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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/unfold/sstas_sstas.ma".
16 include "basic_2/computation/lprs_cprs.ma".
17 include "basic_2/computation/cpxs_cpxs.ma".
18 include "basic_2/computation/cpds.ma".
20 (* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
22 (* Advanced properties ******************************************************)
24 lemma cpds_cprs_trans: ∀h,g,L,T1,T,T2.
25 ⦃h, L⦄ ⊢ T1 •*➡*[g] T → L ⊢ T ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
26 #h #g #L #T1 #T #T2 * #T0 #HT10 #HT0 #HT2
27 lapply (cprs_trans … HT0 … HT2) -T /2 width=3/
30 lemma sstas_cpds_trans: ∀h,g,L,T1,T,T2.
31 ⦃h, L⦄ ⊢ T1 •*[g] T → ⦃h, L⦄ ⊢ T •*➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
32 #h #g #L #T1 #T #T2 #HT1 * #T0 #HT0 #HT02
33 lapply (sstas_trans … HT1 … HT0) -T /2 width=3/
36 (* Advanced inversion lemmas ************************************************)
38 lemma cpds_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1. T1 •*➡*[g] U2 →
39 ∃∃V2,T2. L ⊢ V1 ➡* V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 •*➡*[g] T2 &
41 #h #g #a #L #V1 #T1 #U2 * #X #H1 #H2
42 elim (sstas_inv_bind1 … H1) -H1 #U #HTU1 #H destruct
43 elim (cprs_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HUT2 #H destruct /3 width=5/
46 lemma cpds_inv_abbr_abst: ∀h,g,a1,a2,L,V1,W2,T1,T2. ⦃h, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[g] ⓛ{a2}W2.T2 →
47 ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 •*➡*[g] T & ⇧[0, 1] ⓛ{a2}W2.T2 ≡ T & a1 = true.
48 #h #g #a1 #a2 #L #V1 #W2 #T1 #T2 * #X #H1 #H2
49 elim (sstas_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
50 elim (cprs_inv_abbr1 … H2) -H2 *
51 [ #V2 #U2 #HV12 #HU12 #H destruct
56 (* Advanced forward lemmas **************************************************)
58 lemma cpds_fwd_cpxs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
59 #h #g #L #T1 #T2 * /3 width=3 by cpxs_trans, sstas_cpxs, cprs_cpxs/