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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,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
23 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡* V2 → ⦃G, 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,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡* L2 →
29 ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
31 /3 width=3 by TC_lpx_sn_inv_pair1, lpr_cprs_trans/ qed-.
33 lemma lprs_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡* K2.ⓑ{I}V2 →
34 ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
36 /3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
38 (* Properties on context-sensitive parallel computation for terms ***********)
40 lemma lprs_cpr_trans: ∀G. s_r_trans … (cpr G) (lprs G).
41 /3 width=5 by s_r_trans_TC2, lpr_cprs_trans/ qed-.
43 (* Basic_1: was just: pr3_pr3_pr3_t *)
44 lemma lprs_cprs_trans: ∀G. s_rs_trans … (cpr G) (lprs G).
45 /3 width=5 by s_r_trans_TC1, lprs_cpr_trans/ qed-.
47 lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
48 ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
49 ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
50 #G #L0 #T0 #T1 #HT01 #L1 #H elim H -L1
52 elim (cprs_lpr_conf_dx … HT01 … HL01) -L0 /2 width=3/
53 | #L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
54 elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2 #HT12
55 elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3 #HT03
56 elim (cprs_conf … HT2 … HT3) -T #T #HT2 #HT3
57 lapply (cprs_trans … HT03 … HT3) -T3
58 lapply (cprs_trans … HT12 … HT2) -T2 /2 width=3/
62 lemma lprs_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
63 ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
64 ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
65 /3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
67 lemma lprs_cprs_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
68 ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
69 ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
70 #G #L0 #T0 #T1 #HT01 #L1 #HL01
71 elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01 #T #HT1
72 lapply (lprs_cprs_trans … HT1 … HL01) -HT1 /2 width=3/
75 lemma lprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
76 ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
77 ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
78 /3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
80 lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
81 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
82 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
83 #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
84 lapply (lprs_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
87 (* Inversion lemmas on context-sensitive parallel computation for terms *****)
89 (* Basic_1: was: pr3_gen_abst *)
90 lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
91 ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
93 #a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/
94 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
95 elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
96 lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?) /2 width=1/ -HT02 #HT02
97 lapply (cprs_strap1 … HV10 … HV02) -V0
98 lapply (cprs_trans … HT10 … HT02) -T0 /2 width=5/
101 lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
102 ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
103 #a #G #L #W1 #W2 #T1 #T2 #H
104 elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1/
107 (* Basic_1: was pr3_gen_abbr *)
108 lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
109 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
112 ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
113 #a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
115 [ #V0 #T0 #HV10 #HT10 #H destruct
116 elim (cpr_inv_abbr1 … HU02) -HU02 *
117 [ #V2 #T2 #HV02 #HT02 #H destruct
118 lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) /2 width=1/ -HT02 #HT02
119 lapply (cprs_strap1 … HV10 … HV02) -V0
120 lapply (cprs_trans … HT10 … HT02) -T0 /3 width=5/
122 lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02 /2 width=1/ -V0 #HT02
123 lapply (cprs_trans … HT10 … HT02) -T0 /3 width=3/
126 elim (lift_total U2 0 1) #U #HU2
127 lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0 /2 width=1/ /4 width=3/
131 (* More advanced properties *************************************************)
133 lemma lprs_pair2: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
134 ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
135 /3 width=3 by lprs_pair, lprs_cprs_trans/ qed.