Pairs of Numbers

In an Inductive type definition, each constructor can take any number of arguments — none (as with true and O), one (as with S), or more than one, as in this definition:

This declaration can be read: "There is just one way to construct a pair of numbers: by applying the constructor pair to two arguments of type nat." We can construct an element of natprod like this:

Here are two simple function definitions for extracting the first and second components of a pair. (The definitions also illustrate how to do pattern matching on two-argument constructors.)

Since pairs are used quite a bit, it is nice to be able to write them with the standard mathematical notation (x,y) instead of pair x y. We can tell Coq to allow this with a Notation declaration.

The new notation can be used both in expressions and in pattern matches (indeed, we've seen it already in the previous chapter — this notation is provided as part of the standard library):

Let's try and prove a few simple facts about pairs. If we state the lemmas in a particular (and slightly peculiar) way, we can prove them with just reflexivity (and its built-in simplification):

Note that reflexivity is not enough if we state the lemma in a more natural way:

We have to expose the structure of p so that simpl can perform the pattern match in fst and snd. We can do this with destruct. Notice that, unlike for nats, destruct doesn't generate an extra subgoal here. That's because natprods can only be constructed in one way.

Notice that Coq allows us to use the notation we introduced for pairs in the "as..." pattern that tells it what variables to bind.

Exercise: 1 star (snd_fst_is_swap)

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Exercise: 1 star, optional (fst_swap_is_snd)

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Lists of Numbers

Generalizing the definition of pairs a little, we can describe the type of lists of numbers like this: "A list is either the empty list or else a pair of a number and another list."

For example, here is a three-element list:

As with pairs, it is more convenient to write lists in familiar programming notation. The following two declarations allow us to use :: as an infix cons operator and square brackets as an "outfix" notation for constructing lists.

It is not necessary to fully understand these declarations, but in case you are interested, here is roughly what's going on. The right associativity annotation tells Coq how to parenthesize expressions involving several uses of :: so that, for example, the next three declarations mean exactly the same thing:

The at level 60 part tells Coq how to parenthesize expressions that involve both :: and some other infix operator. For example, since we defined + as infix notation for the plus function at level 50, The + operator will bind tighter than ::, so 1 + 2 :: [3] will be parsed, as we'd expect, as (1 + 2) :: [3] rather than 1 + (2 :: [3]). (By the way, it's worth noting in passing that expressions like "1 + 2 :: [3]" can be a little confusing when you read them in a .v file. The inner brackets, around 3, indicate a list, but the outer brackets, which are invisible in the HTML rendering, are there to instruct the "coqdoc" tool that the bracketed part should be displayed as Coq code rather than running text.) The second and third Notation declarations above introduce the standard square-bracket notation for lists; the right-hand side of the third one illustrates Coq's syntax for declaring n-ary notations and translating them to nested sequences of binary constructors. A number of functions are useful for manipulating lists. For example, the repeat function takes a number n and a count and returns a list of length count where every element is n.

The length function calculates the length of a list.

The app ("append") function concatenates two lists.

Actually, app will be used a lot in some parts of what follows, so it is convenient to have an infix operator for it.

Here are two smaller examples of programming with lists. The hd function returns the first element (the "head") of the list, while tail returns everything but the first element. Of course, the empty list has no first element, so we must pass a default value to be returned in that case.

Exercise: 2 stars (list_funs)

Complete the definitions of nonzeros, oddmembers and countoddmembers below. Have a look at the tests to understand what these functions should do.

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Exercise: 3 stars, advanced (alternate)

Complete the definition of alternate, which "zips up" two lists into one, alternating between elements taken from the first list and elements from the second. See the tests below for more specific examples. Note: one natural and elegant way of writing alternate will fail to satisfy Coq's requirement that all Fixpoint definitions be "obviously terminating." If you find yourself in this rut, look for a slightly more verbose solution that considers elements of both lists at the same time. (One possible solution requires defining a new kind of pairs, but this is not the only way.)

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Bags via Lists

A bag (or multiset) is like a set, but each element can appear multiple times instead of just once. One reasonable implementation of bags is to represent a bag of numbers as a list.

Exercise: 3 stars (bag_functions)

Complete the following definitions for the functions count, sum, add, and member for bags.

All these proofs can be done just by reflexivity.

Multiset sum is similar to set union: sum a b contains all the elements of a and of b. (Mathematicians usually define union on multisets a little bit differently, which is why we don't use that name for this operation.) For sum we're giving you a header that does not give explicit names to the arguments. Moreover, it uses the keyword Definition instead of Fixpoint, so even if you had names for the arguments, you wouldn't be able to process them recursively. The point of stating the question this way is to encourage you to think about whether sum can be implemented in another way — perhaps by using functions that have already been defined.

