helm/software/matita/contribs/procedural/Coq/Arith/Lt.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: Lt.v,v 1.11.2.1 2004/07/16 19:31:00 herbelin Exp $ i*)
  34
  35 include "Arith/Le.ma".
  36
  37 (* UNEXPORTED
  38 Open Local Scope nat_scope.
  39 *)
  40
  41 (* UNEXPORTED
  42 Implicit Types m n p : nat.
  43 *)
  44
  45 (*#* Irreflexivity *)
  46
  47 inline procedural "cic:/Coq/Arith/Lt/lt_irrefl.con" as theorem.
  48
  49 (* UNEXPORTED
  50 Hint Resolve lt_irrefl: arith v62.
  51 *)
  52
  53 (*#* Relationship between [le] and [lt] *)
  54
  55 inline procedural "cic:/Coq/Arith/Lt/lt_le_S.con" as theorem.
  56
  57 (* UNEXPORTED
  58 Hint Immediate lt_le_S: arith v62.
  59 *)
  60
  61 inline procedural "cic:/Coq/Arith/Lt/lt_n_Sm_le.con" as theorem.
  62
  63 (* UNEXPORTED
  64 Hint Immediate lt_n_Sm_le: arith v62.
  65 *)
  66
  67 inline procedural "cic:/Coq/Arith/Lt/le_lt_n_Sm.con" as theorem.
  68
  69 (* UNEXPORTED
  70 Hint Immediate le_lt_n_Sm: arith v62.
  71 *)
  72
  73 inline procedural "cic:/Coq/Arith/Lt/le_not_lt.con" as theorem.
  74
  75 inline procedural "cic:/Coq/Arith/Lt/lt_not_le.con" as theorem.
  76
  77 (* UNEXPORTED
  78 Hint Immediate le_not_lt lt_not_le: arith v62.
  79 *)
  80
  81 (*#* Asymmetry *)
  82
  83 inline procedural "cic:/Coq/Arith/Lt/lt_asym.con" as theorem.
  84
  85 (*#* Order and successor *)
  86
  87 inline procedural "cic:/Coq/Arith/Lt/lt_n_Sn.con" as theorem.
  88
  89 (* UNEXPORTED
  90 Hint Resolve lt_n_Sn: arith v62.
  91 *)
  92
  93 inline procedural "cic:/Coq/Arith/Lt/lt_S.con" as theorem.
  94
  95 (* UNEXPORTED
  96 Hint Resolve lt_S: arith v62.
  97 *)
  98
  99 inline procedural "cic:/Coq/Arith/Lt/lt_n_S.con" as theorem.
 100
 101 (* UNEXPORTED
 102 Hint Resolve lt_n_S: arith v62.
 103 *)
 104
 105 inline procedural "cic:/Coq/Arith/Lt/lt_S_n.con" as theorem.
 106
 107 (* UNEXPORTED
 108 Hint Immediate lt_S_n: arith v62.
 109 *)
 110
 111 inline procedural "cic:/Coq/Arith/Lt/lt_O_Sn.con" as theorem.
 112
 113 (* UNEXPORTED
 114 Hint Resolve lt_O_Sn: arith v62.
 115 *)
 116
 117 inline procedural "cic:/Coq/Arith/Lt/lt_n_O.con" as theorem.
 118
 119 (* UNEXPORTED
 120 Hint Resolve lt_n_O: arith v62.
 121 *)
 122
 123 (*#* Predecessor *)
 124
 125 inline procedural "cic:/Coq/Arith/Lt/S_pred.con" as lemma.
 126
 127 inline procedural "cic:/Coq/Arith/Lt/lt_pred.con" as lemma.
 128
 129 (* UNEXPORTED
 130 Hint Immediate lt_pred: arith v62.
 131 *)
 132
 133 inline procedural "cic:/Coq/Arith/Lt/lt_pred_n_n.con" as lemma.
 134
 135 (* UNEXPORTED
 136 Hint Resolve lt_pred_n_n: arith v62.
 137 *)
 138
 139 (*#* Transitivity properties *)
 140
 141 inline procedural "cic:/Coq/Arith/Lt/lt_trans.con" as theorem.
 142
 143 inline procedural "cic:/Coq/Arith/Lt/lt_le_trans.con" as theorem.
 144
 145 inline procedural "cic:/Coq/Arith/Lt/le_lt_trans.con" as theorem.
 146
 147 (* UNEXPORTED
 148 Hint Resolve lt_trans lt_le_trans le_lt_trans: arith v62.
 149 *)
 150
 151 (*#* Large = strict or equal *)
 152
 153 inline procedural "cic:/Coq/Arith/Lt/le_lt_or_eq.con" as theorem.
 154
 155 inline procedural "cic:/Coq/Arith/Lt/lt_le_weak.con" as theorem.
 156
 157 (* UNEXPORTED
 158 Hint Immediate lt_le_weak: arith v62.
 159 *)
 160
 161 (*#* Dichotomy *)
 162
 163 inline procedural "cic:/Coq/Arith/Lt/le_or_lt.con" as theorem.
 164
 165 inline procedural "cic:/Coq/Arith/Lt/nat_total_order.con" as theorem.
 166
 167 (*#* Comparison to 0 *)
 168
 169 inline procedural "cic:/Coq/Arith/Lt/neq_O_lt.con" as theorem.
 170
 171 (* UNEXPORTED
 172 Hint Immediate neq_O_lt: arith v62.
 173 *)
 174
 175 inline procedural "cic:/Coq/Arith/Lt/lt_O_neq.con" as theorem.
 176
 177 (* UNEXPORTED
 178 Hint Immediate lt_O_neq: arith v62.
 179 *)
 180