matita/matita/contribs/lambdadelta/static_2/syntax/sh_nexts.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/nat_succ_iter.ma".
  16 include "static_2/notation/functions/uparrowstar_2_0.ma".
  17 include "static_2/syntax/sh.ma".
  18
  19 (* SORT HIERARCHY ***********************************************************)
  20
  21 definition sh_nexts (h) (n): nat → nat ≝ ⇡[h]^n.
  22
  23 interpretation
  24   "iterated next sort (sort hierarchy)"
  25   'UpArrowStar_2_0 h n = (sh_nexts h n).
  26
  27 (* Basic constructions *)
  28
  29 lemma sh_nexts_zero (h):
  30       ∀s. s = ⇡*[h,𝟎]s.
  31 // qed.
  32
  33 lemma sh_nexts_unit (h):
  34       ∀s. ⇡[h]s = ⇡*[h,𝟏]s.
  35 // qed.
  36
  37 lemma sh_nexts_succ (h) (n):
  38       ∀s. ⇡[h](⇡*[h,n]s) = ⇡*[h,↑n]s.
  39 #h #n #s @(niter_succ … (⇡[h]))
  40 qed.
  41
  42 (* Advanced constructions ****************************)
  43
  44 lemma sh_nexts_swap (h) (n):
  45       ∀s. ⇡[h](⇡*[h,n]s) = ⇡*[h,n](⇡[h]s).
  46 #h #n #s @(niter_appl … (⇡[h]))
  47 qed.
  48
  49 (* Helper constructions *****************************************************)
  50
  51 lemma sh_nexts_succ_next (h) (n):
  52       ∀s. ⇡*[h,n](⇡[h]s) = ⇡*[h,↑n]s.
  53 // qed.