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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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11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/reduction/lpr_lpr.ma".
16 include "basic_2/computation/cprs_lift.ma".
18 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
20 (* Main properties **********************************************************)
22 (* Basic_1: was: pr3_t *)
23 (* Basic_1: includes: pr1_t *)
24 theorem cprs_trans: ∀L. Transitive … (cprs L).
25 #L #T1 #T #HT1 #T2 @trans_TC @HT1 qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
27 (* Basic_1: was: pr3_confluence *)
28 (* Basic_1: includes: pr1_confluence *)
29 theorem cprs_conf: ∀L. confluent2 … (cprs L) (cprs L).
30 #L @TC_confluent2 /2 width=3 by cpr_conf/ qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
32 theorem cprs_ext_bind: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡* T2 →
33 ∀a,I. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
34 #L #V1 #V2 #H #V #T1 #T2 #HT12 #a #I @(TC_ind_dx … V1 H) -V1 /2 width=3/
35 #V1 #V0 #HV10 #_ #IHV02
36 @(cprs_trans … IHV02) /2 width=1/
39 theorem cprs_bind: ∀a,I,L,V1,V2,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 → L ⊢ V1 ➡* V2 →
40 L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
41 #a #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 /2 width=1/
43 @(cprs_trans … IHV1) -V1 /2 width=1/
46 (* Basic_1: was: pr3_flat *)
47 theorem cprs_flat: ∀I,L,V1,V2,T1,T2. L ⊢ T1 ➡* T2 → L ⊢ V1 ➡* V2 →
48 L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
49 #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 /2 width=1/
51 @(cprs_trans … IHV1) -IHV1 /2 width=1/
54 theorem cprs_beta: ∀a,L,V1,V2,W,T1,T2.
55 L.ⓛW ⊢ T1 ➡* T2 → L ⊢ V1 ➡* V2 →
56 L ⊢ ⓐV1.ⓛ{a}W.T1 ➡* ⓓ{a}V2.T2.
57 #a #L #V1 #V2 #W #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 /2 width=1/
59 @(cprs_trans … IHV1) /2 width=1/
62 theorem cprs_theta_rc: ∀a,L,V1,V,V2,W1,W2,T1,T2.
63 L ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → L.ⓓW1 ⊢ T1 ➡* T2 →
64 L ⊢ W1 ➡* W2 → L ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
65 #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2 /2 width=3/
67 @(cprs_trans … IHW1) /2 width=1/
70 theorem cprs_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2.
71 ⇧[0, 1] V ≡ V2 → L ⊢ W1 ➡* W2 → L.ⓓW1 ⊢ T1 ➡* T2 →
72 L ⊢ V1 ➡* V → L ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
73 #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /2 width=3/
74 #V1 #V0 #HV10 #_ #IHV0
75 @(cprs_trans … IHV0) /2 width=1/
78 (* Properties concerning sn parallel reduction on local environments ********)
80 (* Basic_1: was just: pr3_pr2_pr2_t *)
81 (* Basic_1: includes: pr3_pr0_pr2_t *)
82 lemma lpr_cpr_trans: s_r_trans … cpr lpr.
83 #L2 #T1 #T2 #HT12 elim HT12 -L2 -T1 -T2
85 | #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
86 elim (lpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
87 elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
88 lapply (IHV02 … HK12) -K2 #HV02
89 lapply (cprs_strap2 … HV10 … HV02) -V0 /2 width=6/
90 | #a #I #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
91 lapply (IHT12 (L1.ⓑ{I}V1) ?) -IHT12 /2 width=1/ /3 width=1/
93 | #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
94 lapply (IHT1 (L1.ⓓV2) ?) -IHT1 /2 width=1/ /2 width=3/
95 | #a #L2 #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
96 lapply (IHT12 (L1.ⓛW) ?) -IHT12 /2 width=1/ /3 width=1/
97 | #a #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
98 lapply (IHT12 (L1.ⓓW1) ?) -IHT12 /2 width=1/ /3 width=3/
102 lemma cpr_bind2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡ T2 →
103 ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
104 #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
105 lapply (lpr_cpr_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
108 (* Advanced properties ******************************************************)
110 (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
111 lemma lpr_cprs_trans: s_rs_trans … cpr lpr.
112 /3 width=5 by s_r_trans_TC1, lpr_cpr_trans/ qed-.
114 (* Basic_1: was: pr3_strip *)
115 (* Basic_1: includes: pr1_strip *)
116 lemma cprs_strip: ∀L. confluent2 … (cprs L) (cpr L).
117 #L @TC_strip1 /2 width=3 by cpr_conf/ qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
119 lemma cprs_lpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ➡ L1 →
120 ∃∃T. L1 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
121 #L0 #T0 #T1 #H elim H -T1
122 [ #T1 #HT01 #L1 #HL01
123 elim (lpr_cpr_conf_dx … HT01 … HL01) -L0 /3 width=3/
124 | #T #T1 #_ #HT1 #IHT0 #L1 #HL01
125 elim (IHT0 … HL01) #T2 #HT2 #HT02
126 elim (lpr_cpr_conf_dx … HT1 … HL01) -L0 #T3 #HT3 #HT13
127 elim (cprs_strip … HT2 … HT3) -T #T #HT2 #HT3
128 lapply (cprs_strap2 … HT13 … HT3) -T3
129 lapply (cprs_strap1 … HT02 … HT2) -T2 /2 width=3/
133 lemma cprs_lpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ➡ L1 →
134 ∃∃T. L0 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
135 #L0 #T0 #T1 #HT01 #L1 #HL01
136 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01 #T #HT1
137 lapply (lpr_cprs_trans … HT1 … HL01) -HT1 /2 width=3/
140 lemma cprs_bind2_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 →
141 ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
142 #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
143 lapply (lpr_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/