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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 *)
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11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/computation/scpds_scpds.ma".
16 include "basic_2/equivalence/scpes.ma".
18 (* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
20 (* Advanced inversion lemmas ************************************************)
22 lemma scpes_inv_abst2: ∀h,o,a,G,L,T1,T2,W2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] ⓛ{a}W2.T2 →
23 ∃∃W,T. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d1] ⓛ{a}W.T & ⦃G, L⦄ ⊢ W2 ➡* W &
24 ⦃G, L.ⓛW2⦄ ⊢ T2 •*➡*[h, o, d2] T.
25 #h #o #a #G #L #T1 #T2 #W2 #d1 #d2 * #T0 #HT10 #H
26 elim (scpds_inv_abst1 … H) -H #W #T #HW2 #HT2 #H destruct /2 width=5 by ex3_2_intro/
29 (* Advanced properties ******************************************************)
31 lemma scpes_refl: ∀h,o,G,L,T,d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, o] d1 →
32 ⦃G, L⦄ ⊢ T •*⬌*[h, o, d2, d2] T.
33 #h #o #G #L #T #d1 #d2 #Hd21 #Hd1
34 elim (da_lstas … Hd1 … d2) #U #HTU #_
35 /3 width=3 by scpds_div, lstas_scpds/
38 lemma lstas_scpes_trans: ∀h,o,G,L,T1,d0,d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d0 → d1 ≤ d0 →
39 ∀T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
40 ∀T2,d,d2. ⦃G, L⦄ ⊢ T •*⬌*[h,o,d,d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h,o,d1+d,d2] T2.
41 #h #o #G #L #T1 #d0 #d1 #Hd0 #Hd10 #T #HT1 #T2 #d #d2 *
42 /3 width=3 by scpds_div, lstas_scpds_trans/ qed-.
44 (* Properties on parallel computation for terms *****************************)
46 lemma cprs_scpds_div: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T →
47 ∀d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d →
48 ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*➡*[h, o, d2] T →
49 ⦃G, L⦄⊢ T1 •*⬌*[h, o, 0, d2] T2.
50 #h #o #G #L #T1 #T #HT1 #d #Hd elim (da_lstas … Hd 0)
51 #X1 #HTX1 #_ elim (cprs_strip … HT1 X1) -HT1
52 /3 width=5 by scpds_strap1, scpds_div, lstas_cpr, ex4_2_intro/
55 (* Main properties **********************************************************)
57 theorem scpes_trans: ∀h,o,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d] T →
58 ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
59 #h #o #G #L #T1 #T #d1 #d * #X1 #HT1X1 #HTX1 #T2 #d2 * #X2 #HTX2 #HT2X2
60 elim (scpds_conf_eq … HTX1 … HTX2) -T -d /3 width=5 by scpds_cprs_trans, scpds_div/
63 theorem scpes_canc_sn: ∀h,o,G,L,T,T1,d,d1. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d1] T1 →
64 ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
65 /3 width=4 by scpes_trans, scpes_sym/ qed-.
67 theorem scpes_canc_dx: ∀h,o,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d] T →
68 ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*⬌*[h, o, d2, d] T → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
69 /3 width=4 by scpes_trans, scpes_sym/ qed-.