In this chapter, we take a more serious look at how to use Coq to study other things. Our case study is a simple imperative programming language called Imp, embodying a tiny core fragment of conventional mainstream languages such as C and Java. Here is a familiar mathematical function written in Imp.
       Z := X;
       Y := 1;
       while ~(Z = 0) do
         Y := Y × Z;
         Z := Z - 1
       end

We concentrate here on defining the syntax and semantics of Imp; later chapters in Programming Language Foundations (Software Foundations, volume 2) develop a theory of program equivalence and introduce Hoare Logic, a widely used logic for reasoning about imperative programs.

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

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

In this chapter, we'll mostly 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 lexical analyzer and parser that can perform this translation. You do not need to understand that chapter to understand this one, but if you haven't already 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:
    a := nat
        | a + a
        | a - a
        | a × a

    b := true
        | false
        | a = a
        | a ≤ a
        | ¬b
        | b && b

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 with an infix +) while leaving other aspects of lexical analysis and parsing (like the relative precedence of +, -, and ×, the use of parens to group subexpressions, etc.) unspecified. Some additional information -- and human intelligence -- would be required to turn this description into a formal definition, e.g., for implementing a compiler.
  The Coq version consistently omits all this information and concentrates on the abstract syntax only.
• Conversely, the BNF version is lighter and easier to read. Its informality makes it flexible, a big 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 kind of BNF they're using because there is no need to: a rough-and-ready informal understanding is all that's important.

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.

Similarly, evaluating a boolean expression yields a boolean.

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.

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

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.

Coq Automation

The amount of repetition in this last proof is a little annoying. And if either the language of arithmetic expressions or the optimization being proved sound were significantly more complex, it would start 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 try Tactical

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 (rather than failing).

There is no real reason to use try in completely manual proofs like these, but it is very useful for doing automated proofs in conjunction with the ; tactical, which we show next.

The ; Tactical (Simple Form)

In its most common form, the ; tactical takes two tactics as arguments. The compound tactic T;T' first performs T and then performs T' on each subgoal generated by T. For example, consider the following trivial lemma:

We can simplify this proof using the ; tactical:

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

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. For example, here is an informal proof of the optimization theorem that matches the structure of the formal one: Theorem: For all arithmetic expressions a,
       aeval (optimize_0plus a) = aeval a.

Proof: By induction on a. Most cases follow directly from the IH. The remaining cases are as follows:

• Suppose a = 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 a = APlus a₁ a₂ for some a₁ and a₂. We must show
            aeval (optimize_0plus (APlus a₁ a₂)) = aeval (APlus a₁ a₂).
  Consider the possible forms of a₁. For most of them, optimize_0plus simply calls itself recursively for the subexpressions and rebuilds a new expression of the same form as a₁; in these cases, the result follows directly from the IH.
  The interesting case is when a₁ = ANum n for some n. If n = 0, then
            optimize_0plus (APlus a₁ a₂) = optimize_0plus a₂
  and the IH for a₂ 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. ☐

However, this proof can still be improved: the first case (for a = 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:

The ; Tactical (General Form)

The ; tactical also has a more general form than the simple T;T' we've seen above. 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 shorthand for:
      T; [T' | T' | ... | T']

The repeat Tactical

The repeat tactical takes another tactic and keeps applying this tactic until it fails to make progress. Here is an example showing that 10 is in a long list using repeat.

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

The tactic repeat T also does not have any upper bound on the number of times it applies T. If T is a tactic that always succeeds (and makes progress), then repeat T will loop forever.

While evaluation in Coq's term language, Gallina, is guaranteed to terminate, tactic evaluation is not! This does not affect Coq's logical consistency, however, since the job of repeat and other tactics is to guide Coq in constructing proofs; if the construction process diverges (i.e., it does not terminate), this simply means that we have failed to construct a proof, not that we have constructed a wrong one.

Exercise: 3 stars, standard (optimize_0plus_b_sound)

Since the optimize_0plus transformation 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 that performs this transformation on bexps and prove it is sound. Use the tacticals we've just seen to make the proof as elegant as possible.

Exercise: 4 stars, standard, optional (optimize)

Design exercise: The optimization implemented by our optimize_0plus function is only one of many possible optimizations on arithmetic and boolean expressions. Write a more sophisticated optimizer and prove it correct. (You will probably find it easiest to start small -- add just a single, simple optimization and its correctness proof -- and build up to something more interesting incrementially.)

