Definition injective {X Y:Type} (f:X->Y) :=
forall x1 x2:X, f x1 = f x2 -> x1 = x2.
Definition surjective {X Y:Type} (f:X->Y) :=
forall y:Y, exists x:X, f x = y.
Definition bijective {X Y:Type} (f:X->Y) :=
injective f /\ surjective f.
Inductive invertible {X Y:Type} (f:X->Y) : Prop :=
| intro_invertible: forall g:Y->X,
(forall x:X, g (f x) = x) -> (forall y:Y, f (g y) = y) ->
invertible f.

(* Specifications: *)
Lemma unique_inverse: forall {X Y:Type} (f:X->Y), invertible f ->
exists! g:Y->X, (forall x:X, g (f x) = x) /\
(forall y:Y, f (g y) = y).