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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*)
(*********************************************************)
(** Definitions for the axiomatization *)
(*********************************************************)
Require Export ZArith_base.
Parameter R : Set.
(* Declare Scope positive_scope with Key R *)
Delimit Scope R_scope with R.
(* Automatically open scope R_scope for arguments of type R *)
Bind Scope R_scope with R.
Parameter R0 : R.
Parameter R1 : R.
Parameter Rplus : R -> R -> R.
Parameter Rmult : R -> R -> R.
Parameter Ropp : R -> R.
Parameter Rinv : R -> R.
Parameter Rlt : R -> R -> Prop.
Parameter up : R -> Z.
Infix "+" := Rplus : R_scope.
Infix "*" := Rmult : R_scope.
Notation "- x" := (Ropp x) : R_scope.
Notation "/ x" := (Rinv x) : R_scope.
Infix "<" := Rlt : R_scope.
(*i*******************************************************i*)
(**********)
Definition Rgt (r1 r2:R) : Prop := (r2 < r1)%R.
(**********)
Definition Rle (r1 r2:R) : Prop := (r1 < r2)%R \/ r1 = r2.
(**********)
Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2.
(**********)
Definition Rminus (r1 r2:R) : R := (r1 + - r2)%R.
(**********)
Definition Rdiv (r1 r2:R) : R := (r1 * / r2)%R.
(**********)
Infix "-" := Rminus : R_scope.
Infix "/" := Rdiv : R_scope.
Infix "<=" := Rle : R_scope.
Infix ">=" := Rge : R_scope.
Infix ">" := Rgt : R_scope.
Notation "x <= y <= z" := ((x <= y)%R /\ (y <= z)%R) : R_scope.
Notation "x <= y < z" := ((x <= y)%R /\ (y < z)%R) : R_scope.
Notation "x < y < z" := ((x < y)%R /\ (y < z)%R) : R_scope.
Notation "x < y <= z" := ((x < y)%R /\ (y <= z)%R) : R_scope.
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