| Definitions for the axiomatization |
Require Export ZArith_base.
Parameter R : Set.
Delimit Scope R_scope with 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.
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.