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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id$ i*)
(** * Natural numbers in base 2^31 *)
(**
Author: Arnaud Spiwack
*)
Require Export Int31.
Require Import Z31Z.
Require Import NMake.
Require Import ZnZ.
Open Scope int31_scope.
Module BigN := NMake.Make Int31_words.
Definition bigN := BigN.t.
Delimit Scope bigN_scope with bigN.
Bind Scope bigN_scope with bigN.
Bind Scope bigN_scope with BigN.t.
Bind Scope bigN_scope with BigN.t_.
Notation " i + j " := (BigN.add i j) : bigN_scope.
Notation " i - j " := (BigN.sub i j) : bigN_scope.
Notation " i * j " := (BigN.mul i j) : bigN_scope.
Notation " i / j " := (BigN.div i j) : bigN_scope.
Notation " i ?= j " := (BigN.compare i j) : bigN_scope.
Theorem succ_pred: forall q,
(0 < BigN.to_Z q ->
BigN.to_Z (BigN.succ (BigN.pred q)) = BigN.to_Z q)%Z.
intros q Hq.
rewrite BigN.spec_succ.
rewrite BigN.spec_pred; auto.
generalize Hq; set (a := BigN.to_Z q).
ring_simplify (a - 1 + 1)%Z; auto.
Qed.
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