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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 *)
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15 include "basic_2/dynamic/snv_aaa.ma".
16 include "basic_2/dynamic/snv_scpes.ma".
17 include "basic_2/dynamic/lsubsv_lstas.ma".
19 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
21 (* Properties on sn parallel reduction for local environments ***************)
23 fact lstas_cpr_lpr_aux: ∀h,o,G0,L0,T0.
24 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_lstas h o G1 L1 T1) →
25 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h o G1 L1 T1) →
26 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h o G1 L1 T1) →
27 (∀G1,L1,T1. ⦃G0, L0, T0⦄ >≛[h, o] ⦃G1, L1, T1⦄ → IH_lstas_cpr_lpr h o G1 L1 T1) →
28 ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_lstas_cpr_lpr h o G1 L1 T1.
29 #h #o #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
30 [ #s #_ #_ #_ #_ #d1 #d2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH4 -IH3 -IH2 -IH1
31 >(lstas_inv_sort1 … H2) -X2
32 >(cpr_inv_sort1 … H3) -X3 /2 width=3 by ex2_intro/
33 | #i #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3
34 elim (snv_inv_lref … H1) -H1 #I0 #K0 #X0 #HK0 #HX0
35 elim (da_inv_lref … Hd1) -Hd1 * #K1 [ #V1 | #W1 #d ] #HLK1 [ #HVd1 | #HWd1 #H destruct ]
36 lapply (drop_mono … HK0 … HLK1) -HK0 #H destruct
37 elim (lstas_inv_lref1 … H2) -H2 * #K0 #V0 #X0 [3,6: #d0 ] #HK0 #HVX0 [1,2: #HX02 #H |3,5: #HX02 |4,6: #H1 #H2 ] destruct
38 lapply (drop_mono … HK0 … HLK1) -HK0 #H destruct
39 [ lapply (le_plus_to_le_c … Hd21) -Hd21 #Hd21 |3: -Hd21 ]
40 lapply (fqup_lref … G1 … HLK1) #HKV1
41 elim (lpr_drop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
42 elim (lpr_inv_pair1 … H) -H #K2 [ #W2 | #W2 | #V2 ] #HK12 [ #HW12 | #HW12 | #HV12 ] #H destruct
43 lapply (drop_fwd_drop2 … HLK2) #H2
44 elim (cpr_inv_lref1 … H3) -H3
45 [1,3,5: #H destruct -HLK1
46 |2,4,6: * #K #V #X #H #HVX #HX3
47 lapply (drop_mono … H … HLK1) -H -HLK1 #H destruct
49 [ lapply (IH2 … HWd1 … HW12 … HK12) /2 width=1 by fqup_fpbg/ -IH2 #H
50 elim (da_lstas … H d0) -H
51 elim (IH1 … HWd1 … HVX0 … HW12 … HK12) -IH1 -HVX0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
52 elim (lift_total V2 0 (i+1))
53 /3 width=12 by cpcs_lift, lstas_succ, ex2_intro/
54 | elim (IH1 … HWd1 … HVX0 … HW12 … HK12) -IH1 -HVX0
55 /3 width=5 by fqup_fpbg, lstas_zero, ex2_intro/
56 | elim (IH1 … HVd1 … HVX0 … HV12 … HK12) -IH1 -HVd1 -HVX0 -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, lstas_ldef, ex2_intro/
59 | elim (IH1 … HVd1 … HVX0 … HVX … HK12) -IH1 -HVd1 -HVX0 -HVX -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -d1 #W2 #HXW2 #HW02
60 elim (lift_total W2 0 (i+1))
61 /3 width=12 by cpcs_lift, lstas_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 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
66 elim (snv_inv_bind … H1) -H1 #_ #HT1
67 lapply (da_inv_bind … Hd1) -Hd1 #Hd1
68 elim (lstas_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 … Hd1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hd1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
72 /4 width=5 by cpcs_bind2, lpr_cpr_conf, lstas_bind, ex2_intro/
73 | #T3 #HT13 #HXT3 #H1 #H2 destruct
74 elim (IH1 … Hd1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hd1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
75 elim (lstas_inv_lift1 … HTU3 L2 … HXT3) -T3
