matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr.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/grammar/cl_shift.ma".
  16 include "basic_2/unfold/tpss.ma".
  17 include "basic_2/reducibility/tpr.ma".
  18
  19 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
  20
  21 (* Basic_1: includes: pr2_delta1 *)
  22 definition cpr: lenv → relation term ≝
  23    λL,T1,T2. ∃∃T. T1 ➡ T & L ⊢ T [0, |L|] ▶* T2.
  24
  25 interpretation
  26    "context-sensitive parallel reduction (term)"
  27    'PRed L T1 T2 = (cpr L T1 T2).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T [d, e] ▶* T2 → L ⊢ T1 ➡ T2.
  32 /4 width=3/ qed-.
  33
  34 (* Basic_1: was by definition: pr2_free *)
  35 lemma cpr_tpr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
  36 /2 width=3/ qed.
  37
  38 lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶* T2 → L ⊢ T1 ➡ T2.
  39 /3 width=5/ qed.
  40
  41 lemma cpr_refl: ∀L,T. L ⊢ T ➡ T.
  42 /2 width=1/ qed.
  43
  44 (* Note: new property *)
  45 (* Basic_1: was only: pr2_thin_dx *)
  46 lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
  47                 L ⊢ V1 ➡ V2 → L ⊢ T1 ➡ T2 → L ⊢ ⓕ{I} V1. T1 ➡ ⓕ{I} V2. T2.
  48 #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
  49 qed.
  50
  51 lemma cpr_cast: ∀L,V,T1,T2.
  52                 L ⊢ T1 ➡ T2 → L ⊢ ⓣV. T1 ➡ T2.
  53 #L #V #T1 #T2 * /3 width=3/
  54 qed.
  55
  56 (* Note: it does not hold replacing |L1| with |L2| *)
  57 (* Basic_1: was only: pr2_change *)
  58 lemma cpr_lsubs_conf: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
  59                       ∀L2. L1 [0, |L1|] ≼ L2 → L2 ⊢ T1 ➡ T2.
  60 #L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
  61 lapply (tpss_lsubs_conf … HT2 … HL12) -HT2 -HL12 /3 width=4/
  62 qed.
  63
  64 (* Basic inversion lemmas ***************************************************)
  65
  66 (* Basic_1: was: pr2_gen_csort *)
  67 lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → T1 ➡ T2.
  68 #T1 #T2 * #T #HT normalize #HT2
  69 <(tpss_inv_refl_O2 … HT2) -HT2 //
  70 qed-.
  71
  72 (* Basic_1: was: pr2_gen_sort *)
  73 lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
  74 #L #T2 #k * #X #H
  75 >(tpr_inv_atom1 … H) -H #H
  76 >(tpss_inv_sort1 … H) -H //
  77 qed-.
  78
  79 (* Basic_1: was pr2_gen_abbr *)
  80 lemma cpr_inv_abbr1: ∀L,V1,T1,U2. L ⊢ ⓓV1. T1 ➡ U2 →
  81                      (∃∃V,V2,T2. V1 ➡ V & L ⊢ V [O, |L|] ▶* V2 &
  82                                  L. ⓓV ⊢ T1 ➡ T2 &
  83                                  U2 = ⓓV2. T2
  84                       ) ∨
  85                       ∃∃T. ⇧[0,1] T ≡ T1 & L ⊢ T ➡ U2.
  86 #L #V1 #T1 #Y * #X #H1 #H2
  87 elim (tpr_inv_abbr1 … H1) -H1 *
  88 [ #V #T0 #T #HV1 #HT10 #HT0 #H destruct
  89   elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
  90   lapply (tps_lsubs_conf … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
  91   lapply (tps_weak_all … HT0) -HT0 #HT0
  92   lapply (tpss_lsubs_conf … HT2 (L. ⓓV) ?) -HT2 /2 width=1/ #HT2
  93   lapply (tpss_weak_all … HT2) -HT2 #HT2
  94   lapply (tpss_strap … HT0 HT2) -T /4 width=7/
  95 | /4 width=5/
  96 ]
  97 qed-.
  98
  99 (* Basic_1: was: pr2_gen_cast *)
 100 lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓣV1. T1 ➡ U2 → (
 101                         ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
 102                                  U2 = ⓣV2. T2
 103                      ) ∨ L ⊢ T1 ➡ U2.
 104 #L #V1 #T1 #U2 * #X #H #HU2
 105 elim (tpr_inv_cast1 … H) -H /3 width=3/
 106 * #V #T #HV1 #HT1 #H destruct
 107 elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
 108 qed-.
 109
 110 (* Basic_1: removed theorems 6:
 111             pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
 112             pr2_gen_ctail pr2_ctail
 113    Basic_1: removed local theorems 3:
 114             pr2_free_free pr2_free_delta pr2_delta_delta
 115 *)