# ImpSimple Imperative Programs

(* \$Date: 2012-09-15 09:25:36 -0400 (Sat, 15 Sep 2012) \$ *)

In this chapter, we begin a new direction that will continue for the rest of the course. Up to now we've been mostly studying Coq itself, but from now on we'll mostly be using Coq to formalize other things.
Our first case study is a simple imperative programming language called Imp. Here is a familiar mathematical function written in Imp.
Z ::= X;
Y ::= 1;
WHILE not (Z = 0) DO
Y ::= Y * Z;
Z ::= Z - 1
END
This chapter looks at how to define the syntax and semantics of Imp; the chapters that follow develop a theory of program equivalence and introduce Hoare Logic, the best-known logic for reasoning about imperative programs.

### Sflib

A minor technical point: Instead of asking Coq to import our earlier definitions from chapter Logic, we import a small library called Sflib.v, containing just a few definitions and theorems from earlier chapters that we'll actually use in the rest of the course. This change should be nearly invisible, since most of what's missing from Sflib has identical definitions in the Coq standard library. The main reason for doing it is to tidy the global Coq environment so that, for example, it is easier to search for relevant theorems.

Require Export SfLib.

# Arithmetic and Boolean Expressions

We'll present Imp in three parts: first a core language of arithmetic and boolean expressions, then an extension of these expressions with variables, and finally a language of commands including assignment, conditions, sequencing, and loops.

## Syntax

Module AExp.

These two definitions specify the abstract syntax of arithmetic and boolean expressions.

Inductive aexp : Type :=
| ANum : nat aexp
| APlus : aexp aexp aexp
| AMinus : aexp aexp aexp
| AMult : aexp aexp aexp.

Inductive bexp : Type :=
| BTrue : bexp
| BFalse : bexp
| BEq : aexp aexp bexp
| BLe : aexp aexp bexp
| BNot : bexp bexp
| BAnd : bexp bexp bexp.

In this chapter, we'll elide the translation from the concrete syntax that a programmer would actually write to these abstract syntax trees — the process that, for example, would translate the string "1+2*3" to the AST APlus (ANum 1) (AMult (ANum 2) (ANum 3)). The optional chapter ImpParser develops a simple implementation of a lexical analyzer and parser that can perform this translation. You do not need to understand that file to understand this one, but if you haven't taken a course where these techniques are covered (e.g., a compilers course) you may want to skim it.
For comparison, here's a conventional BNF (Backus-Naur Form) grammar defining the same abstract syntax:
aexp ::= nat
| aexp '+' aexp
| aexp '-' aexp
| aexp '*' aexp

bexp ::= true
| false
| aexp '=' aexp
| aexp '<=' aexp
| bexp 'and' bexp
| 'not' bexp
Compared to the Coq version above...
• The BNF is more informal — for example, it gives some suggestions about the surface syntax of expressions (like the fact that the addition operation is written + and is an infix symbol) while leaving other aspects of lexical analysis and parsing (like the relative precedence of +, -, and *) unspecified. Some additional information — and human intelligence — would be required to turn this description into a formal definition (when implementing a compiler, for example).
The Coq version consistently omits all this information and concentrates on the abstract syntax only.
• On the other hand, the BNF version is lighter and easier to read. Its informality makes it flexible, which is a huge advantage in situations like discussions at the blackboard, where conveying general ideas is more important than getting every detail nailed down precisely.
Indeed, there are dozens of BNF-like notations and people switch freely among them, usually without bothering to say which form of BNF they're using because there is no need to: a rough-and-ready informal understanding is all that's needed.
It's good to be comfortable with both sorts of notations: informal ones for communicating between humans and formal ones for carrying out implementations and proofs.

## Evaluation

Evaluating an arithmetic expression produces a number.

Fixpoint aeval (e : aexp) : nat :=
match e with
| ANum n => n
| APlus a1 a2 => (aeval a1) + (aeval a2)
| AMinus a1 a2 => (aeval a1) - (aeval a2)
| AMult a1 a2 => (aeval a1) * (aeval a2)
end.

Example test_aeval1:
aeval (APlus (ANum 2) (ANum 2)) = 4.
Proof. reflexivity. Qed.

Similarly, evaluating a boolean expression yields a boolean.

Fixpoint beval (e : bexp) : bool :=
match e with
| BTrue => true
| BFalse => false
| BEq a1 a2 => beq_nat (aeval a1) (aeval a2)
| BLe a1 a2 => ble_nat (aeval a1) (aeval a2)
| BNot b1 => negb (beval b1)
| BAnd b1 b2 => andb (beval b1) (beval b2)
end.

## Optimization

We haven't defined very much yet, but we can already get some mileage out of the definitions. Suppose we define a function that takes an arithmetic expression and slightly simplifies it, changing every occurrence of 0+e (i.e., (APlus (ANum 0) e) into just e.

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.

To make sure our optimization is doing the right thing we can test it on some examples and see if the output looks OK.

Example test_optimize_0plus:
optimize_0plus (APlus (ANum 2)
(APlus (ANum 0)
(APlus (ANum 0) (ANum 1))))
= APlus (ANum 2) (ANum 1).
Proof. reflexivity. Qed.

But if we want to be sure the optimization is correct — i.e., that evaluating an optimized expression gives the same result as the original — we should prove it.

Theorem optimize_0plus_sound: e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e. induction e.
Case "ANum". reflexivity.
Case "APlus". destruct e1.
SCase "e1 = ANum n". destruct n.
SSCase "n = 0". simpl. apply IHe2.
SSCase "n <> 0". simpl. rewrite IHe2. reflexivity.
SCase "e1 = APlus e1_1 e1_2".
simpl. simpl in IHe1. rewrite IHe1.
rewrite IHe2. reflexivity.
SCase "e1 = AMinus e1_1 e1_2".
simpl. simpl in IHe1. rewrite IHe1.
rewrite IHe2. reflexivity.
SCase "e1 = AMult e1_1 e1_2".
simpl. simpl in IHe1. rewrite IHe1.
rewrite IHe2. reflexivity.
Case "AMinus".
simpl. rewrite IHe1. rewrite IHe2. reflexivity.
Case "AMult".
simpl. rewrite IHe1. rewrite IHe2. reflexivity. Qed.

