No offense, but this is exactly the sort of nonsense I associate with urbit.
>ideally, all values in its range are fixpoints, f(x) == f(f(x)).
This doesn't make any sense whatsoever. The only function for which this is true is identity, and I assure you it takes more than just the identity function to make a useful program. This isn't even possible unless the function has the type "a -> a".
The only other thing I can see you trying to express here is some butchery of set notation, where by f(x) you actually mean {f(x) | forall x of the correct type}. Even then, what you're saying is still completely nonsensical.
I honestly cannot tell if urbit is some sort of coordinated trolling effort.
Identity is not the only function for which that holds. As an immediate example, constant functions have this property as well. More interesting examples exist as well.
Also what was stated was not an equation for any old fixed point but instead idempotence—a specific kind of fixed point. In particular, this describes precisely the shape of a "validation" function so long as it either succeeds on all inputs or has a notion of failure in either its computational context or it's range ("span" I guess).
The return type isn't the same as the input type there. It only looks that way because of C's limitations.
Put it this way: f(f(x)) never makes sense for those definitions of f; it would always indicate a programming error. And so the type system ought to prevent it.
>ideally, all values in its range are fixpoints, f(x) == f(f(x)).
This doesn't make any sense whatsoever. The only function for which this is true is identity, and I assure you it takes more than just the identity function to make a useful program. This isn't even possible unless the function has the type "a -> a".
The only other thing I can see you trying to express here is some butchery of set notation, where by f(x) you actually mean {f(x) | forall x of the correct type}. Even then, what you're saying is still completely nonsensical.
I honestly cannot tell if urbit is some sort of coordinated trolling effort.