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): ∀s. s = ⇡*[h,𝟎]s.
  30 // qed.
  31
  32 lemma sh_nexts_unit (h): ⇡[h] ≐ ⇡*[h,𝟏].
  33 // qed.
  34
  35 lemma sh_nexts_succ (h) (n): ⇡[h] ∘ (⇡*[h,n]) ≐ ⇡*[h,↑n].
  36 /2 width=1 by niter_succ/ qed.
  37
  38 (* Advanced constructions ****************************)
  39
  40 lemma sh_nexts_swap (h) (n): ⇡[h] ∘ (⇡*[h,n]) ≐ ⇡*[h,n] ∘ ⇡[h].
  41 /2 width=1 by niter_appl/ qed.
  42
  43 (* Helper constructions *****************************************************)
  44
  45 lemma sh_nexts_succ_next (h) (n): ⇡*[h,n] ∘ (⇡[h]) ≐ ⇡*[h,↑n].
  46 // qed.