# Library Coqtail.Reals.Raxioms.Rdefinitions

Definitions for the axiomatization

Require Export ZArith_base.

Parameter R : Set.

Delimit Scope R_scope with R.

Bind Scope R_scope with R.

Open Local Scope R_scope.

Parameter R0 : R.
Parameter R1 : R.
Parameter Rplus : RRR.
Parameter Rmult : RRR.
Parameter Ropp : RR.
WATCH OUT
Parameter Rinv : (r : R), r R0R.
Parameter Rlt : RRProp.
Parameter up : RZ.

Infix "+" := Rplus : R_scope.
Infix "×" := Rmult : R_scope.
Notation "- x" := (Ropp x) : R_scope.
Notation "/ x H" := (Rinv x H) (at level 0) : 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) (H : r2 R0): R := r1 × (Rinv r2 H).

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.