.
remember (fold_constants_bexp b) as b'.
destruct b';
(* If the optimization doesn't eliminate the if, then the result
is easy to prove from the IH and fold_constants_bexp_sound *)

try (apply CIf_congruence; assumption).
SCase "b always true".
apply trans_cequiv with c1; try assumption.
apply IFB_true; assumption.
SCase "b always false".
apply trans_cequiv with c2; try assumption.
apply IFB_false; assumption.
Case "WHILE".
(* FILL IN HERE *) Admitted.

### Soundness of (0 + n) Elimination, Redux

#### Exercise: 4 stars, optional (optimize_0plus)

Recall the definition optimize_0plus from Imp.v:
Fixpoint optimize_0plus (e:aexp) : aexp :=
match e with
| ANum n =>
ANum n
| APlus (ANum 0) e2 =>
optimize_0plus e2
| APlus e1 e2 =>
APlus (optimize_0plus e1) (optimize_0plus e2)
| AMinus e1 e2 =>
AMinus (optimize_0plus e1) (optimize_0plus e2)
| AMult e1 e2 =>
AMult (optimize_0plus e1) (optimize_0plus e2)
end.
Note that this function is defined over the old aexps, without states.
Write a new version of this function that accounts for variables, and analogous ones for bexps and commands:
optimize_0plus_aexp
optimize_0plus_bexp
optimize_0plus_com
Prove that these three functions are sound, as we did for fold_constants_*. Make sure you use the congruence lemmas in the proof of optimize_0plus_com (otherwise it will be long!).
Then define an optimizer on commands that first folds constants (using fold_constants_com) and then eliminates 0 + n terms (using optimize_0plus_com).
• Give a meaningful example of this optimizer's output.
• Prove that the optimizer is sound. (This part should be very easy.)

(* FILL IN HERE *)

# Proving That Programs Are Not Equivalent

Suppose that c1 is a command of the form X ::= a1; Y ::= a2 and c2 is the command X ::= a1; Y ::= a2', where a2' is formed by substituting a1 for all occurrences of X in a2. For example, c1 and c2 might be:
c1  =  (X ::= 42 + 53;
Y ::= Y + X)
c2  =  (X ::= 42 + 53;
Y ::= Y + (42 + 53))
Clearly, this particular c1 and c2 are equivalent. Is this true in general?
We will see in a moment that it is not, but it is worthwhile to pause, now, and see if you can find a counter-example on your own.
Here, formally, is the function that substitutes an arithmetic expression for each occurrence of a given variable in another expression:

Fixpoint subst_aexp (i : id) (u : aexp) (a : aexp) : aexp :=
match a with
| ANum n => ANum n
| AId i' => if beq_id i i' then u else AId i'
| APlus a1 a2 => APlus (subst_aexp i u a1) (subst_aexp i u a2)
| AMinus a1 a2 => AMinus (subst_aexp i u a1) (subst_aexp i u a2)
| AMult a1 a2 => AMult (subst_aexp i u a1) (subst_aexp i u a2)
end.

Example subst_aexp_ex :
subst_aexp X (APlus (ANum 42) (ANum 53)) (APlus (AId Y) (AId X)) =
(APlus (AId Y) (APlus (ANum 42) (ANum 53))).
Proof. reflexivity. Qed.

And here is the property we are interested in, expressing the claim that commands c1 and c2 as described above are always equivalent.

Definition subst_equiv_property := i1 i2 a1 a2,
cequiv (i1 ::= a1; i2 ::= a2)
(i1 ::= a1; i2 ::= subst_aexp i1 a1 a2).

Sadly, the property does not always hold.
Theorem: It is not the case that, for all i1, i2, a1, and a2,
cequiv (i1 ::= a1i2 ::= a2)
(i1 ::= a1i2 ::= subst_aexp i1 a1 a2).
Proof: Suppose, for a contradiction, that for all i1, i2, a1, and a2, we have
cequiv (i1 ::= a1i2 ::= a2
(i1 ::= a1i2 ::= subst_aexp i1 a1 a2).
Consider the following program:
X ::= APlus (AId X) (ANum 1); Y ::= AId X
Note that
(X ::= APlus (AId X) (ANum 1); Y ::= AId X)
/ empty_state  st1,
where st1 = { X |-> 1, Y |-> 1 }.
By our assumption, we know that
cequiv (X ::= APlus (AId X) (ANum 1); Y ::= AId X)
(X ::= APlus (AId X) (ANum 1); Y ::= APlus (AId X) (ANum 1))
so, by the definition of cequiv, we have
(X ::= APlus (AId X) (ANum 1); Y ::= APlus (AId X) (ANum 1))
/ empty_state  st1.
But we can also derive
(X ::= APlus (AId X) (ANum 1); Y ::= APlus (AId X) (ANum 1))
/ empty_state  st2,
where st2 = { X |-> 1, Y |-> 2 }. Note that st1 <> st2; this is a contradiction, since ceval is deterministic!

