 # Rationals

### Related Doc: package rationals

#### object Rationals extends Fractions with Field with OrderedRing with RingWithIntConversions

The theory and field of rational numbers.

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1. Rationals
2. RingWithIntConversions
3. OrderedRing
4. RingWithOrder
5. Field
6. CommutativeRing
7. CommutativePseudoRing
8. Ring
9. Fractions
10. RingWithDivision
11. PseudoRing
12. Theory
13. AnyRef
14. Any
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### 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. #### object Fraction

Extractor for fractions, where numerator and denominator are expressions from the underlying ring

Extractor for fractions, where numerator and denominator are expressions from the underlying ring

Definition Classes
Fractions
5. #### object FractionSort extends ProxySort

Definition Classes
Fractions
6. #### def additiveGroup: Group with Abelian with SymbolicTimes

Addition gives rise to an Abelian group

Addition gives rise to an Abelian group

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

Definition Classes
Any
8. #### val axioms: Formula

Axioms defining the theory; such axioms are simply added as formulae to the problem to be proven, and thus handled using the standard reasoning techniques (including e-matching).

Axioms defining the theory; such axioms are simply added as formulae to the problem to be proven, and thus handled using the standard reasoning techniques (including e-matching).

Definition Classes
FractionsTheory
9. #### def clone(): AnyRef

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( ... )
10. #### val denom: IFunction

Function used internally to represent the unique denominator for all fractions

Function used internally to represent the unique denominator for all fractions

Definition Classes
Fractions
11. #### val dependencies: List[nia.GroebnerMultiplication.type]

Optionally, other theories that this theory depends on.

Optionally, other theories that this theory depends on. Specified dependencies will be loaded before this theory, but the preprocessors of the dependencies will be called after the preprocessor of this theory.

Definition Classes
RationalsTheory
12. #### def div(s: ITerm, t: ITerm): ITerm

Division operation

Division operation

Definition Classes
FractionsRingWithDivision
13. #### val dom: Sort

Domain of the ring

Domain of the ring

Definition Classes
FractionsPseudoRing
14. #### final def eq(arg0: AnyRef): Boolean

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

Definition Classes
AnyRef → Any
16. #### def evalFun(f: IFunApp): Option[ITerm]

Optionally, a function evaluating theory functions applied to concrete arguments, represented as constructor terms.

Optionally, a function evaluating theory functions applied to concrete arguments, represented as constructor terms.

Definition Classes
Theory
17. #### def evalPred(p: IAtom): Option[Boolean]

Optionally, a function evaluating theory predicates applied to concrete arguments, represented as constructor terms.

Optionally, a function evaluating theory predicates applied to concrete arguments, represented as constructor terms.

Definition Classes
Theory
18. #### def extend(order: TermOrder): TermOrder

Add the symbols defined by this theory to the `order`

Add the symbols defined by this theory to the `order`

Definition Classes
Theory
19. #### def finalize(): Unit

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( classOf[java.lang.Throwable] )
20. #### val frac: IFunction

Function to represent fractions, where numerator and denominator are expressions from the underlying ring

Function to represent fractions, where numerator and denominator are expressions from the underlying ring

Definition Classes
Fractions
21. #### val functionPredicateMapping: List[(IFunction, Predicate)]

Mapping of interpreted functions to interpreted predicates, used translating input ASTs to internal ASTs (the latter only containing predicates).

Mapping of interpreted functions to interpreted predicates, used translating input ASTs to internal ASTs (the latter only containing predicates).

Definition Classes
FractionsTheory
22. #### val functionalPredicates: Set[Predicate]

Information which of the predicates satisfy the functionality axiom; at some internal points, such predicates can be handled more efficiently

Information which of the predicates satisfy the functionality axiom; at some internal points, such predicates can be handled more efficiently

Definition Classes
FractionsTheory
23. #### val functions: List[IFunction]

Interpreted functions of the theory

Interpreted functions of the theory

Definition Classes
FractionsTheory
24. #### def generateDecoderData(model: Conjunction): Option[TheoryDecoderData]

If this theory defines any `Theory.Decoder`, which can translate model data into some theory-specific representation, this function can be overridden to pre-compute required data from a model.

