Instance compond_relation_proper {A: Type}: Proper (same_relation A ==> same_relation A ==> same_relation A) compond_relation.

(* Specifications: *)
Lemma compond_assoc: forall {A: Type} (R1 R2 R3: relation A), same_relation _ (compond_relation (compond_relation R1 R2) R3) (compond_relation R1 (compond_relation R2 R3)).
Lemma compond_eq_right: forall {A: Type} (R: relation A), same_relation _(compond_relation R eq) R.
Lemma compond_eq_left: forall {A: Type} (R: relation A), same_relation _(compond_relation eq R) R.
Lemma relation_conjunction_inclusion: forall {A} (R1 R2 R3 R4: relation A), inclusion _ R1 R2 -> inclusion _ R3 R4 -> inclusion _ (relation_conjunction R1 R3) (relation_conjunction R2 R4).