proposition: Lemma in_prod_iff : forall (l:list A)(l':list B)(x:A)(y:B), In (x,y) (list_prod l l') <-> In x l /\ In y l'.
proof: Proof. intros l l' x y; split; [ | intros H; now apply in_prod ]. induction l as [|a l IHl]; cbn; [easy|]. intros [[? [[= -> ->] ?]] %in_map_iff|] %in_app_or; tauto. Qed.
imports: PeanoNat