If this theory defines any `Theory.Decoder`, which can translate model data into some theory-specific representation, this function can be overridden to pre-compute required data from a model.

Definition Classes
Theory
25. #### def geq(s: ITerm, t: ITerm): IFormula

Greater-than-or-equal operator

Greater-than-or-equal operator

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

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

Greater-than operator

Greater-than operator

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

Definition Classes
AnyRef → Any
29. #### def iPostprocess(f: IFormula, signature: Signature): IFormula

Optionally, a post-processor that is applied to formulas output by the prover, for instance to interpolants or the result of quantifier elimination.

Optionally, a post-processor that is applied to formulas output by the prover, for instance to interpolants or the result of quantifier elimination. This method will be applied to the formula after calling `Internal2Inputabsy`.

Definition Classes
Theory
30. #### def iPreprocess(f: IFormula, signature: Signature): (IFormula, Signature)

Optionally, a pre-processor that is applied to formulas over this theory, prior to sending the formula to a prover.

Optionally, a pre-processor that is applied to formulas over this theory, prior to sending the formula to a prover. This method will be applied very early in the translation process.

Definition Classes
FractionsTheory
31. #### def individualsStream: Option[Stream[ITerm]]

Optionally, a stream of the constructor terms for this domain can be defined (e.g., the fractions with co-prime numerator and denominator).

Optionally, a stream of the constructor terms for this domain can be defined (e.g., the fractions with co-prime numerator and denominator).

Attributes
protected
Definition Classes
RationalsFractions
32. #### val int: IFunction

Function to embed integers in the ring of fractions

Function to embed integers in the ring of fractions

Definition Classes
Fractions
33. #### 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
FractionsPseudoRing
34. #### def inverse(s: ITerm): ITerm

Definition Classes
Field
35. #### final def isInstanceOf[T0]: Boolean

Definition Classes
Any
36. #### def isInt(s: ITerm): IFormula

Test whether a rational is integer.

Test whether a rational is integer.

Definition Classes
RationalsRingWithIntConversions
37. #### def isSoundForSat(theories: Seq[Theory], config: Theory.SatSoundnessConfig.Value): Boolean

Check whether we can tell that the given combination of theories is sound for checking satisfiability of a problem, i.e., if proof construction ends up in a dead end, can it be concluded that a problem is satisfiable.

Check whether we can tell that the given combination of theories is sound for checking satisfiability of a problem, i.e., if proof construction ends up in a dead end, can it be concluded that a problem is satisfiable.

Definition Classes
FractionsTheory
38. #### def leq(s: ITerm, t: ITerm): IFormula

Less-than-or-equal operator

Less-than-or-equal operator

Definition Classes
RationalsRingWithOrder
39. #### def lt(s: ITerm, t: ITerm): IFormula

Less-than operator

Less-than operator

Definition Classes
RationalsRingWithOrder
40. #### def minus(s: ITerm): ITerm

Definition Classes
FractionsPseudoRing
41. #### def minus(s: ITerm, t: ITerm): ITerm

Difference between two terms

Difference between two terms

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

Ring multiplication

Ring multiplication

Definition Classes
FractionsPseudoRing
43. #### def multiplicativeGroup: Group with Abelian

Non-zero elements now give rise to an Abelian group

Non-zero elements now give rise to an Abelian group

Definition Classes
Field
44. #### def multiplicativeMonoid: Monoid with Abelian

Multiplication gives rise to an Abelian monoid

Multiplication gives rise to an Abelian monoid

Definition Classes
CommutativeRingRing
45. #### final def ne(arg0: AnyRef): Boolean

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

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

Definition Classes
AnyRef
48. #### val one: ITerm

The one element of this ring

The one element of this ring

Definition Classes
FractionsPseudoRing
49. #### val plugin: None.type

Optionally, a plug-in implementing reasoning in this theory

Optionally, a plug-in implementing reasoning in this theory

Definition Classes
FractionsTheory
50. #### def plus(s: ITerm, t: ITerm): ITerm

Definition Classes
FractionsPseudoRing
51. #### def postSimplifiers: Seq[(IExpression) ⇒ IExpression]

Optionally, simplifiers that are applied to formulas output by the prover, for instance to interpolants or the result of quantifier.