Theorem subst_inequiv :
~ subst_equiv_property.
Proof.
unfold subst_equiv_property.
intros Contra.

(* Here is the counterexample: assuming that subst_equiv_property
holds allows us to prove that these two programs are
equivalent... *)

remember (X ::= APlus (AId X) (ANum 1);
Y ::= AId X)
as c1.
remember (X ::= APlus (AId X) (ANum 1);
Y ::= APlus (AId X) (ANum 1))
as c2.
assert (cequiv c1 c2) by (subst; apply Contra).

(* ... allows us to show that the command c2 can terminate
in two different final states:
st1 = {X |-> 1, Y |-> 1}
st2 = {X |-> 1, Y |-> 2}. *)

remember (update (update empty_state X 1) Y 1) as st1.
remember (update (update empty_state X 1) Y 2) as st2.
assert (H1: c1 / empty_state st1);
assert (H2: c2 / empty_state st2);
try (subst;
apply E_Seq with (st' := (update empty_state X 1));
apply E_Ass; reflexivity).
apply H in H1.

(* Finally, we use the fact that evaluation is deterministic

assert (Hcontra: st1 = st2)
by (apply (ceval_deterministic c2 empty_state); assumption).
assert (Hcontra': st1 Y = st2 Y)
by (rewrite Hcontra; reflexivity).
subst. inversion Hcontra'. Qed.

#### Exercise: 4 stars, optional (better_subst_equiv)

The equivalence we had in mind above was not complete nonsense — it was actually almost right. To make it correct, we just need to exclude the case where the variable X occurs in the right-hand-side of the first assignment statement.

Inductive var_not_used_in_aexp (X:id) : aexp Prop :=
| VNUNum: n, var_not_used_in_aexp X (ANum n)
| VNUId: Y, X <> Y var_not_used_in_aexp X (AId Y)
| VNUPlus: a1 a2,
var_not_used_in_aexp X a1
var_not_used_in_aexp X a2
var_not_used_in_aexp X (APlus a1 a2)
| VNUMinus: a1 a2,
var_not_used_in_aexp X a1
var_not_used_in_aexp X a2
var_not_used_in_aexp X (AMinus a1 a2)
| VNUMult: a1 a2,
var_not_used_in_aexp X a1
var_not_used_in_aexp X a2
var_not_used_in_aexp X (AMult a1 a2).

Lemma aeval_weakening : i st a ni,
var_not_used_in_aexp i a
aeval (update st i ni) a = aeval st a.
Proof.
(* FILL IN HERE *) Admitted.

Using var_not_used_in_aexp, formalize and prove a correct verson of subst_equiv_property.

(* FILL IN HERE *)

#### Exercise: 3 stars, recommended (inequiv_exercise)

Theorem inequiv_exercise:
~ cequiv (WHILE BTrue DO SKIP END) SKIP.
Proof.
(* FILL IN HERE *) Admitted.

# Extended exercise: Non-deterministic Imp

As we have seen (in theorem ceval_deterministic in the Imp chapter), Imp's evaluation relation is deterministic. However, non-determinism is an important part of the definition of many real programming languages. For example, in many imperative languages (such as C and its relatives), the order in which function arguments are evaluated is unspecified. The program fragment
x = 0;
f(++xx);
might call f with arguments (1, 0) or (1, 1), depending how the compiler chooses to order things. This can be a little confusing for programmers, but it gives the compiler writer useful freedom.
In this exercise, we will extend Imp with a simple non-deterministic command and study how this change affects program equivalence. The new command has the syntax HAVOC X, where X is an identifier. The effect of executing HAVOC X is to assign an arbitrary number to the variable X, non-deterministically. For example, after executing the program:
HAVOC Y;
Z ::= Y * 2
the value of Y can be any number, while the value of Z is twice that of Y (so Z is always even). Note that we are not saying anything about the /probabilities/ of the outcomes — just that there are (infinitely) many different outcomes that can possibly happen after executing this non-deterministic code.
In a sense a variable on which we do HAVOC roughly corresponds to an unitialized variable in the C programming language. After the HAVOC the variable holds a fixed but arbitrary number. Most sources of nondeterminism in language definitions are there precisely because programmers don't care which choice is made (and so it is good to leave it open to the compiler to choose whichever will run faster).
We call this new language Himp (``Imp extended with HAVOC'').

