# Mixing Ltac and Gallina for Fun and Profit

One of the most misleading introduction 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. Conversly, 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

## Revisions

This revisions table has been automatically generated
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history
of this website repository, and the change
descriptions may not always be as useful as they
should.

You can consult the source of this file in its current version here.

2020-08-28 | Heavy reworking of the Ltac series | a7e4531 |

2020-07-31 | Make the two articles about Ltac refer to each other | c2bb9b0 |

2020-07-31 | “For Fun and Benefit” was not an idiomatic expression | c0364ff |

2020-07-26 | Initial publication | a254d35 |

## 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 accompagnied 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)

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.
Conversly, there exists ways to construct terms in Gallina when writing a
proof script. Certains tactics takes 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.

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 typecheck it.
As a consequence, wrap_id generates a specialized identity function for
each specific context.

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

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

add = fun _ y : nat => y : nat -> nat -> natThat 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 sens. Dependently-typed programming can help 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 conversly prevent tactics such as auto to generate “incorrect” terms. Interested readers may refer to the dedicated series on this very website.