Optionally, simplifiers that are applied to formulas output by the prover, for instance to interpolants or the result of quantifier. Such simplifiers are invoked by with `ap.parser.Simplifier`.

Definition Classes
Theory
52. #### def postprocess(f: Conjunction, order: TermOrder): Conjunction

Optionally, a post-processor that is applied to formulas output by the prover, for instance to interpolants or the result of quantifier elimination.

Optionally, a post-processor that is applied to formulas output by the prover, for instance to interpolants or the result of quantifier elimination. This method will be applied to the raw formulas, before calling `Internal2Inputabsy`.

Definition Classes
Theory
53. #### val predicateMatchConfig: PredicateMatchConfig

Information how interpreted predicates should be handled for e-matching.

Information how interpreted predicates should be handled for e-matching.

Definition Classes
FractionsTheory
54. #### val predicates: Seq[Predicate]

Interpreted predicates of the theory

Interpreted predicates of the theory

Definition Classes
FractionsTheory
55. #### def preprocess(f: Conjunction, order: TermOrder): Conjunction

Optionally, a pre-processor that is applied to formulas over this theory, prior to sending the formula to a prover.

Optionally, a pre-processor that is applied to formulas over this theory, prior to sending the formula to a prover.

Definition Classes
Theory
56. #### def product(terms: ITerm*): ITerm

N-ary sums

N-ary sums

Definition Classes
PseudoRing
57. #### val reducerPlugin: ReducerPluginFactory

Optionally, a plugin for the reducer applied to formulas both before and during proving.

Optionally, a plugin for the reducer applied to formulas both before and during proving.

Definition Classes
Theory
58. #### def ring2int(s: ITerm): ITerm

Conversion of a rational term to an integer term, the floor operator.

Conversion of a rational term to an integer term, the floor operator.

Definition Classes
RationalsRingWithIntConversions
59. #### def simplifyFraction(n: ITerm, d: ITerm): (ITerm, ITerm)

Method that can be overwritten in sub-classes to term concrete fractions into canonical form.

Method that can be overwritten in sub-classes to term concrete fractions into canonical form.

Attributes
protected
Definition Classes
RationalsFractions
60. #### val singleInstantiationPredicates: Set[Predicate]

When instantiating existentially quantifier formulas, `EX phi`, at most one instantiation is necessary provided that all predicates in `phi` are contained in this set.

When instantiating existentially quantifier formulas, `EX phi`, at most one instantiation is necessary provided that all predicates in `phi` are contained in this set.

Definition Classes
Theory
61. #### def summation(terms: ITerm*): ITerm

N-ary sums

N-ary sums

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

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

`num * s`

`num * s`

Definition Classes
FractionsPseudoRing
64. #### def toString(): String

Definition Classes
FractionsPseudoRing → AnyRef → Any
65. #### val totalityAxioms: Conjunction

Additional axioms that are included if the option `+genTotalityAxioms` is given to Princess.

Additional axioms that are included if the option `+genTotalityAxioms` is given to Princess.

Definition Classes
FractionsTheory
66. #### val triggerRelevantFunctions: Set[IFunction]

A list of functions that should be considered in automatic trigger generation

A list of functions that should be considered in automatic trigger generation

Definition Classes
FractionsTheory
67. #### final def wait(): Unit

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

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

Definition Classes
AnyRef
Annotations
@throws( ... )
70. #### val zero: ITerm

The zero element of this ring

The zero element of this ring

Definition Classes
FractionsPseudoRing