# 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.```
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