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15 include "basic_2/substitution/lpss_ldrop.ma".
16 include "basic_2/static/aaa_lift.ma".
18 (* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
20 (* Properties about sn parallel substitution ********************************)
23 lemma aaa_cpss_lpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. L1 ⊢ T1 ▶* T2 →
24 ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T2 ⁝ A.
25 #L1 #T1 #A #H elim H -L1 -T1 -A
27 >(cpss_inv_sort1 … H) -H //
28 | #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
29 elim (cpss_inv_lref1 … H) -H
31 elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
32 elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
33 | * #Y #Z #V2 #H #HV12 #HV2
34 lapply (ldrop_mono … H … HLK1) -H #H destruct
35 elim (lpss_ldrop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
36 elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
37 lapply (ldrop_fwd_ldrop2 … HLK2) -V0 /3 width=7/
39 | #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
40 elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=2/
41 | #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
42 elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=1/
43 | #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
44 elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=3/
45 | #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
46 elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/
50 lemma aaa_cpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ▶* T2 → L ⊢ T2 ⁝ A.
51 /2 width=5 by aaa_cpss_lpss_conf/ qed-.
53 lemma aaa_lpss_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ▶* L2 → L2 ⊢ T ⁝ A.
54 /2 width=5 by aaa_cpss_lpss_conf/ qed-.