Scheme eq_indd := Induction for eq Sort Prop.(** Equivalent definition of dependent equality as a dependent pair of
      equalities *)

  Inductive eq_dep1 (p:U) (x:P p) (q:U) (y:P q) : Prop :=
    eq_dep1_intro : forall h:q = p, x = rew h in y -> eq_dep1 p x q y.(** Equivalent definition of dependent equality as a dependent pair of
      equalities *)

  Inductive eq_dep1 (p:U) (x:P p) (q:U) (y:P q) : Prop :=
    eq_dep1_intro : forall h:q = p, x = rew h in y -> eq_dep1 p x q y.

(* Specifications: *)
Lemma eq_dep1_dep : forall (p:U) (x:P p) (q:U) (y:P q), eq_dep1 p x q y -> eq_dep p x q y.
Lemma eq_dep_dep1 : forall (p q:U) (x:P p) (y:P q), eq_dep p x q y -> eq_dep1 p x q y.
Lemma eq_sigT_eq_dep : forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q), existT P p x = existT P q y -> eq_dep p x q y.
Lemma eq_dep_eq_sigT : forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q), eq_dep p x q y -> existT P p x = existT P q y.
Lemma eq_sigT_iff_eq_dep : forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q), existT P p x = existT P q y <-> eq_dep p x q y.