matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve_far.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/predstar_7.ma".
  16 include "basic_2/rt_computation/fpbg.ma".
  17 include "basic_2/rt_computation/cpms_fpbs.ma".
  18 include "basic_2/dynamic/cnv.ma".
  19
  20 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
  21
  22 (* Inductive premises for the preservation results **************************)
  23
  24 definition cpsms (n) (h) (o): relation4 genv lenv term term ≝ λG,L,T1,T2.
  25                  ∃∃n1,n2,T. T1 ≛[h,o] T → ⊥ & ⦃G, L⦄ ⊢ T1 ➡[n1,h] T & ⦃G, L⦄ ⊢ T ➡*[n2,h] T2 & n1+n2 = n.
  26
  27 interpretation
  28    "context-sensitive parallel stratified t-bound  rt-computarion (term)"
  29    'PRedStar n h o G L T1 T2 = (cpsms n h o G L T1 T2).
  30
  31 definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
  32                                 λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
  33                                 ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡[n,h] T2 →
  34                                 ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
  35
  36 definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
  37                                  λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
  38                                  ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
  39                                  ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
  40
  41 definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
  42                                λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  43                                ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
  44                                ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  45                                ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  46
  47 definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝
  48                                  λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  49                                  ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
  50                                  ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  51                                  ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  52
  53 definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝
  54                                 λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  55                                 ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 →
  56                                 ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  57                                 ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  58
  59 definition IH_cnv_cpsms_conf_lpr (a) (h) (o): relation3 genv lenv term ≝
  60                                  λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  61                                  ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h,o] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h,o] T2 →
  62                                  ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  63                                  ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  64
  65 (* Properties for preservation **********************************************)
  66
  67 lemma cnv_cpms_trans_lpr_far (a) (h) (o):
  68                              ∀G0,L0,T0.
  69                              (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
  70                              ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
  71 #a #h #o #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
  72 @(cpms_ind_dx … H) -n -T2
  73 /4 width=7 by cpms_fwd_fpbs, fpbg_fpbs_trans/
  74 qed-.
  75
  76 lemma cnv_cpm_conf_lpr_far (a) (h) (o):
  77                            ∀G0,L0,T0.
  78                            (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
  79                            ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
  80 /3 width=8 by cpm_cpms/ qed-.
  81
  82 lemma cnv_cpms_strip_lpr_far (a) (h) (o):
  83                              ∀G0,L0,T0.
  84                              (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
  85                              ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
  86 /3 width=8 by cpm_cpms/ qed-.