Implementing Strongly-Specified Functions with the `refine` Tactic

coq

I started to play with Coq, the interactive theorem prover developed by Inria, a few weeks ago. It is a very powerful tool, yet hard to master. Fortunately, there are some very good readings if you want to learn (I recommend the Coq'Art). This article is not one of them.

In this article, we will see how to implement strongly specified list manipulation functions in Coq. Strong specifications are used to ensure some properties on functions' arguments and return value. It makes Coq type system very expressive. Thus, it is possible to specify in the type of the function pop that the return value is the list passed as an argument in which the first element has been removed, for example.

Is This List Empty?

It's the first question to deal with when manipulating lists. There are some functions that require their arguments not to be empty. It's the case for the pop function, for instance it is not possible to remove the first element of a list that does not have any elements in the first place.

When one wants to answer such a question as “Is this list empty?”, he has to keep in mind that there are two ways to do it: by a predicate or by a boolean function. Indeed, Prop and bool are two different worlds that do not mix easily. One solution is to write two definitions and to prove their equivalence. That is forall args, predicate args <-> bool_function args = true.

Another solution is to use the sumbool type as middlemen. The scheme is the following:

Defining predicate : args → Prop
Defining predicate_dec : args -> { predicate args } + { ~predicate args }
Defining predicate_b:

Definition predicate_b (args) :=
  if predicate_dec args then true else false.

Defining the `empty` Predicate

A list is empty if it is [] (nil). It's as simple as that!

Definition empty {a} (l : list a) : Prop := l = [].

Defining a decidable version of `empty`

A decidable version of empty is a function which takes a list l as its argument and returns either a proof that l is empty, or a proof that l is not empty. This is encoded in the Coq standard library with the sumbool type, and is written as follows: { empty l } + { ~ empty l }.

Definition empty_dec {a} (l : list a)
  : { empty l } + { ~ empty l }.
Proof.
  refine (match l with
          | [] => left _ _
          | _ => right _ _
          end);
    unfold empty; trivial.
  unfold not; intro H; discriminate H.
Defined.

In this example, I decided to use the refine tactic which is convenient when we manipulate the Set and Prop sorts at the same time.

Defining `empty_b`

With empty_dec, we can define empty_b.

Definition empty_b {a} (l : list a) : bool :=
  if empty_dec l then true else false.

Let's try to extract empty_b:

type bool =
| True
| False

type sumbool =
| Left
| Right

type 'a list =
| Nil
| Cons of 'a * 'a list

(** val empty_dec : 'a1 list -> sumbool **)

let empty_dec = function
| Nil -> Left
| Cons (a, l0) -> Right

(** val empty_b : 'a1 list -> bool **)

let empty_b l =
  match empty_dec l with
  | Left -> True
  | Right -> False

In addition to 'a list, Coq has created the sumbool and bool types and empty_b is basically a translation from the former to the latter. We could have stopped with empty_dec, but Left and Right are less readable that True and False. Note that it is possible to configure the Extraction mechanism to use primitive OCaml types instead, but this is out of the scope of this article.

Defining Some Utility Functions

Defining `pop`

There are several ways to write a function that removes the first element of a list. One is to return nil if the given list was already empty:

Definition pop {a} ( l :list a) :=
  match l with
  | _ :: l => l
  | [] => []
  end.

But it's not really satisfying. A pop call over an empty list should not be possible. It can be done by adding an argument to pop: the proof that the list is not empty.

Definition pop {a} (l : list a) (h : ~ empty l)
  : list a.

There are, as usual when it comes to lists, two cases to consider.

l = x :: rst, and therefore pop (x :: rst) h is rst
l = [], which is not possible since we know l is not empty.

The challenge is to convince Coq that our reasoning is correct. There are, again, several approaches to achieve that. We can, for instance, use the refine tactic again, but this time we need to know a small trick to succeed as using a “regular” match will not work.

From the following goal:

  a : Type
  l : list a
  h : ~ empty l
  ============================
  list a

Using the refine tactic naively, for instance, this way:

  refine (match l with
          | _ :: rst => rst
          | [] => _
          end).

leaves us the following goal to prove:

  a : Type
  l : list a
  h : ~ empty l
  ============================
  list a

Nothing has changed! Well, not exactly. See, refine has taken our incomplete Gallina term, found a hole, done some type-checking, found that the type of the missing piece of our implementation is list a and therefore has generated a new goal of this type. What refine has not done, however, is remembering that we are in the case where l = [].

