(* Properties with length for local environments ****************************)
-lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
+lemma append_length: ∀L1,L2. |L1 + L2| = |L1| + |L2|.
#L1 #L2 elim L2 -L2 //
#L2 #I >append_bind >length_bind >length_bind //
qed.
-lemma ltail_length: â\88\80I,L. |â\93\98{I}.L| = ⫯|L|.
+lemma ltail_length: â\88\80I,L. |â\93\98{I}.L| = â\86\91|L|.
#I #L >append_length //
qed.
(* Advanced inversion lemmas on length for local environments ***************)
(* Basic_2A1: was: length_inv_pos_dx_ltail *)
-lemma length_inv_succ_dx_ltail: â\88\80L,n. |L| = ⫯n →
+lemma length_inv_succ_dx_ltail: â\88\80L,n. |L| = â\86\91n →
∃∃I,K. |K| = n & L = ⓘ{I}.K.
#Y #n #H elim (length_inv_succ_dx … H) -H #I #L #Hn #HLK destruct
elim (lenv_case_tail … L) [2: * #K #J ]
qed-.
(* Basic_2A1: was: length_inv_pos_sn_ltail *)
-lemma length_inv_succ_sn_ltail: â\88\80L,n. ⫯n = |L| →
+lemma length_inv_succ_sn_ltail: â\88\80L,n. â\86\91n = |L| →
∃∃I,K. n = |K| & L = ⓘ{I}.K.
#Y #n #H elim (length_inv_succ_sn … H) -H #I #L #Hn #HLK destruct
elim (lenv_case_tail … L) [2: * #K #J ]
(* Inversion lemmas with length for local environments **********************)
(* Basic_2A1: was: append_inj_sn *)
-lemma append_inj_length_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
+lemma append_inj_length_sn: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |K1| = |K2| →
L1 = L2 ∧ K1 = K2.
#K1 elim K1 -K1
[ * /2 width=1 by conj/
(* Note: lemma 750 *)
(* Basic_2A1: was: append_inj_dx *)
-lemma append_inj_length_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
+lemma append_inj_length_dx: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |L1| = |L2| →
L1 = L2 ∧ K1 = K2.
#K1 elim K1 -K1
[ * /2 width=1 by conj/
(* Advanced inversion lemmas ************************************************)
-lemma append_inj_dx: ∀L,K1,K2. L@@K1 = L@@K2 → K1 = K2.
+lemma append_inj_dx: ∀L,K1,K2. L+K1 = L+K2 → K1 = K2.
#L #K1 #K2 #H elim (append_inj_length_dx … H) -H //
qed-.
-lemma append_inv_refl_dx: ∀L,K. L@@K = L → K = ⋆.
+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.
+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-.
(* Basic_1: was: c_tail_ind *)
(* Basic_2A1: was: lenv_ind_alt *)
-lemma lenv_ind_tail: ∀R:predicate lenv.
- R (⋆) → (∀I,L. R L → R (ⓘ{I}.L)) → ∀L. R L.
-#R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * //
+lemma lenv_ind_tail: ∀Q:predicate lenv.
+ Q (⋆) → (∀I,L. Q L → Q (ⓘ{I}.L)) → ∀L. Q L.
+#Q #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * //
#L #I -IH1 #H destruct
elim (lenv_case_tail … L) [2: * #K #J ]
#H destruct /3 width=1 by/