# Mixing Ltac and Gallina for Fun and Profit

One of the most misleading introductions to Coq is to say that “Gallina is for programs, while tactics are for proofs.” Indeed, in Coq we construct terms of given types, always. Terms encodes both programs and proofs about these programs. Gallina is the preferred way to construct programs, and tactics are the preferred way to construct proofs.

The key word here is “preferred.” We do not always need to use tactics to
construct a proof term. Conversely, there are some occasions where
constructing a program with tactics become handy. Furthermore, Coq actually
allows for *mixing together* Ltac and Gallina.

In the previous article of this series, we discuss how Ltac provides two very interesting features:

- With
`match goal with`

it can inspect its context - With
`match type of _ with`

it can pattern matches on types

It turns out these features are more than handy when it comes to metaprogramming (that is, the generation of programs by programs).

## A Tale of Two Worlds, and Some Bridges

Constructing terms proofs directly in Gallina often happens when one is
writing dependently typed definition. For instance, we can write a type-safe
`from_option`

function (inspired by this very nice
write-up ) such that
the option to unwrap shall be accompanied by a proof that said option contains
something. This extra argument is used in the `None`

case to derive a
proof of `False`

, from which we can derive
anything.

```
Definition is_some {α} (x : option α) : bool :=
match x with Some _ => true | None => false end.
Lemma is_some_None {α} (x : option α)
: x = None -> is_some x <> true.
Proof. intros H. rewrite H. discriminate. Qed.
Definition from_option {α}
(x : option α) (some : is_some x = true)
: α :=
match x as y return x = y -> α with
| Some x => fun _ => x
| None => fun equ => False_rect α (is_some_None x equ some)
end eq_refl.
```

In `from_option`

, we construct two proofs without using tactics:

`False_rect α (is_some_None x equ some)`

to exclude the absurd case`eq_refl`

in conjunction with a dependent pattern matching (if you are not familiar with this trick: the main idea is to allow Coq to “remember” that`x = None`

in the second branch)

We can use another approach. We can decide to implement `from_option`

with a proof script.

```
Definition from_option' {α}
(x : option α) (some : is_some x = true)
: α.
Proof.
case_eq x.
+ intros y _.
exact y.
+ intros equ.
rewrite equ in some.
now apply is_some_None in some.
Defined.
```

There is a third approach we can consider: mixing Gallina terms and tactics.
This is possible thanks to the `ltac:()`

feature.

```
Definition from_option'' {α}
(x : option α) (some : is_some x = true)
: α :=
match x as y return x = y -> α with
| Some x => fun _ => x
| None => fun equ => ltac:(rewrite equ in some;
now apply is_some_None in some)
end eq_refl.
```

When Coq encounters `ltac:()`

, it treats it as a hole. It sets up a
corresponding goal, and tries to solve it with the supplied tactic.

Conversely, there exists ways to construct terms in Gallina when writing a proof
script. Certain tactics take such terms as arguments. Besides, Ltac provides
`constr:()`

and `uconstr:()`

which work similarly to
`ltac:()`

. The difference between `constr:()`

and
`uconstr:()`

is that Coq will try to assign a type to the argument of
`constr:()`

, but will leave the argument of `uconstr:()`

untyped.

For instance, consider the following tactic definition.

```
Tactic Notation "wrap_id" uconstr(x) :=
let f := uconstr:(fun x => x) in
exact (f x).
```

Both `x`

the argument of `wrap_id`

and `f`

the anonymous identity function
are not typed. It is only when they are composed together as an argument of
`exact`

(which expects a typed argument, more precisely of the type of the
goal) that Coq actually tries to type check it.

As a consequence, `wrap_id`

generates a specialized identity function for
each specific context.

```
Definition zero : nat := ltac:(wrap_id 0).
```

The generated anonymous identity function is `fun x : nat => x`

.

```
Definition empty_list α : list α := ltac:(wrap_id nil).
```

The generated anonymous identity function is `fun x : list α => x`

.
Besides, we do not need to give more type information about `nil`

. If
`wrap_id`

were to be expecting a typed term, we would have to replace
`nil`

by [(@nil α)].

## Beware the Automation Elephant in the Room

Proofs and computational programs are encoded in Coq as terms, but there is a fundamental difference between them, and it is highlighted by one of the axioms provided by the Coq standard library: proof irrelevance.

Proof irrelevance states that two proofs of the same theorem (i.e., two proof terms which share the same type) are essentially equivalent, and can be substituted without threatening the trustworthiness of the system. From a formal methods point of view, it makes sense. Even if we value “beautiful proofs,” we mostly want correct proofs.

The same reasoning does *not* apply to computational programs. Two functions of
type `nat -> nat -> nat`

are unlikely to be equivalent. For instance,
`add`

, `mul`

or `sub`

share the same type, but computes
totally different results.

Using tactics such as `auto`

to generate terms which do not live inside
`Prop`

is risky, to say the least. For instance,

```
Definition add (x y : nat) : nat := ltac:(auto).
```

This works, but it is certainly not what you would expect:

```
add = fun _ y : nat => y
: nat -> nat -> nat
```

That being said, if we keep that in mind, and assert the correctness of the generated programs (either by providing a proof, or by extensively testing it), there is no particular reason not to use Ltac to generate terms when it makes sense.

Dependently typed programming can help you here. If we decorate the return type of
a function with the expected properties of the result wrt. the function’s
arguments, we can ensure the function is correct, and conversely prevent tactics
such as `auto`

to generate “incorrect” terms. Interested readers may
refer to the dedicated series on this very
website.