Module Himp.

To formalize the language, we first add a clause to the definition of commands.

Inductive com : Type :=
| CSkip : com
| CAss : id aexp com
| CSeq : com com com
| CIf : bexp com com com
| CWhile : bexp com com
| CHavoc : id com. (* <---- new *)

Tactic Notation "com_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "SKIP" | Case_aux c "::=" | Case_aux c ";"
| Case_aux c "IFB" | Case_aux c "WHILE" | Case_aux c "HAVOC" ].

Notation "'SKIP'" :=
CSkip.
Notation "X '::=' a" :=
(CAss X a) (at level 60).
Notation "c1 ; c2" :=
(CSeq c1 c2) (at level 80, right associativity).
Notation "'WHILE' b 'DO' c 'END'" :=
(CWhile b c) (at level 80, right associativity).
Notation "'IFB' e1 'THEN' e2 'ELSE' e3 'FI'" :=
(CIf e1 e2 e3) (at level 80, right associativity).
Notation "'HAVOC' l" := (CHavoc l) (at level 60).

#### Exercise: 2 stars (himp_ceval)

Now, we must extend the operational semantics. We have provided a template for the ceval relation below, specifying the big-step semantics. What rule(s) must be added to the definition of ceval to formalize the behavior of the HAVOC command?

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

Inductive ceval : com state state Prop :=
| E_Skip : st : state, SKIP / st st
| E_Ass : (st : state) (a1 : aexp) (n : nat) (X : id),
aeval st a1 = n (X ::= a1) / st update st X n
| E_Seq : (c1 c2 : com) (st st' st'' : state),
c1 / st st' c2 / st' st'' (c1 ; c2) / st st''
| E_IfTrue : (st st' : state) (b1 : bexp) (c1 c2 : com),
beval st b1 = true
c1 / st st' (IFB b1 THEN c1 ELSE c2 FI) / st st'
| E_IfFalse : (st st' : state) (b1 : bexp) (c1 c2 : com),
beval st b1 = false
c2 / st st' (IFB b1 THEN c1 ELSE c2 FI) / st st'
| E_WhileEnd : (b1 : bexp) (st : state) (c1 : com),
beval st b1 = false (WHILE b1 DO c1 END) / st st
| E_WhileLoop : (st st' st'' : state) (b1 : bexp) (c1 : com),
beval st b1 = true
c1 / st st'
(WHILE b1 DO c1 END) / st' st''
(WHILE b1 DO c1 END) / st st''
(* FILL IN HERE *)

where "c1 '/' st '' st'" := (ceval c1 st st').

Tactic Notation "ceval_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"
(* FILL IN HERE *)
].

As a sanity check, the following claims should be provable for your definition:

Example havoc_example1 : (HAVOC X) / empty_state update empty_state X 0.
Proof.
(* FILL IN HERE *) Admitted.

Example havoc_example2 :
(SKIP; HAVOC Z) / empty_state update empty_state Z 42.
Proof.
(* FILL IN HERE *) Admitted.
Finally, we repeat the definition of command equivalence from above:

Definition cequiv (c1 c2 : com) : Prop := st st' : state,
c1 / st st' c2 / st st'.

This definition still makes perfect sense in the case of always terminating programs, so let's apply it to prove some non-deterministic programs equivalent or non-equivalent.

#### Exercise: 3 stars (havoc_swap)

Are the following two programs equivalent?

Definition pXY :=
HAVOC X; HAVOC Y.

Definition pYX :=
HAVOC Y; HAVOC X.

If you think they are equivalent, prove it. If you think they are not, prove that.

Theorem pXY_cequiv_pYX :
cequiv pXY pYX ~cequiv pXY pYX.
Proof. (* FILL IN HERE *) Admitted.

#### Exercise: 4 stars (havoc_copy)

Are the following two programs equivalent?

Definition ptwice :=
HAVOC X; HAVOC Y.

Definition pcopy :=
HAVOC X; Y ::= AId X.

If you think they are equivalent, then prove it. If you think they are not, then prove that. (Hint: You may find the assert tactic useful.)

