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Suppose I have some locale where a type-class can be inferred from the assumptions.

locale some_locale =
  fixes xs :: "'x list"
  assumes xs_contains_UNIV: "set xs = UNIV"
begin

lemma finite_type: "OFCLASS('x, finite_class)"
proof (intro class.finite.of_class.intro class.finite.intro)
  have "finite (set xs)" ..
  then show "finite (UNIV :: 'x set)" unfolding xs_contains_UNIV .
qed

Can I then instantiate the type-class in some way?

Direct instantiation (instance, instantiation) does not work in locale contexts. I also had no luck with interpretation, as finite has no fixed constants, so I cannot specify the type I want to interpret.

interpretation finite (* subgoal: "class.finite TYPE('a)" for an arbitrary type 'a *)
proof
  show "finite (UNIV :: 'x set)" (* "Failed to refine any pending goal", because of the type mismatch *) sorry
  show "finite (UNIV :: 'a set)" (* would work, but impossible to prove for arbitrary type 'a *) oops

instance 'x :: finite (* not possible in locale context (Bad context for command "instance") *)

I know that I could simply fixes xs :: "'x::finite list", which is probably the solution I will have to accept, but that just seems redundant.

mini
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1 Answers1

2

It's currently (Isabelle2021-1) impossible to instantiate a type class inside a locale. However, you can interpret the local context of the type class inside a locale. This makes all the theorems and definitions available that have been made inside the type class context, but not the definitions and theorems that have been proven outside of this context.

interpretation finite 
proof
  show "finite (UNIV :: 'a set)"

If that is not enough for your use case, then adding the sort constraint to the locale declaration is the way to go.

Note that this only works if the type variable 'x in the locale is renamed to 'a, as the interpretation command for unknown reasons insists on using the type variable 'a.

Andreas Lochbihler
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  • Applying the `interpretation` inside the locale yields `Failed to refine any pending goal`, as the type of `UNIV` in the goal is actually `'a set`. I have edited the code in my question to make this clearer. – mini May 07 '22 at 08:53
  • You're right. Isabelle infers a more general type for the `finite` locale (`'a` instead of `'x`). So this approach works only if you rename the type variable `'x` in your locale to `'a`. For unknown reasons, `interpretation` then still infers the type variable `'a`. I've edited my answer accordingly. – Andreas Lochbihler May 08 '22 at 11:06