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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/rt_computation/cpxs_lfpx.ma".
16 include "basic_2/rt_computation/lfpxs_fqup.ma".
18 (* UNBOUND PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ******)
20 (* Properties with unbound context-sensitive rt-computation for terms *******)
22 (* Basic_2A1: uses: lpxs_pair lpxs_pair_refl *)
23 lemma lfpxs_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
24 ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
25 /2 width=1 by tc_lfxs_pair_refl/ qed.
27 lemma lfpxs_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpxs h G).
28 #h #G @s_r_trans_CTC2 @lfpx_cpxs_trans (**) (* auto fails *)
31 (* Note: lfpxs_cpx_conf does not hold, thus we cannot invoke s_r_trans_CTC1 *)
32 lemma lfpxs_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpxs h G).
33 #h #G @s_r_to_s_rs_trans @s_r_trans_CTC2
34 @s_rs_trans_TC1 /2 width=3 by lfpx_cpxs_trans/ (**) (* full auto too slow *)
37 (* Advanced properties on unbound rt-computation for terms ******************)
39 lemma cpxs_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
40 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
41 ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
42 /4 width=3 by lfpxs_cpxs_trans, lfpxs_pair_refl, cpxs_bind/ qed.
44 (* Advanced inversion lemmas on unbound rt-computation for terms ************)
46 lemma cpxs_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
47 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
49 #h #p #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
50 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
51 elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
52 lapply (lfpxs_cpx_trans … HT02 (L.ⓛV1) ?)
53 /3 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro/
56 lemma cpxs_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → (
57 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
60 ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = true.
61 #h #p #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
63 [ #V0 #T0 #HV10 #HT10 #H destruct
64 elim (cpx_inv_abbr1 … HU02) -HU02 *
65 [ #V2 #T2 #HV02 #HT02 #H destruct
66 lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?)
67 /4 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
69 lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
70 /4 width=3 by lfpxs_pair_refl, cpxs_trans, ex3_intro, or_intror/
73 elim (cpx_lifts_sn … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
74 /4 width=3 by cpxs_strap1, drops_refl, drops_drop, ex3_intro, or_intror/