Library Coq.Logic.ClassicalDescription

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. Require Export Description. 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 }.

Theorem classical_definite_description :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists ! x : A , P x) -> P x }.

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)).

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)).

Compatibility lemmas

Unset Implicit Arguments.

Definition dependent_description := dependent_unique_choice.
Definition description := unique_choice.