Theorem ptwice_cequiv_pcopy :
cequiv ptwice pcopy ~cequiv ptwice pcopy.
Proof. (* FILL IN HERE *) Admitted.
The definition of program equivalence we are using here has some subtle consequences on programs that may loop forever. What cequiv says is that the set of possible terminating outcomes of two equivalent programs is the same. However, in a language with non-determinism, like Himp, some programs always terminate, some programs always diverge, and some programs can non-deterministically terminate in some runs and diverge in others. The final part of the following optional exercise illustrates this phenomenon.

#### Exercise: 5 stars, optional (havoc_diverge)

Prove the following program equivalences and non-equivalences, and try to understand why the cequiv definition has the behavior it has on these examples.

Definition p1 : com :=
WHILE (BNot (BEq (AId X) (ANum 0))) DO
HAVOC Y;
X ::= APlus (AId X) (ANum 1)
END.

Definition p2 : com :=
WHILE (BNot (BEq (AId X) (ANum 0))) DO
SKIP
END.

Theorem p1_p2_equiv : cequiv p1 p2.
Proof. (* FILL IN HERE *) Admitted.

Definition p3 : com :=
Z ::= ANum 1;
WHILE (BNot (BEq (AId X) (ANum 0))) DO
HAVOC X;
HAVOC Z
END.

Definition p4 : com :=
X ::= (ANum 0);
Z ::= (ANum 1).

Theorem p3_p4_inequiv : ~ cequiv p3 p4.
Proof. (* FILL IN HERE *) Admitted.

Definition p5 : com :=
WHILE (BNot (BEq (AId X) (ANum 1))) DO
HAVOC X
END.

Definition p6 : com :=
X ::= ANum 1.

Theorem p5_p6_equiv : cequiv p5 p6.
Proof. (* FILL IN HERE *) Admitted.

End Himp.

# Doing Without Extensionality (Optional)

Purists might object to using the functional_extensionality axiom. In general, it can be quite dangerous to add axioms, particularly several at once (as they may be mutually inconsistent). In fact, functional_extensionality and excluded_middle can both be assumed without any problems, but some Coq users prefer to avoid such "heavyweight" general techniques, and instead craft solutions for specific problems that stay within Coq's standard logic.
For our particular problem here, rather than extending the definition of equality to do what we want on functions representing states, we could instead give an explicit notion of equivalence on states. For example:

Definition stequiv (st1 st2 : state) : Prop :=
(X:id), st1 X = st2 X.

Notation "st1 '~' st2" := (stequiv st1 st2) (at level 30).

It is easy to prove that stequiv is an equivalence (i.e., it is reflexive, symmetric, and transitive), so it partitions the set of all states into equivalence classes.

#### Exercise: 1 star, optional (stequiv_refl)

Lemma stequiv_refl : (st : state),
st ~ st.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 1 star, optional (stequiv_sym)

Lemma stequiv_sym : (st1 st2 : state),
st1 ~ st2
st2 ~ st1.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 1 star, optional (stequiv_trans)

Lemma stequiv_trans : (st1 st2 st3 : state),
st1 ~ st2
st2 ~ st3
st1 ~ st3.
Proof.
(* FILL IN HERE *) Admitted.
Another useful fact...

#### Exercise: 1 star, optional (stequiv_update)

Lemma stequiv_update : (st1 st2 : state),
st1 ~ st2
(X:id) (n:nat),
update st1 X n ~ update st2 X n.
Proof.
(* FILL IN HERE *) Admitted.
It is then straightforward to show that aeval and beval behave uniformly on all members of an equivalence class:

#### Exercise: 2 stars, optional (stequiv_aeval)

Lemma stequiv_aeval : (st1 st2 : state),
st1 ~ st2
(a:aexp), aeval st1 a = aeval st2 a.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 2 stars, optional (stequiv_beval)

Lemma stequiv_beval : (st1 st2 : state),
st1 ~ st2
(b:bexp), beval st1 b = beval st2 b.
Proof.
(* FILL IN HERE *) Admitted.
We can also characterize the behavior of ceval on equivalent states (this result is a bit more complicated to write down because ceval is a relation).

