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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
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(** This module proves some logical properties of the axiomatic of Reals.  - Decidability of arithmetical statements. - Derivability of the Archimedean "axiom". - Decidability of negated formulas. *)
(** * Decidability of arithmetical statements *)
(** One can iterate this lemma and use classical logic to decide any statement in the arithmetical hierarchy. *)
Section Arithmetical_dec.
Variable P : nat -> Prop.
Hypothesis HP : forall n, {P n} + {~P n}.

(* Specifications: *)
Lemma sig_forall_dec : {n | ~P n} + {forall n, P n}.
End Arithmetical_dec.