1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/unfold/lstas_lstas.ma".
16 include "basic_2/computation/lprs_cprs.ma".
17 include "basic_2/computation/cpxs_cpxs.ma".
18 include "basic_2/computation/scpds.ma".
20 (* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
22 (* Advanced properties ******************************************************)
24 lemma scpds_strap2: ∀h,g,G,L,T1,T,T2,l1,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l1+1 →
25 ⦃G, L⦄ ⊢ T1 •[h] T → ⦃G, L⦄ ⊢ T •*➡*[h, g, l] T2 →
26 ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l+1] T2.
27 #h #g #G #L #T1 #T #T2 #l1 #l #Hl1 #HT1 *
28 #T0 #l0 #Hl0 #HTl0 #HT0 #HT02
29 lapply (da_sta_conf … HT1 … Hl1) <minus_plus_m_m #HTl1
30 lapply (da_mono … HTl0 … HTl1) -HTl0 -HTl1 #H destruct
31 /3 width=6 by lstas_step_sn, le_S_S, ex4_2_intro/
34 lemma scpds_cprs_trans: ∀h,g,G,L,T1,T,T2,l.
35 ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T2.
36 #h #g #G #L #T1 #T #T2 #l * /3 width=8 by cprs_trans, ex4_2_intro/
39 lemma lstas_scpds_trans: ∀h,g,G,L,T1,T,T2,l1,l2,l.
40 l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 →
41 ⦃G, L⦄ ⊢ T1 •*[h, l2] T → ⦃G, L⦄ ⊢ T •*➡*[h, g, l] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l2+l] T2.
42 #h #g #G #L #T1 #T #T2 #l1 #l2 #l #Hl21 #HTl1 #HT1 * #T0 #l0 #Hl0 #HTl0 #HT0 #HT02
43 lapply (lstas_da_conf … HT1 … HTl1) #HTl12
44 lapply (da_mono … HTl12 … HTl0) -HTl12 -HTl0 #H destruct
45 lapply (le_minus_to_plus_r … Hl21 Hl0) -Hl21 -Hl0
46 /3 width=7 by lstas_trans, ex4_2_intro/
49 (* Advanced inversion lemmas ************************************************)
51 lemma scpds_inv_abst1: ∀h,g,a,G,L,V1,T1,U2,l. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 •*➡*[h, g, l] U2 →
52 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 •*➡*[h, g, l] T2 &
54 #h #g #a #G #L #V1 #T1 #U2 #l2 * #X #l1 #Hl21 #Hl1 #H1 #H2
55 lapply (da_inv_bind … Hl1) -Hl1 #Hl1
56 elim (lstas_inv_bind1 … H1) -H1 #U #HTU1 #H destruct
57 elim (cprs_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HUT2 #H destruct
58 /3 width=6 by ex4_2_intro, ex3_2_intro/
61 lemma scpds_inv_abbr_abst: ∀h,g,a1,a2,G,L,V1,W2,T1,T2,l. ⦃G, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[h, g, l] ⓛ{a2}W2.T2 →
62 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 •*➡*[h, g, l] T & ⇧[0, 1] ⓛ{a2}W2.T2 ≡ T & a1 = true.
63 #h #g #a1 #a2 #G #L #V1 #W2 #T1 #T2 #l2 * #X #l1 #Hl21 #Hl1 #H1 #H2
64 lapply (da_inv_bind … Hl1) -Hl1 #Hl1
65 elim (lstas_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
66 elim (cprs_inv_abbr1 … H2) -H2 *
67 [ #V2 #U2 #HV12 #HU12 #H destruct
68 | /3 width=6 by ex4_2_intro, ex3_intro/
72 lemma scpds_inv_lstas_eq: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T2 →
73 ∀T. ⦃G, L⦄ ⊢ T1 •*[h, l] T → ⦃G, L⦄ ⊢ T ➡* T2.
74 #h #g #G #L #T1 #T2 #l2 *
75 #T0 #l1 #_ #_ #HT10 #HT02 #T #HT1
76 lapply (lstas_mono … HT10 … HT1) #H destruct //
79 (* Advanced forward lemmas **************************************************)
81 lemma scpds_fwd_cpxs: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
82 #h #g #G #L #T1 #T2 #l * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
85 (* Main properties **********************************************************)
87 theorem scpds_conf_eq: ∀h,g,G,L,T0,T1,l. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, l] T1 →
88 ∀T2. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, l] T2 →
89 ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
90 #h #g #G #L #T0 #T1 #l0 * #U1 #l1 #_ #_ #H1 #HUT1 #T2 * #U2 #l2 #_ #_ #H2 #HUT2 -l1 -l2
91 lapply (lstas_mono … H1 … H2) #H destruct -h -l0 /2 width=3 by cprs_conf/