In this chapter, we take a more serious look at how to use Coq as a tool 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, in Programming Language Foundations (Software Foundations, volume 2), we develop a theory of program equivalence and introduce Hoare Logic, a popular 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 with variables, and finally a language of commands including assignment, conditionals, sequencing, and loops.

Syntax

Abstract syntax trees for arithmetic and boolean expressions:

Evaluation

Evaluating an arithmetic expression produces a number.

Similarly, evaluating a boolean expression yields a boolean.

Optimization

Coq Automation

That last proof was getting a little repetitive. Time to learn a few more Coq tricks...

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).

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:

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.

The repeat Tactical

The repeat tactical takes another tactic and keeps applying this tactic until it fails or until it succeeds but doesn't make any progress. Here is an example proving that 10 is in a long list using repeat.

repeat can loop forever.

Defining New Tactics

Coq also provides ways of "programming" tactic scripts:

• Ltac: scripting language for tactics (good for more sophisticated proof engineering)
• Tactic Notation for defining tactics with custom concrete syntax
• OCaml tactic scripting API (for wizards)

Ltac is all we need for present purposes. For example:

The lia Tactic

The lia tactic implements a decision procedure for integer linear arithmetic, a subset of propositional logic and arithmetic. 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 >), and
• the logical connectives ∧, ∨, ¬, and →,

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

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: Given 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 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 is often more flexible -- is as a relation between expressions and their values. This perspective leads to Inductive definitions like the following...

A standard notation for "evaluates to":

With infix notation:

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. 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) ...can be written like this as an inference rule:

e1 ==> n1
e2 ==> n2	(E_APlus)
APlus e1 e2 ==> n1+n2

There is nothing very deep going on here:

• a group of inference rules corresponds to a single Inductive definition
• each rule's name corresponds to a constructor name
• above the line are premises
• below the line is conclusion
• metavariables like e₁ and n₁ are implicitly universally quantified

For example, we could define ==> as 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:

Again, we can make the proof quite a bit shorter using some tacticals.

Computational vs. Relational Definitions

Sometimes relational definitions are the only reasonable option...

Adding division

What should aeval return for ADiv (ANum 1) (ANum 0)??

Adding division, relationally

Notice that this evaluation relation corresponds to a partial function: There are some inputs for which it does not specify an output.

Adding Nondeterminism

Another example: A nondeterministic number generator:

What should aeval do with nondeterminism??

Adding nondeterminism, relationally

Tradeoffs

Which is better, functional or relational definitions?

• Functional: take advantage of computation.
• Relational: more (easily) expressive.
• Best of both worlds: define both and prove them equivalent.

Expressions With Variables

Let's return to defining Imp, where 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 use total maps from the Maps chapter. A machine state (or just state) represents the current values of all variables at some point in the execution of a program.

Syntax

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

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

Notations

To make Imp programs easier to read and write, we introduce some notations and implicit coercions. (The details are a bit hideous, but also not important to understand.)

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)).

Evaluation

Now we need to add an st parameter to both evaluation functions:

We can use our notation for total maps in the specific case of states -- i.e., we write the empty state as (_ !-> 0).

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

Commands

Now we are ready to define the syntax and behavior of Imp commands (or 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 Formally:

As we did 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 Coq definition. When this command terminates, the variable Y will contain the factorial of the initial value of X.

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 (with a bogus while case).

Nontermination leads to Inconsistency

Consider the following "proof object":

        Fixpoint loop_false (n : nat) : False := loop_false n.

Accepting such a definition would be catastrophic, so Coq conservatively rejects all nonterminating (or potentially non-terminating, or not-obviously-terminating) programs.

Evaluation as a Relation

Here's a better way: define ceval as a relation rather than a function -- i.e., make its result a Prop rather than a state, similar to what we did for aevalR above. 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_Asgn)
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''

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

What sorts of things might we want to prove using these definitions? Here are some simple examples...

Determinism of Evaluation

Finally, we should pause to check that our evaluation relation really is a (partial) function...