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 built-in 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 other books on Coq, 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.

This defines a new tactical called simpl_and_try that takes one tactic c as an argument and is defined to be equivalent to the tactic simpl; try c. Now writing "simpl_and_try reflexivity." in a proof will be the same as writing "simpl; try reflexivity."

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 by William Pugh [Pugh 1991]. 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 ordering (≤), and
• the logical connectives ∧, ∨, ¬, and →,

then invoking omega will either solve the goal or fail, meaning that the goal is actually false. (If the goal is not of this form, omega will also fail.)

(Note the From Coq Require Import Lia. at the top of the file.)

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: For a variable 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 (where x is a variable).
• 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, solve the goal.
• 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 examples of all of these as we go along.

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...

This style gives us more control over the names that Coq chooses during proofs involving aevalR, at the cost of making the definition a little more verbose.

It will be convenient to have an infix notation for aevalR. We'll write e ==> n to mean that arithmetic expression e evaluates to value n.

As we saw in IndProp in our case study of regular expressions, Coq provides a way to use this notation in the definition of aevalR itself.

Inference Rule Notation

In informal discussions, it is convenient to 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 (we have already seen them in the IndProp chapter). For example, the constructor E_APlus...
      | E_APlus : ∀ (e₁ e₂: aexp) (n₁ n₂: nat),
          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 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 (as well as the line itself) as →. All the variables mentioned in the rule (e₁, n₁, etc.) are implicitly bound by universal quantifiers at the beginning. (Such variables are often called metavariables to distinguish them from the variables of the language we are defining. At the moment, our arithmetic expressions don't include variables, but we'll soon be adding them.) The whole collection of rules is understood as being wrapped in an Inductive declaration. In informal prose, 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

Exercise: 1 star, standard, optional (beval_rules)

Here, again, is the Coq definition of the beval function:
  Fixpoint beval (e : bexp) : bool :=
    match e with
    | BTrue ⇒ true
    | BFalse ⇒ false
    | BEq a₁ a₂ ⇒ (aeval a₁) =? (aeval a₂)
    | BLe a₁ a₂ ⇒ (aeval a₁) <=? (aeval a₂)
    | BNot b ⇒ negb (beval b)
    | BAnd b₁ b₂ ⇒ andb (beval b₁) (beval b₂)
    end.

Write out a corresponding definition of boolean evaluation as a relation (in inference rule notation).

Equivalence of the Definitions

It is straightforward to prove that the relational and functional definitions of evaluation agree:

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

Exercise: 3 stars, standard (bevalR)

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

Computational vs. Relational Definitions

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: either way works. However, there are circumstances where relational definitions of evaluation work much better than functional ones.

For example, suppose that we wanted to extend the arithmetic operations with division:

Extending the definition of aeval to handle this new operation would not be straightforward (what should we return as the result of ADiv (ANum 5) (ANum 0)?). But extending aevalR is very easy.

Notice that the evaluation relation has now become partial: There are some inputs for which it simply does not specify an output.

Or suppose that we want to extend the arithmetic operations by a nondeterministic number generator any that, when evaluated, may yield any number. Note that this is not the same as making a probabilistic choice among all possible numbers -- we're not specifying any particular probability distribution for the results, just saying what results are possible.

Again, extending aeval would be tricky, since now evaluation is not a deterministic function from expressions to numbers, but extending aevalR is no problem...

At this point you maybe wondering: which style should I use by default? In the examples we've just seen, relational definitions turned out to be more useful than functional ones. For situations like these, where the thing being defined is not easy to express as a function, or indeed where it is not a function, there is no real choice. But what about when both styles are workable? One point in favor of relational definitions is that they can be more elegant and easier to understand. Another is that Coq automatically generates nice inversion and induction principles from Inductive definitions. On the other hand, functional definitions can often be more convenient:

• Functions are by definition deterministic and defined on all arguments; for a relation we have to prove these properties explicitly if we need them.
• With functions we can also take advantage of Coq's computation mechanism to simplify expressions during proofs.

Furthermore, functions can be directly "extracted" from Gallina to executable code in OCaml or Haskell. Ultimately, the choice often comes down to either the specifics of a particular situation or simply a question of taste. Indeed, in large Coq developments it is common to see a definition given in both functional and relational styles, plus a lemma stating that the two coincide, allowing further proofs to switch from one point of view to the other at will.

