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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) *)
(************************************************************************)
(*********************************************************)
(** Definitions for the axiomatization *)
(*********************************************************)
Declare ML Module "r_syntax_plugin".
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.
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.
(***********************************************************)
(**********)
Definition Rgt (r1 r2:R) : Prop := r2 < r1.
(**********)
Definition Rle (r1 r2:R) : Prop := r1 < r2 \/ r1 = r2.
(**********)
Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2.
(**********)
Definition Rminus (r1 r2:R) : R := r1 + - r2.
(**********)
Definition Rdiv (r1 r2:R) : R := r1 * / r2.
(**********)
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.
(**********************************************************)
(** * Injection from [Z] to [R] *)
(**********************************************************)
(* 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.
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