Definition eq_dec (E : Set) := forall a b : E, {a = b} + {a <> b}.

(* Specifications: *)
Lemma case_eq : forall (E F : Set) (eq_E_dec : eq_dec E) (x : E) (y z : F), match eq_E_dec x x with | left _ => y | right _ => z end = y.
Lemma case_ineq : forall (E F : Set) (eq_E_dec : eq_dec E) (x x' : E) (y z : F), x <> x' -> match eq_E_dec x x' with | left _ => y | right _ => z end = z.
Lemma eq_couple_dec : forall (E F : Set) (eq_E_dec : eq_dec E) (eq_F_dec : eq_dec F) (x1 x2 : E) (y1 y2 : F), {x1 = x2 /\ y1 = y2} + {x1 <> x2 \/ y1 <> y2}.