Question

The title is the question, simple.

The identity function fun x -> x has the type 'a -> 'a.

Are there any other functions with the same type 'a -> 'a?

I can’t think of another.

Answer 1

No.

fun x -> print_endline "foo"; x;;

(failwith "bang" : 'a -> 'a);;

(fun x -> failwith "bang" : 'a -> 'a);;

(fun x -> List.hd [] : 'a -> 'a);;

let rec f (x : 'a) : 'a = f x;;

let counter = ref 0;;
(fun x -> incr counter; x);;

The identity function is the only inhabitant of 'a -> 'a in total programming language with no side-effects whatsoever, including nontermination. Neither OCaml nor Haskell qualify, but some languages used as proof assistants (where this totality is important) do, in particular Coq (which has the impredicative polymorphism used to formulate this type).