Library Coq.Logic.Diaconescu


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 [Carlström04] 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 is equivalent to AC_ext, Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004.

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

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.

Lemma proof_irrel : forall (A:Prop) (a1 a2:A), a1 = a2.

From proof-irrelevance and relational choice, we get guarded relational choice
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')).

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

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 A (fun x => x=a1 \/ x=a2).

Definition a1':A'.
Defined.

Definition a2':A'.
Defined.

By proof-irrelevance, projection is a retraction
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).

Thanks to the axiom of choice, the boolean witnesses move from the propositional world to the relevant world
An alternative more concise proof can be done by directly using the guarded relational choice

Extensional Hilbert's epsilon description operator -> Excluded-Middle

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


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.


Theorem extensional_epsilon_imp_EM : forall P:Prop, P \/ ~ P.

End ExtensionalEpsilon_imp_EM.