# OrderedRing

### Related Doc: package algebra

#### trait OrderedRing extends Ring with RingWithOrder

Ordered rings are rings with ordering relation in which addition, multiplication, and ordering are consistent: `leq(s, t) ==> leq(plus(s, a), plus(t, a))` and `leq(zero, s) & leq(zero, t) ==> leq(zero, mul(s, t))`.

Linear Supertypes
Known Subclasses
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Inherited
1. OrderedRing
2. RingWithOrder
3. Ring
4. PseudoRing
5. AnyRef
6. Any
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### Abstract Value Members

1. #### abstract val dom: Sort

Domain of the ring

Domain of the ring

Definition Classes
PseudoRing
2. #### abstract def int2ring(s: ITerm): ITerm

Conversion of an integer term to a ring term

Conversion of an integer term to a ring term

Definition Classes
PseudoRing
3. #### abstract def leq(s: ITerm, t: ITerm): IFormula

Less-than-or-equal operator

Less-than-or-equal operator

Definition Classes
RingWithOrder
4. #### abstract def lt(s: ITerm, t: ITerm): IFormula

Less-than operator

Less-than operator

Definition Classes
RingWithOrder
5. #### abstract def minus(s: ITerm): ITerm

Definition Classes
PseudoRing
6. #### abstract def mul(s: ITerm, t: ITerm): ITerm

Ring multiplication

Ring multiplication

Definition Classes
PseudoRing
7. #### abstract def one: ITerm

The one element of this ring

The one element of this ring

Definition Classes
PseudoRing
8. #### abstract def plus(s: ITerm, t: ITerm): ITerm

Definition Classes
PseudoRing
9. #### abstract def zero: ITerm

The zero element of this ring

The zero element of this ring

Definition Classes
PseudoRing

### Concrete Value Members

1. #### final def !=(arg0: Any): Boolean

Definition Classes
AnyRef → Any
2. #### final def ##(): Int

Definition Classes
AnyRef → Any
3. #### final def ==(arg0: Any): Boolean

Definition Classes
AnyRef → Any
4. #### def additiveGroup: Group with Abelian with SymbolicTimes

Addition gives rise to an Abelian group

Addition gives rise to an Abelian group

Definition Classes
PseudoRing
5. #### final def asInstanceOf[T0]: T0

Definition Classes
Any
6. #### def clone(): AnyRef

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( ... )
7. #### final def eq(arg0: AnyRef): Boolean

Definition Classes
AnyRef
8. #### def equals(arg0: Any): Boolean

Definition Classes
AnyRef → Any
9. #### def finalize(): Unit

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( classOf[java.lang.Throwable] )
10. #### def geq(s: ITerm, t: ITerm): IFormula

Greater-than-or-equal operator

Greater-than-or-equal operator

Definition Classes
RingWithOrder
11. #### final def getClass(): Class[_]

Definition Classes
AnyRef → Any
12. #### def gt(s: ITerm, t: ITerm): IFormula

Greater-than operator

Greater-than operator

Definition Classes
RingWithOrder
13. #### def hashCode(): Int

Definition Classes
AnyRef → Any
14. #### final def isInstanceOf[T0]: Boolean

Definition Classes
Any
15. #### def minus(s: ITerm, t: ITerm): ITerm

Difference between two terms

Difference between two terms

Definition Classes
PseudoRing
16. #### def multiplicativeMonoid: Monoid

Multiplication gives rise to a monoid

Multiplication gives rise to a monoid

Definition Classes
Ring
17. #### final def ne(arg0: AnyRef): Boolean

Definition Classes
AnyRef
18. #### final def notify(): Unit

Definition Classes
AnyRef
19. #### final def notifyAll(): Unit

Definition Classes
AnyRef
20. #### def product(terms: ITerm*): ITerm

N-ary sums

N-ary sums

Definition Classes
PseudoRing
21. #### def summation(terms: ITerm*): ITerm

N-ary sums

N-ary sums

Definition Classes
PseudoRing
22. #### final def synchronized[T0](arg0: ⇒ T0): T0

Definition Classes
AnyRef
23. #### def times(num: IdealInt, s: ITerm): ITerm

`num * s`

`num * s`

Definition Classes
PseudoRing
24. #### def toString(): String

Definition Classes
PseudoRing → AnyRef → Any
25. #### final def wait(): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )
26. #### final def wait(arg0: Long, arg1: Int): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )
27. #### final def wait(arg0: Long): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )