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+(************************************************************************)
+(*  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        *)
+(************************************************************************)
+
+(** This is a proof in the pure Calculus of Construction that
+    classical logic in Prop + dependent elimination of disjunction entails
+    proof-irrelevance.
+
+    Since, dependent elimination is derivable in the Calculus of
+    Inductive Constructions (CCI), we get proof-irrelevance from classical
+    logic in the CCI.
+
+    Reference:
+
+    - [Coquand] T. Coquand, "Metamathematical Investigations of a
+      Calculus of Constructions", Proceedings of Logic in Computer Science
+      (LICS'90), 1990.
+
+    Proof skeleton: classical logic + dependent elimination of
+    disjunction + discrimination of proofs implies the existence of a
+    retract from Prop into bool, hence inconsistency by encoding any
+    paradox of system U- (e.g. Hurkens' paradox).
+*)
+
+Require Hurkens.
+
+Section Proof_irrelevance_CC.
+
+Variable or : Prop -> Prop -> Prop.
+Variable or_introl : (A,B:Prop)A->(or A B).
+Variable or_intror : (A,B:Prop)B->(or A B).
+Hypothesis or_elim : (A,B:Prop)(C:Prop)(A->C)->(B->C)->(or A B)->C.
+Hypothesis or_elim_redl :
+  (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(a:A)
+  (f a)==(or_elim A B C f g (or_introl A B a)).
+Hypothesis or_elim_redr :
+  (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(b:B)
+  (g b)==(or_elim A B C f g (or_intror A B b)).
+Hypothesis or_dep_elim :
+ (A,B:Prop)(P:(or A B)->Prop)
+ ((a:A)(P (or_introl A B a))) ->
+ ((b:B)(P (or_intror A B b))) -> (b:(or A B))(P b).
+
+Hypothesis em : (A:Prop)(or A ~A).
+Variable B : Prop.
+Variable b1,b2 : B.
+
+(** [p2b] and [b2p] form a retract if [~b1==b2] *)
+
+Definition p2b [A] := (or_elim A ~A B [_]b1 [_]b2 (em A)).
+Definition b2p [b] := b1==b.
+
+Lemma p2p1 : (A:Prop) A -> (b2p (p2b A)).
+Proof.
+  Unfold p2b; Intro A; Apply or_dep_elim with b:=(em A); Unfold b2p; Intros.
+  Apply (or_elim_redl A ~A B [_]b1 [_]b2).
+  NewDestruct (b H).
+Qed.
+Lemma p2p2 :  ~b1==b2->(A:Prop) (b2p (p2b A)) -> A.
+Proof.
+  Intro not_eq_b1_b2.
+  Unfold p2b; Intro A; Apply or_dep_elim with b:=(em A); Unfold b2p; Intros.
+  Assumption.
+  NewDestruct not_eq_b1_b2.
+  Rewrite <- (or_elim_redr A ~A B [_]b1 [_]b2) in H.
+  Assumption.
+Qed.
+
+(** Using excluded-middle a second time, we get proof-irrelevance *)
+
+Theorem proof_irrelevance_cc : b1==b2.
+Proof.
+  Refine (or_elim ? ? ? ? ? (em b1==b2));Intro H.
+    Trivial.
+  Apply (paradox B p2b b2p (p2p2 H) p2p1).
+Qed.
+
+End Proof_irrelevance_CC.
+
+
+(** The Calculus of Inductive Constructions (CCI) enjoys dependent
+    elimination, hence classical logic in CCI entails proof-irrelevance.
+*)
+
+Section Proof_irrelevance_CCI.
+
+Hypothesis em : (A:Prop) A \/ ~A.
+
+Definition or_elim_redl :
+  (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(a:A)
+  (f a)==(or_ind A B C f g (or_introl A B a))
+  := [A,B,C;f;g;a](refl_eqT C (f a)).
+Definition or_elim_redr :
+  (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(b:B)
+  (g b)==(or_ind A B C f g (or_intror A B b))
+  := [A,B,C;f;g;b](refl_eqT C (g b)).
+Scheme or_indd := Induction for or Sort Prop.
+
+Theorem proof_irrelevance_cci : (B:Prop)(b1,b2:B)b1==b2.
+Proof
+  (proof_irrelevance_cc or or_introl or_intror or_ind
+     or_elim_redl or_elim_redr or_indd em).
+
+End Proof_irrelevance_CCI.
+
+(** Remark: in CCI, [bool] can be taken in [Set] as well in the
+    paradox and since [~true=false] for [true] and [false] in
+    [bool], we get the inconsistency of [em : (A:Prop){A}+{~A}] in CCI
+*)