(** Invariance by Substitution of Reflexive Equality Proofs *)

  Definition Eq_rect_eq_on (p : U) (Q : U -> Type) (x : Q p) :=
    forall (h : p = p), x = eq_rect p Q x p h.Definition Eq_rect_eq := forall p Q x, Eq_rect_eq_on p Q x.(** Injectivity of Dependent Equality *)

  Definition Eq_dep_eq_on (P : U -> Type) (p : U) (x : P p) :=
    forall (y : P p), eq_dep p x p y -> x = y.Definition Eq_dep_eq := forall P p x, Eq_dep_eq_on P p x.(** Uniqueness of Identity Proofs (UIP) *)

  Definition UIP_on_ (x y : U) (p1 : x = y) :=
    forall (p2 : x = y), p1 = p2.Definition UIP_ := forall x y p1, UIP_on_ x y p1.(** Uniqueness of Reflexive Identity Proofs *)

  Definition UIP_refl_on_ (x : U) :=
    forall (p : x = x), p = eq_refl x.Definition UIP_refl_ := forall x, UIP_refl_on_ x.(** Streicher's axiom K *)

  Definition Streicher_K_on_ (x : U) (P : x = x -> Prop) :=
    P (eq_refl x) -> forall p : x = x, P p.Definition Streicher_K_ := forall x P, Streicher_K_on_ x P.

Theorem UIP_shift_on (X : Type) (x : X) : UIP_refl_on_ X x -> forall y : x = x, UIP_refl_on_ (x = x) y.