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14
15 include "basic_2/rt_computation/cpxs_lfpx.ma".
16 include "basic_2/rt_computation/lfpxs_fqup.ma".
17
18 (* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
19
20 (* Properties with uncounted context-sensitive rt-computation for terms *****)
21
22 lemma lfpxs_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
23                   ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
24 #h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
25 /3 width=3 by lfpxs_strap1, lfpx_pair/
26 qed.
27
28 (* Basic_2A1: was just: lpxs_cpx_trans *)
29 lemma lfpxs_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpxs h G).
30 /3 width=5 by s_r_trans_LTC2, lfpx_cpxs_trans/ qed-.
31
32 (* Note: lfpxs_cpx_conf does not hold, thus we cannot invoke s_r_trans_LTC1 *)
33 (* Basic_2A1: was just: lpxs_cpxs_trans *)
34 lemma lfpxs_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpxs h G).
35 #h #G @s_r_to_s_rs_trans @s_r_trans_LTC2
36 @s_rs_trans_TC1 /2 width=3 by lfpx_cpxs_trans/ (**) (* full auto too slow *)
37 qed-.
38
39 (* Advanced properties on uncounted rt-computation for terms ****************)
40
41 lemma cpxs_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
42                   ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
43                   ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
44 /4 width=3 by lfpxs_cpxs_trans, lfpxs_pair, cpxs_bind/ qed.
45
46 (* Advanced inversion lemmas on uncounted rt-computation for terms **********)
47
48 lemma cpxs_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
49                       ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
50                                U2 = ⓛ{p}V2.T2.
51 #h #p #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
52 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
53 elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
54 lapply (lfpxs_cpx_trans … HT02 (L.ⓛV1) ?)
55 /3 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
56 qed-.
57
58 lemma cpxs_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → (
59                       ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
60                                U2 = ⓓ{p}V2.T2
61                       ) ∨
62                       ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≡ T2 & p = true.
63 #h #p #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
64 #U0 #U2 #_ #HU02 * *
65 [ #V0 #T0 #HV10 #HT10 #H destruct
66   elim (cpx_inv_abbr1 … HU02) -HU02 *
67   [ #V2 #T2 #HV02 #HT02 #H destruct
68     lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?)
69     /4 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
70   | #T2 #HT02 #HUT2
71     lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
72     /4 width=3 by lfpxs_pair, cpxs_trans, ex3_intro, or_intror/
73   ]
74 | #U1 #HTU1 #HU01
75   elim (cpx_lifts … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
76   /4 width=3 by cpxs_strap1, drops_refl, drops_drop, ex3_intro, or_intror/
77 ]
78 qed-.