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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/unfold/sstas.ma".
16 include "basic_2/computation/cprs.ma".
18 (* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
20 definition dxprs: ∀h. sd h → lenv → relation term ≝ λh,g,L,T1,T2.
21 ∃∃T. ⦃h, L⦄ ⊢ T1 •*[g] T & L ⊢ T ➡* T2.
23 interpretation "decomposed extended parallel computation (term)"
24 'DecomposedPRedStar h g L T1 T2 = (dxprs h g L T1 T2).
26 (* Basic properties *********************************************************)
28 lemma dxprs_refl: ∀h,g,L. reflexive … (dxprs h g L).
31 lemma sstas_dxprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
34 lemma cprs_dxprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
37 lemma dxprs_strap1: ∀h,g,L,T1,T,T2.
38 ⦃h, L⦄ ⊢ T1 •*➡*[g] T → L ⊢ T ➡ T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
39 #h #g #L #T1 #T #T2 * /3 width=5/
42 lemma dxprs_strap2: ∀h,g,L,T1,T,T2,l.
43 ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T⦄ → ⦃h, L⦄ ⊢ T •*➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
44 #h #g #L #T1 #T #T2 #l #HT1 * /3 width=4/
47 lemma ssta_cprs_dxprs: ∀h,g,L,T1,T,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T⦄ →
48 L ⊢ T ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
51 (* Basic inversion lemmas ***************************************************)
53 lemma dxprs_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1. T1 •*➡*[g] U2 →
54 ∃∃V2,T2. L ⊢ V1 ➡* V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 •*➡*[g] T2 &
56 #h #g #a #L #V1 #T1 #U2 * #X #H1 #H2
57 elim (sstas_inv_bind1 … H1) -H1 #U #HTU1 #H destruct
58 elim (cprs_fwd_abst1 … H2 Abst V1) -H2 #V2 #T2 #HV12 #HUT2 #H destruct /3 width=5/