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

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "ground/arith/nat_le.ma".
  16 include "static_2/notation/functions/one_0.ma".
  17 include "static_2/notation/functions/two_0.ma".
  18 include "static_2/notation/functions/omega_0.ma".
  19
  20 (* APPLICABILITY CONDITION **************************************************)
  21
  22 (* applicability condition specification *)
  23 record ac: Type[0] ≝ {
  24 (* applicability domain *)
  25    ad: predicate nat
  26 }.
  27
  28 (* applicability condition postulates *)
  29 record ac_props (a): Prop ≝ {
  30    ac_dec: ∀m. Decidable (∃∃n. ad a n & m ≤ n)
  31 }.
  32
  33 (* Notable specifications ***************************************************)
  34
  35 definition apply_top: predicate nat ≝ λn. ⊤.
  36
  37 definition ac_top: ac ≝ mk_ac apply_top.
  38
  39 interpretation "any number (applicability domain)"
  40   'Omega = (ac_top).
  41
  42 lemma ac_top_props: ac_props ac_top ≝ mk_ac_props ….
  43 /3 width=3 by or_introl, ex2_intro/
  44 qed.
  45
  46 definition ac_eq (k): ac ≝ mk_ac (eq … k).
  47
  48 interpretation "one (applicability domain)"
  49   'Two = (ac_eq (nsucc nzero)).
  50
  51 interpretation "zero (applicability domain)"
  52   'One = (ac_eq nzero).
  53
  54 lemma ac_eq_props (k): ac_props (ac_eq k) ≝ mk_ac_props ….
  55 #m elim (nle_dec m k) #Hm
  56 [ /3 width=3 by or_introl, ex2_intro/
  57 | @or_intror * #n #Hn #Hmn destruct /2 width=1 by/
  58 ]
  59 qed.
  60
  61 definition ac_le (k): ac ≝ mk_ac (λn. n ≤ k).
  62
  63 lemma ac_le_props (k): ac_props (ac_le k) ≝ mk_ac_props ….
  64 #m elim (nle_dec m k) #Hm
  65 [ /3 width=3 by or_introl, ex2_intro/
  66 | @or_intror * #n #Hn #Hmn
  67   /3 width=3 by nle_trans/
  68 ]
  69 qed.