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15 include "basic_2/notation/relations/predsnstaralt_4.ma".
16 include "basic_2/computation/cpxs_cpxs.ma".
17 include "basic_2/computation/lpxs.ma".
19 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
21 (* alternative definition *)
22 definition lpxsa: ∀h. sd h → relation lenv ≝ λh,g. lpx_sn … (cpxs h g).
24 interpretation "extended parallel computation (local environment, sn variant) alternative"
25 'PRedSnStarAlt h g L1 L2 = (lpxsa h g L1 L2).
27 (* Main properties on the alternative definition ****************************)
29 theorem lpxsa_lpxs: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡➡*[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
30 /2 width=1 by lpx_sn_LTC_TC_lpx_sn/ qed-.
32 (* Main inversion lemmas on the alternative definition **********************)
34 theorem lpxs_inv_lpxsa: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → ⦃h, L1⦄ ⊢ ➡➡*[g] L2.
35 /3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpx_cpxs_trans/ qed-.
37 (* Alternative eliminators **************************************************)
39 lemma lpxs_ind_alt: ∀h,g. ∀R:relation lenv.
42 ⦃h, K1⦄ ⊢ ➡*[g] K2 → ⦃h, K1⦄ ⊢ V1 ➡*[g] V2 →
43 R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
45 ∀L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → R L1 L2.
46 /3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ qed-.