helm/software/matita/contribs/procedural/Coq/Logic/ProofIrrelevance.mma

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   4 (*      ||A||       A project by Andrea Asperti                           *)
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  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 include "Coq.ma".
  18
  19 (*#***************************************************************************)
  20
  21 (*                 The Calculus of Inductive Constructions                  *)
  22
  23 (*                                                                          *)
  24
  25 (*                            Projet LogiCal                                *)
  26
  27 (*                                                                          *)
  28
  29 (*                     INRIA                        LRI-CNRS                *)
  30
  31 (*              Rocquencourt                        Orsay                   *)
  32
  33 (*                                                                          *)
  34
  35 (*                              May 29th 2002                               *)
  36
  37 (*                                                                          *)
  38
  39 (*#***************************************************************************)
  40
  41 (*#* This is a proof in the pure Calculus of Construction that
  42     classical logic in Prop + dependent elimination of disjunction entails
  43     proof-irrelevance.
  44
  45     Since, dependent elimination is derivable in the Calculus of
  46     Inductive Constructions (CCI), we get proof-irrelevance from classical
  47     logic in the CCI.
  48
  49     Reference:
  50
  51     - [Coquand] T. Coquand, "Metamathematical Investigations of a
  52       Calculus of Constructions", Proceedings of Logic in Computer Science
  53       (LICS'90), 1990.
  54
  55     Proof skeleton: classical logic + dependent elimination of
  56     disjunction + discrimination of proofs implies the existence of a
  57     retract from [Prop] into [bool], hence inconsistency by encoding any
  58     paradox of system U- (e.g. Hurkens' paradox).
  59 *)
  60
  61 include "Logic/Hurkens.ma".
  62
  63 (* UNEXPORTED
  64 Section Proof_irrelevance_CC
  65 *)
  66
  67 (* UNEXPORTED
  68 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/or.var
  69 *)
  70
  71 (* UNEXPORTED
  72 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/or_introl.var
  73 *)
  74
  75 (* UNEXPORTED
  76 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/or_intror.var
  77 *)
  78
  79 (* UNEXPORTED
  80 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/or_elim.var
  81 *)
  82
  83 (* UNEXPORTED
  84 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/or_elim_redl.var
  85 *)
  86
  87 (* UNEXPORTED
  88 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/or_elim_redr.var
  89 *)
  90
  91 (* UNEXPORTED
  92 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/or_dep_elim.var
  93 *)
  94
  95 (* UNEXPORTED
  96 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/em.var
  97 *)
  98
  99 (* UNEXPORTED
 100 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/B.var
 101 *)
 102
 103 (* UNEXPORTED
 104 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/b1.var
 105 *)
 106
 107 (* UNEXPORTED
 108 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CC/b2.var
 109 *)
 110
 111 (*#* [p2b] and [b2p] form a retract if [~b1=b2] *)
 112
 113 inline procedural "cic:/Coq/Logic/ProofIrrelevance/p2b.con" as definition.
 114
 115 inline procedural "cic:/Coq/Logic/ProofIrrelevance/b2p.con" as definition.
 116
 117 inline procedural "cic:/Coq/Logic/ProofIrrelevance/p2p1.con" as lemma.
 118
 119 inline procedural "cic:/Coq/Logic/ProofIrrelevance/p2p2.con" as lemma.
 120
 121 (*#* Using excluded-middle a second time, we get proof-irrelevance *)
 122
 123 inline procedural "cic:/Coq/Logic/ProofIrrelevance/proof_irrelevance_cc.con" as theorem.
 124
 125 (* UNEXPORTED
 126 End Proof_irrelevance_CC
 127 *)
 128
 129 (*#* The Calculus of Inductive Constructions (CCI) enjoys dependent
 130     elimination, hence classical logic in CCI entails proof-irrelevance.
 131 *)
 132
 133 (* UNEXPORTED
 134 Section Proof_irrelevance_CCI
 135 *)
 136
 137 (* UNEXPORTED
 138 cic:/Coq/Logic/ProofIrrelevance/Proof_irrelevance_CCI/em.var
 139 *)
 140
 141 inline procedural "cic:/Coq/Logic/ProofIrrelevance/or_elim_redl.con" as definition.
 142
 143 inline procedural "cic:/Coq/Logic/ProofIrrelevance/or_elim_redr.con" as definition.
 144
 145 inline procedural "cic:/Coq/Logic/ProofIrrelevance/or_indd.con" as theorem.
 146
 147 inline procedural "cic:/Coq/Logic/ProofIrrelevance/proof_irrelevance_cci.con" as theorem.
 148
 149 (* UNEXPORTED
 150 End Proof_irrelevance_CCI
 151 *)
 152
 153 (*#* Remark: in CCI, [bool] can be taken in [Set] as well in the
 154     paradox and since [~true=false] for [true] and [false] in
 155     [bool], we get the inconsistency of [em : forall A:Prop, {A}+{~A}] in CCI
 156 *)
 157