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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/unfold/delift_lift.ma".
16 include "apps_2/functional/lift.ma".
18 (* CORE SUBSTITUTION ********************************************************)
20 let rec fsubst W d U on U ≝ match U with
21 [ TAtom I ⇒ match I with
23 | LRef i ⇒ tri … i d (#i) (↑[0, i] W) (#(i-1))
26 | TPair I V T ⇒ match I with
27 [ Bind2 I ⇒ ⓑ{I} (fsubst W d V). (fsubst W (d+1) T)
28 | Flat2 I ⇒ ⓕ{I} (fsubst W d V). (fsubst W d T)
32 interpretation "functional core substitution" 'Subst V d T = (fsubst V d T).
34 (* Main properties **********************************************************)
36 theorem fsubst_delift: ∀K,V,T,L,d.
37 ⇩[0, d] L ≡ K. ⓓV → L ⊢ T ▼*[d, 1] ≡ [d ← V] T.
39 [ * #i #L #d #HLK normalize in ⊢ (? ? ? ? ? %); /2 width=3/
40 elim (lt_or_eq_or_gt i d) #Hid
41 [ -HLK >(tri_lt ?????? Hid) /3 width=3/
42 | destruct >tri_eq /4 width=4 by tpss_strap, tps_subst, le_n, ex2_1_intro/ (**) (* too slow without trace *)
43 | -HLK >(tri_gt ?????? Hid) /3 width=3/
45 | * /3 width=1/ /4 width=1/
49 (* Main inversion properties ************************************************)
51 theorem fsubst_inv_delift: ∀K,V,T1,L,T2,d. ⇩[0, d] L ≡ K. ⓓV →
52 L ⊢ T1 ▼*[d, 1] ≡ T2 → [d ← V] T1 = T2.
54 [ * #i #L #T2 #d #HLK #H
55 [ -HLK >(delift_inv_sort1 … H) -H //
56 | elim (lt_or_eq_or_gt i d) #Hid normalize
57 [ -HLK >(delift_inv_lref1_lt … H) -H // /2 width=1/
59 elim (delift_inv_lref1_be … H ? ?) -H // #K0 #V0 #V2 #HLK0
60 lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H >minus_plus <minus_n_n #HV2 #HVT2 destruct
61 >(delift_inv_refl_O2 … HV2) -V >(flift_inv_lift … HVT2) -V2 //
62 | -HLK >(delift_inv_lref1_ge … H) -H // /2 width=1/
64 | -HLK >(delift_inv_gref1 … H) -H //
66 | * #I #V1 #T1 #IHV1 #IHT1 #L #X #d #HLK #H
67 [ elim (delift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
68 <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // /2 width=1/
69 | elim (delift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
70 <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 //