(* 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
+lemma llor_skip: ∀L1,L2,U,l. |L1| = |L2| → yinj (|L1|) ≤ l → L1 ⋓[U, l] L2 ≡ L1.
+#L1 #L2 #U #l #HL12 #Hl @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
qed.
(* Note: lemma 1400 concludes the "big tree" theorem *)
-lemma llor_total: ∀L1,L2,T,d. |L1| = |L2| → ∃L. L1 ⋓[T, d] L2 ≡ L.
+lemma llor_total: ∀L1,L2,T,l. |L1| = |L2| → ∃L. L1 ⋓[T, l] 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
+[ #L2 #T #l #H >(length_inv_zero_sn … H) -L2 /2 width=2 by ex_intro/
+| #I1 #L1 #V1 #IHL1 #Y #T #l >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
+ elim (ylt_split l (|ⓑ{I1}V1.L1|))
+ [ elim (frees_dec (ⓑ{I1}V1.L1) T l (|L1|)) #HnU
+ elim (IHL1 L2 T l) // -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/
]