Inductive remove_rel {A: Type}: A -> list A -> list A -> Prop :=
| remove_rel_nil: forall a, remove_rel a nil nil
| remove_rel_cons_eq: forall a xs ys, remove_rel a xs ys -> remove_rel a (a :: xs) ys
| remove_rel_cons_neq: forall a b xs ys, a <> b -> remove_rel a xs ys -> remove_rel a (b :: xs) (b :: ys).

Lemma remove_rel_In: forall (A : Type) (l1 l2 : list A) (x : A), remove_rel x l1 l2 -> ~ In x l2.