Definition conditional_eq {A} (x y : A) := eq x y.

(* Specifications: *)
Lemma simplification_existT2 A (P : A -> Type) B (p : A) (x y : P p) : (x = y -> B) -> (conditional_eq (existT P p x) (existT P p y) -> B).
Lemma simplification_existT1 A (P : A -> Type) B (p q : A) (x : P p) (y : P q) : (p = q -> conditional_eq (existT P p x) (existT P q y) -> B) -> (existT P p x = existT P q y -> B).
Lemma simplification_K A (x : A) (B : x = x -> Type) : B eq_refl -> (forall p : x = x, B p).