Module DecidableEqDep (M:DecidableType).
(** Invariance by Substitution of Reflexive Equality Proofs *)
Proof eq_rect_eq_dec eq_dec.
(** Injectivity of Dependent Equality *)
Proof (eq_rect_eq__eq_dep_eq U eq_rect_eq).
(** Uniqueness of Identity Proofs (UIP) *)
Proof (eq_dep_eq__UIP U eq_dep_eq).
(** Uniqueness of Reflexive Identity Proofs *)
Proof (UIP__UIP_refl U UIP).
(** Streicher's axiom K *)
Proof (K_dec_type eq_dec).
(** Injectivity of equality on dependent pairs in [Type] *)
Proof eq_dep_eq__inj_pairT2 U eq_dep_eq.
(** Proof-irrelevance on subsets of decidable sets *)

Lemma inj_pairP2 :
forall (P:U -> Prop) (x:U) (p q:P x),
ex_intro P x p = ex_intro P x q -> p = q.