matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_etc.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/rt_computation/fpbg.ma".
  16 include "basic_2/rt_computation/cpms_fpbs.ma".
  17 include "basic_2/dynamic/cnv.ma".
  18
  19 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
  20
  21 (* Inductive premises for the preservation results **************************)
  22
  23 definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
  24                                 λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
  25                                 ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡[n,h] T2 →
  26                                 ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
  27
  28 definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
  29                                  λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
  30                                  ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
  31                                  ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
  32
  33 definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
  34                                λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  35                                ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
  36                                ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  37                                ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  38
  39 definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝
  40                                  λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  41                                  ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
  42                                  ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  43                                  ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  44
  45 definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝
  46                                 λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  47                                 ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 →
  48                                 ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  49                                 ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  50
  51 (* Properties for preservation **********************************************)
  52
  53 lemma cnv_cpms_trans_lpr_far (a) (h) (o):
  54                              ∀G0,L0,T0.
  55                              (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
  56                              ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
  57 #a #h #o #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
  58 @(cpms_ind_dx … H) -n -T2
  59 /4 width=7 by cpms_fwd_fpbs, fpbg_fpbs_trans/
  60 qed-.
  61
  62 lemma cnv_cpms_strip_lpr_far (a) (h) (o):
  63                              ∀G0,L0,T0.
  64                              (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
  65                              ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
  66 /3 width=8 by cpm_cpms/ qed-.
  67
  68 (*
  69 fact cnv_cpms_strip_lpr_aux (a) (h) (o):
  70                             ∀G0,L0,T0.
  71                             (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1) →
  72                             ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
  73 #a #h #o #G0 #L0 #T0 #IH0 #G #L #T #H0 #HT #n1 #T1 #H
  74 generalize in match HT; generalize in match H0; -H0 -HT
  75 @(cpms_ind_sn … H) -n1 -T [ /2 width=8 by/ ]
  76 #n1 #n2 #T #X #HTX #HXT1 #IH #H0 #HT #n2 #T2 #HT2 #L1 #HL1 #L2 #HL2
  77 elim (IH0 … HTX … HT2 … HL1 … HL2) // -L -T #T0 #HXT0 #HT20
  78
  79   @(IH … 0 T … HT2 … HL1 … HL2) // -L -IH
  80   #T0 #HT20 #HT0
  81 *)