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14
15 include "basic_2/static/aaa_drops.ma".
16 include "basic_2/static/lsuba_aaa.ma".
17 include "basic_2/rt_transition/lfpx_fqup.ma".
18
19 (* UNBOUND PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *******)
20
21 (* Properties with atomic arity assignment for terms ************************)
22
23 lemma cpx_aaa_conf_lfpx: ∀h,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
24                          ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
25                          ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
26 #h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
27 [ #G #L1 #s #X2 #HX
28   elim (cpx_inv_sort1 … HX) -HX //
29 | #I #G #L1 #V1 #B #_ #IH #X2 #HX #Y #HY
30   elim (lfpx_inv_zero_pair_sn … HY) -HY #L2 #V2 #HL12 #HV12 #H destruct
31   elim (cpx_inv_zero1_pair … HX) -HX
32   [ #H destruct /3 width=1 by aaa_zero/
33   | -HV12 * /4 width=7 by aaa_lifts, drops_refl, drops_drop/
34   ]
35 | #I #G #L1 #B #i #_ #IH #X2 #HX #Y #HY
36   elim (lfpx_inv_lref_bind_sn … HY) -HY #I2 #L2 #HL12 #H destruct
37   elim (cpx_inv_lref1_bind … HX) -HX
38   [ #H destruct /3 width=1 by aaa_lref/
39   | * /4 width=7 by aaa_lifts, drops_refl, drops_drop/
40   ]
41 | #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #HX #L2 #HL12
42   elim (lfpx_inv_bind … HL12) -HL12 #HV #HT
43   elim (cpx_inv_abbr1 … HX) -HX *
44   [ #V2 #T2 #HV12 #HT12 #H destruct
45     /5 width=2 by lfpx_bind_repl_dx, aaa_abbr, ext2_pair/
46   | #T2 #HT12 #HXT2 #H destruct -IHV
47     /4 width=7 by aaa_inv_lifts, drops_drop, drops_refl/
48   ]
49 | #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #H #L2 #HL12
50   elim (lfpx_inv_bind … HL12) -HL12 #HV #HT
51   elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
52   /5 width=2 by lfpx_bind_repl_dx, aaa_abst, ext2_pair/
53 | #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X2 #H #L2 #HL12
54   elim (lfpx_inv_flat … HL12) -HL12 #HV #HT
55   elim (cpx_inv_appl1 … H) -H *
56   [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
57   | #q #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
58     lapply (IHV … HV12 … HV) -IHV -HV12 -HV #HV2
59     lapply (IHT (ⓛ{q}W2.U2) … HT) -IHT -HT /2 width=1 by cpx_bind/ -L1 #H
60     elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
61     /5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
62   | #q #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
63     lapply (aaa_lifts G L2 … B … (L2.ⓓW2) … HV2) -HV2 /3 width=2 by drops_drop, drops_refl/ #HV2
64     lapply (IHT (ⓓ{q}W2.U2) … HT) -IHT -HT /2 width=1 by cpx_bind/ -L1 #H
65     elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
66   ]
67 | #G #L1 #V1 #T1 #A #_ #_ #IHV #IHT #X2 #HX #L2 #HL12
68   elim (lfpx_inv_flat … HL12) -HL12 #HV #HT
69   elim (cpx_inv_cast1 … HX) -HX
70   [ * #V2 #T2 #HV12 #HT12 #H destruct ] /3 width=1 by aaa_cast/
71 ]
72 qed-.
73
74 lemma lfpx_aaa_conf: ∀h,G,T. Conf3 … (λL. aaa G L T) (lfpx h G T).
75 /2 width=6 by cpx_aaa_conf_lfpx/ qed-.