(**************************************************************************)
(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
(**************************************************************************)

(* This file was automatically generated: do not edit *********************)

include "Coq.ma".

include "Init/Prelude.ma".

(*#***********************************************************************)

(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)

(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)

(*   \VV/  **************************************************************)

(*    //   *      This file is distributed under the terms of the       *)

(*         *       GNU Lesser General Public License Version 2.1        *)

(*#***********************************************************************)

(*i $Id: Eqdep_dec.v,v 1.14.2.1 2004/07/16 19:31:06 herbelin Exp $ i*)

(*#* We prove that there is only one proof of [x=x], i.e [(refl_equal ? x)].
   This holds if the equality upon the set of [x] is decidable.
   A corollary of this theorem is the equality of the right projections
   of two equal dependent pairs.

   Author:   Thomas Kleymann |<tms@dcs.ed.ac.uk>| in Lego
             adapted to Coq by B. Barras

   Credit:   Proofs up to [K_dec] follows an outline by Michael Hedberg
*)

(*#* We need some dependent elimination schemes *)

(* UNEXPORTED
Set Implicit Arguments.
*)

(*#* Bijection between [eq] and [eqT] *)

inline procedural "cic:/Coq/Logic/Eqdep_dec/eq2eqT.con" as definition.

inline procedural "cic:/Coq/Logic/Eqdep_dec/eqT2eq.con" as definition.

inline procedural "cic:/Coq/Logic/Eqdep_dec/eq_eqT_bij.con" as lemma.

inline procedural "cic:/Coq/Logic/Eqdep_dec/eqT_eq_bij.con" as lemma.

(* UNEXPORTED
Section DecidableEqDep
*)

(* UNEXPORTED
cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/A.var
*)

(* UNAVAILABLE OBJECT: cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/comp.con ***)

inline procedural "cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/comp.con" "DecidableEqDep__" as definition.

inline procedural "cic:/Coq/Logic/Eqdep_dec/trans_sym_eqT.con" as remark.

(* UNEXPORTED
cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/eq_dec.var
*)

(* UNEXPORTED
cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/x.var
*)

(* UNAVAILABLE OBJECT: cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/nu.con *****)

inline procedural "cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/nu.con" "DecidableEqDep__" as definition.

(* UNAVAILABLE OBJECT: cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/nu_constant.con *)

inline procedural "cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/nu_constant.con" "DecidableEqDep__" as definition.

(* UNAVAILABLE OBJECT: cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/nu_inv.con *)

inline procedural "cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/nu_inv.con" "DecidableEqDep__" as definition.

inline procedural "cic:/Coq/Logic/Eqdep_dec/nu_left_inv.con" as remark.

inline procedural "cic:/Coq/Logic/Eqdep_dec/eq_proofs_unicity.con" as theorem.

inline procedural "cic:/Coq/Logic/Eqdep_dec/K_dec.con" as theorem.

(*#* The corollary *)

(* UNAVAILABLE OBJECT: cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/proj.con ***)

inline procedural "cic:/Coq/Logic/Eqdep_dec/DecidableEqDep/proj.con" "DecidableEqDep__" as definition.

inline procedural "cic:/Coq/Logic/Eqdep_dec/inj_right_pair.con" as theorem.

(* UNEXPORTED
End DecidableEqDep
*)

(*#* We deduce the [K] axiom for (decidable) Set *)

inline procedural "cic:/Coq/Logic/Eqdep_dec/K_dec_set.con" as theorem.