Fixpoint functions in Type Class Instances - typeclass

Question

I am trying to use a fixpoint style function in the context of a type class instance but it doesn't seem to work. Is there something extra I have to do to make this work? For the time being I've used a hack of moving the function outside the type class and explicitly declaring it Fixpoint. This seems awful, however.
Here's the short example:
Inductive cexp : Type :=
| CAnd : cexp -> cexp -> cexp
| COr : cexp -> cexp -> cexp
| CProp : bool -> cexp.
Class Propable ( A : Type ) := { compile : A -> Prop }.
Instance: Propable cexp :=
{ compile c :=
match c with
| CAnd x y => (compile x) /\ (compile y)
| COr x y => (compile x) \/ (compile y)
| CProp _ => False
end
}.
This fails with:
Error: Unable to satisfy the following constraints:
In environment:
c, x, y : cexp
?Propable : "Propable cexp"
What does one have to do to make this work?

Answer 1

You can use fix to do that:
Instance: Propable cexp :=
{ compile := fix compile c :=
match c with
| CAnd x y => (compile x) /\ (compile y)
| COr x y => (compile x) \/ (compile y)
| CProp _ => False
end
}.
Let me illustrate how one can come up with it. Let's take the following piece of code:
Fixpoint silly n :=
match n with
| 0 => 0
| S n => silly n
end.
Fixpoint here is a vernacular command which makes the definition a little bit easier on the eyes, but it conceals what is going on here. It turns out that what Coq actually does is something like this:
Definition silly' :=
fix self n :=
match n with
| 0 => 0
| S n => self n
end.
You can verify it by using Print silly. after the definition.