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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/notation/relations/predsnstar_5.ma".
16 include "basic_2/reduction/lpx.ma".
17 include "basic_2/computation/lprs.ma".
19 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
21 definition lpxs: ∀h. sd h → relation3 genv lenv lenv ≝
22 λh,g,G. TC … (lpx h g G).
24 interpretation "extended parallel computation (local environment, sn variant)"
25 'PRedSnStar h g G L1 L2 = (lpxs h g G L1 L2).
27 (* Basic eliminators ********************************************************)
29 lemma lpxs_ind: ∀h,g,G,L1. ∀R:predicate lenv. R L1 →
30 (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → R L → R L2) →
31 ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L2.
32 #h #g #G #L1 #R #HL1 #IHL1 #L2 #HL12
33 @(TC_star_ind … HL1 IHL1 … HL12) //
36 lemma lpxs_ind_dx: ∀h,g,G,L2. ∀R:predicate lenv. R L2 →
37 (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → R L → R L1) →
38 ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1.
39 #h #g #G #L2 #R #HL2 #IHL2 #L1 #HL12
40 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
43 (* Basic properties *********************************************************)
45 lemma lprs_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
48 lemma lpx_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
51 lemma lpxs_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡*[h, g] L.
54 lemma lpxs_strap1: ∀h,g,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
57 lemma lpxs_strap2: ∀h,g,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
60 lemma lpxs_pair_refl: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V.
61 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
63 (* Basic inversion lemmas ***************************************************)
65 lemma lpxs_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡*[h, g] L2 → L2 = ⋆.
66 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
68 lemma lpxs_inv_atom2: ∀h,g,G,L1. ⦃G, L1⦄ ⊢ ➡*[h, g] ⋆ → L1 = ⋆.
69 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
71 (* Basic forward lemmas *****************************************************)
73 lemma lpxs_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → |L1| = |L2|.
74 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.