Library Coqtail.Reals.Raxioms.Rdefinitions

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 : R → R → R.
Parameter Rmult : R → R → R.
Parameter Ropp : R → R.

Parameter Rinv : ∀ (r : R), r ≠ R0 → 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 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.