matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfsx.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/predtysnstrong_5.ma".
  16 include "basic_2/static/lfdeq.ma".
  17 include "basic_2/rt_transition/lfpx.ma".
  18
  19 (* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
  20
  21 definition lfsx: ∀h. sd h → relation3 term genv lenv ≝
  22                  λh,o,T,G. SN … (lfpx h G T) (lfdeq h o T).
  23
  24 interpretation
  25    "strong normalization for uncounted context-sensitive parallel rt-transition on referred entries (local environment)"
  26    'PRedTySNStrong h o T G L = (lfsx h o T G L).
  27
  28 (* Basic eliminators ********************************************************)
  29
  30 (* Basic_2A1: uses: lsx_ind *)
  31 lemma lfsx_ind: ∀h,o,G,T. ∀R:predicate lenv.
  32                 (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
  33                       (∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → (L1 ≛[h, o, T] L2 → ⊥) → R L2) →
  34                       R L1
  35                 ) →
  36                 ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → R L.
  37 #h #o #G #T #R #H0 #L1 #H elim H -L1
  38 /5 width=1 by SN_intro/
  39 qed-.
  40
  41 (* Basic properties *********************************************************)
  42
  43 (* Basic_2A1: uses: lsx_intro *)
  44 lemma lfsx_intro: ∀h,o,G,L1,T.
  45                   (∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → (L1 ≛[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
  46                   G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
  47 /5 width=1 by SN_intro/ qed.
  48
  49 (* Basic_2A1: uses: lsx_sort *)
  50 lemma lfsx_sort: ∀h,o,G,L,s. G ⊢ ⬈*[h, o, ⋆s] 𝐒⦃L⦄.
  51 #h #o #G #L1 #s @lfsx_intro
  52 #L2 #H #Hs elim Hs -Hs elim (lfpx_inv_sort … H) -H *
  53 [ #H1 #H2 destruct //
  54 | #I1 #I2 #K1 #K2 #HK12 #H1 #H2 destruct
  55   /4 width=4 by lfdeq_sort, lfxs_isid, frees_sort, frees_inv_sort/
  56 ]
  57 qed.
  58
  59 (* Basic_2A1: uses: lsx_gref *)
  60 lemma lfsx_gref: ∀h,o,G,L,p. G ⊢ ⬈*[h, o, §p] 𝐒⦃L⦄.
  61 #h #o #G #L1 #s @lfsx_intro
  62 #L2 #H #Hs elim Hs -Hs elim (lfpx_inv_gref … H) -H *
  63 [ #H1 #H2 destruct //
  64 | #I1 #I2 #K1 #K2 #HK12 #H1 #H2 destruct
  65   /4 width=4 by lfdeq_gref, lfxs_isid, frees_gref, frees_inv_gref/
  66 ]
  67 qed.
  68
  69 lemma lfsx_unit: ∀h,o,I,G,L. G ⊢ ⬈*[h, o, #0] 𝐒⦃L.ⓤ{I}⦄.
  70 #h #o #I #G #L1 @lfsx_intro
  71 #Y #HY #HnY elim HnY -HnY /2 width=2 by lfxs_unit_sn/
  72 qed.
  73
  74 (* Basic forward lemmas *****************************************************)
  75
  76 fact lfsx_fwd_pair_aux: ∀h,o,G,L. G ⊢ ⬈*[h, o, #0] 𝐒⦃L⦄ →
  77                         ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
  78 #h #o #G #L #H
  79 @(lfsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
  80 /5 width=5 by lfpx_pair, lfsx_intro, lfdeq_fwd_zero_pair/
  81 qed-.
  82
  83 lemma lfsx_fwd_pair: ∀h,o,I,G,K,V.
  84                      G ⊢ ⬈*[h, o, #0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
  85 /2 width=4 by lfsx_fwd_pair_aux/ qed-.
  86
  87 (* Basic_2A1: removed theorems 9:
  88               lsx_ge_up lsx_ge
  89               lsxa_ind lsxa_intro lsxa_lleq_trans lsxa_lpxs_trans lsxa_intro_lpx lsx_lsxa lsxa_inv_lsx
  90 *)