Booleans

It's worth taking a look at some of the datatypes in Coq's standard library. Note that all of these are manually defined: None of them are primitive. Let's start with booleans, which should remind us of the coin type in the last chapter.

Although we are rolling our own booleans here for the sake of building up everything from scratch, Coq does, of course, provide a default implementation of the booleans, together with a multitude of useful functions and lemmas. (Take a look at Coq.Init.Datatypes in the Coq library documentation if you're interested.) Whenever possible, we'll name our own definitions and theorems so that they exactly coincide with the ones in the standard library. Functions over booleans can be defined in the same way as above:

The last two of these illustrate Coq's syntax for multi-argument function definitions. The corresponding multi-argument application syntax is illustrated by the following "unit tests," which constitute a complete specification — a truth table — for the orb function:

We can also introduce some familiar syntax for the boolean operations we have just defined. The Notation command defines a new symbolic notation for an existing definition.

A note on notation: In .v files, we use square brackets to delimit fragments of Coq code within comments; this convention, also used by the coqdoc documentation tool, keeps them visually separate from the surrounding text. In the HTML version of the files, these pieces of text appear in a different font. The command Admitted can be used as a placeholder for an incomplete proof. We'll use it in exercises, to indicate the parts that we're leaving for you — i.e., your job is to replace Admitteds with real proofs.

Exercise: 1 star (nandb)

Remove "Admitted." and complete the definition of the following function; then make sure that the Example assertions below can each be verified by Coq. (I.e., fill in each proof, following the model of the orb tests above.) The function should return true if either or both of its inputs are false.

It is sometimes useful to invoke destruct inside a subgoal, generating yet more proof obligations. In this case, we use different kinds of bullets to mark goals on different "levels." For example:

Modules

Coq provides a module system, to aid in organizing large developments. In this course we won't need most of its features, but one is useful: If we enclose a collection of declarations between Module X and End X markers, then, in the remainder of the file after the End, these definitions are referred to by names like X.foo instead of just foo. We will use this feature to introduce the definition of the type nat in an inner module so that it does not interfere with the one from the standard library (which we want to use in the rest because it comes with a tiny bit of convenient special notation).

Numbers

The types we have defined so far, "enumerated types" such as coin, play and bool share the property that each type has a finite set of values. The natural numbers are an infinite set, and we need to represent all of them in a datatype with a finite number of constructors. There are many representations of numbers to choose from. We are most familiar with decimal notation (base 10), using the digits 0 through 9, for example, to form the number 123. You may have encountered hexadecimal notation (base 16), in which the same number is represented as 7B, or octal (base 8), where it is 173, or binary (base 2), where it is 1111011. Using an enumerated type to represent digits, we could use any of these to represent natural numbers. There are circumstances where each of these choices can be useful. Binary is valuable in computer hardware because it can in turn be represented with two voltage levels, resulting in simple circuitry. Analogously, we wish here to choose a representation that makes proofs simpler. Indeed, there is a representation of numbers that is even simpler than binary, namely unary (base 1), in which only a single digit is used (as one might do while counting days in prison by scratching on the walls). To represent unary with a Coq datatype, we use two constructors. The capital-letter O constructor represents zero. When the S constructor is applied to the representation of the natural number n, the result is the representation of n+1. (S stands for "successor", or "scratch" if one is in prison.) Here is the complete datatype definition.

With this definition, 0 is represented by O, 1 by S O, 2 by S (S O), and so on. The clauses of this definition can be read:

• O is a natural number (note that this is the letter "O," not the numeral "0").
• S can be put in front of a natural number to yield another one — if n is a natural number, then S n is too.

Again, let's look at this in a little more detail. The definition of nat says how expressions in the set nat can be built:

• O and S are constructors;
• the expression O belongs to the set nat;
• if n is an expression belonging to the set nat, then S n is also an expression belonging to the set nat; and
• expressions formed in these two ways are the only ones belonging to the set nat.

The same rules apply for our definitions of coin, bool, play, etc. The above conditions are the precise force of the Inductive declaration. They imply that the expression O, the expression S O, the expression S (S O), the expression S (S (S O)), and so on all belong to the set nat, while other expressions built from data constructors, like true, andb true false, S (S false), and O (O (O S)) do not. A critical point here is that what we've done so far is just to define a representation of numbers: a way of writing them down. The names O and S are arbitrary, and at this point they have no special meaning — they are just two different marks that we can use to write down numbers (together with a rule that says any nat will be written as some string of S marks followed by an O). If we like, we can write essentially the same definition this way:

The interpretation of these marks comes from how we use them to compute. We can do this by writing functions that pattern match on representations of natural numbers just as we did above with booleans and days — for example, here is the predecessor function:

The second branch can be read: "if n has the form S n' for some n', then return n'."

