Definition lex_pair {A B} (Ra: A -> A -> Prop) (Rb: B -> B -> Prop)
            (x: A*B) (y: A*B) : Prop :=
 Ra (fst x) (fst y) \/ (fst x = fst y /\ Rb (snd x) (snd y)).

(* Specifications: *)
Lemma well_founded_incl: forall A (Rs Rt: A -> A -> Prop), inclusion _ Rt Rs -> well_founded Rs -> well_founded Rt.
Lemma well_founded_image: forall A B (f: A -> B) (Rb: B -> B -> Prop), well_founded Rb -> well_founded (fun x y => Rb (f x) (f y)).
Lemma well_founded_lex_pair: forall A B (Ra: A -> A -> Prop) (Rb: B -> B -> Prop), well_founded Ra -> well_founded Rb -> well_founded (lex_pair Ra Rb).
Lemma well_founded_trans: forall A (Ra: A -> A -> Prop), well_founded Ra <-> well_founded (clos_trans _ Ra).