Trait

ap.algebra

OrderedRing

Related Doc: package algebra

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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)).

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

  1. abstract val dom: Sort

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    Domain of the ring

    Domain of the ring

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

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    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

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    Less-than-or-equal operator

    Less-than-or-equal operator

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

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    Less-than operator

    Less-than operator

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

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    Additive inverses

    Additive inverses

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

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    Ring multiplication

    Ring multiplication

    Definition Classes
    PseudoRing
  7. abstract def one: ITerm

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    The one element of this ring

    The one element of this ring

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

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    Ring addition

    Ring addition

    Definition Classes
    PseudoRing
  9. abstract def zero: ITerm

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    The zero element of this ring

    The zero element of this ring

    Definition Classes
    PseudoRing

Concrete Value Members

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

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

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

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

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    Addition gives rise to an Abelian group

    Addition gives rise to an Abelian group

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

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    Definition Classes
    Any
  6. def clone(): AnyRef

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

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

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

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

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    Greater-than-or-equal operator

    Greater-than-or-equal operator

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

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

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    Greater-than operator

    Greater-than operator

    Definition Classes
    RingWithOrder
  13. def hashCode(): Int

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

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

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    Difference between two terms

    Difference between two terms

    Definition Classes
    PseudoRing
  16. def multiplicativeMonoid: Monoid

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    Multiplication gives rise to a monoid

    Multiplication gives rise to a monoid

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

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

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

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

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    N-ary sums

    N-ary sums

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

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    N-ary sums

    N-ary sums

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

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

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    num * s

    num * s

    Definition Classes
    PseudoRing
  24. def toString(): String

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

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

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

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    Definition Classes
    AnyRef
    Annotations
    @throws( ... )

Inherited from RingWithOrder

Inherited from Ring

Inherited from PseudoRing

Inherited from AnyRef

Inherited from Any

Ungrouped