Because natural numbers are such a pervasive form of data, Coq provides a tiny bit of built-in magic for parsing and printing them: ordinary decimal numerals can be used as an alternative to the "unary" notation defined by the constructors S and O. Coq prints numbers in decimal form by default:

The constructor S has the type nat → nat, just like pred and functions like minustwo:

These are all things that can be applied to a number to yield a number. However, there is a fundamental difference between the first one and the other two: functions like pred and minustwo come with computation rules — e.g., the definition of pred says that pred 2 can be simplified to 1 — while the definition of S has no such behavior attached. Although it is like a function in the sense that it can be applied to an argument, it does not do anything at all! It is just a way of writing down numbers. (Think about standard decimal numerals: the numeral 1 is not a computation; it's a piece of data. When we write 111 to mean the number one hundred and eleven, we are using 1, three times, to write down a concrete representation of a number.) For most function definitions over numbers, just pattern matching is not enough: we also need recursion. For example, to check that a number n is even, we may need to recursively check whether n-2 is even. To write such functions, we use the keyword Fixpoint.

We can define oddb by a similar Fixpoint declaration, but here is a simpler definition:

Naturally, we can also define multi-argument functions by recursion.

Adding three to two now gives us five, as we'd expect.

The simplification that Coq performs to reach this conclusion can be visualized as follows:

As a notational convenience, if two or more arguments have the same type, they can be written together. In the following definition, (n m : nat) means just the same as if we had written (n : nat) (m : nat).

You can match two expressions at once by putting a comma between them:

Exercise: 1 star (factorial)

Recall the standard mathematical factorial function:

       factorial(0)  =  1
       factorial(n)  =  n * factorial(n-1)     (if n>0)

Translate this into Coq.

(The level, associativity, and nat_scope annotations control how these notations are treated by Coq's parser. The details are not important for our purposes, but interested readers can refer to the "More on Notation" section at the end of this chapter.) Note that these do not change the definitions we've already made: they are simply instructions to the Coq parser to accept x + y in place of plus x y and, conversely, to the Coq pretty-printer to display plus x y as x + y. When we say that Coq comes with almost nothing built-in, we really mean it: even equality testing is a user-defined operation! Here is a function eqb, which tests natural numbers for equality, yielding a boolean. Note the use of nested matches (we could also have used a simultaneous match, as we did in minus.)

Similarly, the leb function tests whether its first argument is less than or equal to its second argument, yielding a boolean.

Since we'll be using these (especially eqb) a lot, let's give them infix notations.

Exercise: 1 star (ltb)

The ltb function tests natural numbers for less-than, yielding a boolean. Instead of making up a new Fixpoint for this one, define it in terms of a previously defined function. (It can be done with just one previously defined function, but you can use two if you need to.)

Proof by Simplification

Now that we've defined a few datatypes and functions, let's turn to stating and proving properties of their behavior. Recall that Coq does quite a bit of simplification for us. For instance, we were able to use apply same_tails to prove that flip (flip tails) = tails. However, explicit simplification via simpl can be useful for both writing and reading proofs. This sort of "proof by simplification" can be used to prove more interesting properties as well. For example, the fact that 0 is a "neutral element" for + on the left can be proved just by observing that 0 + n reduces to n no matter what n is, a fact that can be read directly off the definition of plus.

(You may notice that the above statement looks different in the .v file in your IDE than it does in the HTML rendition in your browser, if you are viewing both. In .v files, we write the ∀universal quantifier using the reserved identifier "forall." When the .v files are converted to HTML, this gets transformed into an upside-down-A symbol.) The form of the theorem we just stated and its proof are almost exactly the same as the simpler examples we saw earlier; there are just a few differences. First, we've used the keyword Theorem instead of Example. This difference is mostly a matter of style; the keywords Example and Theorem (and a few others, including Lemma, Fact, and Remark) mean pretty much the same thing to Coq. Second, we've added the quantifier ∀n:nat, so that our theorem talks about all natural numbers n. Informally, to prove theorems of this form, we generally start by saying "Suppose n is some number..." Formally, this is achieved in the proof by intros n, which moves n from the quantifier in the goal to a context of current assumptions. The keywords intros, simpl, and reflexivity are examples of tactics. A tactic is a command that is used between Proof and Qed to guide the process of checking some claim we are making. We will see several more tactics in the rest of this chapter and yet more in future chapters. Other similar theorems can be proved with the same pattern.