76 /5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, drop_drop, ex2_intro/
78 | #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
79 elim (snv_inv_appl … H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
80 lapply (da_inv_flat … Hd1) -Hd1 #Hd1
81 elim (lstas_inv_appl1 … H2) -H2 #X #HT1U #H destruct
82 elim (cpr_inv_appl1 … H3) -H3 *
83 [ #V2 #T2 #HV12 #HT12 #H destruct -a -d0 -W1 -U1 -HV1 -IH4 -IH3 -IH2
84 elim (IH1 … Hd1 … HT1U … HT12 … HL12) -IH1 -Hd1 -HT1U
85 /4 width=5 by fqup_fpbg, cpcs_flat, lpr_cpr_conf, lstas_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 … Hd1) -Hd1 #Hd1
89 elim (lstas_inv_bind1 … HT1U) -HT1U #U #HT2U #H destruct
90 elim (scpds_inv_abst1 … HTU1) -HTU1 #W0 #U0 #HW20 #_ #H destruct -U0 -d0
91 elim (snv_fwd_da … HW2) #d0 #HW2d
92 lapply (cprs_scpds_div … HW20 … HW2d … HVW1) -W0 #H21
93 elim (snv_fwd_da … HV1) #d #HV1d
94 elim (da_scpes_aux … IH4 IH3 IH2 … HW2d … HV1d … H21) /2 width=1 by fqup_fpbg/ #_ #H
95 <minus_n_O #H0 destruct >(plus_minus_k_k d 1) in HV1d; // -H #HV1d
96 lapply (scpes_cpr_lpr_aux … IH2 IH1 … H21 … HW23 … HV12 … HL12) -H21 /2 width=1 by fqup_fpbg/ #H32
97 lapply (IH3 … HW23 … HL12) /2 width=1 by fqup_fpbg/ #HW3
98 lapply (IH3 … HV12 … HL12) /2 width=1 by fqup_fpbg/ #HV2
99 lapply (IH2 … HW2d … HW23 … HL12) /2 width=1 by fqup_fpbg/ -HW2 -HW2d #HW3d
100 lapply (IH2 … HV1d … HV12 … HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV1d #HV2d
101 elim (IH1 … Hd1 … HT2U … HT23 (L2.ⓛW3)) -HT2U /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
102 elim (lsubsv_lstas_trans … o … HTU3 … Hd21 … (L2.ⓓⓝW3.V2)) -HTU3
103 [ #U4 #HT3U4 #HU43 -IH1 -IH2 -IH3 -IH4 -d -d1 -HW3 -HV2 -HT2
104 @(ex2_intro … (ⓓ{b}ⓝW3.V2.U4)) /2 width=1 by lstas_bind/ -HT3U4
105 @(cpcs_canc_dx … (ⓓ{b}ⓝW3.V2.U3)) /2 width=1 by cpcs_bind_dx/ -HU43
106 @(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
107 /4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
109 @(lsubsv_beta … (d-1)) /3 width=7 by fqup_fpbg/
110 @shnv_cast [1,2: // ] #d0 #Hd0
111 lapply (scpes_le_aux … IH4 IH3 IH2 IH1 … HW3d … HV2d … d0 … H32) -IH4 -IH3 -IH2 -IH1 -HW3d -HV2d -H32
112 /3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
113 | -IH1 -IH3 -IH4 /3 width=9 by fqup_fpbg, lpr_pair/
115 | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -d0 -W1 -HV1 -IH4 -IH3 -IH2
116 elim (snv_inv_bind … HT1) -HT1 #_ #HT0
117 lapply (da_inv_bind … Hd1) -Hd1 #Hd1
118 elim (lstas_inv_bind1 … HT1U) -HT1U #U0 #HTU0 #H destruct
119 elim (IH1 … Hd1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hd1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
120 lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
121 lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
122 lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
123 lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
124 lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?)
125 /4 width=3 by lstas_appl, lstas_bind, cpr_theta, ex2_intro/
127 | #W1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
128 elim (snv_inv_cast … H1) -H1 #U1 #_ #HT1 #_ #_ -U1
129 lapply (da_inv_flat … Hd1) -Hd1 #Hd1
130 lapply (lstas_inv_cast1 … H2) -H2 #HTU1
131 elim (cpr_inv_cast1 … H3) -H3
132 [ * #U2 #T2 #_ #HT12 #H destruct
133 elim (IH1 … Hd1 … HTU1 … HT12 … HL12) -IH1 -Hd1 -HTU1 -HL12
134 /3 width=3 by fqup_fpbg, lstas_cast, ex2_intro/
135 | #HT1X3 elim (IH1 … Hd1 … HTU1 … HT1X3 … HL12) -IH1 -Hd1 -HTU1 -HL12
136 /2 width=3 by fqup_fpbg, ex2_intro/