matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.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/reduction/cpx_cpys.ma".
  16 include "basic_2/computation/lpxs_llneq.ma".
  17 include "basic_2/computation/csx_alt.ma".
  18 include "basic_2/computation/lsx_lpxs.ma".
  19
  20 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
  21
  22 (* Advanced properties ******************************************************)
  23
  24 lemma lpxs_cpys_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
  25                          ∀T.  ⦃G, L1⦄ ⊢ T ▶*[0, ∞] U →
  26                          G ⊢ ⋕⬊*[h, g, T] L1.
  27 #h #g #G #L1 #U #H @(csx_ind_alt … H) -U
  28 #U #_ #IHU #T #HTU @lsx_intro
  29 #L2 #HL02 #HnT elim (lpxs_nlleq_fwd_cpxs … HL02 HnT) -HnT
  30 #U0 #U2 #H0 #H2 #HU02 #HnU02 elim (cpys_conf_eq … HTU … H0)
  31 #X #HUX #H0X elim (eq_term_dec U X) #HnUX destruct
  32 [ -HUX
  33 | -HnU02 @(lsx_lpxs_trans … HL02) @(IHU … HnUX)
  34   [ /3 width=3 by cpys_cpx, cpx_cpxs/
  35   | /2 width=3 by cpys_trans_eq/
  36   ]
  37 ]
  38
  39 lemma lpxs_cpys_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
  40                          ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀T.  ⦃G, L2⦄ ⊢ T ▶*[0, ∞] U →
  41                          G ⊢ ⋕⬊*[h, g, T] L2.
  42 #h #g #G #L1 #U #H @(csx_ind_alt … H) -U
  43 #U #_ #IHU #L0 #HL10 #T #HTU @lsx_intro
  44 #L2 #HL02 #HnT elim (lpxs_nlleq_fwd_cpxs … HL02 HnT) -HnT
  45 #U0 #U2 #H0 #H2 #HU02 #HnU02 elim (cpys_conf_eq … HTU … H0)
  46 #X #HUX #H0X elim (eq_term_dec U X) #HnUX destruct
  47 [ -HUX
  48 | -HnU02 @(IHU … HnUX)
  49
  50
  51 -HnT /4 width=9 by lpxs_trans, lpxs_cpxs_trans, cpx_cpye_fwd_lpxs/
  52 ]
  53
  54
  55
  56
  57
  58
  59
  60 include "basic_2/reduction/cpx_cpys.ma".
  61 include "basic_2/computation/lpxs_cpye.ma".
  62 include "basic_2/computation/csx_alt.ma".
  63 include "basic_2/computation/lsx_lpxs.ma".
  64
  65 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
  66
  67 (* Advanced properties ******************************************************)
  68
  69 axiom lpxs_cpye_csx_lsx: ∀h,g,G,L1,U. ⦃G, L1⦄ ⊢ ⬊*[h, g] U →
  70                          ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀T.  ⦃G, L2⦄ ⊢ T ▶*[0, ∞] 𝐍⦃U⦄ →
  71                          G ⊢ ⋕⬊*[h, g, T] L2.
  72 (*
  73 #h #g #G #L1 #U #H @(csx_ind_alt … H) -U
  74 #U0 #_ #IHU0 #L0 #HL10 #T #H0 @lsx_intro
  75 #L2 #HL02 #HnT elim (cpye_total G L2 T 0 (∞))
  76 #U2 #H2 elim (eq_term_dec U0 U2) #H destruct
  77 [ -IHU0
  78 | -HnT /4 width=9 by lpxs_trans, lpxs_cpxs_trans, cpx_cpye_fwd_lpxs/
  79 ]
  80 *)
  81 (* Main properties **********************************************************)
  82
  83 lemma csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → G ⊢ ⋕⬊*[h, g, T] L.
  84 #h #g #G #L #T #HT elim (cpye_total G L T 0 (∞))
  85 #U #HTU elim HTU
  86 /4 width=5 by lpxs_cpye_csx_lsx, csx_cpx_trans, cpys_cpx/
  87 qed.