proposition: Lemma infseq_if_all_seq_inf: forall a, all_seq_inf a -> infseq a.
proof: Proof. intros a0 ALL0. exists all_seq_inf; split; auto. intros a1 ALL1. destruct (ALL1 a1) as [a2 R12]. constructor. exists a2; split; auto. intros a3 S23. destruct (ALL1 a3) as [a4 R23]. apply star_step with a2; auto. exists a4; auto. Qed.