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Exercise: 3 stars, optional (bag_more_functions)

Here are some more bag functions for you to practice with.

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Exercise: 3 stars (bag_theorem)

Write down an interesting theorem about bags involving the functions count and add, and prove it. Note that, since this problem is somewhat open-ended, it's possible that you may come up with a theorem which is true, but whose proof requires techniques you haven't learned yet. Feel free to ask for help if you get stuck!

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Reasoning About Lists

Just as with numbers, simple facts about list-processing functions can sometimes be proved entirely by simplification. For example, the simplification performed by reflexivity is enough for this theorem...

... because the [] is substituted into the match position in the definition of app, allowing the match itself to be simplified. Also, as with numbers, it is sometimes helpful to perform case analysis on the possible shapes (empty or non-empty) of an unknown list.

Here, the nil case works because we've chosen to define tail nil = nil. Notice that the as annotation on the destruct tactic here introduces two names, n and l', corresponding to the fact that the cons constructor for lists takes two arguments (the head and tail of the list it is constructing). Usually, though, interesting theorems about lists require induction for their proofs.

Micro-Sermon

Simply reading example proofs will not get you very far! It is very important to work through the details of each one, using Coq and thinking about what each step of the proof achieves. Otherwise it is more or less guaranteed that the exercises will make no sense.

Induction on Lists

Proofs by induction over datatypes like natlist are perhaps a little less familiar than standard natural number induction, but the basic idea is equally simple. Each Inductive declaration defines a set of data values that can be built up from the declared constructors: a boolean can be either true or false; a number can be either O or S applied to a number; a list can be either nil or cons applied to a number and a list. Moreover, applications of the declared constructors to one another are the only possible shapes that elements of an inductively defined set can have, and this fact directly gives rise to a way of reasoning about inductively defined sets: a number is either O or else it is S applied to some smaller number; a list is either nil or else it is cons applied to some number and some smaller list; etc. So, if we have in mind some proposition P that mentions a list l and we want to argue that P holds for all lists, we can reason as follows:

• First, show that P is true of l when l is nil.
• Then show that P is true of l when l is cons n l' for some number n and some smaller list l', assuming that P is true for l'.

Since larger lists can only be built up from smaller ones, eventually reaching nil, these two things together establish the truth of P for all lists l. Here's a concrete example:

Again, this Coq proof is not especially illuminating as a static written document — it is easy to see what's going on if you are reading the proof in an interactive Coq session and you can see the current goal and context at each point, but this state is not visible in the written-down parts of the Coq proof. So a natural-language proof — one written for human readers — will need to include more explicit signposts; in particular, it will help the reader stay oriented if we remind them exactly what the induction hypothesis is in the second case. Theorem: For all lists l1, l2, and l3, (l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3). Proof: By induction on l1.

• First, suppose l1 = []. We must show
    ([] ++ l2) ++ l3 = [] ++ (l2 ++ l3),
  which follows directly from the definition of ++.
• Next, suppose l1 = n::l1', with
    (l1' ++ l2) ++ l3 = l1' ++ (l2 ++ l3)
  (the induction hypothesis). We must show
    ((n :: l1') ++ l2) ++ l3 = (n :: l1') ++ (l2 ++ l3).
  By the definition of ++, this follows from
    n :: ((l1' ++ l2) ++ l3) = n :: (l1' ++ (l2 ++ l3)),
  which is immediate from the induction hypothesis. ☐

Here is a similar example to be worked together in class:

For a slightly more involved example of an inductive proof over lists, suppose we define a "cons on the right" function snoc like this...

... and use it to define a list-reversing function rev like this:

Now let's prove some more list theorems using our newly defined snoc and rev. For something a little more challenging than the inductive proofs we've seen so far, let's prove that reversing a list does not change its length. Our first attempt at this proof gets stuck in the successor case...

So let's take the equation about snoc that would have enabled us to make progress and prove it as a separate lemma.

Now we can complete the original proof.

For comparison, here are informal proofs of these two theorems: Theorem: For all numbers n and lists l, length (snoc l n) = S (length l). Proof: By induction on l.

• First, suppose l = []. We must show
    length (snoc [] n) = S (length []),
  which follows directly from the definitions of length and snoc.
• Next, suppose l = n'::l', with
    length (snoc l' n) = S (length l').
  We must show
    length (snoc (n' :: l') n) = S (length (n' :: l')).
  By the definitions of length and snoc, this follows from
    S (length (snoc l' n)) = S (S (length l')),
  which is immediate from the induction hypothesis. ☐

Theorem: For all lists l, length (rev l) = length l. Proof: By induction on l.

