Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Logic.ClassicalUniqueChoice

This file provides classical logic and unique choice; this is weaker than providing iota operator and classical logic as the definite descriptions provided by the axiom of unique choice can be used only in a propositional context (especially, they cannot be used to build functions outside the scope of a theorem proof)
Classical logic and unique choice, as shown in [ChicliPottierSimpson02], implies the double-negation of excluded-middle in Set, hence it implies a strongly classical world. Especially it conflicts with the impredicativity of Set.
[ChicliPottierSimpson02] Laurent Chicli, Loïc Pottier, Carlos Simpson, Mathematical Quotients and Quotient Types in Coq, Proceedings of TYPES 2002, Lecture Notes in Computer Science 2646, Springer Verlag.
Require Export Classical.
Axiom
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)).
Unique choice reifies functional relations into functions
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)).
The following proof comes from [ChicliPottierSimpson02]
Require Import Setoid.
Theorem classic_set_in_prop_context :
forall C:Prop, ((forall P:Prop, {P} + {~ P}) -> C) -> C.
Corollary not_not_classic_set :
((forall P:Prop, {P} + {~ P}) -> False) -> False.
Notation classic_set := not_not_classic_set (only parsing).