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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/computation/fpbs_fpbs.ma".
16 include "basic_2/dynamic/snv_lift.ma".
17 include "basic_2/dynamic/snv_cpcs.ma".
18 include "basic_2/dynamic/lsubsv_snv.ma".
20 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
22 (* Properties on context-free parallel reduction for local environments *****)
24 fact snv_cpr_lpr_aux: ∀h,g,G0,L0,T0.
25 (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
26 (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
27 (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
28 (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
29 ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g G1 L1 T1.
30 #h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
31 [ #k #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
32 >(cpr_inv_sort1 … H2) -X //
33 | #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
34 elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
35 elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
36 elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
37 lapply (fqup_lref … G1 … HLK1) #HKL
38 elim (cpr_inv_lref1 … H2) -H2
39 [ #H destruct -HLK1 /4 width=10 by fqup_fpbr, snv_lref/
40 | * #K0 #V0 #W0 #H #HVW0 #W0 -HV12
41 lapply (ldrop_mono … H … HLK1) -HLK1 -H #H destruct
42 lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 /4 width=7 by fqup_fpbr, snv_lift/
44 | #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
45 elim (snv_inv_gref … H1)
46 | #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
47 elim (snv_inv_bind … H1) -H1 #HV1 #HT1
48 elim (cpr_inv_bind1 … H2) -H2 *
49 [ #V2 #T2 #HV12 #HT12 #H destruct /4 width=8 by fqup_fpbr, snv_bind, lpr_pair/
50 | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
51 /4 width=10 by fqup_fpbr, snv_inv_lift, lpr_pair, ldrop_ldrop/
53 | #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
54 elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU1
55 elim (cpr_inv_appl1 … H2) -H2 *
56 [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
57 lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbr/ #HV2
58 lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbr/ #HT2
59 lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbr/ #H2l0
60 elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) -Hl0 -HVW1 -HV12 /2 width=1 by fqup_fpbr/ -HV1 #W2 #HVW2 #HW12
61 elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU1 … HT12 … HL12) /2 width=1 by fqup_fpbr/ -HT12 -HTU1 #X #HTU2 #H
62 elim (cprs_inv_abst1 … H) -H #W20 #U2 #HW120 #_ #H destruct
63 lapply (lpr_cprs_conf … HL12 … HW10) -L1 #HW10
64 lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
65 lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
66 elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
67 lapply (cpds_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?)
68 /2 width=7 by snv_appl, cprs_bind/
69 | #b #V2 #W20 #W2 #T20 #T2 #HV12 #HW202 #HT202 #H1 #H2 destruct
70 elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
71 elim (cpds_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
72 lapply (cprs_div … HW10 … HW230) -W30 #HW120
73 lapply (snv_ssta_aux … IH4 … Hl0 … HVW1) /2 width=1 by fqup_fpbr/ #HW10
74 lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #HlW10
75 elim (snv_fwd_da … HW20) #l #Hl
76 lapply (da_cpcs_aux … IH1 IH2 … HlW10 … Hl … HW120) // -HlW10
77 /3 width=5 by fpbr_fpbs_trans, fqup_fpbr, ssta_fpbs/ #H destruct
78 lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbr/ #HlV2
