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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/static/aaa_lift.ma".
16 include "basic_2/static/lsuba_aaa.ma".
17 include "basic_2/reduction/lpx_ldrop.ma".
19 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
21 (* Properties on atomic arity assignment for terms **************************)
24 lemma aaa_cpx_lpx_conf: ∀h,g,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
25 ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
26 ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
27 #h #g #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
29 elim (cpx_inv_sort1 … H) -H // * //
30 | #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
31 elim (cpx_inv_lref1 … H) -H
33 elim (lpx_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
34 elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=6/
35 | * #J #Y #Z #V2 #H #HV12 #HV2
36 lapply (ldrop_mono … H … HLK1) -H #H destruct
37 elim (lpx_ldrop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
38 elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
39 lapply (ldrop_fwd_ldrop2 … HLK2) -V0 /3 width=7/
41 | #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
42 elim (cpx_inv_abbr1 … H) -H *
43 [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=2/
44 | #T2 #HT12 #HT2 #H destruct -IHV1
45 @(aaa_inv_lift … (L2.ⓓV1) … HT2) -HT2 /2 width=1/ /3 width=1/
47 | #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
48 elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=1/
49 | #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
50 elim (cpx_inv_appl1 … H) -H *
51 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3/
52 | #b #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
53 lapply (IHV1 … HV12 … HL12) -IHV1 -HV12 #HV2
54 lapply (IHT1 (ⓛ{b}W2.U2) … HL12) -IHT1 /2 width=1/ -L1 #H
55 elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
56 lapply (lsuba_aaa_trans … HU2 (L2.ⓓⓝW2.V2) ?) -HU2 /3 width=3/
57 | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
58 lapply (aaa_lift G L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=1/ #HV2
59 lapply (IHT1 (ⓓ{b}W2.U2) … HL12) -IHT1 /2 width=1/ -L1 #H
60 elim (aaa_inv_abbr … H) -H /3 width=3/
62 | #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
63 elim (cpx_inv_cast1 … H) -H
64 [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1/
71 lemma aaa_cpx_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
72 /2 width=7 by aaa_cpx_lpx_conf/ qed-.
74 lemma aaa_lpx_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
75 /2 width=7 by aaa_cpx_lpx_conf/ qed-.
77 lemma aaa_cpr_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
78 /3 width=5 by aaa_cpx_conf, cpr_cpx/ qed-.
80 lemma aaa_lpr_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
81 /3 width=5 by aaa_lpx_conf, lpr_lpx/ qed-.