Inductive compond_relation {A: Type} (R1 R2: relation A) : relation A :=
| compond_intro: forall x y z, R1 x y -> R2 y z -> compond_relation R1 R2 x z.

(* Specifications: *)
Lemma compond_relation_spec: forall {A} (R1 R2: relation A) x z,
compond_relation R1 R2 x z ->
exists y, R1 x y /\ R2 y z.
Lemma compond_relation_inclusion: forall {A} (R1 R2 R3 R4: relation A),
inclusion _ R1 R2 ->
inclusion _ R3 R4 ->
inclusion _ (compond_relation R1 R3) (compond_relation R2 R4).