1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/substitution/ltps_ldrop.ma".
16 include "basic_2/static/aaa_lift.ma".
18 (* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
20 (* Properties about parallel substitution ***********************************)
23 lemma aaa_ltps_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ÷ A → ∀L2,d,e. L1 [d, e] ▶ L2 →
24 ∀T2. L2 ⊢ T1 [d, e] ▶ T2 → L2 ⊢ T2 ÷ A.
25 #L1 #T1 #A #H elim H -L1 -T1 -A
26 [ #L1 #k #L2 #d #e #_ #T2 #H
27 >(tps_inv_sort1 … H) -H //
28 | #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H
29 elim (tps_inv_lref1 … H) -H
31 elim (lt_or_ge i d) #Hdi
32 [ elim (ltps_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
33 elim (ltps_inv_tps11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
35 | elim (lt_or_ge i (d + e)) #Hide
36 [ elim (ltps_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
37 elim (ltps_inv_tps21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
40 lapply (ltps_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide #HLK2
44 | * #K2 #V2 #Hdi #Hide #HLK2 #HVT2
45 elim (ltps_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
46 elim (ltps_inv_tps21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
47 lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
48 lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 /3 width=7/
50 | #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
51 elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
52 | #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
53 elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
54 | #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
55 elim (tps_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
56 | #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
57 elim (tps_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
61 lemma aaa_ltps_conf: ∀L1,T,A. L1 ⊢ T ÷ A → ∀L2,d,e. L1 [d, e] ▶ L2 → L2 ⊢ T ÷ A.
64 lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ÷ A → ∀T2,d,e. L ⊢ T1 [d, e] ▶ T2 → L ⊢ T2 ÷ A.