Expressions With Variables

Now we return 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.

States

Since we'll want to look variables up to find out their current values, we'll reuse maps from the Maps chapter, and strings will be used to represent variables in Imp. A machine state (or just state) represents the current values of all variables at some point in the execution of a program. For simplicity, we assume that the state is defined for all variables, even though any given program is only going to mention a finite number of them. The state captures all of the information stored in memory. For Imp programs, because each variable stores a natural number, we can represent the state as a mapping from strings to nat, and will use 0 as default value in the store. For more complex programming languages, the state might have more structure.

Syntax

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

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

(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 the chapters developed to Imp, this overloading should not cause confusion.) The definition of bexps is unchanged (except that it now refers to the new aexps):

Notations

To make Imp programs easier to read and write, we introduce some notations and implicit coercions. You do not need to understand exactly what these declarations do. Briefly, though:

• The Coercion declaration stipulates that a function (or constructor) can be implicitly used by the type system to coerce a value of the input type to a value of the output type. For instance, the coercion declaration for AId allows us to use plain strings when an aexp is expected; the string will implicitly be wrapped with AId.
• Declare Custom Entry com tells Coq to create a new "custom grammar" for parsing Imp expressions and programs. The first notation declaration after this tells Coq that anything between <{ and }> should be parsed using the Imp grammar. Again, it is not necessary to understand the details, but it is important to recognize that we are defining new interpretations for some familiar operators like +, -, ×, =, ≤, etc., when they occur between <{ and }>.

We can now write 3 + (X × 2) instead of APlus 3 (AMult X 2), and true && ~(X ≤ 4) instead of BAnd true (BNot (BLe X 4)).

One downside of these and notation tricks -- coercions in particular -- is that they can make it a little harder for humans to calculate the types of expressions. If you ever find yourself confused, try doing Set Printing Coercions to see exactly what is going on.

Evaluation

The arith and boolean evaluators are extended to handle variables in the obvious way, taking a state as an extra argument:

We specialize our notation for total maps to the specific case of states, i.e. using (_ !-> 0) as empty state.

Now we can add a notation for a "singleton state" with just one variable bound to a value.

Commands

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

Syntax

Informally, commands c are described by the following BNF grammar.
     c := skip | x := a | c ; c | if b then c else c end
         | while b do c end

Here is the formal definition of the abstract syntax of commands:

As for expressions, we can use a few Notation declarations to make reading and writing Imp programs more convenient.

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

Desugaring notations

Coq offers a rich set of features to manage the increasing complexity of the objects we work with, such as coercions and notations. However, their heavy usage can make for quite overwhelming syntax. It is often instructive to "turn off" those features to get a more elementary picture of things, using the following commands:

• Unset Printing Notations (undo with Set Printing Notations)
• Set Printing Coercions (undo with Unset Printing Coercions)
• Set Printing All (undo with Unset Printing All)

These commands can also be used in the middle of a proof, to elaborate the current goal and context.

The Locate command

Finding notations

When faced with unknown notation, use Locate with a string containing one of its symbols to see its possible interpretations.

Finding identifiers

When used with an identifier, the command Locate prints the full path to every value in scope with the same name. This is useful to troubleshoot problems due to variable shadowing.

More Examples

Assignment:

Loops

An infinite loop:

Evaluating Commands

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 as a Function (Failed Attempt)

Here's an attempt at defining an evaluation function for commands, omitting the while case. The following declaration is needed to be able to use the notations in match patterns.

In a traditional functional programming language like OCaml or Haskell we could add the while case as follows:

        Fixpoint ceval_fun (st : state) (c : com) : state :=
          match c with
            ...
            | while b do c end =>
                if (beval st b)
                  then ceval_fun st (c ; while b do c 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, it doesn't always terminate: for example, 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!) 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 (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, of ceval_fun cannot be written in Coq -- at least not without additional tricks and workarounds (see chapter ImpCEvalFun if you're curious about what those might be).

