1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/computation/dxprs_dxprs.ma".
16 include "basic_2/dynamic/snv_lift.ma".
17 include "basic_2/dynamic/snv_cpcs.ma".
19 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
21 (* Properties on context-free parallel reduction for local environments *****)
23 fact snv_cpr_lpr_aux: ∀h,g,L0,T0.
24 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
25 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
26 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
27 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
28 ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g L1 T1.
29 #h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L1 * * [||||*]
30 [ #k #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
31 >(cpr_inv_sort1 … H2) -X //
32 | #i #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
33 elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
34 elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
35 elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
36 lapply (fsupp_lref … HLK1) #HKL
37 elim (cpr_inv_lref1 … H2) -H2
39 lapply (IH1 … HV12 … HK12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HKL /2 width=5/
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 #HLK2
43 lapply (IH1 … HVW0 … HK12) -IH1 -HVW0 -HK12 // -HV1 [ /2 width=1/ ] -HKL /3 width=7/
45 | #p #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
46 elim (snv_inv_gref … H1)
47 | #a #I #V1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
48 elim (snv_inv_bind … H1) -H1 #HV1 #HT1
49 elim (cpr_inv_bind1 … H2) -H2 *
50 [ #V2 #T2 #HV12 #HT12 #H destruct
51 lapply (IH1 … HV12 … HL12) // -HV1 [ /2 width=1/ ] #HV2
52 lapply (IH1 … HT12 (L2.ⓑ{I}V2) ?) // -IH1 /2 width=1/
53 | #T2 #HT12 #HXT2 #H1 #H2 destruct -HV1
54 lapply (IH1 … HT1 … HT12 (L2.ⓓV1) ?) -IH1 [1,2: /2 width=1/ ] -L1 -T1 #HT2
55 lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
57 | #V1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
58 elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
59 elim (cpr_inv_appl1 … H2) -H2 *
60 [ #V2 #T2 #HV12 #HT12 #H destruct -IH4
61 lapply (IH1 … HV1 … HV12 … HL12) [ /2 width=1/ ] #HV2
62 lapply (IH1 … HT1 … HT12 … HL12) [ /2 width=1/ ] #HT2
63 elim (IH3 … HVW1 … HV12 … HL12) -HVW1 -HV12 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
64 elim (dxprs_cprs_lpr_aux … IH2 IH1 IH3 … HTU1 … T2 … HL12) // [2,3: /2 width=1/ ] -IH2 -IH1 -IH3 -HT1 -HT12 -HTU1 #X #HTU2 #H
65 elim (cprs_fwd_abst1 … H Abst W1) -H #W20 #U2 #HW120 #_ #H destruct
66 lapply (cprs_lpr_conf … HL12 … HW10) -L1 #HW10
67 lapply (cpcs_cprs_strap1 … HW10 … HW120) -W1 #HW120
68 lapply (cpcs_canc_sn … HW12 HW120) -W10 #HW20
69 elim (cpcs_inv_cprs … HW20) -HW20 #W0 #HW20 #HW200
70 lapply (dxprs_cprs_trans … (ⓛ{a}W0.U2) HTU2 ?) [ /2 width=1/ ] -HW200 -HTU2 /2 width=8/
71 | #b #V2 #W20 #T20 #T2 #HV12 #HT202 #H1 #H2 destruct
72 elim (snv_inv_bind … HT1) -HT1 #HW20 #HT20
73 elim (dxprs_inv_abst1 … HTU1) -HTU1 #W30 #T30 #HW230 #_ #H destruct -T30
74 lapply (cprs_div … HW10 … HW230) -W30 #HW120
75 lapply (cpcs_lpr_conf … HL12 … HW120) -HW120 #HW120
76 elim (IH3 … HVW1 … HV12 … HL12) // [2: /2 width=1/ ] -HVW1 #W2 #HVW2 #HW102
77 lapply (cpcs_canc_sn … HW102 … HW120) -W10 #HW220
78 lapply (IH1 … HV12 … HL12) // [ /2 width=1/ ] #HV2 -HV1
79 lapply (IH1 … HW20 … W20 … HL12) // [ /2 width=1/ ] -HW20 #HW20
80 lapply (IH1 … HT20 … HT202 … (L2.ⓛW20) ?) [1,2: /2 width=1/ ] -HT20 #HT2
81 lapply (IH2 … HVW2) //
82 [ @(ygt_yprs_trans … L1 L1 … V1) (**) (* auto /4 width=5/ is a bit slow even with trace *)
83 [ /2 width=1 by fsupp_ygt/
84 | /3 width=1 by cprs_lpr_yprs, cpr_cprs/
87 elim (snv_fwd_ssta … HW20) #l0 #U20 #HWU20
88 elim (ssta_fwd_correct … HVW2) <minus_plus_m_m #U2 #HWU2
89 elim (ssta_cpcs_lpr_aux … IH1 IH3 … HWU2 … HWU20 … HW220 … L2) // -IH3
90 [2: /4 width=4 by ygt_yprs_trans, fsupp_ygt, ypr_yprs, ypr_lpr/
91 |3: @(ygt_yprs_trans … L1 L2 … V2) (**) (* auto not tried *)
92 [ @(ygt_yprs_trans … L1 L1 … V1)
93 [ /2 width=1 by fsupp_ygt/
94 | /3 width=1 by cprs_lpr_yprs, cpr_cprs/
96 | /3 width=2 by ypr_ssta, ypr_yprs/
98 ] #H #_ destruct -IH2 -U2
99 lapply (IH4 … HT2 (L2.ⓓV2) ?)
