(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
include "basic_2/static/da.ma".
(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
(* Properties on relocation *************************************************)
lemma da_lift: ∀h,g,G,L1,T1,l. ⦃G, L1⦄ ⊢ T1 ▪[h, g] l →
- ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
+ ∀L2,s,d,e. ⇩[s, d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
⦃G, L2⦄ ⊢ T2 ▪[h, g] l.
#h #g #G #L1 #T1 #l #H elim H -G -L1 -T1 -l
-[ #G #L1 #k #l #Hkl #L2 #d #e #_ #X #H
- >(lift_inv_sort1 … H) -X /2 width=1/
-| #G #L1 #K1 #V1 #i #l #HLK1 #_ #IHV1 #L2 #d #e #HL21 #X #H
+[ #G #L1 #k #l #Hkl #L2 #s #d #e #_ #X #H
+ >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/
+| #G #L1 #K1 #V1 #i #l #HLK1 #_ #IHV1 #L2 #s #d #e #HL21 #X #H
elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=7/
- | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=7/
+ [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=9 by da_ldef/
+ | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=8 by da_ldef, ldrop_inv_gen/
]
-| #G #L1 #K1 #W1 #i #l #HLK1 #_ #IHW1 #L2 #d #e #HL21 #X #H
+| #G #L1 #K1 #W1 #i #l #HLK1 #_ #IHW1 #L2 #s #d #e #HL21 #X #H
elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
- /3 width=7/
- | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=7/
+ [ elim (ldrop_trans_le … HL21 … HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+ /3 width=8 by da_ldec/
+ | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=8 by da_ldec, ldrop_inv_gen/
]
-| #a #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #d #e #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct /4 width=4/
-| #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #d #e #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct /3 width=4/
+| #a #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+ elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+ /4 width=5 by da_bind, ldrop_skip/
+| #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+ elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+ /3 width=5 by da_flat/
]
qed.
(* Inversion lemmas on relocation *******************************************)
lemma da_inv_lift: ∀h,g,G,L2,T2,l. ⦃G, L2⦄ ⊢ T2 ▪[h, g] l →
- ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
+ ∀L1,s,d,e. ⇩[s, d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
⦃G, L1⦄ ⊢ T1 ▪[h, g] l.
#h #g #G #L2 #T2 #l #H elim H -G -L2 -T2 -l
-[ #G #L2 #k #l #Hkl #L1 #d #e #_ #X #H
- >(lift_inv_sort2 … H) -X /2 width=1/
-| #G #L2 #K2 #V2 #i #l #HLK2 #HV2 #IHV2 #L1 #d #e #HL21 #X #H
+[ #G #L2 #k #l #Hkl #L1 #s #d #e #_ #X #H
+ >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/
+| #G #L2 #K2 #V2 #i #l #HLK2 #HV2 #IHV2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HV2 | -IHV2 ]
- [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // /3 width=7/
- | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // /2 width=4/
+ [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/
+ | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/
]
-| #G #L2 #K2 #W2 #i #l #HLK2 #HW2 #IHW2 #L1 #d #e #HL21 #X #H
+| #G #L2 #K2 #W2 #i #l #HLK2 #HW2 #IHW2 #L1 #s #d #e #HL21 #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct [ -HW2 | -IHW2 ]
- [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 // /3 width=7/
- | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 // /2 width=4/
+ [ elim (ldrop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/
+ | lapply (ldrop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/
]
-| #a #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #d #e #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct /4 width=4/
-| #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #d #e #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct /3 width=4/
+| #a #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /4 width=5 by da_bind, ldrop_skip/
+| #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=5 by da_flat/
]
qed-.