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4 (* ||A|| A project by Andrea Asperti *)
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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_abst *)
83 lemma cprs_inv_abst1: ∀a,L,W1,T1,U2. L ⊢ ⓛ{a}W1.T1 ➡* U2 →
84 ∃∃W2,T2. L ⊢ W1 ➡* W2 & L.ⓛW1 ⊢ T1 ➡* T2 &
86 #a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/
87 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
88 elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
89 lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?) /2 width=1/ -HT02 #HT02
90 lapply (cprs_strap1 … HV10 … HV02) -V0
91 lapply (cprs_trans … HT10 … HT02) -T0 /2 width=5/
94 lemma cprs_inv_abst: ∀a,L,W1,W2,T1,T2. L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
95 L ⊢ W1 ➡* W2 ∧ L.ⓛW1 ⊢ T1 ➡* T2.
96 #a #L #W1 #W2 #T1 #T2 #H
97 elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1/
100 (* Basic_1: was pr3_gen_abbr *)
101 lemma cprs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1.T1 ➡* U2 → (
102 ∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 &
105 ∃∃T2. L. ⓓV1 ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
106 #a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
108 [ #V0 #T0 #HV10 #HT10 #H destruct
109 elim (cpr_inv_abbr1 … HU02) -HU02 *
110 [ #V2 #T2 #HV02 #HT02 #H destruct
111 lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) /2 width=1/ -HT02 #HT02
112 lapply (cprs_strap1 … HV10 … HV02) -V0
113 lapply (cprs_trans … HT10 … HT02) -T0 /3 width=5/
115 lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02 /2 width=1/ -V0 #HT02
116 lapply (cprs_trans … HT10 … HT02) -T0 /3 width=3/
119 elim (lift_total U2 0 1) #U #HU2
120 lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0 /2 width=1/ /4 width=3/
124 (* More advanced properties *************************************************)
126 lemma lprs_pair2: ∀I,L1,L2. L1 ⊢ ➡* L2 → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
127 L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
128 /3 width=3 by lprs_pair, lprs_cprs_trans/ qed.