| GRef _ ⇒ U
]
| TPair I V T ⇒ match I with
- [ Bind2 I ⇒ ⓑ{I} (fsubst W d V). (fsubst W (d+1) T)
- | Flat2 I ⇒ ⓕ{I} (fsubst W d V). (fsubst W d T)
+ [ Bind2 a I ⇒ ⓑ{a,I} (fsubst W d V). (fsubst W (d+1) T)
+ | Flat2 I ⇒ ⓕ{I} (fsubst W d V). (fsubst W d T)
]
].
(* Main properties **********************************************************)
theorem fsubst_delift: ∀K,V,T,L,d.
- ⇩[0, d] L ≡ K. ⓓV → L ⊢ T [d, 1] ≡ [d ← V] T.
+ ⇩[0, d] L ≡ K. ⓓV → L ⊢ ▼*[d, 1] T ≡ [d ← V] T.
#K #V #T elim T -T
[ * #i #L #d #HLK normalize in ⊢ (? ? ? ? ? %); /2 width=3/
elim (lt_or_eq_or_gt i d) #Hid
[ -HLK >(tri_lt ?????? Hid) /3 width=3/
- | destruct >tri_eq /4 width=4 by tpss_strap, tps_subst, le_n, ex2_1_intro/ (**) (* too slow without trace *)
+ | destruct >tri_eq /4 width=4 by tpss_strap2, tps_subst, le_n, ex2_1_intro/ (**) (* too slow without trace *)
| -HLK >(tri_gt ?????? Hid) /3 width=3/
]
| * /3 width=1/ /4 width=1/
(* Main inversion properties ************************************************)
theorem fsubst_inv_delift: ∀K,V,T1,L,T2,d. ⇩[0, d] L ≡ K. ⓓV →
- L ⊢ T1 [d, 1] ≡ T2 → [d ← V] T1 = T2.
+ L ⊢ ▼*[d, 1] T1 ≡ T2 → [d ← V] T1 = T2.
#K #V #T1 elim T1 -T1
[ * #i #L #T2 #d #HLK #H
- [ -HLK >(delift_fwd_sort1 … H) -H //
+ [ -HLK >(delift_inv_sort1 … H) -H //
| elim (lt_or_eq_or_gt i d) #Hid normalize
- [ -HLK >(delift_fwd_lref1_lt … H) -H // /2 width=1/
+ [ -HLK >(delift_inv_lref1_lt … H) -H // /2 width=1/
| destruct
- elim (delift_fwd_lref1_be … H ? ?) -H // #K0 #V0 #V2 #HLK0
+ elim (delift_inv_lref1_be … H ? ?) -H // #K0 #V0 #V2 #HLK0
lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H >minus_plus <minus_n_n #HV2 #HVT2 destruct
>(delift_inv_refl_O2 … HV2) -V >(flift_inv_lift … HVT2) -V2 //
- | -HLK >(delift_fwd_lref1_ge … H) -H // /2 width=1/
+ | -HLK >(delift_inv_lref1_ge … H) -H // /2 width=1/
]
- | -HLK >(delift_fwd_gref1 … H) -H //
+ | -HLK >(delift_inv_gref1 … H) -H //
]
-| * #I #V1 #T1 #IHV1 #IHT1 #L #X #d #HLK #H
- [ elim (delift_fwd_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #L #X #d #HLK #H
+ [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
<(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // /2 width=1/
- | elim (delift_fwd_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
<(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 //
]
]