(* Basic_1: was: csubc_drop_conf_O *)
(* Note: the constant 0 can not be generalized *)
-lemma lsubc_ldrop_O1_trans: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ⊑[RP] K2.
-#RP #L1 #L2 #H elim H -L1 -L2
-[ #X #e #H
- >(ldrop_inv_atom1 … H) -H /2 width=3/
+lemma lsubc_ldrop_O1_trans: ∀RP,G,L1,L2. G ⊢ L1 ⊑[RP] L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. ⇩[0, e] L1 ≡ K1 & G ⊢ K1 ⊑[RP] K2.
+#RP #G #L1 #L2 #H elim H -L1 -L2
+[ #X #e #H elim (ldrop_inv_atom1 … H) -H /2 width=3/
| #I #L1 #L2 #V #_ #IHL12 #X #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H destruct
- [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=3/
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #H destruct
+ [ elim (IHL12 L2 0) -IHL12 // #X #H <(ldrop_inv_O2 … H) -H /3 width=3/
| elim (IHL12 … H) -L2 /3 width=3/
]
-| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #X #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H destruct
- [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=7/
+| #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #e #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #H destruct
+ [ elim (IHL12 L2 0) -IHL12 // #X #H <(ldrop_inv_O2 … H) -H /3 width=8/
| elim (IHL12 … H) -L2 /3 width=3/
]
qed-.
(* Basic_1: was: csubc_drop_conf_rev *)
lemma ldrop_lsubc_trans: ∀RR,RS,RP.
- acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 ⊑[RP] K2 →
- ∃∃L2. L1 ⊑[RP] L2 & ⇩[d, e] L2 ≡ K2.
-#RR #RS #RP #Hacp #Hacr #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #X #H
- >(lsubc_inv_atom1 … H) -H /2 width=3/
+ acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
+ ∀G,L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. G ⊢ K1 ⊑[RP] K2 →
+ ∃∃L2. G ⊢ L1 ⊑[RP] L2 & ⇩[d, e] L2 ≡ K2.
+#RR #RS #RP #Hacp #Hacr #G #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #X #H elim (lsubc_inv_atom1 … H) -H /2 width=3/
| #L1 #I #V1 #X #H
elim (lsubc_inv_pair1 … H) -H *
[ #K1 #HLK1 #H destruct /3 width=3/
- | #K1 #W1 #A #HV1 #HW1 #HLK1 #H1 #H2 destruct /3 width=3/
+ | #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct /3 width=4/
]
| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
elim (IHLK1 … HK12) -K1 /3 width=5/
elim (lsubc_inv_pair1 … H) -H *
[ #K2 #HK12 #H destruct
elim (IHLK1 … HK12) -K1 /3 width=5/
- | #K2 #W2 #A #HV2 #HW2 #HK12 #H1 #H2 destruct
+ | #K2 #V #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
+ elim (lift_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
elim (IHLK1 … HK12) #K #HL1K #HK2
lapply (aacr_acr … Hacp Hacr A) -Hacp -Hacr #HA
- lapply (s7 … HA … HV2 … HLK1 HV21) -HV2
- elim (lift_total W2 d e) /4 width=9/
+ lapply (s8 … HA … HV2 … HLK1 HV3) -HV2
+ lapply (s8 … HA … H1W2 … HLK1 HW23) -H1W2 /4 width=10/
]
]
qed-.