In this chapter, we'll take a more serious look at how to use Coq to study interesting things outside of itself. 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. This chapter looks at how to define the syntax and semantics of Imp; further 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 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 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 chapter 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: 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 *, the use of parens to explicitly group subexpressions, etc.) unspecified. Some additional information (and human intelligence) would be required to turn this description into a formal definition, for example when implementing a compiler.
  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, 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 form 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 (instead of 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, 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 = ANum 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 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:

The repeat Tactical

The repeat tactical takes another tactic and keeps applying this tactic until it fails. 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, then repeat T will loop forever (e.g., repeat simpl loops forever, since simpl always succeeds). 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, this simply means that we have failed to construct a proof, not that we have constructed a wrong one.

Exercise: 3 stars (optimize_0plus_b)

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 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.

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 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 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 in 1991 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 inequality (≤), and
• the logical connectives ∧, ∨, ¬, and →,

then invoking omega will either solve the goal or tell you that it is actually false.

(Note the Require Import Coq.omega.Omega. 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: 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 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 examples 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...

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.

In fact, Coq provides a way to use this notation in the definition of aevalR itself. This reduces confusion by avoiding 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.

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... ...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

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 (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 by considering also a division operation:

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 straightforward.

Suppose, instead, 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 distribution of results, but 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? The examples above show that relational definitions are fundamentally more powerful 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 choice. But what about when both styles are workable? One point in favor of relational definitions is that some people feel they are 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 show 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" 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

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.

States

Since we'll want to look variables up to find out their current values, we'll reuse the type id from the Maps chapter for the type of 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 identifiers to nat. 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 devoped to Imp, this overloading should not cause confusion.) The definition of bexps is unchanged (except for using the new aexps):

Evaluation

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

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. (We choose this slightly awkward concrete syntax for the sake of being able to define Imp syntax using Coq's Notation mechanism. In particular, we use IFB to avoid conflicting with the if notation from the standard library.) For example, here's factorial in Imp: When this command terminates, the variable Y will contain the factorial of the initial value of X. Here is the formal definition of the abstract syntax of commands:

As usual, we can use a few Notation declarations to make things more readable. To avoid conflicts with Coq's built-in notations, we keep this light — in particular, we don't introduce any notations for aexps and bexps to avoid confusion with the numeric and boolean operators we've already defined.

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

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.

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 c / st \\ st' for the ceval relation: 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'".

Operational Semantics

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

	(E_Skip)
SKIP / st \\ st

aeval st a1 = n	(E_Ass)
x := a1 / st \\ (t_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 b = false	(E_WhileFalse)
WHILE b DO c END / st \\ st

beval st b = true
c / st \\ st'
WHILE b DO c END / st' \\ st''	(E_WhileTrue)
WHILE b DO c END / st \\ 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 (ceval_example2)

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 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.

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, recommendedM (XtimesYinZ_spec)

State and prove a specification of XtimesYinZ.

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).

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.

Additional Exercises

Exercise: 3 stars (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 (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 task 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.
• 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 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 what we do, since our compiler will never emit such a malformed program.

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: 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.

Next, we need to define the behavior of BREAK. Informally, whenever BREAK is executed in a sequence of commands, it stops the execution of that sequence and signals that the innermost enclosing loop should terminate. (If there aren't any enclosing loops, then the whole program simply terminates.) The final state should be the same as the one in which the BREAK statement was executed. One important point is what to do when there are multiple loops enclosing a given BREAK. In those cases, BREAK should only terminate the innermost loop. Thus, after executing the following... ... the value of X should be 1, and not 0. One way of expressing this behavior is to add another parameter to the evaluation relation that specifies whether evaluation of a command executes a BREAK statement:

Intuitively, c / st \\ s / st' means that, if c is started in state st, then it terminates in state st' and either signals that the innermost surrounding loop (or the whole program) should exit immediately (s = SBreak) or that execution should continue normally (s = SContinue). The definition of the "c / st \\ s / st'" relation is very similar to the one we gave above for the regular evaluation relation (c / st \\ st') — we just need to handle the termination signals appropriately:

• If the command is SKIP, then the state doesn't change and execution of any enclosing loop can continue normally.
• If the command is BREAK, the state stays unchanged but we signal a SBreak.
• If the command is an assignment, then we update the binding for that variable in the state accordingly and signal that execution can continue normally.
• If the command is of the form IFB b THEN c₁ ELSE c₂ FI, then the state is updated as in the original semantics of Imp, except that we also propagate the signal from the execution of whichever branch was taken.
• If the command is a sequence c₁ ;; c₂, we first execute c₁. If this yields a SBreak, we skip the execution of c₂ and propagate the SBreak signal to the surrounding context; the resulting state is the same as the one obtained by executing c₁ alone. Otherwise, we execute c₂ on the state obtained after executing c₁, and propagate the signal generated there.
• Finally, for a loop of the form WHILE b DO c END, the semantics is almost the same as before. The only difference is that, when b evaluates to true, we execute c and check the signal that it raises. If that signal is SContinue, then the execution proceeds as in the original semantics. Otherwise, we stop the execution of the loop, and the resulting state is the same as the one resulting from the execution of the current iteration. In either case, since BREAK only terminates the innermost loop, WHILE signals SContinue.

Based on the above description, complete the definition of the ceval relation.

Now prove the following properties of your definition of ceval:

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.)