 # 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. #### val RatDivZero: IFunction

Uninterpreted function representing the SMT-LIB rational division by zero.

8. #### def additiveGroup: Group with Abelian with SymbolicTimes

Addition gives rise to an Abelian group

Addition gives rise to an Abelian group

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

Definition Classes
Any
10. #### 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
11. #### def clone(): AnyRef

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( ... )
12. #### 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
13. #### 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
14. #### def div(s: ITerm, t: ITerm): ITerm

Division operation

Division operation

Definition Classes
FractionsRingWithDivision
15. #### def divWithSpecialZero(s: ITerm, t: ITerm): ITerm

Division, assuming SMT-LIB semantics for division by zero.

16. #### val dom: Sort

Domain of the ring

Domain of the ring

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

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

Definition Classes
AnyRef → Any
19. #### 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
20. #### 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
21. #### 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
22. #### def finalize(): Unit

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( classOf[java.lang.Throwable] )
23. #### 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
24. #### 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
25. #### 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
26. #### val functions: List[IFunction]

Interpreted functions of the theory

Interpreted functions of the theory

Definition Classes
FractionsTheory
27. #### 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
28. #### def geq(s: ITerm, t: ITerm): IFormula

Greater-than-or-equal operator

Greater-than-or-equal operator

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

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

Greater-than operator

Greater-than operator

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

Definition Classes
AnyRef → Any
32. #### 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
33. #### 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
34. #### 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
35. #### val int: IFunction

Function to embed integers in the ring of fractions

Function to embed integers in the ring of fractions

Definition Classes
Fractions
36. #### 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
37. #### def inverse(s: ITerm): ITerm

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

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

Test whether a rational is integer.

Test whether a rational is integer.

Definition Classes
RationalsRingWithIntConversions
40. #### 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
41. #### def leq(s: ITerm, t: ITerm): IFormula

Less-than-or-equal operator

Less-than-or-equal operator

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

Less-than operator

Less-than operator

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

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

Difference between two terms

Difference between two terms

Definition Classes
PseudoRing
45. #### val modelGenPredicates: Set[Predicate]

Optionally, a set of predicates used by the theory to tell the `PresburgerModelFinder` about terms that will be handled exclusively by this theory.

Optionally, a set of predicates used by the theory to tell the `PresburgerModelFinder` about terms that will be handled exclusively by this theory. If a proof goal in model generation mode contains an atom `p(x)`, for `p` in this set, then the `PresburgerModelFinder` will ignore `x` when assigning concrete values to symbols.

Definition Classes
Theory
46. #### def mul(s: ITerm, t: ITerm): ITerm

Ring multiplication

Ring multiplication

Definition Classes
FractionsPseudoRing
47. #### 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
48. #### def multiplicativeMonoid: Monoid with Abelian

Multiplication gives rise to an Abelian monoid

Multiplication gives rise to an Abelian monoid

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

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

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

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

The one element of this ring

The one element of this ring

Definition Classes
FractionsPseudoRing
53. #### 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
54. #### def plus(s: ITerm, t: ITerm): ITerm

Definition Classes
FractionsPseudoRing
55. #### 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
56. #### 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
57. #### 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
58. #### val predicates: Seq[Predicate]

Interpreted predicates of the theory

Interpreted predicates of the theory

Definition Classes
FractionsTheory
59. #### 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
60. #### def product(terms: ITerm*): ITerm

N-ary sums

N-ary sums

Definition Classes
PseudoRing
61. #### 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
62. #### 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
63. #### 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
64. #### 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
65. #### def summation(terms: ITerm*): ITerm

N-ary sums

N-ary sums

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

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

`num * s`

`num * s`

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

Definition Classes
FractionsPseudoRing → AnyRef → Any
69. #### 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
70. #### 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
71. #### final def wait(): Unit

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

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

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

The zero element of this ring

The zero element of this ring

Definition Classes
FractionsPseudoRing