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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \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 *)
Delimits 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.
V8Infix "+" Rplus : R_scope.
V8Infix "*" Rmult : R_scope.
V8Notation "- x" := (Ropp x) : R_scope.
V8Notation "/ x" := (Rinv x) : R_scope.
V8Infix "<" Rlt : R_scope.
(*i*******************************************************i*)
(**********)
Definition Rgt:R->R->Prop:=[r1,r2:R](Rlt r2 r1).
(**********)
Definition Rle:R->R->Prop:=[r1,r2:R]((Rlt r1 r2)\/(r1==r2)).
(**********)
Definition Rge:R->R->Prop:=[r1,r2:R]((Rgt r1 r2)\/(r1==r2)).
(**********)
Definition Rminus:R->R->R:=[r1,r2:R](Rplus r1 (Ropp r2)).
(**********)
Definition Rdiv:R->R->R:=[r1,r2:R](Rmult r1 (Rinv r2)).
V8Infix "-" Rminus : R_scope.
V8Infix "/" Rdiv : R_scope.
V8Infix "<=" Rle : R_scope.
V8Infix ">=" Rge : R_scope.
V8Infix ">" Rgt : R_scope.
V8Notation "x <= y <= z" := (Rle x y)/\(Rle y z) : R_scope.
V8Notation "x <= y < z" := (Rle x y)/\(Rlt y z) : R_scope.
V8Notation "x < y < z" := (Rlt x y)/\(Rlt y z) : R_scope.
V8Notation "x < y <= z" := (Rlt x y)/\(Rle y z) : R_scope.
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