# Coq Automation

The repetition in this last proof is starting to be a little annoying. If either the language of arithmetic expressions or the optimization being proved sound were significantly more complex, it would begin to be a real problem.
So far, we've been doing all our proofs using just a small handful of Coq's tactics and completely ignoring its powerful facilities for constructing parts of proofs automatically. This section introduces some of these facilities, and we will see more over the next several chapters. Getting used to them will take some energy — Coq's automation is a power tool — but it will allow us to scale up our efforts to more complex definitions and more interesting properties without becoming overwhelmed by boring, repetitive, low-level details.

## Tacticals

Tacticals is Coq's term for tactics that take other tactics as arguments — "higher-order tactics," if you will.

### The repeat Tactical

The repeat tactical takes another tactic and keeps applying this tactic until the tactic fails. Here is an example showing that 100 is even using repeat.

Theorem ev100 : ev 100.
Proof.
repeat (apply ev_SS). (* applies ev_SS 50 times,
then apply ev_SS fails *)

apply ev_0.
Qed.
(* Print ev100. *)

The repeat T tactic never fails; if the tactic T doesn't apply to the original goal, then repeat still succeeds without changing the original goal (it repeats zero times).

Theorem ev100' : ev 100.
Proof.
repeat (apply ev_0). (* doesn't fail, applies ev_0 zero times *)
repeat (apply ev_SS). apply ev_0. (* we can continue the proof *)
Qed.

The repeat T tactic does not have any bound on the number of times it applies T. If T is a tactic that always succeeds then repeat T will loop forever (e.g. repeat simpl loops forever since simpl always succeeds). While Coq's term language is guaranteed to terminate, Coq's tactic language is not.

### The try Tactical

A similar tactical is try: If T is a tactic, then try T is a tactic that is just like T except that, if T fails, try T successfully does nothing at all (instead of failing).

Theorem silly1 : ae, aeval ae = aeval ae.
Proof. try reflexivity. (* this just does reflexivity *) Qed.

Theorem silly2 : (P : Prop), P P.
Proof.
intros P HP.
try reflexivity. (* just reflexivity would have failed *)
apply HP. (* we can still finish the proof in some other way *)
Qed.

Using try in a completely manual proof is a bit silly, but we'll see below that try is very useful for doing automated proofs in conjunction with the ; tactical.

### The ; Tactical (Simple Form)

In its simplest and most commonly used form, the ; tactical takes 2 tactics as argument: T;T' first performs the tactic T and then performs the tactic T' on each subgoal generated by T.
For example, consider the following trivial lemma:

Lemma foo : n, ble_nat 0 n = true.
Proof.
intros.
destruct n.
(* Leaves two subgoals...  *)
Case "n=0". simpl. reflexivity.
Case "n=Sn'". simpl. reflexivity.
(* ... which are discharged similarly *)
Qed.

We can simplify this proof using the ; tactical:

Lemma foo' : n, ble_nat 0 n = true.
Proof.
intros.
destruct n; (* destruct the current goal *)
simpl; (* then simpl each resulting subgoal *)
reflexivity. (* then do reflexivity on each resulting subgoal *)
Qed.

Using try and ; together, we can get rid of the repetition in the proof that was bothering us a little while ago.

Theorem optimize_0plus_sound': e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e.
induction e;
(* Most cases follow directly by the IH *)
try (simpl; rewrite IHe1; rewrite IHe2; reflexivity).
(* The remaining cases -- ANum and APlus -- are more
interesting... *)

Case "ANum". reflexivity.
Case "APlus".
destruct e1;
(* Again, most cases follow directly by the IH *)
try (simpl; simpl in IHe1; rewrite IHe1;
rewrite IHe2; reflexivity).
(* The interesting case, on which the try... does nothing,
is when e1 = ANum n. In this case, we have to destruct
n (to see whether the optimization applies) and rewrite
with the induction hypothesis. *)

SCase "e1 = ANum n". destruct n;
simpl; rewrite IHe2; reflexivity. Qed.

Coq experts often use this "...; try... " idiom after a tactic like induction to take care of many similar cases all at once. Naturally, this practice has an analog in informal proofs. Here is an informal proof of this theorem that matches the structure of the formal one:
Theorem: For all arithmetic expressions e,
aeval (optimize_0plus e) = aeval e.
Proof: By induction on e. The AMinus and AMult cases follow directly from the IH. The remaining cases are as follows:
• Suppose e = ANum n for some n. We must show
aeval (optimize_0plus (ANum n)) = aeval (ANum n).
This is immediate from the definition of optimize_0plus.
• Suppose e = APlus e1 e2 for some e1 and e2. We must show
aeval (optimize_0plus (APlus e1 e2))
aeval (APlus e1 e2).
Consider the possible forms of e1. For most of them, optimize_0plus simply calls itself recursively for the subexpressions and rebuilds a new expression of the same form as e1; in these cases, the result follows directly from the IH.
The interesting case is when e1 = ANum n for some n. If n = ANum 0, then
optimize_0plus (APlus e1 e2) = optimize_0plus e2
and the IH for e2 is exactly what we need. On the other hand, if n = S n' for some n', then again optimize_0plus simply calls itself recursively, and the result follows from the IH.
This proof can still be improved: the first case (for e = ANum n) is very trivial — even more trivial than the cases that we said simply followed from the IH — yet we have chosen to write it out in full. It would be better and clearer to drop it and just say, at the top, "Most cases are either immediate or direct from the IH. The only interesting case is the one for APlus..." We can make the same improvement in our formal proof too. Here's how it looks:

Theorem optimize_0plus_sound'': e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e.
induction e;
(* Most cases follow directly by the IH *)
try (simpl; rewrite IHe1; rewrite IHe2; reflexivity);
(* ... or are immediate by definition *)
try reflexivity.
(* The interesting case is when e = APlus e1 e2. *)
Case "APlus".
destruct e1; try (simpl; simpl in IHe1; rewrite IHe1;
rewrite IHe2; reflexivity).
SCase "e1 = ANum n". destruct n;
simpl; rewrite IHe2; reflexivity. Qed.

