Library Coq.Logic.SetoidChoice

This module states the functional form of the axiom of choice over setoids, commonly called extensional axiom of choice [Carlström04], [Martin-Löf05]. This is obtained by a decomposition of the axiom into the following components:
  • classical logic
  • relational axiom of choice
  • axiom of unique choice
  • a limited form of functional extensionality
Among other results, it entails:
  • proof irrelevance
  • choice of a representative in equivalence classes
[Carlström04] Jesper Carlström, EM + Ext + AC_int is equivalent to AC_ext, Mathematical Logic Quaterly, vol 50(3), pp 236-240, 2004.
[Martin-Löf05] Per Martin-Löf, 100 years of Zermelo’s axiom of choice: what was the problem with it?, lecture notes for KTH/SU colloquium, 2005.

Require Export ClassicalChoice. Require Export ExtensionalFunctionRepresentative.

Require Import ChoiceFacts.
Require Import ClassicalFacts.
Require Import RelationClasses.

Theorem setoid_choice :
  forall A B,
  forall R : A -> A -> Prop,
  forall T : A -> B -> Prop,
  Equivalence R ->
  (forall x x' y, R x x' -> T x y -> T x' y) ->
  (forall x, exists y, T x y) ->
  exists f : A -> B, forall x : A, T x (f x) /\ (forall x' : A, R x x' -> f x = f x').

Theorem representative_choice :
  forall A (R:A->A->Prop), (Equivalence R) ->
  exists f : A->A, forall x : A, R x (f x) /\ forall x', R x x' -> f x = f x'.