79 lapply (IH2 … Hl … HW202 … HL12) /2 width=1 by fqup_fpbr/ #HlW2
80 elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HVW1 … HV12 … HL12) /2 width=1 by fqup_fpbr/ #W3 #HV2W3 #HW103
81 lapply (ssta_da_conf … HV2W3 … HlV2) <minus_plus_m_m #HlW3
82 lapply (cpcs_cpr_strap1 … HW120 … HW202) -HW120 #HW102
83 lapply (lpr_cpcs_conf … HL12 … HW102) -HW102 #HW102
84 lapply (cpcs_canc_sn … HW103 … HW102) -W10 #HW32
85 lapply (IH1 … HV12 … HL12) /2 width=1 by fqup_fpbr/ -HV1 #HV2
86 lapply (IH1 … HW202 … HL12) /2 width=1 by fqup_fpbr/ -HW20 #HW2
87 lapply (IH1 … HT20 … HT202 … (L2.ⓛW2) ?) /2 width=1 by fqup_fpbr, lpr_pair/ -HT20 #HT2
88 lapply (snv_ssta_aux … IH4 … HlV2 … HV2W3)
89 /3 width=5 by fpbr_fpbs_trans, fqup_fpbr, cpr_lpr_fpbs/ #HW3
90 lapply (lsubsv_snv_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /3 width=3 by snv_bind, snv_cast/
91 @(lsubsv_abbr … l) /3 width=7 by fqup_fpbr/ #W #W0 #l0 #Hl0 #HV2W #HW20
92 lapply (lsstas_ssta_conf_pos … HV2W3 … HV2W) -HV2W #HW3W
93 @(lsstas_cpcs_lpr_aux … IH1 IH2 IH3 … HlW3 … HW3W … HlW2 … HW20 … HW32) //
94 [ /3 width=9 by fpbr_fpbs_trans, fqup_fpbr, cpr_lpr_ssta_fpbs/
95 | /3 width=5 by fpbr_fpbs_trans, fqup_fpbr, cpr_lpr_fpbs/
97 | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
98 elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
99 elim (cpds_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
100 elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
101 lapply (lpr_cprs_conf … HL12 … HW10) -HW10 #HW10
102 elim (cpds_cpr_lpr_aux … IH2 IH3 … HTU0 … HT02 (L2.ⓓW2)) /2 width=1 by fqup_fpbr, lpr_pair/ -HTU0 #X #HTU2 #H
103 elim (cprs_inv_abst1 … H) -H #W #U2 #HW1 #_ #H destruct -U3
104 elim (ssta_cpr_lpr_aux … IH3 … HVW1 … HV10 … HL12) /2 width=2 by fqup_fpbr/ -IH3 -HVW1 #X #H1 #H2
105 lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
106 elim (lift_total X 0 1) #W20 #H3
107 lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1 by ldrop_ldrop/ -H1 #HVW20
108 lapply (cpcs_lift … (L2.ⓓW2) … H3 … HW13 H2) /2 width=1 by ldrop_ldrop/ -HW13 -H3 -H2 #HW320
109 lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
110 elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
111 lapply (cpds_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) /2 width=1 by cprs_bind/ -HW3 -HTU2 #HTU2
112 lapply (IH2 … Hl0 … HV10 … HL12) /2 width=1 by fqup_fpbr/ -IH2 -Hl0 #Hl0
113 lapply (da_lift … Hl0 (L2.ⓓW2) … HV02) /2 width=1 by ldrop_ldrop/ -Hl0 #Hl0
114 lapply (IH1 … HW02 … HL12) /2 width=1 by fqup_fpbr/ -HW0 #HW2
115 lapply (IH1 … HV10 … HL12) /2 width=1 by fqup_fpbr/ -HV1 -HV10 #HV0
116 lapply (IH1 … HT02 (L2.ⓓW2) ?) /2 width=1 by fqup_fpbr, lpr_pair/ -L1 #HT2
117 lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /3 width=7 by snv_bind, snv_appl, ldrop_ldrop/
119 | #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4
120 elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #Hl0 #HTU1 #HUW1
121 elim (cpr_inv_cast1 … H2) -H2
122 [ * #W2 #T2 #HW12 #HT12 #H destruct
123 lapply (cpcs_cprs_strap1 … HUW1 W2 ?) /2 width=1 by cpr_cprs/ -HUW1 #H1
124 lapply (IH1 … HW12 … HL12) /2 width=1 by fqup_fpbr/ -HW1 -HW12 #HW2
125 lapply (IH1 … HT12 … HL12) /2 width=1 by fqup_fpbr/ -IH1 #HT2
126 elim (ssta_cpr_lpr_aux … IH3 … Hl0 … HTU1 … HT12 … HL12) /2 width=2 by fqup_fpbr/ -IH3 -HTU1 #U2 #HTU2 #HU12
127 lapply (IH2 … Hl0 … HT12 … HL12) /2 width=1 by fqup_fpbr/ -IH2 -HT1 -HT12 -Hl0 #Hl0
128 /4 width=7 by snv_cast, lpr_cpcs_conf, cpcs_canc_sn/
129 | #H -IH3 -IH2 -HW1 -HTU1 -HUW1
130 lapply (IH1 … H … HL12) /2 width=1 by fqup_fpbr/