+qed-.
+
+lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
+#L #K #H
+elim (append_inj_dx … (⋆) … H ?) //
+qed-.
+
+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 ?) //
+qed-.
+
+lemma length_inv_pos_dx_append: ∀d,L. |L| = d + 1 →
+ ∃∃I,K,V. |K| = d & L = ⋆.ⓑ{I}V @@ K.
+#d @(nat_ind_plus … d) -d
+[ #L #H
+ elim (length_inv_pos_dx … H) -H #I #K #V #H
+ >(length_inv_zero_dx … H) -H #H destruct
+ @ex2_3_intro [4: /2 width=2/ |5: // |1,2,3: skip ] (**) (* /3/ does not work *)
+| #d #IHd #L #H
+ elim (length_inv_pos_dx … H) -H #I #K #V #H
+ elim (IHd … H) -IHd -H #I0 #K0 #V0 #H1 #H2 #H3 destruct
+ @(ex2_3_intro … (K0.ⓑ{I}V)) //
+]
+qed-.
+
+(* Basic_eliminators ********************************************************)
+
+fact lenv_ind_dx_aux: ∀R:predicate lenv. R ⋆ →
+ (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
+ ∀d,L. |L| = d → R L.
+#R #Hatom #Hpair #d @(nat_ind_plus … d) -d
+[ #L #H >(length_inv_zero_dx … H) -H //
+| #d #IH #L #H
+ elim (length_inv_pos_dx_append … H) -H #I #K #V #H1 #H2 destruct /3 width=1/
+]
+qed-.
+
+lemma lenv_ind_dx: ∀R:predicate lenv. R ⋆ →
+ (∀I,L,V. R L → R (⋆.ⓑ{I}V @@ L)) →
+ ∀L. R L.
+/3 width=2 by lenv_ind_dx_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma length_inv_pos_sn_append: ∀d,L. 1 + d = |L| →
+ ∃∃I,K,V. d = |K| & L = ⋆. ⓑ{I}V @@ K.
+#d >commutative_plus @(nat_ind_plus … d) -d
+[ #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
+ >(length_inv_zero_sn … H1) -K
+ @(ex2_3_intro … (⋆)) // (**) (* explicit constructor *)
+| #d #IHd #L #H elim (length_inv_pos_sn … H) -H #I #K #V #H1 #H2 destruct
+ >H1 in IHd; -H1 #IHd
+ elim (IHd K ?) -IHd // #J #L #W #H1 #H2 destruct
+ @(ex2_3_intro … (L.ⓑ{I}V)) // (**) (* explicit constructor *)
+ >append_length /2 width=1/
+]
+qed-.