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

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