+#f #t #k @conj
+#p * #q #Hq #H0 destruct
+[ @(ex2_intro … (𝗱k◗𝗺◗q))
+ /2 width=1 by in_comp_iref_hd/
+| elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
+ /2 width=1 by in_comp_unwind2_path_term/
+]
+qed.
+
+lemma unwind2_term_abst (f) (t):
+ (𝛌.▼[⫯f]t) ⇔ ▼[f]𝛌.t.
+#f #t @conj #p #Hp
+[ elim (in_comp_inv_abst … Hp) -Hp #q #H1 * #r #Hr #H2 destruct
+ /3 width=1 by in_comp_unwind2_path_term, in_comp_abst_hd/
+| elim Hp -Hp #q #Hq #H0 destruct
+ elim (in_comp_inv_abst … Hq) -Hq #r #H0 #Hr destruct
+ /3 width=1 by in_comp_unwind2_path_term, in_comp_abst_hd/
+]
+qed.
+
+lemma unwind2_term_appl (f) (v) (t):
+ @▼[f]v.▼[f]t ⇔ ▼[f]@v.t.
+#f #v #t @conj #p #Hp
+[ elim (in_comp_inv_appl … Hp) -Hp * #q #H1 * #r #Hr #H2 destruct
+ /3 width=1 by in_comp_unwind2_path_term, in_comp_appl_sd, in_comp_appl_hd/
+| elim Hp -Hp #q #Hq #H0 destruct
+ elim (in_comp_inv_appl … Hq) -Hq * #r #H0 #Hr destruct
+ /3 width=1 by in_comp_unwind2_path_term, in_comp_appl_sd, in_comp_appl_hd/
+]