• First, suppose l = []. We must show
    length (rev []) = length [],
  which follows directly from the definitions of length and rev.
• Next, suppose l = n::l', with
    length (rev l') = length l'.
  We must show
    length (rev (n :: l')) = length (n :: l').
  By the definition of rev, this follows from
    length (snoc (rev l') n) = S (length l')
  which, by the previous lemma, is the same as
    S (length (rev l')) = S (length l').
  This is immediate from the induction hypothesis. ☐

Obviously, the style of these proofs is rather longwinded and pedantic. After the first few, we might find it easier to follow proofs that give fewer details (since we can easily work them out in our own minds or on scratch paper if necessary) and just highlight the non-obvious steps. In this more compressed style, the above proof might look more like this: Theorem: For all lists l, length (rev l) = length l. Proof: First, observe that for any l. This follows by a straightforward induction on l. The main property now follows by another straightforward induction on l, using the observation together with the induction hypothesis in the case where l = n'::l'. ☐ Which style is preferable in a given situation depends on the sophistication of the expected audience and on how similar the proof at hand is to ones that the audience will already be familiar with. The more pedantic style is a good default for present purposes.

SearchAbout

We've seen that proofs can make use of other theorems we've already proved, using rewrite, and later we will see other ways of reusing previous theorems. But in order to refer to a theorem, we need to know its name, and remembering the names of all the theorems we might ever want to use can become quite difficult! It is often hard even to remember what theorems have been proven, much less what they are named. Coq's SearchAbout command is quite helpful with this. Typing SearchAbout foo will cause Coq to display a list of all theorems involving foo. For example, try uncommenting the following to see a list of theorems that we have proved about rev:

Keep SearchAbout in mind as you do the following exercises and throughout the rest of the course; it can save you a lot of time! Also, if you are using ProofGeneral, you can run SearchAbout with C-c C-a C-a. Pasting its response into your buffer can be accomplished with C-c C-;.

List Exercises, Part 1

Exercise: 3 stars (list_exercises)

More practice with lists.

There is a short solution to the next exercise. If you find yourself getting tangled up, step back and try to look for a simpler way.

An exercise about your implementation of nonzeros:

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List Exercises, Part 2

Exercise: 2 stars (list_design)

Design exercise:

• Write down a non-trivial theorem involving cons (::), snoc, and app (++).
• Prove it.

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Exercise: 3 stars, advanced (bag_proofs)

Here are a couple of little theorems to prove about your definitions about bags in the previous problem.

The following lemma about ble_nat might help you in the next proof.

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Exercise: 3 stars, optional (bag_count_sum)

Write down an interesting theorem about bags involving the functions count and sum, and prove it.

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Exercise: 4 stars, advanced (rev_injective)

Prove that the rev function is injective, that is, There is a hard way and an easy way to solve this exercise.

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Options

Here is another type definition that is often useful in day-to-day programming:

One use of natoption is as a way of returning "error codes" from functions. For example, suppose we want to write a function that returns the nth element of some list. If we give it type nat → natlist → nat, then we'll have to return some number when the list is too short!

On the other hand, if we give it type nat → natlist → natoption, then we can return None when the list is too short and Some a when the list has enough members and a appears at position n.

This example is also an opportunity to introduce one more small feature of Coq's programming language: conditional expressions...

Coq's conditionals are exactly like those found in any other language, with one small generalization. Since the boolean type is not built in, Coq actually allows conditional expressions over any inductively defined type with exactly two constructors. The guard is considered true if it evaluates to the first constructor in the Inductive definition and false if it evaluates to the second. The function below pulls the nat out of a natoption, returning a supplied default in the None case.

Exercise: 2 stars (hd_opt)

Using the same idea, fix the hd function from earlier so we don't have to pass a default element for the nil case.

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Exercise: 1 star, optional (option_elim_hd)

This exercise relates your new hd_opt to the old hd.

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Exercise: 2 stars (beq_natlist)

Fill in the definition of beq_natlist, which compares lists of numbers for equality. Prove that beq_natlist l l yields true for every list l.

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Dictionaries

As a final illustration of how fundamental data structures can be defined in Coq, here is the declaration of a simple dictionary data type, using numbers for both the keys and the values stored under these keys. (That is, a dictionary represents a finite map from numbers to numbers.)

This declaration can be read: "There are two ways to construct a dictionary: either using the constructor empty to represent an empty dictionary, or by applying the constructor record to a key, a value, and an existing dictionary to construct a dictionary with an additional key to value mapping."

Here is a function find that searches a dictionary for a given key. It evaluates evaluates to None if the key was not found and Some val if the key was mapped to val in the dictionary. If the same key is mapped to multiple values, find will return the first one it finds.

Exercise: 1 star (dictionary_invariant1)

Complete the following proof.

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Exercise: 1 star (dictionary_invariant2)

Complete the following proof.

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