(** We can also write _parameterized_ propositions -- that is,
    functions that take arguments of some type and return a
    proposition. *)

(** For instance, the following function takes a number
    and returns a proposition asserting that this number is equal to
    three: *)

Definition is_three (n : nat) : Prop :=
  n = 3.(* ===> nat -> Prop *)

(** In Coq, functions that return propositions are said to define
    _properties_ of their arguments.

    For instance, here's a (polymorphic) property defining the
    familiar notion of an _injective function_. *)

Definition injective {A B} (f : A -> B) :=
  forall x y : A, f x = f y -> x = y.

(* Specifications: *)
Lemma succ_inj : injective S.