### The ; Tactical (General Form)

The ; tactical has a more general than the simple T;T' we've seen above, and which is sometimes also useful. If T, T1, ..., Tn are tactics, then
T; [T1 | T2 | ... | Tn
is a tactic that first performs T and then performs T1 on the first subgoal generated by T, performs T2 on the second subgoal, etc.
So T;T' is just special notation for the case when all of the Ti's are the same tactic; i.e. T;T' is just a shorthand for:
T; [T' | T' | ... | T'
The form T;T' is used most often in practice.

## Defining New Tactic Notations

Coq also provides several ways of "programming" tactic scripts.
• The Tactic Notation idiom illustrated below gives a handy way to define "shorthand tactics" that bundle several tactics into a single command.
• For more sophisticated programming, Coq offers a small built-in programming language called Ltac with primitives that can examine and modify the proof state. The details are a bit too complicated to get into here (and it is generally agreed that Ltac is not the most beautiful part of Coq's design!), but they can be found in the reference manual, and there are many examples of Ltac definitions in the Coq standard library that you can use as examples.
• There is also an OCaml API, which can be used to build tactics that access Coq's internal structures at a lower level, but this is seldom worth the trouble for ordinary Coq users.
The Tactic Notation mechanism is the easiest to come to grips with, and it offers plenty of power for many purposes. Here's an example.

Tactic Notation "simpl_and_try" tactic(c) :=
simpl;
try c.

This defines a new tactical called simpl_and_try which takes one tactic c as an argument, and is defined to be equivalent to the tactic simpl; try c. For example, writing "simpl_and_try reflexivity." in a proof would be the same as writing "simpl; try reflexivity."
The next subsection gives a more sophisticated use of this feature...

### Bulletproofing Case Analyses

Being able to deal with most of the cases of an induction or destruct all at the same time is very convenient, but it can also be a little confusing. One problem that often comes up is that maintaining proofs written in this style can be difficult. For example, suppose that, later, we extended the definition of aexp with another constructor that also required a special argument. The above proof might break because Coq generated the subgoals for this constructor before the one for APlus, so that, at the point when we start working on the APlus case, Coq is actually expecting the argument for a completely different constructor. What we'd like is to get a sensible error message saying "I was expecting the AFoo case at this point, but the proof script is talking about APlus." Here's a nice trick (due to Aaron Bohannon) that smoothly achieves this.

Tactic Notation "aexp_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "ANum" | Case_aux c "APlus"
| Case_aux c "AMinus" | Case_aux c "AMult" ].

(Case_aux implements the common functionality of Case, SCase, SSCase, etc. For example, Case "foo" is defined as Case_aux Case "foo".)
For example, if e is a variable of type aexp, then doing
aexp_cases (induction eCase
will perform an induction on e (the same as if we had just typed induction e) and also add a Case tag to each subgoal generated by the induction, labeling which constructor it comes from. For example, here is yet another proof of optimize_0plus_sound, using aexp_cases:

Theorem optimize_0plus_sound''': e,
aeval (optimize_0plus e) = aeval e.
Proof.
intros e.
aexp_cases (induction e) Case;
try (simpl; rewrite IHe1; rewrite IHe2; reflexivity);
try reflexivity.
(* At this point, there is already an "APlus" case name
in the context.  The Case "APlus" here in the proof
text has the effect of a sanity check: if the "Case"
string in the context is anything _other_ than "APlus"
(for example, because we added a clause to the definition
of aexp and forgot to change the proof) we'll get a
helpful error at this point telling us that this is now
the wrong case. *)

Case "APlus".
aexp_cases (destruct e1) SCase;
try (simpl; simpl in IHe1; rewrite IHe1; rewrite IHe2; reflexivity).
SCase "ANum". destruct n;
simpl; rewrite IHe2; reflexivity. Qed.

#### Exercise: 3 stars (optimize_0plus_b)

Since the optimize_0plus tranformation doesn't change the value of aexps, we should be able to apply it to all the aexps that appear in a bexp without changing the bexp's value. Write a function which performs that transformation on bexps, and prove it is sound. Use the tacticals we've just seen to make the proof as elegant as possible.

(* FILL IN HERE *)

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

Design exercise: The optimization implemented by our optimize_0plus function is only one of many imaginable optimizations on arithmetic and boolean expressions. Write a more sophisticated optimizer and prove it correct.
(* FILL IN HERE *)

## The omega Tactic

The omega tactic implements a decision procedure for a subset of first-order logic called Presburger arithmetic. It is based on the Omega algorithm invented in 1992 by William Pugh.
If the goal is a universally quantified formula made out of
• numeric constants, addition (+ and S), subtraction (- and pred), and multiplication by constants (this is what makes it Presburger arithmetic),
• equality (= and <>) and inequality (<=), and
• the logical connectives , , ~, and ,
then invoking omega will either solve the goal or tell you that it is actually false.

Example silly_presburger_example : m n o p,
m + n <= n + o o + 3 = p + 3
m <= p.
Proof.
intros. omega.
Qed.

Andrew Appel calls this the "Santa Claus tactic." We'll see examples of its use below.

## A Few More Handy Tactics

Finally, here are some miscellaneous tactics that you may find convenient.
• clear H: Delete hypothesis H from the context.
• subst x: Find an assumption x = e or e = x in the context, replace x with e throughout the context and current goal, and clear the assumption.
• subst: Substitute away all assumptions of the form x = e or e = x.
• rename... into...: Change the name of a hypothesis in the proof context. For example, if the context includes a variable named x, then rename x into y will change all occurrences of x to y.
• assumption: Try to find a hypothesis H in the context that exactly matches the goal; if one is found, behave just like apply H.
• contradiction: Try to find a hypothesis H in the current context that is logically equivalent to False. If one is found, solve the goal.
• constructor: Try to find a constructor c (from some Inductive definition in the current environment) that can be applied to solve the current goal. If one is found, behave like apply c.
We'll see many examples of these in the proofs below.

# Evaluation as a Relation

We have presented aeval and beval as functions defined by Fixpoints. Another way to think about evaluation, one that we will see is often more flexible, is as a relation between expressions and their values. This leads naturally to Inductive definitions like the following one for arithmetic expressions...

Module aevalR_first_try.