101 | @(ygt_yprs_trans … (L1.ⓛW20) … T2) (**) (* auto /5 width=5/ is too slow even with trace timeout=35 *)
102 [ /4 width=4 by ygt_yprs_trans, fsupp_ygt, ypr_yprs, ypr_cpr/
103 | /4 width=1 by ypr_yprs, ypr_lpr, lpr_pair/
105 ] -L1 -V1 -W2 -T20 -U20 -W20 -l0 /2 width=1/
106 | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
107 elim (snv_inv_bind … HT1) -HT1 #HW0 #HT0
108 elim (dxprs_inv_abbr_abst … HTU1) -HTU1 #X #HTU0 #HX #H destruct
109 elim (lift_inv_bind1 … HX) -HX #W3 #U3 #HW13 #_ #H destruct
110 lapply (cprs_lpr_conf … HL12 … HW10) -HW10 #HW10
111 elim (dxprs_cprs_lpr_aux … IH2 IH1 IH3 … HTU0 T2 … (L2.ⓓW2) ?) // [2,3,4: /2 width=1/ ] -IH2 -HTU0 #X #HTU2 #H
112 elim (cprs_fwd_abst1 … H Abst W3) -H #W #U2 #HW1 #_ #H destruct -U3
113 elim (IH3 … HVW1 … HV10 … HL12) // /2 width=1/ -IH3 -HVW1 #X #H1 #H2
114 lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
115 elim (lift_total X 0 1) #W20 #H3
116 lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1/ -H1 #HVW20
117 lapply (cpcs_lift (L2.ⓓW2) … H3 … HW13 H2) /2 width=1/ -HW13 -H3 -H2 #HW320
118 lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
119 elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
120 lapply (dxprs_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) [ /2 width=1/ ] -HW3 -HTU2 #HTU2
121 lapply (IH1 … HW02 … HL12) // [ /2 width=1/ ] -HW0 #HW2
122 lapply (IH1 … HV10 … HL12) // [ /2 width=1/ ] -HV1 -HV10 #HV0
123 lapply (IH1 … HT02 (L2.ⓓW2) ?) // [1,2: /2 width=1/ ] -L1 #HT2
124 lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /2 width=1/ -HV0 -HV02 /3 width=8/
126 | #W1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH2
127 elim (snv_inv_cast … H1) -H1 #U1 #l #HW1 #HT1 #HTU1 #HUW1
128 elim (cpr_inv_cast1 … H2) -H2
129 [ * #W2 #T2 #HW12 #HT12 #H destruct
130 lapply (cpcs_cprs_strap1 … HUW1 W2 ?) [ /2 width=1/ ] -HUW1 #H1
131 lapply (IH1 … HW12 … HL12) // /2 width=1/ -HW1 -HW12 #HW2
132 lapply (IH1 … HT12 … HL12) // /2 width=1/ -IH1 #HT2
133 elim (IH3 … HTU1 … HT12 … HL12) // /2 width=1/ -IH3 -HT1 -HT12 -HTU1 #U2 #HTU2 #HU12
134 lapply (cpcs_lpr_conf … HL12 … H1) -L1 #H1
135 lapply (cpcs_canc_sn … HU12 H1) -U1 /2 width=4/
136 | #H -IH3 -HW1 -HTU1 -HUW1
137 lapply (IH1 … H … HL12) // /2 width=1/