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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/relocation/ldrop_ldrop.ma".
16 include "basic_2/relocation/fsupq.ma".
17 include "basic_2/reduction/cpx.ma".
19 (* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
21 (* Relocation properties ****************************************************)
23 lemma cpx_lift: ∀h,g. l_liftable (cpx h g).
24 #h #g #K #T1 #T2 #H elim H -K -T1 -T2
25 [ #I #K #L #d #e #_ #U1 #H1 #U2 #H2
26 >(lift_mono … H1 … H2) -H1 -H2 //
27 | #K #k #l #Hkl #L #d #e #_ #U1 #H1 #U2 #H2
28 >(lift_inv_sort1 … H1) -U1
29 >(lift_inv_sort1 … H2) -U2 /2 width=2/
30 | #I #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #d #e #HLK #U1 #H #U2 #HWU2
31 elim (lift_inv_lref1 … H) * #Hid #H destruct
32 [ elim (lift_trans_ge … HVW2 … HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
33 elim (ldrop_trans_le … HLK … HKV) -K /2 width=2/ #X #HLK #H
34 elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #K #Y #HKV #HVY #H destruct /3 width=9/
35 | lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 // /2 width=1/ >plus_plus_comm_23 #HVU2
36 lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=7/
38 | #a #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
39 elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
40 elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=5/
41 | #I #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #d #e #HLK #U1 #H1 #U2 #H2
42 elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
43 elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
44 | #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #H #U2 #HTU2
45 elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
46 elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=5/
47 | #K #V #T1 #T2 #_ #IHT12 #L #d #e #HLK #U1 #H #U2 #HTU2
48 elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
49 | #K #V1 #V2 #T #_ #IHV12 #L #d #e #HLK #U1 #H #U2 #HVU2
50 elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=5/
51 | #a #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #d #e #HLK #X1 #HX1 #X2 #HX2
52 elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
53 elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
54 elim (lift_inv_bind1 … HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
55 elim (lift_inv_flat1 … HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=5/
56 | #a #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #d #e #HLK #X1 #HX1 #X2 #HX2
57 elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
58 elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
59 elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
60 elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
61 elim (lift_trans_ge … HV2 … HV3) -V2 // /4 width=5/
65 lemma cpx_inv_lift1: ∀h,g. l_deliftable_sn (cpx h g).
66 #h #g #L #U1 #U2 #H elim H -L -U1 -U2
67 [ * #L #i #K #d #e #_ #T1 #H
68 [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
69 | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
70 | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
72 | #L #k #l #Hkl #K #d #e #_ #T1 #H
73 lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3/
74 | #I #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #d #e #HLK #T1 #H
75 elim (lift_inv_lref2 … H) -H * #Hid #H destruct
76 [ elim (ldrop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
77 elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
78 elim (lift_trans_le … HUV2 … HVW2) -V2 // >minus_plus <plus_minus_m_m // -Hid /3 width=9/
79 | elim (le_inv_plus_l … Hid) #Hdie #Hei
80 lapply (ldrop_conf_ge … HLK … HLV ?) -L // #HKLV
81 elim (lift_split … HVW2 d (i - e + 1)) -HVW2 [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
82 #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O /3 width=9/
84 | #a #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
85 elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
86 elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
87 elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=5/
88 | #I #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H
89 elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
90 elim (IHV12 … HLK … HWV1) -V1
91 elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5/
92 | #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #X #H
93 elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
94 elim (IHU1 (K.ⓓW1) … HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
95 elim (lift_div_le … HU2 … HTU) -U // /3 width=5/
96 | #L #V #U1 #U2 #_ #IHU12 #K #d #e #HLK #X #H
97 elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
98 elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3/
99 | #L #V1 #V2 #U1 #_ #IHV12 #K #d #e #HLK #X #H
100 elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
101 elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3/
102 | #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #d #e #HLK #X #HX
103 elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
104 elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
105 elim (IHV12 … HLK … HV01) -V1 #V3 #HV32 #HV03
106 elim (IHT12 (K.ⓛW0) … HT01) -T1 /2 width=1/ #T3 #HT32 #HT03
107 elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
108 @ex2_intro [2: /3 width=2/ | skip |3: /2 width=1/ ] (**) (* /4 width=6/ is slow *)
109 | #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #d #e #HLK #X #HX
110 elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
111 elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
112 elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
113 elim (IHT12 (K.ⓓW0) … HT01) -T1 /2 width=1/ #T3 #HT32 #HT03
114 elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
115 elim (lift_trans_le … HV3 … HV2) -V // #V #HV3 #HV2
116 @ex2_intro [2: /3 width=2/ | skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
120 (* Properties on supclosure *************************************************)
122 lemma fsupq_cpx_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
123 ∀U2. ⦃h, L2⦄ ⊢ T2 ➡[g] U2 →
124 ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
125 #h #g #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1: /2 width=3/ |3,4,5: /3 width=3/ ]
126 [ #I #L1 #V2 #U2 #HVU2
127 elim (lift_total U2 0 1) /4 width=9/
128 | #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2
129 elim (IHT12 … HTU2) -IHT12 -HTU2 #T #HT1 #HT2
130 elim (lift_total T d e) #U #HTU
131 lapply (cpx_lift … HT1 … HLK1 … HTU1 … HTU) -HT1 -HTU1 /3 width=11/
135 lemma fsupq_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
136 ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
137 ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
138 /3 width=4 by fsupq_cpx_trans, ssta_cpx/ qed-.
140 lemma fsup_cpx_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
141 ∀U2. ⦃h, L2⦄ ⊢ T2 ➡[g] U2 →
142 ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
143 /3 width=3 by fsupq_cpx_trans, fsup_fsupq/ qed-.
145 lemma fsup_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
146 ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
147 ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
148 /3 width=4 by fsupq_ssta_trans, fsup_fsupq/ qed-.