helm/software/matita/contribs/procedural/Coq/Arith/Mult.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: Mult.v,v 1.21.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
  34
  35 include "Arith/Plus.ma".
  36
  37 include "Arith/Minus.ma".
  38
  39 include "Arith/Lt.ma".
  40
  41 include "Arith/Le.ma".
  42
  43 (* UNEXPORTED
  44 Open Local Scope nat_scope.
  45 *)
  46
  47 (* UNEXPORTED
  48 Implicit Types m n p : nat.
  49 *)
  50
  51 (*#* Zero property *)
  52
  53 inline procedural "cic:/Coq/Arith/Mult/mult_0_r.con" as lemma.
  54
  55 inline procedural "cic:/Coq/Arith/Mult/mult_0_l.con" as lemma.
  56
  57 (*#* Distributivity *)
  58
  59 inline procedural "cic:/Coq/Arith/Mult/mult_plus_distr_r.con" as lemma.
  60
  61 (* UNEXPORTED
  62 Hint Resolve mult_plus_distr_r: arith v62.
  63 *)
  64
  65 inline procedural "cic:/Coq/Arith/Mult/mult_plus_distr_l.con" as lemma.
  66
  67 inline procedural "cic:/Coq/Arith/Mult/mult_minus_distr_r.con" as lemma.
  68
  69 (* UNEXPORTED
  70 Hint Resolve mult_minus_distr_r: arith v62.
  71 *)
  72
  73 (*#* Associativity *)
  74
  75 inline procedural "cic:/Coq/Arith/Mult/mult_assoc_reverse.con" as lemma.
  76
  77 (* UNEXPORTED
  78 Hint Resolve mult_assoc_reverse: arith v62.
  79 *)
  80
  81 inline procedural "cic:/Coq/Arith/Mult/mult_assoc.con" as lemma.
  82
  83 (* UNEXPORTED
  84 Hint Resolve mult_assoc: arith v62.
  85 *)
  86
  87 (*#* Commutativity *)
  88
  89 inline procedural "cic:/Coq/Arith/Mult/mult_comm.con" as lemma.
  90
  91 (* UNEXPORTED
  92 Hint Resolve mult_comm: arith v62.
  93 *)
  94
  95 (*#* 1 is neutral *)
  96
  97 inline procedural "cic:/Coq/Arith/Mult/mult_1_l.con" as lemma.
  98
  99 (* UNEXPORTED
 100 Hint Resolve mult_1_l: arith v62.
 101 *)
 102
 103 inline procedural "cic:/Coq/Arith/Mult/mult_1_r.con" as lemma.
 104
 105 (* UNEXPORTED
 106 Hint Resolve mult_1_r: arith v62.
 107 *)
 108
 109 (*#* Compatibility with orders *)
 110
 111 inline procedural "cic:/Coq/Arith/Mult/mult_O_le.con" as lemma.
 112
 113 (* UNEXPORTED
 114 Hint Resolve mult_O_le: arith v62.
 115 *)
 116
 117 inline procedural "cic:/Coq/Arith/Mult/mult_le_compat_l.con" as lemma.
 118
 119 (* UNEXPORTED
 120 Hint Resolve mult_le_compat_l: arith.
 121 *)
 122
 123 inline procedural "cic:/Coq/Arith/Mult/mult_le_compat_r.con" as lemma.
 124
 125 inline procedural "cic:/Coq/Arith/Mult/mult_le_compat.con" as lemma.
 126
 127 inline procedural "cic:/Coq/Arith/Mult/mult_S_lt_compat_l.con" as lemma.
 128
 129 (* UNEXPORTED
 130 Hint Resolve mult_S_lt_compat_l: arith.
 131 *)
 132
 133 inline procedural "cic:/Coq/Arith/Mult/mult_lt_compat_r.con" as lemma.
 134
 135 inline procedural "cic:/Coq/Arith/Mult/mult_S_le_reg_l.con" as lemma.
 136
 137 (*#* n|->2*n and n|->2n+1 have disjoint image *)
 138
 139 inline procedural "cic:/Coq/Arith/Mult/odd_even_lem.con" as theorem.
 140
 141 (*#* Tail-recursive mult *)
 142
 143 (*#* [tail_mult] is an alternative definition for [mult] which is
 144     tail-recursive, whereas [mult] is not. This can be useful
 145     when extracting programs. *)
 146
 147 inline procedural "cic:/Coq/Arith/Mult/mult_acc.con" as definition.
 148
 149 inline procedural "cic:/Coq/Arith/Mult/mult_acc_aux.con" as lemma.
 150
 151 inline procedural "cic:/Coq/Arith/Mult/tail_mult.con" as definition.
 152
 153 inline procedural "cic:/Coq/Arith/Mult/mult_tail_mult.con" as lemma.
 154
 155 (*#* [TailSimpl] transforms any [tail_plus] and [tail_mult] into [plus]
 156     and [mult] and simplify *)
 157
 158 (* UNEXPORTED
 159 Ltac tail_simpl :=
 160   repeat rewrite <- plus_tail_plus; repeat rewrite <- mult_tail_mult;
 161    simpl in |- *.
 162 *)
 163