matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.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/computation/cprs.ma".
  16 include "basic_2/equivalence/cpcs.ma".
  17
  18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
  19
  20 (* Properties about context sensitive computation on terms ******************)
  21
  22 (* Basic_1: was: pc3_pr3_r *)
  23 lemma cpcs_cprs_dx: ∀L,T1,T2. L ⊢ T1 ➡* T2 → L ⊢ T1 ⬌* T2.
  24 #L #T1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /3 width=3/
  25 qed.
  26
  27 (* Basic_1: was: pc3_pr3_x *)
  28 lemma cpcs_cprs_sn: ∀L,T1,T2. L ⊢ T2 ➡* T1 → L ⊢ T1 ⬌* T2.
  29 #L #T1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /3 width=3/
  30 qed.
  31
  32 lemma cpcs_cprs_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
  33 #L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /2 width=3/
  34 qed.
  35
  36 lemma cpcs_cprs_strap2: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  37 #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /width=1/ /2 width=3/
  38 qed.
  39
  40 lemma cpcs_cprs_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
  41 #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /2 width=3/
  42 qed.
  43
  44 (* Basic_1: was: pc3_pr3_conf *)
  45 lemma cpcs_cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  46 #L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /width=1/ /2 width=3/
  47 qed.
  48
  49 (* Basic_1: was: pc3_pr3_t *)
  50 (* Basic_1: note: pc3_pr3_t should be renamed *)
  51 lemma cprs_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
  52 #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=1/ /2 width=3/
  53 qed.
  54
  55 lemma cprs_cpr_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
  56 /3 width=5 by cpr_cprs, cprs_div/ qed-.
  57
  58 lemma cpr_cprs_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
  59 /3 width=3 by cpr_cprs, cprs_div/ qed-.