Notation Inj_dep_pairS := Inj_dep_pair.Notation Inj_dep_pairT := Inj_dep_pair.Notation eq_dep_eq__inj_pairT2 := eq_dep_eq__inj_pair2.

(* Specifications: *)
Lemma eq_rect_eq : forall (p:U) (Q:U -> Type) (x:Q p) (h:p = p), x = eq_rect p Q x p h.
Lemma eq_rec_eq : forall (p:U) (Q:U -> Set) (x:Q p) (h:p = p), x = eq_rec p Q x p h.
Lemma eq_dep_eq : forall (P:U->Type) (p:U) (x y:P p), eq_dep p x p y -> x = y.
Lemma UIP : forall (x y:U) (p1 p2:x = y), p1 = p2.
Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x.
Lemma Streicher_K : forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
Lemma inj_pair2 : forall (U:Type) (P:U -> Type) (p:U) (x y:P p), existT P p x = existT P p y -> x = y.