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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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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Definitions for the axiomatization

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Require Export ZArith_base.

Parameter R : Set.

(* Declare primitive numeral notations for Scope R_scope *)
Declare Scope R_scope.
Declare ML Module "r_syntax_plugin".

(* Declare Scope R_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.

Local Open Scope R_scope.

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.

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Definition Rgt (r1 r2:R) : Prop := r2 < r1.

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Definition Rle (r1 r2:R) : Prop := r1 < r2 \/ r1 = r2.

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Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2.

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Definition Rminus (r1 r2:R) : R := r1 + - r2.

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Definition Rdiv (r1 r2:R) : R := r1 * / r2.

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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 /\ y <= z) : R_scope.
Notation "x <= y < z"  := (x <= y /\ y <  z) : R_scope.
Notation "x < y < z"   := (x <  y /\ y <  z) : R_scope.
Notation "x < y <= z"  := (x <  y /\ y <= z) : R_scope.

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Injection from Z to R

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(* compact representation for 2*p *)
Fixpoint IPR_2 (p:positive) : R :=
  match p with
  | xH => R1 + R1
  | xO p => (R1 + R1) * IPR_2 p
  | xI p => (R1 + R1) * (R1 + IPR_2 p)
  end.

Definition IPR (p:positive) : R :=
  match p with
  | xH => R1
  | xO p => IPR_2 p
  | xI p => R1 + IPR_2 p
  end.
Arguments IPR p%positive : simpl never.

(**********)
Definition IZR (z:Z) : R :=
  match z with
  | Z0 => R0
  | Zpos n => IPR n
  | Zneg n => - IPR n
  end.
Arguments IZR z%Z : simpl never.