1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/unfold/tpss_tpss.ma".
16 include "basic_2/unfold/ltpss_tpss.ma".
17 include "basic_2/static/sta_lift.ma".
19 (* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
21 (* Properties about parallel unfold *****************************************)
23 lemma sta_ltpss_tpss_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 →
24 ∀L2,d,e. L1 ▶* [d, e] L2 →
25 ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
26 ∃∃U2. ⦃h, L2⦄ ⊢ T2 • U2 & L2 ⊢ U1 ▶* [d, e] U2.
27 #h #L1 #T1 #U #H elim H -L1 -T1 -U
28 [ #L1 #k1 #L2 #d #e #_ #T2 #H
29 >(tpss_inv_sort1 … H) -H /2 width=3/
30 | #L1 #K1 #V1 #W1 #U1 #i #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
31 elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
33 elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
34 [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
35 elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
36 elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
37 lapply (ldrop_fwd_ldrop2 … HLK2) #H
38 elim (lift_total W2 0 (i+1)) #U2 #HWU2
39 lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
40 >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
41 | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
42 [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
43 elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
44 elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
45 lapply (ldrop_fwd_ldrop2 … HLK2) #H
46 elim (lift_total W2 0 (i+1)) #U2 #HWU2
47 lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
48 lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
49 | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
52 | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
53 elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
54 elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
55 lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
56 lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
57 lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
58 elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
59 elim (lift_total V2 0 (i+1)) #U2 #HVU2
60 lapply (sta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
61 lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
62 lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
64 | #L1 #K1 #W1 #V1 #U1 #i #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
65 elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
67 elim (lt_or_ge i d) #Hdi [ -HWV1 ]
68 [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
69 elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
70 elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
71 lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
72 elim (lift_total W2 0 (i+1)) #U2 #HWU2
73 lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
74 >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
75 | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
76 [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
77 elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
78 elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
79 lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
80 elim (lift_total W2 0 (i+1)) #U2 #HWU2
81 lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
82 lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
83 | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
86 | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
87 elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
88 elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
89 lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
91 | #I #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
92 elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
93 elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
94 | #L1 #V1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
95 elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
96 elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
97 | #L1 #W1 #T1 #U1 #_ #IHTU1 #L2 #d #e #HL12 #X #H
98 elim (tpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
99 elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=3/
103 lemma sta_ltpss_tps_conf: ∀h,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 • U1 →
104 ∀L2,d,e. L1 ▶* [d, e] L2 →
105 ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
106 ∃∃U2. ⦃h, L2⦄ ⊢ T2 • U2 & L2 ⊢ U1 ▶* [d, e] U2.
109 lemma sta_ltpss_conf: ∀h,L1,T,U1. ⦃h, L1⦄ ⊢ T • U1 →
110 ∀L2,d,e. L1 ▶* [d, e] L2 →
111 ∃∃U2. ⦃h, L2⦄ ⊢ T • U2 & L2 ⊢ U1 ▶* [d, e] U2.
114 lemma sta_tpss_conf: ∀h,L,T1,U1. ⦃h, L⦄ ⊢ T1 • U1 →
115 ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
116 ∃∃U2. ⦃h, L⦄ ⊢ T2 • U2 & L ⊢ U1 ▶* [d, e] U2.
119 lemma sta_tps_conf: ∀h,L,T1,U1. ⦃h, L⦄ ⊢ T1 • U1 →
120 ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
121 ∃∃U2. ⦃h, L⦄ ⊢ T2 • U2 & L ⊢ U1 ▶* [d, e] U2.