Inductive aevalR : aexp nat Prop :=
| E_ANum : (n: nat),
aevalR (ANum n) n
| E_APlus : (e1 e2: aexp) (n1 n2: nat),
aevalR e1 n1
aevalR e2 n2
aevalR (APlus e1 e2) (n1 + n2)
| E_AMinus: (e1 e2: aexp) (n1 n2: nat),
aevalR e1 n1
aevalR e2 n2
aevalR (AMinus e1 e2) (n1 - n2)
| E_AMult : (e1 e2: aexp) (n1 n2: nat),
aevalR e1 n1
aevalR e2 n2
aevalR (AMult e1 e2) (n1 * n2).

As is often the case with relations, we'll find it convenient to define infix notation for aevalR. We'll write e n to mean that arithmetic expression e evaluates to value n. (This notation is one place where the limitation to ASCII symbols becomes a little bothersome. The standard notation for the evaluation relation is a double down-arrow. We'll typeset it like this in the HTML version of the notes and use a double vertical bar as the closest approximation in .v files.)

Notation "e '' n" := (aevalR e n) : type_scope.

End aevalR_first_try.

In fact, Coq provides a way to use this notation in the definition of aevalR itself. This avoids situations where we're working on a proof involving statements in the form e n but we have to refer back to a definition written using the form aevalR e n.
We do this by first "reserving" the notation, then giving the definition together with a declaration of what the notation means.

Reserved Notation "e '' n" (at level 50, left associativity).

Inductive aevalR : aexp nat Prop :=
| E_ANum : (n:nat),
(ANum n) n
| E_APlus : (e1 e2: aexp) (n1 n2 : nat),
(e1 n1) (e2 n2) (APlus e1 e2) (n1 + n2)
| E_AMinus : (e1 e2: aexp) (n1 n2 : nat),
(e1 n1) (e2 n2) (AMinus e1 e2) (n1 - n2)
| E_AMult : (e1 e2: aexp) (n1 n2 : nat),
(e1 n1) (e2 n2) (AMult e1 e2) (n1 * n2)

where "e '' n" := (aevalR e n) : type_scope.

Tactic Notation "aevalR_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "E_ANum" | Case_aux c "E_APlus"
| Case_aux c "E_AMinus" | Case_aux c "E_AMult" ].

It is straightforward to prove that the relational and functional definitions of evaluation agree on all possible arithmetic expressions...

Theorem aeval_iff_aevalR : a n,
(a n) aeval a = n.
Proof.
split.
Case "".
intros H.
aevalR_cases (induction H) SCase; simpl.
SCase "E_ANum".
reflexivity.
SCase "E_APlus".
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
SCase "E_AMinus".
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
SCase "E_AMult".
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
Case "".
generalize dependent n.
aexp_cases (induction a) SCase;
simpl; intros; subst.
SCase "ANum".
apply E_ANum.
SCase "APlus".
apply E_APlus.
apply IHa1. reflexivity.
apply IHa2. reflexivity.
SCase "AMinus".
apply E_AMinus.
apply IHa1. reflexivity.
apply IHa2. reflexivity.
SCase "AMult".
apply E_AMult.
apply IHa1. reflexivity.
apply IHa2. reflexivity.
Qed.

We can make the proof quite a bit shorter by making more aggressive use of tacticals...

Theorem aeval_iff_aevalR' : a n,
(a n) aeval a = n.
Proof.
(* WORKED IN CLASS *)
split.
Case "".
intros H; induction H; subst; reflexivity.
Case "".
generalize dependent n.
induction a; simpl; intros; subst; constructor;
try apply IHa1; try apply IHa2; reflexivity.
Qed.

#### Exercise: 3 stars (bevalR)

Write a relation bevalR in the same style as aevalR, and prove that it is equivalent to beval.

(*
Inductive bevalR:
(* FILL IN HERE *)
*)

For the definitions of evaluation for arithmetic and boolean expressions, the choice of whether to use functional or relational definitions is mainly a matter of taste. In general, Coq has somewhat better support for working with relations. On the other hand, in some sense function definitions carry more information, because functions are necessarily deterministic and defined on all arguments; for a relation we have to show these properties explicitly if we need them. Functions also take advantage of Coq's computational mechanism.
However, there are circumstances where relational definitions of evaluation are greatly preferable to functional ones, as we'll see shortly.

## Inference Rule Notation

In informal discussions, it is convenient write the rules for aevalR and similar relations in the more readable "graphical" form of inference rules, where the premises above the line justify the conclusion below the line. For example, the constructor E_APlus...
| E_APlus :  (e1 e2aexp) (n1 n2nat),
aevalR e1 n1
aevalR e2 n2
aevalR (APlus e1 e2) (n1 + n2)
...would be written like this as an inference rule:
 e1 ⇓ n1 e2 ⇓ n2 (E_APlus) APlus e1 e2 ⇓ n1+n2
Formally, there is nothing very deep about inference rules: they are just implications. You can read the rule name on the right as the name of the constructor and read each of the linebreaks between the premises above the line and the line itself as . All the variables mentioned in the rule (e1, n1, etc.) are implicitly bound by universal quantifiers at the beginning. The whole collection of rules is understood as being wrapped in an Inductive declaration (informally, this is either elided or else indicated by saying something like "Let aevalR be the smallest relation closed under the following rules...").
For example, is the smallest relation closed under these rules:
 (E_ANum) ANum n ⇓ n
 e1 ⇓ n1 e2 ⇓ n2 (E_APlus) APlus e1 e2 ⇓ n1+n2
 e1 ⇓ n1 e2 ⇓ n2 (E_AMinus) AMinus e1 e2 ⇓ n1-n2
 e1 ⇓ n1 e2 ⇓ n2 (E_AMult) AMult e1 e2 ⇓ n1*n2

End AExp.

# Expressions With Variables

Let's turn our attention back to defining Imp. The next thing we need to do is to enrich our arithmetic and boolean expressions with variables. To keep things simple, we'll assume that all variables are global and that they only hold numbers.

## Identifiers

To begin, we'll need to formalize identifiers such as program variables. We could use strings for this — or, in a real compiler, fancier structures like pointers into a symbol table. But for simplicity let's just use natural numbers as identifiers.
(We hide this section in a module because these definitions are actually in SfLib, but we want to repeat them here so that we can explain them.)

