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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/cpds_cpds.ma".
16 include "basic_2/dynamic/snv_aaa.ma".
17 include "basic_2/dynamic/snv_cpcs.ma".
18 include "basic_2/dynamic/lsubsv_lsstas.ma".
20 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
22 (* Properties on sn parallel reduction for local environments ***************)
24 fact lsstas_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_snv_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_lsstas_cpr_lpr h g G1 L1 T1) →
29 ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_lsstas_cpr_lpr h g G1 L1 T1.
30 #h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
31 [ #k #_ #_ #_ #_ #l1 #l2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH4 -IH3 -IH2 -IH1
32 >(lsstas_inv_sort1 … H2) -X2
33 >(cpr_inv_sort1 … H3) -X3 /2 width=3 by cpr_atom, ex2_intro/
34 | #i #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus … l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3
35 [ lapply (lsstas_inv_O … H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ]
36 elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HK0 #HX0
37 elim (da_inv_lref … Hl1) -Hl1 * #K1 [ #V1 | #W1 #l0 ] #HLK1 [ #HVl1 | #HWl1 #H destruct ]
38 lapply (ldrop_mono … HK0 … HLK1) -HK0 #H destruct
39 elim (lsstas_inv_lref1 … H2) -H2 * #K0 #V0 #W0 [2,4: #l ] #HK0 [1,2: #Hl ] #HW0 #HX2
40 lapply (ldrop_mono … HK0 … HLK1) -HK0 #H destruct
41 [ lapply (da_mono … Hl … HWl1) -Hl #H destruct
42 lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21
44 lapply (fqup_lref … G1 … HLK1) #HKV1
45 elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
46 elim (lpr_inv_pair1 … H) -H #K2 [ #W2 | #V2 ] #HK12 [ #HW12 | #HV12 ] #H destruct
47 lapply (ldrop_fwd_drop2 … HLK2) #H2
48 elim (cpr_inv_lref1 … H3) -H3
49 [1,3: #H destruct -HLK1
50 |2,4: * #K0 #V0 #X0 #H #HVX0 #HX0
51 lapply (ldrop_mono … H … HLK1) -H -HLK1 #H destruct
53 [ elim (IH1 … HWl1 … HW0 … HW12 … HK12) -IH1 -HW0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
54 elim (lift_total V2 0 (i+1))
55 /6 width=12 by fqup_fpbg, cpcs_lift, lsstas_ldec, ex2_intro/
56 | elim (IH1 … HVl1 … HW0 … HV12 … HK12) -IH1 -HVl1 -HW0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbg/ #W2 #HVW2 #HW02
57 elim (lift_total W2 0 (i+1))
58 /4 width=12 by cpcs_lift, lsstas_ldef, ex2_intro/
59 | elim (IH1 … HVl1 … HW0 … HVX0 … HK12) -IH1 -HVl1 -HW0 -HVX0 -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -l1 #W2 #HXW2 #HW02
60 elim (lift_total W2 0 (i+1))
61 /3 width=12 by cpcs_lift, lsstas_lift, ex2_intro/
63 | #p #_ #_ #HT0 #H1 destruct -IH4 -IH3 -IH2 -IH1
64 elim (snv_inv_gref … H1)
65 | #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
66 elim (snv_inv_bind … H1) -H1 #_ #HT1
67 lapply (da_inv_bind … Hl1) -Hl1 #Hl1
68 elim (lsstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
69 elim (cpr_inv_bind1 … H3) -H3 *
70 [ #V2 #T2 #HV12 #HT12 #H destruct
71 elim (IH1 … Hl1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
72 /4 width=5 by cpcs_bind2, lpr_cpr_conf, lsstas_bind, ex2_intro/
73 | #T3 #HT13 #HXT3 #H1 #H2 destruct
74 elim (IH1 … Hl1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
75 elim (lsstas_inv_lift1 … HTU3 L2 … HXT3) -T3
76 /5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, ldrop_drop, ex2_intro/
78 | #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
79 elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HTU10
80 lapply (da_inv_flat … Hl1) -Hl1 #Hl1
81 elim (lsstas_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
82 elim (cpr_inv_appl1 … H3) -H3 *
83 [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH4 -IH3 -IH2
84 elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1
