From 9043add656177eeac1491a73d2f3ab92bec0013c Mon Sep 17 00:00:00 2001
From: Benjamin Barenblat <bbaren@debian.org>
Date: Sat, 29 Dec 2018 14:31:27 -0500
Subject: Imported Upstream version 8.8.2

---
 theories/Logic/SetoidChoice.v | 62 +++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 62 insertions(+)
 create mode 100644 theories/Logic/SetoidChoice.v

(limited to 'theories/Logic/SetoidChoice.v')

diff --git a/theories/Logic/SetoidChoice.v b/theories/Logic/SetoidChoice.v
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+(************************************************************************)
+(*         *   The Coq Proof Assistant / The Coq Development Team       *)
+(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
+(* <O___,, *       (see CREDITS file for the list of authors)           *)
+(*   \VV/  **************************************************************)
+(*    //   *    This file is distributed under the terms of the         *)
+(*         *     GNU Lesser General Public License Version 2.1          *)
+(*         *     (see LICENSE file for the text of the license)         *)
+(************************************************************************)
+
+(** 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. (* classical logic, relational choice, unique choice *)
+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').
+Proof.
+  apply setoid_functional_choice_first_characterization. split; [|split].
+  - exact choice.
+  - exact extensional_function_representative.
+  - exact classic.
+Qed.
+
+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'.
+Proof.
+  apply setoid_fun_choice_imp_repr_fun_choice.
+  exact setoid_choice.
+Qed.
-- 
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