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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/equivalence/lsubss_ssta.ma".
16 include "basic_2/dynamic/snv_cpcs.ma".
18 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
20 (* Properties on sn parallel reduction for local environments ***************)
22 fact ssta_cpr_lpr_aux: ∀h,g,L0,T0.
23 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
24 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
25 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
26 ∀L1,T1. L0 = L1 → T0 = T1 → IH_ssta_cpr_lpr h g L1 T1.
27 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [|||| *]
28 [ #k #_ #_ #_ #X2 #l #H2 #X3 #H3 #L2 #HL12 -IH3 -IH2 -IH1
29 elim (ssta_inv_sort1 … H2) -H2 #Hkl #H destruct
30 >(cpr_inv_sort1 … H3) -X3 /4 width=6/
31 | #i #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
32 elim (snv_inv_lref … H1) -H1 #I0 #K0 #V0 #H #HV1
33 elim (ssta_inv_lref1 … H2) -H2 * #K1
34 #V1 #W1 [| #l0 ] #HLK1 #HVW1 #HWU1 [| #H destruct ]
35 lapply (ldrop_mono … H … HLK1) -H #H destruct
36 lapply (fsupp_lref … HLK1) #HKV1
37 elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
38 elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
39 lapply (ldrop_fwd_ldrop2 … HLK2) #H2
40 elim (cpr_inv_lref1 … H3) -H3
41 [1,3: #H destruct -HLK1
42 |2,4: * #K0 #V0 #W0 #H #HVW0 #HW0
43 lapply (ldrop_mono … H … HLK1) -H -HLK1 #H destruct
45 [ elim (IH1 … HVW1 … HV12 … HK12) -IH1 -HVW1 // [2: /2 width=1/ ] -K1 -V1 #W2 #HVW2 #HW12
46 elim (lift_total W2 0 (i+1)) #U2 #HWU2
47 lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
48 | elim (IH1 … HVW1 … HV12 … HK12) -IH1 -HVW1 // [2: /2 width=1/ ] #W2 #HVW2 #_
49 elim (lift_total V2 0 (i+1)) #U2 #HVU2
50 lapply (lpr_cpr_conf … HK12 … HV12) -HV12 #HV12
51 lapply (cpcs_lift … H2 … HWU1 … HVU2 HV12) -H2 -V1 /3 width=6/
52 | elim (IH1 … HVW1 … HVW0 … HK12) -IH1 -HVW1 // [2: /2 width=1/ ] -K1 -V1 #W2 #HVW2 #HW12
53 elim (lift_total W2 0 (i+1)) #U2 #HWU2
54 lapply (ssta_lift … HVW2 … H2 … HW0 … HWU2) -HVW2 -HW0
55 lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
57 | #p #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
58 elim (snv_inv_gref … H1)
59 | #a #I #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
60 elim (snv_inv_bind … H1) -H1 #_ #HT1
61 elim (ssta_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
62 elim (cpr_inv_bind1 … H3) -H3 *
63 [ #V2 #T2 #HV12 #HT12 #H destruct
64 elim (IH1 … HTU1 … HT12 (L2.ⓑ{I}V2)) // [2,3: /2 width=1/ ] -T1
65 lapply (lpr_cpr_conf … HL12 … HV12) -L1 /3 width=5/
66 | #T2 #HT12 #HT2 #H1 #H2 destruct
67 elim (IH1 … HTU1 … HT12 (L2.ⓓV1)) -IH1 -HTU1 // [2,3: /2 width=1/ ] -T1 -HL12 #U2 #HTU2 #HU12
68 elim (ssta_inv_lift1 … HTU2 … HT2) -T2 [3: /2 width=1/ |2: skip ] #U #HXU #HU2
69 lapply (cpcs_bind1 true … V1 V1 … HU12) // -HU12 #HU12
70 lapply (cpcs_cpr_strap1 … HU12 U ?) -HU12 /2 width=3/
