+lemma lifts_push_zero (f): ⬆*[⫯f]#O ≘ #0.
+/2 width=1 by lifts_lref/ qed.
+
+lemma lifts_push_lref (f) (i1) (i2): ⬆*[f]#i1 ≘ #i2 → ⬆*[⫯f]#(↑i1) ≘ #(↑i2).
+#f1 #i1 #i2 #H
+elim (lifts_inv_lref1 … H) -H #j #Hij #H destruct
+/3 width=7 by lifts_lref, at_push/
+qed.
+