Pattern Matching on Types and Contexts
In the a previous article of our series on
Ltac, we have shown how tactics allow for constructing Coq terms incrementally.
Ltac programs (“proof scripts”) generate terms, and the shape of said terms can
be very different regarding the initial context. For instance,
induction x will refine the current goal by using an inductive principle dedicated to
the type of
This is possible because Ltac allows for pattern matching on types and contexts. In this article, we give a short introduction on this feature of key importance.
lazymatch or to
Gallina provides one pattern matching construction, whose semantics is always the same: for a given term, the first pattern to match will always be selected. On the contrary, Ltac provides several pattern matching constructions with different semantics. This key difference has probably been motivated because Ltac is not a total language: a tactic may not always succeed.
Ltac programmers can use
lazymatch. One the one hand,
match, if one pattern matches, but the branch body fails, Coq will
backtrack and try the next branch. On the other hand,
stop with an error.
So, for instance, the two following tactics will print two different messages:
Ltac match_failure := match goal with | _ => fail "fail in first branch" | _ => fail "fail in second branch" end. Ltac lazymatch_failure := lazymatch goal with | _ => fail "fail in first branch" | _ => fail "fail in second branch" end.
We can try that quite easily by starting a dumb goal (eg.
and call our tactic. For
match_failure, we get:
Ltac call to "match_failure" failed. Error: Tactic failure: fail in second branch.
On the other hand, for
lazymatch_failure, we get:
Ltac call to "match_failure'" failed. Error: Tactic failure: fail in first branch.
Moreover, pattern matching allows for matching over patterns, not just constants. Here, Ltac syntax differs from Gallina’s. In Gallina, if a variable name is used in a pattern, Gallina creates a new variable in the scope of the related branch. If a variable with the same name already existed, it is shadowed. On the contrary, in Ltac, using a variable name as-is in a pattern implies an equality check.
To illustrate this difference, we can take the example of the following tactic.
Ltac successive x y := lazymatch y with | S x => idtac | _ => fail end.
successive x y will fail if
y is not the successor of
x. On the contrary, the “syntactically equivalent” function in Gallina
will exhibit a totally different behavior.
Definition successor (x y : nat) : bool := match y with | S x => true | _ => false end.
Here, the first branch of the pattern match is systematically selected when
y is not
O, and in this branch, the argument
x is shadowed by the
For Ltac to adopt a behavior similar to Gallina, the
? prefix shall be
used. For instance, the following tactic will check whether its argument
has a known successor, prints it if it does, or fail otherwise.
Ltac print_pred_or_zero x := match x with | S ?x => idtac x | _ => fail end.
On the one hand,
print_pred_or_zero 3 will print
2. On the
other hand, if there exists a variable
x : nat in the context, calling
print_pred_or_zero x will fail, since the exact value of
Pattern Matching on Types with
type of operator can be used in conjunction to pattern matching to
generate different terms according to the type of a variable. We could
leverage this to reimplement
induction for instance.
As an example, we propose the following
not_nat tactic which, given an
x, fails if
x is of type
Ltac not_nat x := lazymatch type of x with | nat => fail "argument is of type nat" | _ => idtac end.
With this definition,
not_nat true succeeds since
true is of
not_nat O since
O encodes in
We can also use the
? prefix to write true pattern. For instance, the
following tactic will fail if the type of its supplied argument has at least
Ltac not_param_type x := lazymatch type of x with | ?f ?a => fail "argument is of type with parameter" | _ => idtac end.
not_param_type (@nil nat) of type
list nat and
@eq_refl nat 0 of type
0 = 0 fail, but
not_param_type 0 of type
nat succeeds. *)
Pattern Matching on the Context with
Lastly, we discuss how Ltac allows for inspecting the context (i.e., the
hypotheses and the goal) using the
For instance, we propose a naive implementation of the
Ltac subst' := repeat match goal with | H : ?x = _ |- context[?x] => repeat rewrite H; clear H end.
goal, patterns are of the form
H : (pattern), ... |- (pattern).
- At the left side of
|-, we match on hypotheses. Beware that contrary to variable names in pattern, hypothesis names behaves as in Gallina (i.e., fresh binding, shadowing, etc.). In the branch, we are looking for equations, i.e., a hypothesis of the form
?x = _.
- At the right side of
|-, we match on the goal.
In both cases, Ltac makes available an interesting operator,
context[(pattern)], which is satisfied if
(pattern) appears somewhere in
the object we are pattern matching against. So, the branch of the
reads as follows: we are looking for an equation
H which specifies the
value of an object
x which appears in the goal. If such an equation
subst' tries to
rewrite x as many times as possible.
This implementation of
subst' is very fragile, and will not work if the
equation is of the form
_ = ?x, and it may behave poorly if we have
“transitive equations”, such as there exists hypotheses
?x = ?y and
?y = _. Motivated readers may be interested in proposing a more robust