+
+(* Properties with iterated push ********************************************)
+
+lemma isid_pushs: ∀n,f. 𝐈❪f❫ → 𝐈❪⫯*[n]f❫.
+#n elim n -n /3 width=3 by isid_push/
+qed.
+
+(* Inversion lemmas with iterated push **************************************)
+
+lemma isid_inv_pushs: ∀n,g. 𝐈❪⫯*[n]g❫ → 𝐈❪g❫.
+#n elim n -n /3 width=3 by isid_inv_push/
+qed.
+
+(* Properties with tail *****************************************************)
+
+lemma isid_tl: ∀f. 𝐈❪f❫ → 𝐈❪⫱f❫.
+#f cases (pn_split f) * #g * -f #H
+[ /2 width=3 by isid_inv_push/
+| elim (isid_inv_next … H) -H //
+]
+qed.
+
+(* Properties with iterated tail ********************************************)
+
+lemma isid_tls: ∀n,g. 𝐈❪g❫ → 𝐈❪⫱*[n]g❫.
+#n elim n -n /3 width=1 by isid_tl/
+qed.