helm/software/matita/contribs/procedural/Coq/Logic/Diaconescu.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: Diaconescu.v,v 1.5.2.3 2004/08/01 09:36:44 herbelin Exp $ i*)
  34
  35 (*#* R. Diaconescu [Diaconescu] showed that the Axiom of Choice in Set Theory
  36    entails Excluded-Middle; S. Lacas and B. Werner [LacasWerner]
  37    adapted the proof to show that the axiom of choice in equivalence
  38    classes entails Excluded-Middle in Type Theory.
  39
  40    This is an adaptatation of the proof by Hugo Herbelin to show that
  41    the relational form of the Axiom of Choice + Extensionality for
  42    predicates entails Excluded-Middle
  43
  44    [Diaconescu] R. Diaconescu, Axiom of Choice and Complementation, in
  45    Proceedings of AMS, vol 51, pp 176-178, 1975.
  46
  47    [LacasWerner] S. Lacas, B Werner, Which Choices imply the excluded middle?,
  48    preprint, 1999.
  49
  50 *)
  51
  52 (* UNEXPORTED
  53 Section PredExt_GuardRelChoice_imp_EM
  54 *)
  55
  56 (*#* The axiom of extensionality for predicates *)
  57
  58 inline procedural "cic:/Coq/Logic/Diaconescu/PredicateExtensionality.con" as definition.
  59
  60 (*#* From predicate extensionality we get propositional extensionality
  61    hence proof-irrelevance *)
  62
  63 include "Logic/ClassicalFacts.ma".
  64
  65 (* UNEXPORTED
  66 cic:/Coq/Logic/Diaconescu/PredExt_GuardRelChoice_imp_EM/pred_extensionality.var
  67 *)
  68
  69 inline procedural "cic:/Coq/Logic/Diaconescu/prop_ext.con" as lemma.
  70
  71 inline procedural "cic:/Coq/Logic/Diaconescu/proof_irrel.con" as lemma.
  72
  73 (*#* From proof-irrelevance and relational choice, we get guarded
  74    relational choice *)
  75
  76 include "Logic/ChoiceFacts.ma".
  77
  78 (* UNEXPORTED
  79 cic:/Coq/Logic/Diaconescu/PredExt_GuardRelChoice_imp_EM/rel_choice.var
  80 *)
  81
  82 inline procedural "cic:/Coq/Logic/Diaconescu/guarded_rel_choice.con" as lemma.
  83
  84 (*#* The form of choice we need: there is a functional relation which chooses
  85     an element in any non empty subset of bool *)
  86
  87 include "Bool/Bool.ma".
  88
  89 inline procedural "cic:/Coq/Logic/Diaconescu/AC.con" as lemma.
  90
  91 (*#* The proof of the excluded middle *)
  92
  93 (*#* Remark: P could have been in Set or Type *)
  94
  95 inline procedural "cic:/Coq/Logic/Diaconescu/pred_ext_and_rel_choice_imp_EM.con" as theorem.
  96
  97 (* UNEXPORTED
  98 End PredExt_GuardRelChoice_imp_EM
  99 *)
 100