Instance relation_conjunction_proper {A: Type}: Proper (same_relation A ==> same_relation A ==> same_relation A) relation_conjunction.

(* Specifications: *)
Lemma relation_conjunction_iff: forall {A} (R R': relation A) x y, relation_conjunction R R' x y <-> R x y /\ R' x y.
Lemma relation_disjunction_iff: forall {A} (R R': relation A) x y, relation_disjunction R R' x y <-> R x y \/ R' x y.
Lemma relation_disjunction_inclusion_left: forall {A} (R R': relation A), inclusion _ R (relation_disjunction R R').
Lemma relation_disjunction_inclusion_right: forall {A} (R R': relation A), inclusion _ R' (relation_disjunction R R').