-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/
-]
+(* Basic_1: was: c_tail_ind *)
+lemma lenv_ind_alt: ∀R:predicate lenv.
+ R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
+ ∀L. R L.
+#R #IH1 #IH2 #L @(f_ind … length … L) -L #n #IHn * // -IH1
+#L #I #V normalize #H destruct elim (lpair_ltail L I V) /3 width=1 by/