qed.
(* Note: the constant 0 cannot be generalized *)
-lemma lsubd_ldrop_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.
+lemma lsubd_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 (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K1 #s #e #H
- elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_abbr, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK1) -L1 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_abbr, 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 lsubd_ldrop_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.
+lemma lsubd_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 (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K2 #s #e #H
- elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
+ elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
- <(ldrop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_abbr, ldrop_pair, ex2_intro/
- | elim (IHL12 … HLK2) -L2 /3 width=3 by ldrop_drop_lt, ex2_intro/
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_abbr, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
]
qed-.