(**************************************************************************)
include "basic_2/multiple/frees_lift.ma".
-include "basic_2/multiple/llor.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.
+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 (ldrop_mono … HLK … HLK1) -HLK #H destruct
/5 width=3 by ylt_yle_trans, ylt_inj, or3_intro0, and3_intro/
qed.
-axiom llor_total: ∀L1,L2,T,d. |L1| ≤ |L2| → ∃L. L1 ⩖[T, d] L2 ≡ L.
+(* 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-.