induction 1 as [x y| | |].
- constructor 2 with y; auto.
constructor 1.
- constructor 1.
- apply clos_rst1n_sym; auto.
- eapply clos_rst1n_trans; eauto.
Qed.

(* Specifications: *)
Lemma clos_rst_rst1n_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_1n R x y.