Library Coq.Reals.Rdefinitions

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.