matita/matita/contribs/lambdadelta/static_2/syntax/sh_lt.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 "static_2/syntax/sort.ma".
  16
  17 (* SORT HIERARCHY ***********************************************************)
  18
  19 record is_lt (h): Prop ≝
  20 {
  21   next_lt: ∀s. s < ⫯[h]s (* strict monotonicity condition *)
  22 }.
  23
  24 (* Basic properties *********************************************************)
  25
  26 lemma nexts_le (h): is_lt h → ∀s,n. s ≤ (next h)^n s.
  27 #h #Hh #s #n elim n -n [ // ] normalize #n #IH
  28 lapply (next_lt … Hh ((next h)^n s)) #H
  29 lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
  30 qed.
  31
  32 lemma nexts_lt (h): is_lt h → ∀s,n. s < (next h)^(↑n) s.
  33 #h #Hh #s #n normalize
  34 lapply (nexts_le … Hh s n) #H
  35 @(le_to_lt_to_lt … H) /2 width=1 by next_lt/
  36 qed.