--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/multiple/frees_lift.ma".
+include "basic_2/multiple/llor_alt.ma".
+
+(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma llor_skip: ∀L1,L2,U,d. |L1| = |L2| → yinj (|L1|) ≤ d → L1 ⩖[U, d] L2 ≡ L1.
+#L1 #L2 #U #d #HL12 #Hd @and3_intro // -HL12
+#I1 #I2 #I #K1 #K2 #K #W1 #W2 #W #i #HLK1 #_ #HLK -L2 -K2
+lapply (drop_mono … HLK … HLK1) -HLK #H destruct
+lapply (drop_fwd_length_lt2 … HLK1) -K1
+/5 width=3 by ylt_yle_trans, ylt_inj, or3_intro0, and3_intro/
+qed.
+
+(* Note: lemma 1400 concludes the "big tree" theorem *)
+lemma llor_total: ∀L1,L2,T,d. |L1| = |L2| → ∃L. L1 ⩖[T, d] L2 ≡ L.
+#L1 @(lenv_ind_alt … L1) -L1
+[ #L2 #T #d #H >(length_inv_zero_sn … H) -L2 /2 width=2 by ex_intro/
+| #I1 #L1 #V1 #IHL1 #Y #T #d >ltail_length #H
+ elim (length_inv_pos_sn_ltail … H) -H #I2 #L2 #V2 #HL12 #H destruct
+ elim (ylt_split d (|ⓑ{I1}V1.L1|))
+ [ elim (frees_dec (ⓑ{I1}V1.L1) T d (|L1|)) #HnU
+ elim (IHL1 L2 T d) // -IHL1 -HL12
+ [ #L #HL12 >ltail_length /4 width=2 by llor_tail_frees, ylt_fwd_succ2, ex_intro/
+ | /4 width=2 by llor_tail_cofrees, ex_intro/
+ ]
+ | -IHL1 /4 width=3 by llor_skip, plus_to_minus, ex_intro/
+ ]
+]
+qed-.