+
+theorem tr_xap_eq_repl (i):
+ stream_eq_repl … (λf1,f2. f1@❨i❩ = f2@❨i❩).
+#i #f1 #f2 #Hf
+<tr_xap_unfold <tr_xap_unfold
+/3 width=1 by tr_push_eq_repl, tr_nap_eq_repl/
+qed.
+
+lemma tr_nap_plus (f) (m) (n):
+ ⇂*[↑n]f@❨m❩+f@§❨n❩ = f@§❨m+n❩.
+/2 width=1 by eq_inv_nsucc_bi/
+qed.