Module Id.

We define a new inductive datatype Id so that we won't confuse identifiers and numbers.

Inductive id : Type :=
Id : nat id.

Definition beq_id X1 X2 :=
match (X1, X2) with
(Id n1, Id n2) => beq_nat n1 n2
end.

After we "wrap" numbers as identifiers in this way, it is convenient to recapitulate a few properties of numbers as analogous properties of identifiers, so that we can work with identifiers in definitions and proofs abstractly, without unwrapping them to expose the underlying numbers. Since all we need to know about identifiers is whether they are the same or different, just a few basic facts are all we need.

Theorem beq_id_refl : X,
true = beq_id X X.
Proof.
intros. destruct X.
apply beq_nat_refl. Qed.

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

For this and the following exercises, do not use induction, but rather apply similar results already proved for natural numbers. Some of the tactics mentioned above may prove useful.
Theorem beq_id_eq : i1 i2,
true = beq_id i1 i2 i1 = i2.
Proof.
(* FILL IN HERE *) Admitted.

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

Theorem beq_id_false_not_eq : i1 i2,
beq_id i1 i2 = false i1 <> i2.
Proof.
(* FILL IN HERE *) Admitted.

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

Theorem not_eq_beq_id_false : i1 i2,
i1 <> i2 beq_id i1 i2 = false.
Proof.
(* FILL IN HERE *) Admitted.

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

Theorem beq_id_sym: i1 i2,
beq_id i1 i2 = beq_id i2 i1.
Proof.
(* FILL IN HERE *) Admitted.

End Id.

## States

A state represents the current values of all the variables at some point in the execution of a program. For simplicity (to avoid dealing with partial functions), we let the state be defined for all variables, even though any given program is only going to mention a finite number of them.

Definition state := id nat.

Definition empty_state : state :=
fun _ => 0.

Definition update (st : state) (X:id) (n : nat) : state :=
fun X' => if beq_id X X' then n else st X'.

For proofs involving states, we'll need several simple properties of update.

#### Exercise: 1 star (update_eq)

Theorem update_eq : n X st,
(update st X n) X = n.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 1 star (update_neq)

Theorem update_neq : X2 X1 n st,
beq_id X2 X1 = false
(update st X2 n) X1 = (st X1).
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 1 star (update_example)

Before starting to play with tactics, make sure you understand exactly what the theorem is saying!

Theorem update_example : (n:nat),
(update empty_state (Id 2) n) (Id 3) = 0.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 1 star, recommended (update_shadow)

Theorem update_shadow : x1 x2 k1 k2 (f : state),
(update (update f k2 x1) k2 x2) k1 = (update f k2 x2) k1.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 2 stars (update_same)

Theorem update_same : x1 k1 k2 (f : state),
f k1 = x1
(update f k1 x1) k2 = f k2.
Proof.
(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars (update_permute)

Theorem update_permute : x1 x2 k1 k2 k3 f,
beq_id k2 k1 = false
(update (update f k2 x1) k1 x2) k3 = (update (update f k1 x2) k2 x1) k3.
Proof.
(* FILL IN HERE *) Admitted.

## Syntax

We can add variables to the arithmetic expressions we had before by simply adding one more constructor:

Inductive aexp : Type :=
| ANum : nat aexp
| AId : id aexp (* <----- NEW *)
| APlus : aexp aexp aexp
| AMinus : aexp aexp aexp
| AMult : aexp aexp aexp.

Tactic Notation "aexp_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "ANum" | Case_aux c "AId" | Case_aux c "APlus"
| Case_aux c "AMinus" | Case_aux c "AMult" ].

Defining a few variable names as notational shorthands will make examples easier to read:

Definition X : id := Id 0.
Definition Y : id := Id 1.
Definition Z : id := Id 2.

(This convention for naming program variables (X, Y, Z) clashes a bit with our earlier use of uppercase letters for types. Since we're not using polymorphism heavily in this part of the course, this overloading should not cause confusion.)
The definition of bexps is the same as before (using the new aexps):

Inductive bexp : Type :=
| BTrue : bexp
| BFalse : bexp
| BEq : aexp aexp bexp
| BLe : aexp aexp bexp
| BNot : bexp bexp
| BAnd : bexp bexp bexp.

Tactic Notation "bexp_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "BTrue" | Case_aux c "BFalse" | Case_aux c "BEq"
| Case_aux c "BLe" | Case_aux c "BNot" | Case_aux c "BAnd" ].

## Evaluation

The arith and boolean evaluators can be extended to handle variables in the obvious way:

Fixpoint aeval (st : state) (e : aexp) : nat :=
match e with
| ANum n => n
| AId X => st X (* <----- NEW *)
| APlus a1 a2 => (aeval st a1) + (aeval st a2)
| AMinus a1 a2 => (aeval st a1) - (aeval st a2)
| AMult a1 a2 => (aeval st a1) * (aeval st a2)
end.

Fixpoint beval (st : state) (e : bexp) : bool :=
match e with
| BTrue => true
| BFalse => false
| BEq a1 a2 => beq_nat (aeval st a1) (aeval st a2)
| BLe a1 a2 => ble_nat (aeval st a1) (aeval st a2)
| BNot b1 => negb (beval st b1)
| BAnd b1 b2 => andb (beval st b1) (beval st b2)
end.

Example aexp1 :
aeval (update empty_state X 5)
(APlus (ANum 3) (AMult (AId X) (ANum 2)))
= 13.
Proof. reflexivity. Qed.

Example bexp1 :
beval (update empty_state X 5)
(BAnd BTrue (BNot (BLe (AId X) (ANum 4))))
= true.
Proof. reflexivity. Qed.

# Commands

Now we are ready define the syntax and behavior of Imp commands (or statements).

