path: root/theories/Logic/Diaconescu.v

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(** Diaconescu showed that the Axiom of Choice entails Excluded-Middle
   in topoi [Diaconescu75]. Lacas and Werner adapted the proof to show
   that the axiom of choice in equivalence classes entails
   Excluded-Middle in Type Theory [LacasWerner99].

   Three variants of Diaconescu's result in type theory are shown below.

   A. A proof that the relational form of the Axiom of Choice +
      Extensionality for Predicates entails Excluded-Middle (by Hugo
      Herbelin)

   B. A proof that the relational form of the Axiom of Choice + Proof
      Irrelevance entails Excluded-Middle for Equality Statements (by
      Benjamin Werner)

   C. A proof that extensional Hilbert epsilon's description operator
      entails excluded-middle (taken from Bell [Bell93])

   See also [Carlström] for a discussion of the connection between the
   Extensional Axiom of Choice and Excluded-Middle

   [Diaconescu75] Radu Diaconescu, Axiom of Choice and Complementation,
   in Proceedings of AMS, vol 51, pp 176-178, 1975.

   [LacasWerner99] Samuel Lacas, Benjamin Werner, Which Choices imply
   the excluded middle?, preprint, 1999.

   [Bell93] John L. Bell, Hilbert's epsilon operator and classical
   logic, Journal of Philosophical Logic, 22: 1-18, 1993

   [Carlström04] Jesper Carlström, EM + Ext_ + AC_int <-> AC_ext,
   Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004.
*)

(**********************************************************************)
(** * Pred. Ext. + Rel. Axiom of Choice -> Excluded-Middle       *)

Section PredExt_RelChoice_imp_EM.

(** The axiom of extensionality for predicates *)

Definition PredicateExtensionality :=
  forall P Q:bool -> Prop, (forall b:bool, P b <-> Q b) -> P = Q.

(** From predicate extensionality we get propositional extensionality
   hence proof-irrelevance *)

Require Import ClassicalFacts.

Variable pred_extensionality : PredicateExtensionality.

Lemma prop_ext : forall A B:Prop, (A <-> B) -> A = B.
Proof.
  intros A B H.
  change ((fun _ => A) true = (fun _ => B) true) in |- *.
  rewrite
   pred_extensionality with (P := fun _:bool => A) (Q := fun _:bool => B).
    reflexivity.
    intros _; exact H.
Qed.

Lemma proof_irrel : forall (A:Prop) (a1 a2:A), a1 = a2.
Proof.
  apply (ext_prop_dep_proof_irrel_cic prop_ext).
Qed.

(** From proof-irrelevance and relational choice, we get guarded
   relational choice *)

Require Import ChoiceFacts.

Variable rel_choice : RelationalChoice.

Lemma guarded_rel_choice : GuardedRelationalChoice.
Proof.
 apply
  (rel_choice_and_proof_irrel_imp_guarded_rel_choice rel_choice proof_irrel).
Qed.

(** The form of choice we need: there is a functional relation which chooses
    an element in any non empty subset of bool *)

Require Import Bool.

Lemma AC_bool_subset_to_bool :
  exists R : (bool -> Prop) -> bool -> Prop,
   (forall P:bool -> Prop,
      (exists b : bool, P b) ->
       exists b : bool, P b /\ R P b /\ (forall b':bool, R P b' -> b = b')).
Proof.
  destruct (guarded_rel_choice _ _
   (fun Q:bool -> Prop =>  exists y : _, Q y)
   (fun (Q:bool -> Prop) (y:bool) => Q y)) as (R,(HRsub,HR)).
    exact (fun _ H => H).
  exists R; intros P HP.
  destruct (HR P HP) as (y,(Hy,Huni)).
  exists y; firstorder.
Qed.

(** The proof of the excluded middle *)
(** Remark: P could have been in Set or Type *)

Theorem pred_ext_and_rel_choice_imp_EM : forall P:Prop, P \/ ~ P.
Proof.
intro P.

(** first we exhibit the choice functional relation R *)
destruct AC_bool_subset_to_bool as [R H].

set (class_of_true := fun b => b = true \/ P).
set (class_of_false := fun b => b = false \/ P).

(** the actual "decision": is (R class_of_true) = true or false? *)
destruct (H class_of_true) as [b0 [H0 [H0' H0'']]].
exists true; left; reflexivity.
destruct H0.

(** the actual "decision": is (R class_of_false) = true or false? *)
destruct (H class_of_false) as [b1 [H1 [H1' H1'']]].
exists false; left; reflexivity.
destruct H1.

(** case where P is false: (R class_of_true)=true /\ (R class_of_false)=false *)
right.
intro HP.
assert (Hequiv : forall b:bool, class_of_true b <-> class_of_false b).
intro b; split.
unfold class_of_false in |- *; right; assumption.
unfold class_of_true in |- *; right; assumption.
assert (Heq : class_of_true = class_of_false).
apply pred_extensionality with (1 := Hequiv).
apply diff_true_false.
rewrite <- H0.
rewrite <- H1.
rewrite <- H0''. reflexivity.
rewrite Heq.
assumption.

(** cases where P is true *)
left; assumption.
left; assumption.

Qed.

End PredExt_RelChoice_imp_EM.

(**********************************************************************)
(** * B. Proof-Irrel. + Rel. Axiom of Choice -> Excl.-Middle for Equality *)

(** This is an adaptation of Diaconescu's theorem, exploiting the
    form of extensionality provided by proof-irrelevance *)

Section ProofIrrel_RelChoice_imp_EqEM.

Variable rel_choice : RelationalChoice.

