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15 include "basic_2/unfold/ltpss_sn_tps.ma".
17 (* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
19 (* Properties concerning partial unfold on terms ****************************)
21 lemma ltpss_sn_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
22 ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
23 L1 ⊢ T2 ▶* [d2, e2] U2.
24 #L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
26 lapply (ltpss_sn_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
29 lemma ltpss_sn_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
30 ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
31 ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
32 L0 ⊢ U2 ▶* [d1, e1] T.
33 #L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
35 | #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
36 elim (lt_or_ge i2 d1) #Hi2d1
37 [ elim (ltpss_sn_ldrop_conf_le … HL01 … HLK0 ?) /2 width=2/ #X #H #HLK1
38 elim (ltpss_sn_inv_tpss11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
39 lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
40 elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
41 lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // >minus_plus <plus_minus_m_m // /3 width=4/
42 | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
43 [ elim (ltpss_sn_ldrop_conf_be … HL01 … HLK0 ? ?) -HL01 // /2 width=2/ #X #H #HLK1
44 elim (ltpss_sn_inv_tpss21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
45 lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK0
46 elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
47 lapply (tpss_lift_ge … HV01 … HLK0 HVW0 … HVW1) -V0 -HLK0 // normalize #HW01
48 lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
49 | lapply (ltpss_sn_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 -HLK0 // /3 width=4/
52 | #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
53 elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
54 elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 #T #HT2 #H
55 lapply (tpss_lsubr_trans … H (L0.ⓑ{I}V) ?) -H /2 width=1/ /3 width=5/
56 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
57 elim (IHVW2 … HL01) -IHVW2
58 elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
62 lemma ltpss_sn_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
63 ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
64 L1 ⊢ T2 ▶* [d2, e2] U2.
65 #L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
67 lapply (ltpss_sn_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
70 lemma ltpss_sn_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
71 ∀L1,d1,e1. L1 ⊢ ▶* [d1, e1] L0 →
72 ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
73 L1 ⊢ T ▶* [d1, e1] U2.
74 #L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
76 | #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
77 elim (lt_or_ge i2 d1) #Hi2d1
78 [ elim (ltpss_sn_ldrop_trans_le … HL10 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
79 elim (ltpss_sn_inv_tpss12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
80 lapply (ldrop_fwd_ldrop2 … HLK1) #H
81 elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
82 lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
83 | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
84 [ elim (ltpss_sn_ldrop_trans_be … HL10 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
85 elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
86 lapply (ldrop_fwd_ldrop2 … HLK1) #H
87 elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
88 lapply (tpss_lift_ge … HV01 … H HVW1 … HVW0) -V0 -H // normalize #HW01
89 lapply (tpss_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
90 | lapply (ltpss_sn_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
93 | #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
94 elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
95 elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 #T #HT2 #H
96 lapply (tpss_lsubr_trans … H (L1.ⓑ{I}W2) ?) -H /2 width=1/ /3 width=5/
97 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
98 elim (IHVW2 … HL10) -IHVW2
99 elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/