## Syntax

Informally, commands are described by the following BNF grammar:
com ::= 'SKIP'
| X '::=' aexp
| com ';' com
| 'WHILE' bexp 'DO' com 'END'
| 'IFB' bexp 'THEN' com 'ELSE' com 'FI'
For example, here's the factorial function in Imp.
Z ::= X;
Y ::= 1;
WHILE not (Z = 0) DO
Y ::= Y * Z;
Z ::= Z - 1
END
When this command terminates, the variable Y will contain the factorial of the initial value of X.
Here is the formal definition of the syntax of commands:

Inductive com : Type :=
| CSkip : com
| CAss : id aexp com
| CSeq : com com com
| CIf : bexp com com com
| CWhile : bexp com com.

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" ].

As usual, we can use a few Notation declarations to make things more readable. We need to be a bit careful to avoid conflicts with Coq's built-in notations, so we'll keep this light — in particular, we won't introduce any notations for aexps and bexps to avoid confusion with the numerical and boolean operators we've already defined. We use the keyword IFB for conditionals instead of IF, for similar reasons.

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).

For example, here is the factorial function again, written as a formal definition to Coq:

Definition fact_in_coq : com :=
Z ::= AId X;
Y ::= ANum 1;
WHILE BNot (BEq (AId Z) (ANum 0)) DO
Y ::= AMult (AId Y) (AId Z);
Z ::= AMinus (AId Z) (ANum 1)
END.

## Examples

Assignment:

Definition plus2 : com :=
X ::= (APlus (AId X) (ANum 2)).

Definition XtimesYinZ : com :=
Z ::= (AMult (AId X) (AId Y)).

Definition subtract_slowly_body : com :=
Z ::= AMinus (AId Z) (ANum 1) ;
X ::= AMinus (AId X) (ANum 1).

Loops:

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

Definition subtract_3_from_5_slowly : com :=
X ::= ANum 3 ;
Z ::= ANum 5 ;
subtract_slowly.

An infinite loop:

Definition loop : com :=
WHILE BTrue DO
SKIP
END.

Factorial again (broken up into smaller pieces this time, for convenience when we come back to proving things about it later).

Definition fact_body : com :=
Y ::= AMult (AId Y) (AId Z) ;
Z ::= AMinus (AId Z) (ANum 1).

Definition fact_loop : com :=
WHILE BNot (BEq (AId Z) (ANum 0)) DO
fact_body
END.

Definition fact_com : com :=
Z ::= AId X ;
Y ::= ANum 1 ;
fact_loop.

# Evaluation

Next we need to define what it means to evaluate an Imp command. The fact that WHILE loops don't necessarily terminate makes defining an evaluation function tricky ...

## Evaluation Function (Failed Attempt)

Here's an attempt at defining an evaluation function for commands, omitting the WHILE case.

Fixpoint ceval_fun_no_while (st : state) (c : com) : state :=
match c with
| SKIP =>
st
| l ::= a1 =>
update st l (aeval st a1)
| c1 ; c2 =>
let st' := ceval_fun_no_while st c1 in
ceval_fun_no_while st' c2
| IFB b THEN c1 ELSE c2 FI =>
if (beval st b)
then ceval_fun_no_while st c1
else ceval_fun_no_while st c2
| WHILE b1 DO c1 END =>
st (* bogus *)
end.

In a traditional functional programming language like ML or Haskell we could write the WHILE case as follows:
```Fixpoint ceval_fun (st : state) (c : com) : state :=
match c with
...
| WHILE b1 DO c1 END =>
if (beval st b1)
then ceval_fun st (c1; WHILE b1 DO c1 END)
else st
end.
```
Coq doesn't accept such a definition (Error: Cannot guess decreasing argument of fix) because the function we want to define is not guaranteed to terminate. Indeed, the full version of the ceval_fun function applied to the loop program above would never terminate. Since Coq is not just a functional programming language, but also a consistent logic, any potentially non-terminating function needs to be rejected. Here is an invalid(!) Coq program showing what would go wrong if Coq allowed non-terminating recursive functions:
```     Fixpoint loop_false (n : nat) : False := loop_false n.
```
That is, propositions like False would become provable (e.g. loop_false 0 would be a proof of False), which would be a disaster for Coq's logical consistency.
Thus, because it doesn't terminate on all inputs, the full version of ceval_fun cannot be written in Coq — at least not without additional tricks, which make everything much more complicated (see chapter ImpCEvalFun if curious).

## Evaluation as a Relation

Here's a better way: we define ceval as a relation rather than a function — i.e., we define it in Prop instead of Type, as we did for aevalR and bevalR above.
This is an important change. Besides freeing us from the hacks that would be needed to define an evaluation function, it gives us a lot more flexibility in the definition. For example, if we added concurrency features to the language, we'd want the definition of evaluation to be non-deterministic — i.e., not only would it not be total, it would not even be a partial function!
We'll use the notation c / st st' for our ceval relation, that is c / st st' means that executing program c in a starting state st results in an ending state st'. This can be pronounced "c takes state st to st'".
 (E_Skip) SKIP / st ⇓ st
 aeval st a1 = n (E_Ass) X := a1 / st ⇓ (update st X n)
 c1 / st ⇓ st' c2 / st' ⇓ st'' (E_Seq) c1;c2 / st ⇓ st''
 beval st b1 = true c1 / st ⇓ st' (E_IfTrue) IF b1 THEN c1 ELSE c2 FI / st ⇓ st'
 beval st b1 = false c2 / st ⇓ st' (E_IfFalse) IF b1 THEN c1 ELSE c2 FI / st ⇓ st'
 beval st b1 = false (E_WhileEnd) WHILE b1 DO c1 END / st ⇓ st
 beval st b1 = true c1 / st ⇓ st' WHILE b1 DO c1 END / st' ⇓ st'' (E_WhileLoop) WHILE b1 DO c1 END / st ⇓ st''
Here is the formal definition. (Make sure you understand how it corresponds to the inference rules.)

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

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

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" ].

The cost of defining evaluation as a relation instead of a function is that we now need to construct proofs that some program evaluates to some result state, rather than just letting Coq's computation mechanism do it for us.

Example ceval_example1:
(X ::= ANum 2;
IFB BLe (AId X) (ANum 1)
THEN Y ::= ANum 3
ELSE Z ::= ANum 4
FI)
/ empty_state
(update (update empty_state X 2) Z 4).
Proof.
(* We must supply the intermediate state *)
apply E_Seq with (update empty_state X 2).
Case "assignment command".
apply E_Ass. reflexivity.
Case "if command".
apply E_IfFalse.
reflexivity.
apply E_Ass. reflexivity. Qed.