72 | #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct
73 elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
74 elim (ssta_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
75 elim (cpr_inv_appl1 … H3) -H3 *
76 [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH3 -IH2
77 elim (IH1 … HTU1 … HT12 … HL12) -IH1 -HTU1 // [2: /2 width=1/ ] -T1 #U2 #HTU2 #HU12
78 lapply (lpr_cpr_conf … HL12 … HV12) -L1 /3 width=5/
79 | #b #V2 #W #T2 #T20 #HV12 #HT20 #H1 #H2 destruct
80 elim (snv_inv_bind … HT1) -HT1 #HW #HT2
81 elim (ssta_inv_bind1 … HTU1) -HTU1 #U2 #HTU2 #H destruct
82 elim (dxprs_inv_abst1 … HTU10) -HTU10 #W0 #U0 #HW0 #_ #H destruct
83 lapply (cprs_div … HW10 … HW0) -W0 #HW1W
84 elim (ssta_fwd_correct … HVW1) <minus_plus_m_m #X1 #HWX1
85 elim (snv_fwd_ssta … HW) #l1 #V #HWV
86 lapply (IH3 … HVW1) -IH3 // [ /2 width=1/ ] #HW1
87 elim (ssta_cpcs_lpr_aux … IH2 IH1 … HWX1 … HWV … L1) -IH2 -HWX1 //
89 |3: /4 width=4 by ygt_yprs_trans, fsupp_ygt, sstas_yprs, ssta_sstas/
91 elim (IH1 … HVW1 … HV12 … HL12) -HVW1 // -HV1 [2: /2 width=1/ ] #W2 #HVW2 #HW12
92 elim (IH1 … HWV W … HL12) -HWV // -HW [2: /2 width=1/ ] #V0 #HWV0 #_
93 elim (IH1 … HTU2 … HT20 (L2.ⓛW)) -IH1 -HTU2 -HT20 // [2,3: /2 width=1/ ] -HT2 #U20 #HTU20 #HU20
94 lapply (cpcs_lpr_conf … HL12 … HW1W) #HW1W
95 lapply (lpr_cpr_conf … HL12 … HV12) -L1 #HV12
96 lapply (cpcs_ext_bind … HV12 … HU20 b Abbr) -HV12 -HU20 #HU20
97 lapply (cpcs_canc_sn … HW12 HW1W) -W1 #HW2
98 elim (lsubss_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
99 [ #U #HTU20 #HUU20 -HWV0 -W2
100 lapply (cpcs_bind1 b … V2 V2 … HUU20) // -HUU20 #HUU20
101 lapply (cpcs_canc_dx … HU20 … HUU20) -U20 #HU2
102 lapply (cpcs_cpr_strap2 … (ⓐV1.ⓛ{b}W.U2) … HU2) [ /2 width=1/ ] /3 width=3/
103 | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20 /2 width=6/
105 | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH3 -IH2
106 elim (ssta_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
107 elim (snv_inv_bind … HT1) -HT1 #_ #HT0
108 elim (IH1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -HTU0 // [2,3: /2 width=1/ ] -T0 #U2 #HTU2 #HU02
109 lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
110 lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
111 lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
112 lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
113 lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?) [ /2 width=3/ ] -V0 /4 width=3/
115 | #U0 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
116 elim (snv_inv_cast … H1) -H1 #T0 #l0 #_ #HT1 #HT10 #_
117 lapply (ssta_inv_cast1 … H2) -H2 #HTU1
118 elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
119 elim (cpr_inv_cast1 … H3) -H3
120 [ * #U2 #T2 #_ #HT12 #H destruct
121 elim (IH1 … HTU1 … HT12 … HL12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -T1 -U0 /3 width=3/
123 elim (IH1 … HTU1 … HT1X3 … HL12) -IH1 -HTU1 -HL12 // [2: /2 width=1/ ] -U0 -T1 /2 width=3/