matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_alt.ma

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  14
  15 include "basic_2/notation/relations/predsnstaralt_5.ma".
  16 include "basic_2/computation/cpxs_cpxs.ma".
  17 include "basic_2/computation/lpxs.ma".
  18
  19 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
  20
  21 (* alternative definition *)
  22 definition lpxsa: ∀h. sd h → relation3 genv lenv lenv ≝
  23                   λh,g,G. lpx_sn … (cpxs h g G).
  24
  25 interpretation "extended parallel computation (local environment, sn variant) alternative"
  26    'PRedSnStarAlt h g G L1 L2 = (lpxsa h g G L1 L2).
  27
  28 (* Main properties on the alternative definition ****************************)
  29
  30 theorem lpxsa_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡➡*[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
  31 /2 width=1 by lpx_sn_LTC_TC_lpx_sn/ qed-.
  32
  33 (* Main inversion lemmas on the alternative definition **********************)
  34
  35 theorem lpxs_inv_lpxsa: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ ➡➡*[h, g] L2.
  36 /3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpx_cpxs_trans/ qed-.
  37
  38 (* Alternative eliminators **************************************************)
  39
  40 lemma lpxs_ind_alt: ∀h,g,G. ∀R:relation lenv.
  41                     R (⋆) (⋆) → (
  42                        ∀I,K1,K2,V1,V2.
  43                        ⦃G, K1⦄ ⊢ ➡*[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 →
  44                        R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
  45                     ) →
  46                     ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1 L2.
  47 /3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ qed-.