Isabelle 'fun' without 'where'?

Question

Isabelle doesn't let me write just fun f :: "nat list => nat"; I have to add at least one defining equation, e.g. where "f [] = 5". But since it's fine to leave some constructors undefined, why can't I simply leave all constructors undefined? Then, a fun f without a where would be a handy alternative to the consts f declaration.

Answer 1

Firstly, I do not see why fun f without where is a handy alternative to consts f. You have the latter at your disposal (but also have to specify a type), use it if you must. Moreover, consts is more primitive than commands like definition, fun, and -- in my opinion -- shouldn't be used in production code.

As for your comment on leaving constructors undefined inside a function declaration. The wording "undefined" is misleading here, since, e.g., after

fun f :: "'a list => nat"
where
  "f [] = 0"

you can prove that

lemma "f (x#xs) = undefined"
  by (cases "x#xs" rule: f.elims) simp

and thus f (x#xs) is, in some sense, defined to be the value undefined. In contrast, after

consts f :: "'a list => nat"

you cannot proof anything about f. You just told the system that there is a constant f of a certain type, but without saying anything further about it.