Lemma stequiv_ceval: (st1 st2 : state),
st1 ~ st2
(c: com) (st1': state),
(c / st1 st1')
st2' : state,
((c / st2 st2') st1' ~ st2').
Proof.
intros st1 st2 STEQV c st1' CEV1. generalize dependent st2.
induction CEV1; intros st2 STEQV.
Case "SKIP".
st2. split.
constructor.
assumption.
Case ":=".
(update st2 X n). split.
constructor. rewrite H. symmetry. apply stequiv_aeval.
assumption. apply stequiv_update. assumption.
Case ";".
destruct (IHCEV1_1 st2 STEQV) as [st2' [P1 EQV1]].
destruct (IHCEV1_2 st2' EQV1) as [st2'' [P2 EQV2]].
st2''. split.
apply E_Seq with st2'; assumption.
assumption.
Case "IfTrue".
destruct (IHCEV1 st2 STEQV) as [st2' [P EQV]].
st2'. split.
apply E_IfTrue. rewrite H. symmetry. apply stequiv_beval.
assumption. assumption. assumption.
Case "IfFalse".
destruct (IHCEV1 st2 STEQV) as [st2' [P EQV]].
st2'. split.
apply E_IfFalse. rewrite H. symmetry. apply stequiv_beval.
assumption. assumption. assumption.
Case "WhileEnd".
st2. split.
apply E_WhileEnd. rewrite H. symmetry. apply stequiv_beval.
assumption. assumption.
Case "WhileLoop".
destruct (IHCEV1_1 st2 STEQV) as [st2' [P1 EQV1]].
destruct (IHCEV1_2 st2' EQV1) as [st2'' [P2 EQV2]].
st2''. split.
apply E_WhileLoop with st2'. rewrite H. symmetry.
apply stequiv_beval. assumption. assumption. assumption.
assumption.
Qed.

Now we need to redefine cequiv to use ~ instead of =. It is not completely trivial to do this in a way that keeps the definition simple and symmetric, but here is one approach (thanks to Andrew McCreight). We first define a looser variant of that "folds in" the notion of equivalence.

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

Inductive ceval' : com state state Prop :=
| E_equiv : c st st' st'',
c / st st'
st' ~ st''
c / st ' st''
where "c1 '/' st ''' st'" := (ceval' c1 st st').

Now the revised definition of cequiv' looks familiar:

Definition cequiv' (c1 c2 : com) : Prop :=
(st st' : state),
(c1 / st ' st') (c2 / st ' st').

A sanity check shows that the original notion of command equivalence is at least as strong as this new one. (The converse is not true, naturally.)

Lemma cequiv__cequiv' : (c1 c2: com),
cequiv c1 c2 cequiv' c1 c2.
Proof.
unfold cequiv, cequiv'; split; intros.
inversion H0 ; subst. apply E_equiv with st'0.
apply (H st st'0); assumption. assumption.
inversion H0 ; subst. apply E_equiv with st'0.
apply (H st st'0). assumption. assumption.
Qed.

#### Exercise: 2 stars, optional (identity_assignment')

Finally, here is our example once more... (You can complete the proof.)

Example identity_assignment' :
cequiv' SKIP (X ::= AId X).
Proof.
unfold cequiv'. intros. split; intros.
Case "".
inversion H; subst; clear H. inversion H0; subst.
apply E_equiv with (update st'0 X (st'0 X)).
constructor. reflexivity. apply stequiv_trans with st'0.
unfold stequiv. intros. apply update_same.
reflexivity. assumption.
Case "".
(* FILL IN HERE *) Admitted.
On the whole, this explicit equivalence approach is considerably harder to work with than relying on functional extensionality. (Coq does have an advanced mechanism called "setoids" that makes working with equivalences somewhat easier, by allowing them to be registered with the system so that standard rewriting tactics work for them almost as well as for equalities.) But it is worth knowing about, because it applies even in situations where the equivalence in question is not over functions. For example, if we chose to represent state mappings as binary search trees, we would need to use an explicit equivalence of this kind.

#### Exercise: 4 stars, optional (for_while_equiv)

This exercise extends the optional add_for_loop exercise from Imp.v, where you were asked to extend the language of commands with C-style for loops. Prove that the command:
for (c1 ; b ; c2) {
c3
}
is equivalent to:
c1 ;
WHILE b DO
c3 ;
c2
END
(* FILL IN HERE *)

#### Exercise: 3 stars, optional (swap_noninterfering_assignments)

Theorem swap_noninterfering_assignments: l1 l2 a1 a2,
l1 <> l2
var_not_used_in_aexp l1 a2
var_not_used_in_aexp l2 a1
cequiv
(l1 ::= a1; l2 ::= a2)
(l2 ::= a2; l1 ::= a1).
Proof.
(* Hint: You'll need functional_extensionality *)
(* FILL IN HERE *) Admitted.