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