matita/matita/contribs/lambda/paths/alternative_standard_order.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 "paths/standard_precedence.ma".
  16
  17 (* ALTERNATIVE STANDARD ORDER ***********************************************)
  18
  19 (* Note: this is p < q *)
  20 definition slt: relation path ≝ TC … sprec.
  21
  22 interpretation "standard 'less than' on paths"
  23    'lt p q = (slt p q).
  24
  25 lemma slt_step_rc: ∀p,q. p ≺ q → p < q.
  26 /2 width=1/
  27 qed.
  28
  29 lemma slt_nil: ∀o,p. ◊ < o::p.
  30 /2 width=1/
  31 qed.
  32
  33 lemma slt_skip: dx::◊ < ◊.
  34 /2 width=1/
  35 qed.
  36
  37 lemma slt_comp: ∀o,p,q. p < q → o::p < o::q.
  38 #o #p #q #H elim H -q /3 width=1/ /3 width=3/
  39 qed.
  40
  41 theorem slt_trans: transitive … slt.
  42 /2 width=3/
  43 qed-.
  44
  45 lemma slt_refl: ∀p. p < p.
  46 #p elim p -p /2 width=1/
  47 @(slt_trans … (dx::◊)) //
  48 qed.