#
Strongly-Specified Functions in Coq, part 1: using the
`
refine
`
Tactic

This is the first article (initially published on January 11, 2015) of a series of two on
how to write strongly-specified functions in Coq. You can read the next part
here.
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 implement strongly-specified list manipulation functions
in Coq, where 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 in argument in which the
first element has been removed for example.
- Is this list empty?
- Defining some utility functions
- Defining push
- Defining head
- Conclusion & Moving Forward

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## Is this list empty?

Since we will manipulate lists in this article, we first enable several notations of the standard library.
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 middleman. The
scheme is the following:
###
Defining the

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

- 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!
###
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

With empty_dec, we can define empty_b.

###
Defining
`
empty_b
`

With empty_dec, we can define empty_b.
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 -> FalseIn addition to

`list 'a`, 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:
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.

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.

a : Type l : list a h : ~ empty l ============================ list aUsing the refine tactic naively, for instance this way:

leaves us the following goal to prove:

a : Type l : list a h : ~ empty l ============================ list aNothing 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 remember 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 using 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 value of l for a given case (that is, either x :: rst or []). Of course, 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 aWe 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 that 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.

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) -> l0If one tries to call pop nil, the assert ensures the call fails. Extra information given by the sigma-type have 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 miseuse 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.
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 firt value of l.
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 are 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 & Moving Forward

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.