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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
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(** 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
*)