+(* Basic properties *********************************************************)
+
+lemma pushs_O: ∀f. f = ⫯*[𝟎] f.
+// qed.
+
+lemma pushs_S: ∀f,n. ⫯⫯*[n] f = ⫯*[↑n] f.
+#f #n @(niter_succ … push)
+qed.
+
+lemma pushs_eq_repl: ∀n. eq_repl (λf1,f2. ⫯*[n] f1 ≡ ⫯*[n] f2).
+#n @(nat_ind_succ … n) -n /3 width=5 by eq_push/
+qed.
+
+(* Advanced properties ******************************************************)
+
+lemma push_swap (n) (f): ⫯⫯*[n] f = ⫯*[n] ⫯f.
+#n #f @(niter_appl … push)
+qed.
+
+lemma pushs_xn: ∀n,f. ⫯*[n] ⫯f = ⫯*[↑n] f.
+// qed.
+