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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_2A/static/aaa_lift.ma".
16 include "basic_2A/static/lsuba_aaa.ma".
17 include "basic_2A/reduction/lpx_drop.ma".
19 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
21 (* Properties on atomic arity assignment for terms **************************)
24 lemma cpx_lpx_aaa_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_drop_conf … HLK1 … HL12) -L1 #X #H #HLK2
34 elim (lpx_inv_pair1 … H) -H
35 #K2 #V2 #HK12 #HV12 #H destruct /3 width=6 by aaa_lref/
36 | * #J #Y #Z #V2 #H #HV12 #HV2
37 lapply (drop_mono … H … HLK1) -H #H destruct
38 elim (lpx_drop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
39 elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
40 /3 width=8 by aaa_lift, drop_fwd_drop2/
42 | #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
43 elim (cpx_inv_abbr1 … H) -H *
44 [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=2 by lpx_pair, aaa_abbr/
45 | #T2 #HT12 #HT2 #H destruct -IHV1
46 /4 width=8 by lpx_pair, aaa_inv_lift, drop_drop/
48 | #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
49 elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
50 /4 width=1 by lpx_pair, aaa_abst/
51 | #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
52 elim (cpx_inv_appl1 … H) -H *
53 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
54 | #b #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
55 lapply (IHV1 … HV12 … HL12) -IHV1 -HV12 #HV2
56 lapply (IHT1 (ⓛ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
57 elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
58 /5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
59 | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
60 lapply (aaa_lift G L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=2 by drop_drop/ #HV2
61 lapply (IHT1 (ⓓ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
62 elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
64 | #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
65 elim (cpx_inv_cast1 … H) -H
66 [ * #V2 #T2 #HV12 #HT12 #H destruct /3 width=1 by aaa_cast/
67 | -IHV1 /2 width=1 by/
68 | -IHT1 /2 width=1 by/
73 lemma cpx_aaa_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
74 /2 width=7 by cpx_lpx_aaa_conf/ qed-.
76 lemma lpx_aaa_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
77 /2 width=7 by cpx_lpx_aaa_conf/ qed-.
79 lemma cpr_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
80 /3 width=5 by cpx_aaa_conf, cpr_cpx/ qed-.
82 lemma lpr_aaa_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
83 /3 width=5 by lpx_aaa_conf, lpr_lpx/ qed-.