85 /4 width=5 by fqup_fpbg, cpcs_flat, lpr_cpr_conf, lsstas_appl, ex2_intro/
86 | #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
87 elim (snv_inv_bind … HT1) -HT1 #HW2 #HT2
88 lapply (da_inv_bind … Hl1) -Hl1 #Hl1
89 elim (lsstas_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
90 elim (cpds_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW20 #_ #H destruct
91 lapply (cprs_div … HW10 … HW20) -W0 #HW12
92 lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #H
93 elim (snv_fwd_da … HW2) #l #Hl
94 lapply (IH4 … HV1 … 1 … Hl0 W1 ?) /2 width=1 by fqup_fpbg, ssta_lsstas/ #HW1
95 lapply (da_cpcs_aux … IH3 IH2 … H … Hl … HW12) // -H
96 /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, ssta_fpbs/ #H destruct
97 lapply (IH3 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
98 lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2l
99 elim (IH1 … 1 … Hl0 … W1 … HV12 … HL12) /2 width=1 by fqup_fpbg, ssta_lsstas/ -HVW1 #W4 #H #HW14
100 lapply (lsstas_inv_SO … H) #HV2W4
101 lapply (ssta_da_conf … HV2W4 … HV2l) <minus_plus_m_m #HW4l
102 lapply (IH4 … HV2 … HV2l … H) -H /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/ #HW4
103 lapply (IH3 … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3
104 lapply (IH2 … Hl … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3l
105 elim (IH1 … Hl1 … HTU2 … HT23 (L2.ⓛW3)) -HTU2 /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
106 lapply (cpcs_cpr_strap1 … HW12 … HW23) #H
107 lapply (lpr_cpcs_conf … HL12 … H) -H #H
108 lapply (cpcs_canc_sn … HW14 H) -H #HW43
109 elim (lsubsv_lsstas_trans … HTU3 … Hl21 … (L2.ⓓⓝW3.V2)) -HTU3
110 [ #U4 #HT3U4 #HU43 -HW12 -HW3 -HW3l -W4 -IH2 -IH3 -IH4
111 @(ex2_intro … (ⓓ{b}ⓝW3.V2.U4)) /2 width=1 by lsstas_bind/ -HT3U4
112 @(cpcs_canc_dx … (ⓓ{b}ⓝW3.V2.U3)) /2 width=1 by cpcs_bind_dx/ -HU43
113 @(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
114 /4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
116 @(lsubsv_abbr … l) /3 width=7 by fqup_fpbg/
117 #W #W0 #l0 #Hl0 #HV2W #HW30
118 lapply (lsstas_ssta_conf_pos … HV2W4 … HV2W) -HV2W #HW4W
119 @(lsstas_cpcs_lpr_aux … IH3 IH2 IH1 … Hl0 … HW4W … Hl0 … HW30 … HW43) //
120 [ /3 width=9 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_ssta_fpbs/
121 | /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs/
123 | -IH1 -IH3 -IH4 /3 width=9 by fqup_fpbg, lpr_pair/
125 | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH4 -IH3 -IH2
126 elim (snv_inv_bind … HT1) -HT1 #_ #HT0
127 lapply (da_inv_bind … Hl1) -Hl1 #Hl1
128 elim (lsstas_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
129 elim (IH1 … Hl1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
130 lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
131 lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
132 lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
133 lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
134 lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?)
135 /4 width=3 by lsstas_appl, lsstas_bind, cpr_theta, ex2_intro/
137 | #W1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 @(nat_ind_plus … l2) -l2 [ #_ | #l2 #_ #Hl21 ] #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
138 [ lapply (lsstas_inv_O … H2) -H2 #H destruct -IH1 -H1 -l1 /4 width=5 by lpr_cpcs_conf, cpr_cpcs_dx, ex2_intro/ ]
139 elim (snv_inv_cast … H1) -H1 #U1 #l #_ #HT1 #_ #_ #_ -U1 -l
140 lapply (da_inv_flat … Hl1) -Hl1 #Hl1
141 lapply (lsstas_inv_cast1 … H2) -H2 #HTU1
142 elim (cpr_inv_cast1 … H3) -H3
143 [ * #U2 #T2 #_ #HT12 #H destruct
144 elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1 -HL12
145 /3 width=3 by fqup_fpbg, lsstas_cast, ex2_intro/
147 elim (IH1 … Hl1 … HTU1 … HT1X3 … HL12) -IH1 -Hl1 -HTU1 -HL12
148 /2 width=3 by fqup_fpbg, ex2_intro/