We need to generate a goal from a hole wherein this information is available. It is possible to use a long form of match. The general approach is this: rather than returning a value of type list a, our match will return a function of type l = ?l' -> list a, where ?l is a value of l for a given case (that is, either x :: rst or []). Of course and as a consequence, the type of the match in now a function which awaits a proof to return the expected result. Fortunately, this proof is trivial: it is eq_refl.

  refine (match l as l'
                return l = l' -> list a
          with
          | _ :: rst => fun _ => rst
          | [] => fun equ => _
          end eq_refl).

For us to conclude the proof, this is way better.

  a : Type
  l : list a
  h : ~ empty l
  equ : l = []
  ============================
  list a

We conclude the proof, and therefore the definition of pop.

  rewrite equ in h.
  exfalso.
  now apply h.
Defined.

It's better and yet it can still be improved. Indeed, according to its type, pop returns “some list.” As a matter of fact, pop returns “the same list without its first argument.” It is possible to write such precise definition thanks to sigma types, defined as:

Inductive sig (A : Type) (P : A -> Prop) : Type :=
  exist : forall (x : A), P x -> sig P.

Rather than sig A p, sigma-types can be written using the notation { a | P }. They express subsets, and can be used to constraint arguments and results of functions.

We finally propose a strongly specified definition of pop.

Definition pop {a} (l : list a | ~ empty l)
  : { l' | exists a, proj1_sig l = cons a l' }.

If you think the previous use of match term was ugly, brace yourselves.

  refine (match proj1_sig l as l'
                return proj1_sig l = l'
                       -> { l' | exists a, proj1_sig l = cons a l' }
          with
          | [] => fun equ => _
          | (_ :: rst) => fun equ => exist _ rst _
          end eq_refl).

This leaves us two goals to tackle.

First, we need to discard the case where l is the empty list.

  a : Type
  l : {l : list a | ~ empty l}
  equ : proj1_sig l = []
  ============================
  {l' : list a | exists a0 : a, proj1_sig l = a0 :: l'}

  + destruct l as [l nempty]; cbn in *.
    rewrite equ in nempty.
    exfalso.
    now apply nempty.

Then, we need to prove that the result we provide (rst) when the list is not empty is correct with respect to the specification of pop.

  a : Type
  l : {l : list a | ~ empty l}
  a0 : a
  rst : list a
  equ : proj1_sig l = a0 :: rst
  ============================
  exists a1 : a, proj1_sig l = a1 :: rst

  + destruct l as [l nempty]; cbn in *.
    rewrite equ.
    now exists a0.
Defined.

Let's have a look at the extracted code:

(** val pop : 'a1 list -> 'a1 list **)

let pop = function
| Nil -> assert false (* absurd case *)
| Cons (a, l0) -> l0

If one tries to call pop nil, the assert ensures the call fails. Extra information given by the sigma type has been stripped away. It can be confusing, and in practice it means that, we you rely on the extraction mechanism to provide a certified OCaml module, you cannot expose strongly specified functions in its public interface because nothing in the OCaml type system will prevent a misuse which will in practice leads to an assert false. *)

Defining `push`

It is possible to specify push the same way pop has been. The only difference is push accepts lists with no restriction at all. Thus, its definition is a simpler, and we can write it without refine.

Definition push {a} (l : list a) (x : a)
  : { l' | l' = x :: l } :=
  exist _ (x :: l) eq_refl.

And the extracted code is just as straightforward.

let push l a =
  Cons (a, l)

Defining `head`

Same as pop and push, it is possible to add extra information in the type of head, namely the returned value of head is indeed the first value of l.

Definition head {a} (l : list a | ~ empty l)
  : { x | exists r, proj1_sig l = x :: r }.

It's not a surprise its definition is very close to pop.

  refine (match proj1_sig l as l'
                return proj1_sig l = l' -> _
          with
          | [] => fun equ => _
          | x :: _ => fun equ => exist _ x _
          end eq_refl).

The proof is also very similar, and are left to read as an exercise for passionate readers.

  + destruct l as [l falso]; cbn in *.
    rewrite equ in falso.
    exfalso.
    now apply falso.
  + exists l0.
    now rewrite equ.
Defined.

Finally, the extracted code is as straightforward as it can get.

let head = function
| Nil -> assert false (* absurd case *)
| Cons (a, l0) -> a

Conclusion

Writing strongly specified functions allows for reasoning about the result correctness while computing it. This can help in practice. However, writing these functions with the refine tactic does not enable a very idiomatic Coq code.

To improve the situation, the Program framework distributed with the Coq standard library helps, but it is better to understand what Program achieves under its hood, which is basically what we have done in this article.

Implementing Strongly-Specified Functions with the refine Tactic

Is This List Empty?

Defining the empty Predicate

Defining a decidable version of empty

Defining empty_b