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15 include "basic_2/notation/relations/predsnstaralt_2.ma".
16 include "basic_2/computation/cprs_cprs.ma".
17 include "basic_2/computation/lprs.ma".
19 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
21 (* alternative definition *)
22 definition lprsa: relation lenv ≝ lpx_sn … cprs.
24 interpretation "parallel computation (local environment, sn variant) alternative"
25 'PRedSnStarAlt L1 L2 = (lprsa L1 L2).
27 (* Main properties on the alternative definition ****************************)
29 theorem lprsa_lprs: ∀L1,L2. L1 ⊢ ➡➡* L2 → L1 ⊢ ➡* L2.
30 /2 width=1 by lpx_sn_LTC_TC_lpx_sn/ qed-.
32 (* Main inversion lemmas on the alternative definition **********************)
34 theorem lprs_inv_lprsa: ∀L1,L2. L1 ⊢ ➡* L2 → L1 ⊢ ➡➡* L2.
35 /3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpr_cprs_trans/ qed-.
37 (* Alternative eliminators **************************************************)
39 lemma lprs_ind_alt: ∀R:relation lenv.
42 K1 ⊢ ➡* K2 → K1 ⊢ V1 ➡* V2 →
43 R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
45 ∀L1,L2. L1 ⊢ ➡* L2 → R L1 L2.
46 /3 width=4 by TC_lpx_sn_ind, lpr_cprs_trans/ qed-.