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15 include "basic_2/computation/cpxs_cpxs.ma".
16 include "basic_2/computation/lpxs.ma".
18 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
20 (* Advanced properties ******************************************************)
22 lemma lpxs_pair: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
23 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h, g] V2 →
24 ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
25 /2 width=1 by TC_lpx_sn_pair/ qed.
27 (* Advanced inversion lemmas ************************************************)
29 lemma lpxs_inv_pair1: ∀h,g,I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
30 ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L2 = K2.ⓑ{I}V2.
31 /3 width=3 by TC_lpx_sn_inv_pair1, lpx_cpxs_trans/ qed-.
33 lemma lpxs_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
34 ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
35 /3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
37 (* Properties on context-sensitive extended parallel computation for terms **)
39 lemma lpxs_cpx_trans: ∀h,g,G. s_r_trans … (cpx h g G) (lpxs h g G).
40 /3 width=5 by s_r_trans_TC2, lpx_cpxs_trans/ qed-.
42 lemma lpxs_cpxs_trans: ∀h,g,G. s_rs_trans … (cpx h g G) (lpxs h g G).
43 /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ qed-.
45 lemma cpxs_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
46 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
47 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
48 /4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed.
50 (* Inversion lemmas on context-sensitive ext parallel computation for terms *)
52 lemma cpxs_inv_abst1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
53 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
55 #h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
56 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
57 elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
58 lapply (lpxs_cpx_trans … HT02 (L.ⓛV1) ?)
59 /3 width=5 by lpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
62 lemma cpxs_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
63 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
66 ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
67 #h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
69 [ #V0 #T0 #HV10 #HT10 #H destruct
70 elim (cpx_inv_abbr1 … HU02) -HU02 *
71 [ #V2 #T2 #HV02 #HT02 #H destruct
72 lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?)
73 /4 width=5 by lpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
75 lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
76 /4 width=3 by lpxs_pair, cpxs_trans, ex3_intro, or_intror/
79 elim (lift_total U2 0 1) #U #HU2
80 /6 width=11 by cpxs_strap1, cpx_lift, ldrop_ldrop, ex3_intro, or_intror/
84 (* More advanced properties *************************************************)
86 lemma lpxs_pair2: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
87 ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
88 /3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
90 (* Properties on supclosure *************************************************)
92 lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
93 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
94 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
95 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
96 [ /2 width=5 by ex3_2_intro/
97 | #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
98 lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
99 elim (lpx_fquq_trans … HT2 … HK1) -K
100 /3 width=7 by lpxs_strap2, cpxs_strap1, ex3_2_intro/
104 lemma lpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
105 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
106 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
107 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
108 #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
109 #L0 #T0 #HT10 #HT0 #HL0 elim (lpxs_fquq_trans … H2 … HL0) -L
110 #L #T3 #HT3 #HT32 #HL2 elim (fqus_cpxs_trans … HT3 … HT0) -T
111 /3 width=7 by cpxs_trans, fqus_strap1, ex3_2_intro/
114 lemma lpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
115 ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
116 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
117 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
118 #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
119 #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
120 #L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
121 /3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/