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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** This file provides classical logic and definite description, which is
equivalent to providing classical logic and Church's iota operator *)
(** Classical logic and definite descriptions implies excluded-middle
in [Set] and leads to a classical world populated with non
computable functions. It conflicts with the impredicativity of
[Set] *)
Set Implicit Arguments.
Require Export Classical. (* Axiomatize classical reasoning *)
Require Export Description. (* Axiomatize constructive form of Church's iota *)
Require Import ChoiceFacts.
Local Notation inhabited A := A (only parsing).
(** The idea for the following proof comes from [ChicliPottierSimpson02] *)
Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}.
Proof.
apply
(constructive_definite_descr_excluded_middle
constructive_definite_description classic).
Qed.
Theorem classical_definite_description :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists! x : A, P x) -> P x }.
Proof.
intros A P i.
destruct (excluded_middle_informative (exists! x, P x)) as [Hex|HnonP].
apply constructive_definite_description with (P:= fun x => (exists! x : A, P x) -> P x).
destruct Hex as (x,(Hx,Huni)).
exists x; split.
intros _; exact Hx.
firstorder.
exists i; tauto.
Qed.
(** Church's iota operator *)
Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A
:= proj1_sig (classical_definite_description P i).
Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) :
(exists! x:A, P x) -> P (iota i P)
:= proj2_sig (classical_definite_description P i).
(** Axiom of unique "choice" (functional reification of functional relations) *)
Theorem dependent_unique_choice :
forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop),
(forall x:A, exists! y : B x, R x y) ->
(exists f : (forall x:A, B x), forall x:A, R x (f x)).
Proof.
intros A B R H.
assert (Hexuni:forall x, exists! y, R x y).
intro x. apply H.
exists (fun x => proj1_sig (constructive_definite_description (R x) (Hexuni x))).
intro x.
apply (proj2_sig (constructive_definite_description (R x) (Hexuni x))).
Qed.
Theorem unique_choice :
forall (A B:Type) (R:A -> B -> Prop),
(forall x:A, exists! y : B, R x y) ->
(exists f : A -> B, forall x:A, R x (f x)).
Proof.
intros A B.
apply dependent_unique_choice with (B:=fun _:A => B).
Qed.
(** Compatibility lemmas *)
Unset Implicit Arguments.
Definition dependent_description := dependent_unique_choice.
Definition description := unique_choice.
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