helm/software/matita/contribs/procedural/Coq/Arith/Div2.mma

   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 (* This file was automatically generated: do not edit *********************)
  16
  17 include "Coq.ma".
  18
  19 (*#***********************************************************************)
  20
  21 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
  22
  23 (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
  24
  25 (*   \VV/  **************************************************************)
  26
  27 (*    //   *      This file is distributed under the terms of the       *)
  28
  29 (*         *       GNU Lesser General Public License Version 2.1        *)
  30
  31 (*#***********************************************************************)
  32
  33 (*i $Id: Div2.v,v 1.15.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
  34
  35 include "Arith/Lt.ma".
  36
  37 include "Arith/Plus.ma".
  38
  39 include "Arith/Compare_dec.ma".
  40
  41 include "Arith/Even.ma".
  42
  43 (* UNEXPORTED
  44 Open Local Scope nat_scope.
  45 *)
  46
  47 (* UNEXPORTED
  48 Implicit Type n : nat.
  49 *)
  50
  51 (*#* Here we define [n/2] and prove some of its properties *)
  52
  53 inline procedural "cic:/Coq/Arith/Div2/div2.con" as definition.
  54
  55 (*#* Since [div2] is recursively defined on [0], [1] and [(S (S n))], it is
  56     useful to prove the corresponding induction principle *)
  57
  58 inline procedural "cic:/Coq/Arith/Div2/ind_0_1_SS.con" as lemma.
  59
  60 (*#* [0 <n  =>  n/2 < n] *)
  61
  62 inline procedural "cic:/Coq/Arith/Div2/lt_div2.con" as lemma.
  63
  64 (* UNEXPORTED
  65 Hint Resolve lt_div2: arith.
  66 *)
  67
  68 (*#* Properties related to the parity *)
  69
  70 inline procedural "cic:/Coq/Arith/Div2/even_odd_div2.con" as lemma.
  71
  72 (*#* Specializations *)
  73
  74 inline procedural "cic:/Coq/Arith/Div2/even_div2.con" as lemma.
  75
  76 inline procedural "cic:/Coq/Arith/Div2/div2_even.con" as lemma.
  77
  78 inline procedural "cic:/Coq/Arith/Div2/odd_div2.con" as lemma.
  79
  80 inline procedural "cic:/Coq/Arith/Div2/div2_odd.con" as lemma.
  81
  82 (* UNEXPORTED
  83 Hint Resolve even_div2 div2_even odd_div2 div2_odd: arith.
  84 *)
  85
  86 (*#* Properties related to the double ([2n]) *)
  87
  88 inline procedural "cic:/Coq/Arith/Div2/double.con" as definition.
  89
  90 (* UNEXPORTED
  91 Hint Unfold double: arith.
  92 *)
  93
  94 inline procedural "cic:/Coq/Arith/Div2/double_S.con" as lemma.
  95
  96 inline procedural "cic:/Coq/Arith/Div2/double_plus.con" as lemma.
  97
  98 (* UNEXPORTED
  99 Hint Resolve double_S: arith.
 100 *)
 101
 102 inline procedural "cic:/Coq/Arith/Div2/even_odd_double.con" as lemma.
 103
 104 (*#* Specializations *)
 105
 106 inline procedural "cic:/Coq/Arith/Div2/even_double.con" as lemma.
 107
 108 inline procedural "cic:/Coq/Arith/Div2/double_even.con" as lemma.
 109
 110 inline procedural "cic:/Coq/Arith/Div2/odd_double.con" as lemma.
 111
 112 inline procedural "cic:/Coq/Arith/Div2/double_odd.con" as lemma.
 113
 114 (* UNEXPORTED
 115 Hint Resolve even_double double_even odd_double double_odd: arith.
 116 *)
 117
 118 (*#* Application:
 119     - if [n] is even then there is a [p] such that [n = 2p]
 120     - if [n] is odd  then there is a [p] such that [n = 2p+1]
 121
 122     (Immediate: it is [n/2]) *)
 123
 124 inline procedural "cic:/Coq/Arith/Div2/even_2n.con" as lemma.
 125
 126 inline procedural "cic:/Coq/Arith/Div2/odd_S2n.con" as lemma.
 127