#### Exercise: 2 stars (ceval_example2)

Example ceval_example2:
(X ::= ANum 0; Y ::= ANum 1; Z ::= ANum 2) / empty_state
(update (update (update empty_state X 0) Y 1) Z 2).
Proof.
(* FILL IN HERE *) Admitted.

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

Write an Imp program that sums the numbers from 1 to X (inclusive: 1 + 2 + ... + X) in the variable Y. Prove that this program executes as intended for X = 2 (this latter part is trickier than you might expect).

Definition pup_to_n : com :=
(* FILL IN HERE *) admit.

Theorem pup_to_2_ceval :
pup_to_n / (update empty_state X 2)
update (update (update (update (update (update empty_state
X 2) Y 0) Y 2) X 1) Y 3) X 0.
Proof.
(* FILL IN HERE *) Admitted.

## Determinism of Evaluation

Changing from a computational to a relational definition of evaluation is a good move because it allows us to escape from the artificial requirement (imposed by Coq's restrictions on Fixpoint definitions) that evaluation should be a total function. But it also raises a question: Is the second definition of evaluation actually a partial function? That is, is it possible that, beginning from the same state st, we could evaluate some command c in different ways to reach two different output states st' and st''?
In fact, this cannot happen: ceval is a partial function. Here's the proof:

Theorem ceval_deterministic: c st st1 st2,
c / st st1
c / st st2
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2.
ceval_cases (induction E1) Case;
intros st2 E2; inversion E2; subst.
Case "E_Skip". reflexivity.
Case "E_Ass". reflexivity.
Case "E_Seq".
assert (st' = st'0) as EQ1.
SCase "Proof of assertion". apply IHE1_1; assumption.
subst st'0.
apply IHE1_2. assumption.
Case "E_IfTrue".
SCase "b1 evaluates to true".
apply IHE1. assumption.
SCase "b1 evaluates to false (contradiction)".
rewrite H in H5. inversion H5.
Case "E_IfFalse".
SCase "b1 evaluates to true (contradiction)".
rewrite H in H5. inversion H5.
SCase "b1 evaluates to false".
apply IHE1. assumption.
Case "E_WhileEnd".
SCase "b1 evaluates to true".
reflexivity.
SCase "b1 evaluates to false (contradiction)".
rewrite H in H2. inversion H2.
Case "E_WhileLoop".
SCase "b1 evaluates to true (contradiction)".
rewrite H in H4. inversion H4.
SCase "b1 evaluates to false".
assert (st' = st'0) as EQ1.
SSCase "Proof of assertion". apply IHE1_1; assumption.
subst st'0.
apply IHE1_2. assumption. Qed.

We'll get much deeper into systematic techniques for reasoning about Imp programs in the following chapters, but we can do quite a bit just working with the bare definitions. This section explores some examples.

## Basic Examples

Theorem plus2_spec : st n st',
st X = n
plus2 / st st'
st' X = n + 2.
Proof.
intros st n st' HX Heval.
(* Inverting Heval essentially forces Coq to expand one
step of the ceval computation - in this case revealing
that st' must be st extended with the new value of X,
since plus2 is an assignment *)

inversion Heval. subst. clear Heval. simpl.
apply update_eq. Qed.

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

State and prove a specification of XtimesYinZ.

(* FILL IN HERE *)

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

Theorem loop_never_stops : st st',
~(loop / st st').
Proof.
intros st st' contra. unfold loop in contra.
remember (WHILE BTrue DO SKIP END) as loopdef.
(* Proceed by induction on the assumed derivation showing that
loopdef terminates.  Most of the cases are immediately
contradictory (and so can be solved in one step with
inversion). *)

(* FILL IN HERE *) Admitted.

Fixpoint no_whiles (c : com) : bool :=
match c with
| SKIP => true
| _ ::= _ => true
| c1 ; c2 => andb (no_whiles c1) (no_whiles c2)
| IFB _ THEN ct ELSE cf FI => andb (no_whiles ct) (no_whiles cf)
| WHILE _ DO _ END => false
end.

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

The no_whiles property yields true on just those programs that have no while loops. Using Inductive, write a property no_whilesR such that no_whilesR c is provable exactly when c is a program with no while loops. Then prove its equivalence with no_whiles.

Inductive no_whilesR: com Prop :=
(* FILL IN HERE *)
.

Theorem no_whiles_eqv:
c, no_whiles c = true no_whilesR c.
Proof.
(* FILL IN HERE *) Admitted.

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

Imp programs that don't involve while loops always terminate. State and prove a theorem that says this. (Use either no_whiles or no_whilesR, as you prefer.)

(* FILL IN HERE *)

## Proving a Program Correct (Optional)

Recall the factorial program:

Print fact_body. Print fact_loop. Print fact_com.

Here is an alternative "mathematical" definition of the factorial function:

Fixpoint real_fact (n:nat) : nat :=
match n with
| O => 1
| S n' => n * (real_fact n')
end.

We would like to show that they agree — if we start fact_com in a state where variable X contains some number x, then it will terminate in a state where variable Y contains the factorial of x.
To show this, we rely on the critical idea of a loop invariant.

Definition fact_invariant (x:nat) (st:state) :=
(st Y) * (real_fact (st Z)) = real_fact x.

Theorem fact_body_preserves_invariant: st st' x,
fact_invariant x st
st Z <> 0
fact_body / st st'
fact_invariant x st'.
Proof.
unfold fact_invariant, fact_body.
intros st st' x Hm HZnz He.
inversion He; subst; clear He.
inversion H1; subst; clear H1.
inversion H4; subst; clear H4.
unfold update. simpl.
(* Show that st Z = S z' for some z' *)
destruct (st Z) as [| z'].
apply ex_falso_quodlibet. apply HZnz. reflexivity.
rewrite Hm. rewrite mult_assoc.
replace (S z' - 1) with z' by omega.
reflexivity. Qed.

