helm/software/matita/contribs/procedural/Coq/Arith/Wf_nat.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: Wf_nat.v,v 1.16.2.1 2004/07/16 19:31:01 herbelin Exp $ i*)
  34
  35 (*#* Well-founded relations and natural numbers *)
  36
  37 include "Arith/Lt.ma".
  38
  39 (* UNEXPORTED
  40 Open Local Scope nat_scope.
  41 *)
  42
  43 (* UNEXPORTED
  44 Implicit Types m n p : nat.
  45 *)
  46
  47 (* UNEXPORTED
  48 Section Well_founded_Nat
  49 *)
  50
  51 (* UNEXPORTED
  52 cic:/Coq/Arith/Wf_nat/Well_founded_Nat/A.var
  53 *)
  54
  55 (* UNEXPORTED
  56 cic:/Coq/Arith/Wf_nat/Well_founded_Nat/f.var
  57 *)
  58
  59 inline procedural "cic:/Coq/Arith/Wf_nat/ltof.con" as definition.
  60
  61 inline procedural "cic:/Coq/Arith/Wf_nat/gtof.con" as definition.
  62
  63 inline procedural "cic:/Coq/Arith/Wf_nat/well_founded_ltof.con" as theorem.
  64
  65 inline procedural "cic:/Coq/Arith/Wf_nat/well_founded_gtof.con" as theorem.
  66
  67 (*#* It is possible to directly prove the induction principle going
  68    back to primitive recursion on natural numbers ([induction_ltof1])
  69    or to use the previous lemmas to extract a program with a fixpoint
  70    ([induction_ltof2])
  71
  72 the ML-like program for [induction_ltof1] is : [[
  73    let induction_ltof1 F a = indrec ((f a)+1) a
  74    where rec indrec =
  75         function 0    -> (function a -> error)
  76                |(S m) -> (function a -> (F a (function y -> indrec y m)));;
  77 ]]
  78
  79 the ML-like program for [induction_ltof2] is : [[
  80    let induction_ltof2 F a = indrec a
  81    where rec indrec a = F a indrec;;
  82 ]] *)
  83
  84 inline procedural "cic:/Coq/Arith/Wf_nat/induction_ltof1.con" as theorem.
  85
  86 inline procedural "cic:/Coq/Arith/Wf_nat/induction_gtof1.con" as theorem.
  87
  88 inline procedural "cic:/Coq/Arith/Wf_nat/induction_ltof2.con" as theorem.
  89
  90 inline procedural "cic:/Coq/Arith/Wf_nat/induction_gtof2.con" as theorem.
  91
  92 (*#* If a relation [R] is compatible with [lt] i.e. if [x R y => f(x) < f(y)]
  93     then [R] is well-founded. *)
  94
  95 (* UNEXPORTED
  96 cic:/Coq/Arith/Wf_nat/Well_founded_Nat/R.var
  97 *)
  98
  99 (* UNEXPORTED
 100 cic:/Coq/Arith/Wf_nat/Well_founded_Nat/H_compat.var
 101 *)
 102
 103 inline procedural "cic:/Coq/Arith/Wf_nat/well_founded_lt_compat.con" as theorem.
 104
 105 (* UNEXPORTED
 106 End Well_founded_Nat
 107 *)
 108
 109 inline procedural "cic:/Coq/Arith/Wf_nat/lt_wf.con" as lemma.
 110
 111 inline procedural "cic:/Coq/Arith/Wf_nat/lt_wf_rec1.con" as lemma.
 112
 113 inline procedural "cic:/Coq/Arith/Wf_nat/lt_wf_rec.con" as lemma.
 114
 115 inline procedural "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con" as lemma.
 116
 117 inline procedural "cic:/Coq/Arith/Wf_nat/gt_wf_rec.con" as lemma.
 118
 119 inline procedural "cic:/Coq/Arith/Wf_nat/gt_wf_ind.con" as lemma.
 120
 121 inline procedural "cic:/Coq/Arith/Wf_nat/lt_wf_double_rec.con" as lemma.
 122
 123 inline procedural "cic:/Coq/Arith/Wf_nat/lt_wf_double_ind.con" as lemma.
 124
 125 (* UNEXPORTED
 126 Hint Resolve lt_wf: arith.
 127 *)
 128
 129 (* UNEXPORTED
 130 Hint Resolve well_founded_lt_compat: arith.
 131 *)
 132
 133 (* UNEXPORTED
 134 Section LT_WF_REL
 135 *)
 136
 137 (* UNEXPORTED
 138 cic:/Coq/Arith/Wf_nat/LT_WF_REL/A.var
 139 *)
 140
 141 (* UNEXPORTED
 142 cic:/Coq/Arith/Wf_nat/LT_WF_REL/R.var
 143 *)
 144
 145 (* Relational form of inversion *)
 146
 147 (* UNEXPORTED
 148 cic:/Coq/Arith/Wf_nat/LT_WF_REL/F.var
 149 *)
 150
 151 inline procedural "cic:/Coq/Arith/Wf_nat/inv_lt_rel.con" as definition.
 152
 153 (* UNEXPORTED
 154 cic:/Coq/Arith/Wf_nat/LT_WF_REL/F_compat.var
 155 *)
 156
 157 inline procedural "cic:/Coq/Arith/Wf_nat/acc_lt_rel.con" as remark.
 158
 159 inline procedural "cic:/Coq/Arith/Wf_nat/well_founded_inv_lt_rel_compat.con" as theorem.
 160
 161 (* UNEXPORTED
 162 End LT_WF_REL
 163 *)
 164
 165 inline procedural "cic:/Coq/Arith/Wf_nat/well_founded_inv_rel_inv_lt_rel.con" as lemma.
 166