(* WARNING: Not sure simple_lexprod is useful! *)
Definition simple_lexprod {A: Type} (R1: A -> A -> Prop) (R2: A -> A -> Prop)
            (x y: A) : Prop :=
  R1 x y \/ x=y /\ R2 x y.

(* Specifications: *)
Lemma simple_lexprod_eq: forall A R1 R2 x y, @simple_lexprod A R1 R2 x y <-> lexprod A (fun _ => A) R1 (fun _ => R2) (existT (fun _:A => A) x x) (existT (fun _:A => A) y y).
Lemma wellfounded_simple_lexprod: forall A (R1: A -> A -> Prop) (R2: A -> A -> Prop), well_founded R1 -> well_founded R2 -> well_founded (simple_lexprod R1 R2).