Theorem fact_loop_preserves_invariant : st st' x,
fact_invariant x st
fact_loop / st st'
fact_invariant x st'.
Proof.
intros st st' x H Hce.
remember fact_loop as c.
ceval_cases (induction Hce) Case;
inversion Heqc; subst; clear Heqc.
Case "E_WhileEnd".
(* trivial when the loop doesn't run... *)
assumption.
Case "E_WhileLoop".
(* if the loop does run, we know that fact_body preserves
fact_invariant -- we just need to assemble the pieces *)

apply IHHce2.
apply fact_body_preserves_invariant with st;
try assumption.
intros Contra. simpl in H0; subst.
rewrite Contra in H0. inversion H0.
reflexivity. Qed.

Theorem guard_false_after_loop: b c st st',
(WHILE b DO c END) / st st'
beval st' b = false.
Proof.
intros b c st st' Hce.
remember (WHILE b DO c END) as cloop.
ceval_cases (induction Hce) Case;
inversion Heqcloop; subst; clear Heqcloop.
Case "E_WhileEnd".
assumption.
Case "E_WhileLoop".
apply IHHce2. reflexivity. Qed.

Patching it all together...

Theorem fact_com_correct : st st' x,
st X = x
fact_com / st st'
st' Y = real_fact x.
Proof.
intros st st' x HX Hce.
inversion Hce; subst; clear Hce.
inversion H1; subst; clear H1.
inversion H4; subst; clear H4.
inversion H1; subst; clear H1.
rename st' into st''. simpl in H5.
(* The invariant is true before the loop runs... *)
remember (update (update st Z (st X)) Y 1) as st'.
assert (fact_invariant (st X) st').
subst. unfold fact_invariant, update. simpl. omega.
(* ...so when the loop is done running, the invariant
is maintained *)

assert (fact_invariant (st X) st'').
apply fact_loop_preserves_invariant with st'; assumption.
unfold fact_invariant in H0.
(* Finally, if the loop terminated, then Z is 0; so Y must be
factorial of X *)

apply guard_false_after_loop in H5. simpl in H5.
destruct (st'' Z).
Case "st'' Z = 0". simpl in H0. omega.
Case "st'' Z > 0 (impossible)". inversion H5.
Qed.

One might wonder whether all this work with poking at states and unfolding definitions could be ameliorated with some more powerful lemmas and/or more uniform reasoning principles... Indeed, this is exactly the topic of the upcoming Hoare chapter!

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

Prove a specification for subtract_slowly, using the above specification of fact_com and the invariant below as guides.

Definition ss_invariant (x:nat) (z:nat) (st:state) :=
minus (st Z) (st X) = minus z x.

(* FILL IN HERE *)

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

Add C-style for loops to the language of commands, update the ceval definition to define the semantics of for loops, and add cases for for loops as needed so that all the proofs in this file are accepted by Coq.
A for loop should be parameterized by (a) a statement executed initially, (b) a test that is run on each iteration of the loop to determine whether the loop should continue, (c) a statement executed at the end of each loop iteration, and (d) a statement that makes up the body of the loop. (You don't need to worry about making up a concrete Notation for for loops, but feel free to play with this too if you like.)

(* FILL IN HERE *)

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

Most modern programming languages use a "short-circuit" evaluation rule for boolean and: to evaluate BAnd b1 b2, first evaluate b1. If it evaluates to false, then the entire BAnd expression evaluates to false immediately, without evaluating b2. Otherwise, b2 is evaluated to determine the result of the BAnd expression.
Write an alternate version of beval that performs short-circuit evaluation of BAnd in this manner, and prove that it is equivalent to beval.

(* FILL IN HERE *)

#### Exercise: 4 stars, recommended (stack_compiler)

HP Calculators, programming languages like Forth and Postscript, and abstract machines like the Java Virtual Machine all evaluate arithmetic expressions using a stack. For instance, the expression
```   (2*3)+(3*(4-2))
```
would be entered as
```   2 3 * 3 4 2 - * +
```
and evaluated like this:
```  []            |    2 3 * 3 4 2 - * +
[2]           |    3 * 3 4 2 - * +
[3, 2]        |    * 3 4 2 - * +
[6]           |    3 4 2 - * +
[3, 6]        |    4 2 - * +
[4, 3, 6]     |    2 - * +
[2, 4, 3, 6]  |    - * +
[2, 3, 6]     |    * +
[6, 6]        |    +
[12]          |
```
The task of this exercise is to write a small compiler that translates aexps into stack machine instructions, and to prove its correctness.
The instruction set for our stack language will consist of the following instructions:
• SPush n: Push the number n on the stack.
• SLoad X: Load the identifier X from the store and push it on the stack
• SPlus: Pop the two top numbers from the stack, add them, and push the result onto the stack.
• SMinus: Similar, but subtract.
• SMult: Similar, but multiply.

Inductive sinstr : Type :=
| SPush : nat sinstr
| SPlus : sinstr
| SMinus : sinstr
| SMult : sinstr.

Write a function to evaluate programs in the stack language. It takes as input a state, a stack represented as a list of numbers (top stack item is the head of the list), and a program represented as a list of instructions, and returns the stack after executing the program. Test your function on the examples below.
Note that the specification leaves unspecified what to do when encountering an SPlus, SMinus, or SMult instruction if the stack contains less than two elements. In a sense it is immaterial, since our compiler will never emit such a malformed program. However, when you do the correctness proof you may find some choices makes the proof easier than others.

Fixpoint s_execute (st : state) (stack : list nat)
(prog : list sinstr)
: list nat :=
(* FILL IN HERE *) admit.

Example s_execute1 :
s_execute empty_state []
[SPush 5, SPush 3, SPush 1, SMinus]
= [2, 5].
(* FILL IN HERE *) Admitted.

Example s_execute2 :
s_execute (update empty_state X 3) [3,4]
[SPush 4, SLoad X, SMult, SPlus]
= [15, 4].
(* FILL IN HERE *) Admitted.

Next, write a function which compiles an aexp into a stack machine program. The effect of running the program should be the same as pushing the value of the expression on the stack.

Fixpoint s_compile (e : aexp) : list sinstr :=
(* FILL IN HERE *) admit.

(*
Example s_compile1 :
s_compile (AMinus (AId X) (AMult (ANum 2) (AId Y)))