+lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+ ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡* T2 → ⦃G, L1⦄ ⊢ T1 ➡* T2.
+/3 width=6 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+ ∀K1,s,e. ⬇[s, 0, e] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, e] L2 ≡ K2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #l1 #HWV #HW #HVl1 #HWl1 #_ #IHL12 #K1 #s #e #H
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubsv_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 →
+ ∀K2,s, e. ⬇[s, 0, e] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⫃¡[h, g] K2 & ⬇[s, 0, e] L1 ≡ K1.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #l1 #HWV #HW #HVl1 #HWl1 #_ #IHL12 #K2 #s #e #H
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+]