proposition: Lemma path_to_mult alpha s i : nf alpha -> path_to zero s (T1.phi0 alpha) -> path_to (T1.cons alpha i zero) s (T1.cons alpha (S i) zero).
proof: Proof. intros Hnf H ; assert (H1:= path_to_not_In_zero H). rewrite path_to_path_toS_iff; auto. apply path_toS_mult; auto. now rewrite <- path_to_path_toS_iff. Qed.
imports: hydras(DecPreOrder Canon MoreLists First_toggle OrdNotations) Coq(Lia)