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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/notation/relations/predsnstaralt_3.ma".
  16 include "basic_2/computation/cprs_cprs.ma".
  17 include "basic_2/computation/lprs.ma".
  18
  19 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
  20
  21 (* alternative definition *)
  22 definition lprsa: relation3 genv lenv lenv ≝
  23            λG. lpx_sn … (cprs G).
  24
  25 interpretation "parallel computation (local environment, sn variant) alternative"
  26    'PRedSnStarAlt G L1 L2 = (lprsa G L1 L2).
  27
  28 (* Main properties on the alternative definition ****************************)
  29
  30 theorem lprsa_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡➡* L2 → ⦃G, L1⦄ ⊢ ➡* 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 lprs_inv_lprsa: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡➡* L2.
  36 /3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpr_cprs_trans/ qed-.
  37
  38 (* Alternative eliminators **************************************************)
  39
  40 lemma lprs_ind_alt: ∀G. ∀R:relation lenv.
  41                     R (⋆) (⋆) → (
  42                        ∀I,K1,K2,V1,V2.
  43                        ⦃G, K1⦄ ⊢ ➡* K2 → ⦃G, K1⦄ ⊢ V1 ➡* V2 →
  44                        R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
  45                     ) →
  46                     ∀L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L1 L2.
  47 /3 width=4 by TC_lpx_sn_ind, lpr_cprs_trans/ qed-.