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/computation/cprs_cprs.ma".
16 include "basic_2/computation/lprs.ma".
18 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
20 (* Advanced properties ******************************************************)
22 lemma lprs_pair: ∀I,L1,L2. L1 ⊢ ➡* L2 → ∀V1,V2. L1 ⊢ V1 ➡* V2 →
23 L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
24 /2 width=1 by TC_lpx_sn_pair/ qed.
26 (* Advanced inversion lemmas ************************************************)
28 lemma lprs_inv_pair1: ∀I,K1,L2,V1. K1. ⓑ{I} V1 ⊢ ➡* L2 →
29 ∃∃K2,V2. K1 ⊢ ➡* K2 & K1 ⊢ V1 ➡* V2 & L2 = K2. ⓑ{I} V2.
30 /3 width=3 by TC_lpx_sn_inv_pair1, lpr_cprs_trans/ qed-.
32 lemma lprs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➡* K2. ⓑ{I} V2 →
33 ∃∃K1,V1. K1 ⊢ ➡* K2 & K1 ⊢ V1 ➡* V2 & L1 = K1. ⓑ{I} V1.
34 /3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
36 (* Properties on context-sensitive parallel computation for terms ***********)
38 lemma lprs_cpr_trans: s_r_trans … cpr lprs.
39 /3 width=5 by s_r_trans_TC2, lpr_cprs_trans/ qed-.
41 (* Basic_1: was just: pr3_pr3_pr3_t *)
42 lemma lprs_cprs_trans: s_rs_trans … cpr lprs.
43 /3 width=5 by s_r_trans_TC1, lprs_cpr_trans/ qed-.
45 lemma lprs_cprs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ➡* L1 →
46 ∃∃T. L1 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
47 #L0 #T0 #T1 #HT01 #L1 #H elim H -L1
49 elim (cprs_lpr_conf_dx … HT01 … HL01) -L0 /2 width=3/
50 | #L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
51 elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2 #HT12
52 elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3 #HT03
53 elim (cprs_conf … HT2 … HT3) -T #T #HT2 #HT3
54 lapply (cprs_trans … HT03 … HT3) -T3
55 lapply (cprs_trans … HT12 … HT2) -T2 /2 width=3/
59 lemma lprs_cpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡* L1 →
60 ∃∃T. L1 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
61 /3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
63 lemma lprs_cprs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ➡* L1 →
64 ∃∃T. L0 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
65 #L0 #T0 #T1 #HT01 #L1 #HL01
66 elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01 #T #HT1
67 lapply (lprs_cprs_trans … HT1 … HL01) -HT1 /2 width=3/
70 lemma lprs_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡* L1 →
71 ∃∃T. L0 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
72 /3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
74 lemma cprs_bind2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 →
75 ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
76 #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
77 lapply (lprs_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
80 (* Inversion lemmas on context-sensitive parallel computation for terms *****)
82 (* Basic_1: was pr3_gen_abbr *)
83 lemma cprs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1.T1 ➡* U2 → (
84 ∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 &
87 ∃∃T2. L. ⓓV1 ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
88 #a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
90 [ #V0 #T0 #HV10 #HT10 #H destruct
91 elim (cpr_inv_abbr1 … HU02) -HU02 *
92 [ #V2 #T2 #HV02 #HT02 #H destruct
93 lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) /2 width=1/ -HT02 #HT02
94 lapply (cprs_strap1 … HV10 … HV02) -V0
95 lapply (cprs_trans … HT10 … HT02) -T0 /3 width=5/
97 lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02 /2 width=1/ -V0 #HT02
98 lapply (cprs_trans … HT10 … HT02) -T0 /3 width=3/
101 elim (lift_total U2 0 1) #U #HU2
102 lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0 /2 width=1/ /4 width=3/
106 (* More advanced properties *************************************************)
108 lemma lprs_pair2: ∀I,L1,L2. L1 ⊢ ➡* L2 → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
109 L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
110 /3 width=3 by lprs_pair, lprs_cprs_trans/ qed.