Evaluation as a Relation

Here's a better way: define ceval as a relation rather than a function -- i.e., define it in Prop instead of Type, as we did for aevalR above. This is an important change. Besides freeing us from awkward workarounds, it gives us a lot more flexibility in the definition. For example, if we add nondeterministic features like any to the language, we want the definition of evaluation to be nondeterministic -- i.e., not only will it not be total, it will not even be a function! We'll use the notation st =[ c ]=> st' for the ceval relation: st =[ c ]=> 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'".

Operational Semantics

Here is an informal definition of evaluation, presented as inference rules for readability:

	(E_Skip)
st =[ skip ]=> st

aeval st a = n	(E_Ass)
st =[ x := a ]=> (x !-> n ; st)

st =[ c1 ]=> st'
st' =[ c2 ]=> st''	(E_Seq)
st =[ c1;c2 ]=> st''

beval st b = true
st =[ c1 ]=> st'	(E_IfTrue)
st =[ if b then c1 else c2 end ]=> st'

beval st b = false
st =[ c2 ]=> st'	(E_IfFalse)
st =[ if b then c1 else c2 end ]=> st'

beval st b = false	(E_WhileFalse)
st =[ while b do c end ]=> st

beval st b = true
st =[ c ]=> st'
st' =[ while b do c end ]=> st''	(E_WhileTrue)
st =[ while b do c end ]=> st''

Here is the formal definition. Make sure you understand how it corresponds to the inference rules.

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.

Exercise: 2 stars, standard (ceval_example2)

Exercise: 3 stars, standard, optional (pup_to_n)

Write an Imp program that sums the numbers from 1 to X (inclusive: 1 + 2 + ... + X) in the variable Y. Your program should update the state as shown in theorem pup_to_2_ceval, which you can reverse-engineer to discover the program you should write. The proof of that theorem will be somewhat lengthy.

Determinism of Evaluation

Changing from a computational to a relational definition of evaluation is a good move because it frees us from the artificial requirement that evaluation should be a total function. But it also raises a question: Is the second definition of evaluation really a partial function? Or 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:

Reasoning About Imp Programs

We'll get deeper into more systematic and powerful techniques for reasoning about Imp programs in Programming Language Foundations, but we can get some distance just working with the bare definitions. This section explores some examples.

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.

Exercise: 3 stars, standard, optional (XtimesYinZ_spec)

State and prove a specification of XtimesYinZ.

Exercise: 3 stars, standard, especially useful (loop_never_stops)

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 discriminate).

Exercise: 3 stars, standard (no_whiles_eqv)

Consider the following function:

This predicate yields true just on 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.

Exercise: 4 stars, standard (no_whiles_terminating)

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

Additional Exercises

Exercise: 3 stars, standard (stack_compiler)

Old 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 written as

      2 3 * 3 4 2 - * +

and evaluated like this (where we show the program being evaluated on the right and the contents of the stack on the left):

      [ ]           |    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 goal of this exercise is to write a small compiler that translates aexps into stack machine instructions. 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 the first number from the second.
• SMult: Similar, but multiply.

Write a function to evaluate programs in the stack language. It should take 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 it should return the stack after executing the program. Test your function on the examples below. Note that it is unspecified what to do when encountering an SPlus, SMinus, or SMult instruction if the stack contains fewer than two elements. In a sense, it is immaterial what we do, since a correct compiler will never emit such a malformed program. But for sake of later exercises, it would be best to skip the offending instruction and continue with the next one.

Next, write a function that 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.

After you've defined s_compile, prove the following to test that it works.

Exercise: 3 stars, standard (execute_app)

Execution can be decomposed in the following sense: executing stack program p₁ ++ p₂ is the same as executing p₁, taking the resulting stack, and executing p₂ from that stack. Prove that fact.

Exercise: 3 stars, standard (stack_compiler_correct)

Now we'll prove the correctness of the compiler implemented in the previous exercise. Begin by proving the following lemma. If it becomes difficult, consider whether your implementation of s_execute or s_compile could be simplified.

The main theorem should be a very easy corollary of that lemma.

Exercise: 3 stars, standard, 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. (N.b. This is only true because expression evaluation in Imp is rather simple. In a bigger language where evaluating an expression might diverge, the short-circuiting BAnd would not be equivalent to the original, since it would make more programs terminate.)

Exercise: 4 stars, advanced (break_imp)

Imperative languages like C and Java often include a break or similar statement for interrupting the execution of loops. In this exercise we consider how to add break to Imp. First, we need to enrich the language of commands with an additional case.