# Fractions

### Related Doc: package rationals

#### class Fractions extends Theory with RingWithDivision

The theory of fractions `s / t`, with `s, t` taken from some ring. The theory uses an encoding in which the same (fixed, but arbitrary) denominator is used for all expressions. The range of considered denominators is described by the `denomConstraint` argument over the variable `_0`.

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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

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

11. #### val dependencies: Iterable[Theory]

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
Theory
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

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. #### final def getClass(): Class[_]

Definition Classes
AnyRef → Any
26. #### def hashCode(): Int

Definition Classes
AnyRef → Any
27. #### 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
28. #### 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
29. #### 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
30. #### val int: IFunction

Function to embed integers in the ring of fractions

31. #### 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
32. #### final def isInstanceOf[T0]: Boolean

Definition Classes
Any
33. #### 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
34. #### def minus(s: ITerm): ITerm

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

Difference between two terms

Difference between two terms

Definition Classes
PseudoRing
36. #### 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
37. #### def mul(s: ITerm, t: ITerm): ITerm

Ring multiplication

Ring multiplication

Definition Classes
FractionsPseudoRing
38. #### final def ne(arg0: AnyRef): Boolean

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

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

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

The one element of this ring

The one element of this ring

Definition Classes
FractionsPseudoRing
42. #### 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
43. #### def plus(s: ITerm, t: ITerm): ITerm

Definition Classes
FractionsPseudoRing
44. #### 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
45. #### 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
46. #### 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
47. #### val predicates: Seq[Predicate]

Interpreted predicates of the theory

Interpreted predicates of the theory

Definition Classes
FractionsTheory
48. #### 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
49. #### def product(terms: ITerm*): ITerm

N-ary sums

N-ary sums

Definition Classes
PseudoRing
50. #### 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
51. #### 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
52. #### 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
53. #### def summation(terms: ITerm*): ITerm

N-ary sums

N-ary sums

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

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

`num * s`

`num * s`

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

Definition Classes
FractionsPseudoRing → AnyRef → Any
57. #### 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
58. #### 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
59. #### final def wait(): Unit

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

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

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

The zero element of this ring

The zero element of this ring

Definition Classes
FractionsPseudoRing