Section Dependent_Equality.
Variable U : Type.
Variable P : U -> Type.
(** Dependent equality *)
Inductive eq_dep (p:U) (x:P p) : forall q:U, P q -> Prop :=
eq_dep_intro : eq_dep p x p x.
Hint Constructors eq_dep: core.
Proof eq_dep_intro.
Hint Immediate eq_dep_sym: core.

Lemma eq_dep_trans :
forall (p q r:U) (x:P p) (y:P q) (z:P r),
eq_dep p x q y -> eq_dep q y r z -> eq_dep p x r z.