Variable proof_irrelevance : forall P:Prop , forall x y:P, x=y.

(** Let [a1] and [a2] be two elements in some type [A] *)

Variable A :Type.
Variables a1 a2 : A.

(** We build the subset [A'] of [A] made of [a1] and [a2] *)

Definition A' := sigT (fun x => x=a1 \/ x=a2).

Definition a1':A'.
exists a1 ; auto.
Defined.

Definition a2':A'.
exists a2 ; auto.
Defined.

(** By proof-irrelevance, projection is a retraction *)

Lemma projT1_injective : a1=a2 -> a1'=a2'.
Proof.
  intro Heq ; unfold a1', a2', A'.
  rewrite Heq.
  replace (or_introl (a2=a2) (refl_equal a2))
     with (or_intror (a2=a2) (refl_equal a2)).
  reflexivity.
  apply proof_irrelevance.
Qed.

(** But from the actual proofs of being in [A'], we can assert in the
    proof-irrelevant world the existence of relevant boolean witnesses *)

Lemma decide : forall x:A', exists y:bool ,
  (projT1 x = a1 /\ y = true ) \/ (projT1 x = a2 /\ y = false).
Proof.
  intros [a [Ha1|Ha2]]; [exists true | exists false]; auto.
Qed.

(** Thanks to the axiom of choice, the boolean witnesses move from the
    propositional world to the relevant world *)

Theorem proof_irrel_rel_choice_imp_eq_dec : a1=a2 \/ ~a1=a2.
Proof.
  destruct
    (rel_choice A' bool
       (fun x y =>  projT1 x = a1 /\ y = true \/ projT1 x = a2 /\ y = false))
    as (R,(HRsub,HR)).
    apply decide.
  destruct (HR a1') as (b1,(Ha1'b1,_Huni1)).
  destruct (HRsub a1' b1 Ha1'b1) as [(_, Hb1true)|(Ha1a2, _Hb1false)].
  destruct (HR a2') as (b2,(Ha2'b2,Huni2)).
  destruct (HRsub a2' b2 Ha2'b2) as [(Ha2a1, _Hb2true)|(_, Hb2false)].
    left; symmetry; assumption.
    right; intro H.
    subst b1; subst b2.
    rewrite (projT1_injective H) in Ha1'b1.
    assert (false = true) by auto using Huni2.
    discriminate.
  left; assumption.
Qed.

(** An alternative more concise proof can be done by directly using
    the guarded relational choice *)

Declare Implicit Tactic auto.

Lemma proof_irrel_rel_choice_imp_eq_dec' : a1=a2 \/ ~a1=a2.
Proof.
  assert (decide: forall x:A, x=a1 \/ x=a2 ->
                exists y:bool, x=a1 /\ y=true \/ x=a2 /\ y=false).
    intros a [Ha1|Ha2]; [exists true | exists false]; auto.
  assert (guarded_rel_choice :=
          rel_choice_and_proof_irrel_imp_guarded_rel_choice
	  rel_choice
	  proof_irrelevance).
  destruct
    (guarded_rel_choice A bool
      (fun x => x=a1 \/ x=a2)
      (fun x y => x=a1 /\ y=true \/ x=a2 /\ y=false))
      as (R,(HRsub,HR)).
      apply decide.
  destruct (HR a1) as (b1,(Ha1b1,_Huni1)). left; reflexivity.
  destruct (HRsub a1 b1 Ha1b1) as [(_, Hb1true)|(Ha1a2, _Hb1false)].
  destruct (HR a2) as (b2,(Ha2b2,Huni2)). right; reflexivity.
  destruct (HRsub a2 b2 Ha2b2) as [(Ha2a1, _Hb2true)|(_, Hb2false)].
    left; symmetry; assumption.
    right; intro H.
    subst b1; subst b2; subst a1.
    assert (false = true) by auto using Huni2, Ha1b1.
    discriminate.
  left; assumption.
Qed.

End ProofIrrel_RelChoice_imp_EqEM.

(**********************************************************************)
(** * Extensional Hilbert's epsilon description operator -> Excluded-Middle *)

(** Proof sketch from Bell [Bell93] (with thanks to P. Castéran) *)

Notation Local inhabited A := A (only parsing).

Section ExtensionalEpsilon_imp_EM.

Variable epsilon : forall A : Type, inhabited A -> (A -> Prop) -> A.

Hypothesis epsilon_spec :
  forall (A:Type) (i:inhabited A) (P:A->Prop),
  (exists x, P x) -> P (epsilon A i P).

Hypothesis epsilon_extensionality :
  forall (A:Type) (i:inhabited A) (P Q:A->Prop),
  (forall a, P a <-> Q a) -> epsilon A i P = epsilon A i Q.

Notation Local eps := (epsilon bool true) (only parsing).

Theorem extensional_epsilon_imp_EM : forall P:Prop, P \/ ~ P.
Proof.
  intro P.
  pose (B := fun y => y=false \/ P).
  pose (C := fun y => y=true  \/ P).
  assert (B (eps B)) as [Hfalse|HP]
    by (apply epsilon_spec; exists false; left; reflexivity).
  assert (C (eps C)) as [Htrue|HP]
    by (apply epsilon_spec; exists true; left; reflexivity).
    right; intro HP.
    assert (forall y, B y <-> C y) by (intro y; split; intro; right; assumption).
    rewrite epsilon_extensionality with (1:=H) in Hfalse.
    rewrite Htrue in Hfalse.
    discriminate.
  auto.
  auto.
Qed.

End ExtensionalEpsilon_imp_EM.