The _l suffix in the names of these theorems is pronounced "on the left." It is worth stepping through these proofs to observe how the context and the goal change. You may want to add calls to simpl before reflexivity to see the simplifications that Coq performs on the terms before checking that they are equal.

Exercise: 2 stars (mult_S_1)

Case Analysis on Inductive Types

Sometimes we will need to do case analysis on non-simple inductive types, like the natural numbers. In this case, we will get two subcases: n = 0 and n = S x for some x. In this case, we will want to name any new variables that appear.

Before closing the chapter, let's mention one final convenience. As you may have noticed, many proofs perform case analysis on a variable right after introducing it: This pattern is so common that Coq provides a shorthand for it: we can perform case analysis on a variable when introducing it by using an intro pattern instead of a variable name. For instance, here is a shorter proof of the plus_1_neq_0 theorem above. (You'll also note one downside of this shorthand: we lose the equation recording the assumption we are making in each subgoal, which we previously got from the eqn:E annotation.)

Exercise: 1 star (zero_nbeq_plus_1)

More on Notation (Optional)

(In general, sections marked Optional are not needed to follow the rest of the book, except possibly other Optional sections. On a first reading, you might want to skim these sections so that you know what's there for future reference.) Recall the notation definitions for infix plus and times:

For each notation symbol in Coq, we can specify its precedence level and its associativity. The precedence level n is specified by writing at level n; this helps Coq parse compound expressions. The associativity setting helps to disambiguate expressions containing multiple occurrences of the same symbol. For example, the parameters specified above for + and * say that the expression 1+2*3*4 is shorthand for (1+((2*3)*4)). Coq uses precedence levels from 0 to 100, and left, right, or no associativity. We will see more examples of this later, e.g., in the Lists chapter. Each notation symbol is also associated with a notation scope. Coq tries to guess what scope is meant from context, so when it sees S(O*O) it guesses nat_scope, but when it sees the cartesian product (tuple) type bool*bool (which we'll see in later chapters) it guesses type_scope. Occasionally, it is necessary to help it out with percent-notation by writing (x*y)%nat, and sometimes in what Coq prints it will use %nat to indicate what scope a notation is in. Notation scopes also apply to numeral notation (3, 4, 5, etc.), so you may sometimes see 0%nat, which means O (the natural number 0 that we're using in this chapter), or 0%Z, which means the Integer zero (which comes from a different part of the standard library). Pro tip: Coq's notation mechanism is not especially powerful. Don't expect too much from it!

Fixpoints and Structural Recursion (Optional)

Here is a copy of the definition of addition:

When Coq checks this definition, it notes that plus' is "decreasing on 1st argument." What this means is that we are performing a structural recursion over the argument n — i.e., that we make recursive calls only on strictly smaller values of n. This implies that all calls to plus' will eventually terminate. Coq demands that some argument of every Fixpoint definition is "decreasing." This requirement is a fundamental feature of Coq's design: In particular, it guarantees that every function that can be defined in Coq will terminate on all inputs. However, because Coq's "decreasing analysis" is not very sophisticated, it is sometimes necessary to write functions in slightly unnatural ways.

Exercise: 2 stars, optional (decreasing)

To get a concrete sense of this, find a way to write a sensible Fixpoint definition (of a simple function on numbers, say) that does terminate on all inputs, but that Coq will reject because of this restriction. (If you choose to turn in this optional exercise as part of a homework assignment, make sure you comment out your solution so that it doesn't cause Coq to reject the whole file!)

Optional Exercises

Exercise: 2 stars, optional (andb_true_elim2)

Prove the following claim using rewrite and destruct

(a) Complete the definitions below of an increment function incr for binary numbers, and a function bin_to_nat to convert binary numbers to unary numbers.

(b) Write five unit tests test_bin_incr1, test_bin_incr2, etc. for your increment and binary-to-unary functions. (A "unit test" in Coq is a specific Example that can be proved with just reflexivity, as we've done for several of our definitions.) Notice that incrementing a binary number and then converting it to unary should yield the same result as first converting it to unary and then incrementing.