Section Arithmetical_dec.
Variable P : nat -> Prop.
Hypothesis HP : forall n, {P n} + {~P n}.
End Arithmetical_dec.
(** * Derivability of the Archimedean axiom *)
(** This is a standard proof (it has been taken from PlanetMath). It is formulated negatively so as to avoid the need for classical logic. Using a proof of [{n | ~P n}+{forall n, P n}], we can in principle also derive [up] and its specification. The proof above cannot be used for that purpose, since it relies on the [archimed] axiom. *)

Theorem not_not_archimedean :
forall r : R, ~ (forall n : nat, (INR n <= r)%R).