(* Basic_2A1: was: length_inv_pos_dx_ltail *)
lemma length_inv_succ_dx_ltail: ∀L,n. |L| = ↑n →
(* Basic_2A1: was: length_inv_pos_dx_ltail *)
lemma length_inv_succ_dx_ltail: ∀L,n. |L| = ↑n →
#Y #n #H elim (length_inv_succ_dx … H) -H #I #L #Hn #HLK destruct
elim (lenv_case_tail … L) [2: * #K #J ]
#H destruct /2 width=4 by ex2_2_intro/
#Y #n #H elim (length_inv_succ_dx … H) -H #I #L #Hn #HLK destruct
elim (lenv_case_tail … L) [2: * #K #J ]
#H destruct /2 width=4 by ex2_2_intro/
(* Basic_2A1: was: length_inv_pos_sn_ltail *)
lemma length_inv_succ_sn_ltail: ∀L,n. ↑n = |L| →
(* Basic_2A1: was: length_inv_pos_sn_ltail *)
lemma length_inv_succ_sn_ltail: ∀L,n. ↑n = |L| →
#Y #n #H elim (length_inv_succ_sn … H) -H #I #L #Hn #HLK destruct
elim (lenv_case_tail … L) [2: * #K #J ]
#H destruct /2 width=4 by ex2_2_intro/
#Y #n #H elim (length_inv_succ_sn … H) -H #I #L #Hn #HLK destruct
elim (lenv_case_tail … L) [2: * #K #J ]
#H destruct /2 width=4 by ex2_2_intro/
-lemma append_inv_pair_dx: ∀I,L,K,V. L+K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
-#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
+lemma append_inv_pair_dx: ∀I,L,K,V. L+K = L.ⓑ[I]V → K = ⋆.ⓑ[I]V.
+#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ[I]V) … H) //
(* Basic_1: was: c_tail_ind *)
(* Basic_2A1: was: lenv_ind_alt *)
lemma lenv_ind_tail: ∀Q:predicate lenv.
(* Basic_1: was: c_tail_ind *)
(* Basic_2A1: was: lenv_ind_alt *)
lemma lenv_ind_tail: ∀Q:predicate lenv.