(* 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.

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).