Packages

  • package root
    Definition Classes
    root
  • package provingground

    This is work towards automated theorem proving based on learning, using homotopy type theory (HoTT) as foundations and natural language processing.

    This is work towards automated theorem proving based on learning, using homotopy type theory (HoTT) as foundations and natural language processing.

    The implementation of homotopy type theory is split into:

    • the object HoTT with terms, types, functions and dependent functions, pairs etc
    • the package induction with general inductive types and recursion/induction on these.

    The learning package has the code for learning.

    Scala code, including the spire library, is integrated with homotopy type theory in the scalahott package

    We have implemented a functor based approach to translation in the translation package, used for nlp as well as serialization and parsing.

    The library package is contains basic structures implemented in HoTT.

    Definition Classes
    root
  • package induction

    Much of the richness of HoTT is in the definitions of Inductive types (and their indexed counterparts) and of (dependent) functions on these by recursion and induction These are implemented using several layers of recursive definitions and diagonals (i.e., fixed points).

    Much of the richness of HoTT is in the definitions of Inductive types (and their indexed counterparts) and of (dependent) functions on these by recursion and induction These are implemented using several layers of recursive definitions and diagonals (i.e., fixed points). In HoTT, recursion and induction are applications of (dependent) functions rec_W,X and ind_W, Xs to the definition data.

    It is useful to capture information regarding inductive types and the recursion and induction functions in scala types. Our implementation is designed to do this.

    Inductive Type Definitions

    Inductive types are specified by introduction rules. Each introduction rule is specified in ConstructorShape (without specifying the type) and ConstructorTL including the specific type. The full definition is in ConstructorSeqTL.

    Recursion and Induction functions

    These are defined recursively, first for each introduction rule and then for the inductive type as a whole. A subtlety is that the scala type of the rec_W,X and induc_W, Xs functions depends on the scala type of the codomain X (or family Xs). To make these types visible, some type level calculations using implicits are done, giving traits ConstructorPatternMap and ConstructorSeqMap that have recursive definition of the recursion and induction functions, the former for the case of a single introduction rule. Traits ConstructorSeqMapper and ConstructorPatternMapper provide the lifts.

    Indexed Versions

    There are indexed versions of all these definitions, to work with indexed inductive type families.

    Definition Classes
    provingground
  • package coarse

    an earlier implementation of induction, without clean separation of codomain scala types; use induction instead

    an earlier implementation of induction, without clean separation of codomain scala types; use induction instead

    Definition Classes
    induction
  • class IndexedConstructorPatterns[C <: Term with Subs[C], H <: Term with Subs[H], F <: Term with Subs[F]] extends AnyRef

    Definition Classes
    coarse
  • CnstDepFuncPtn
  • CnstFncPtn
  • DepFuncPtn
  • Family
  • FuncPtn
  • PartialiConstructorSeq
  • RecursiveiConstructorPattern
  • iConstructor
  • iConstructorDefn
  • iConstructorPattern
  • iConstructorSeq
  • iConstructorTyp
  • iW

case class DepFuncPtn[U <: Term with Subs[U], V <: Term with Subs[V], VV <: Term with Subs[VV], W <: Term with Subs[W], TF <: Term with Subs[TF]](tail: IterFuncPtn[H, C, TF], tailIndex: Ind, index: Ind, headfibre: (TF) => iConstructorPattern[U] { ... /* 2 definitions in type refinement */ }) extends RecursiveiConstructorPattern[TF, U, FuncLike[TF, U]] with Product with Serializable

Self Type
DepFuncPtn[U, V, VV, W, TF]
Linear Supertypes
Serializable, Product, Equals, RecursiveiConstructorPattern[TF, U, FuncLike[TF, U]], iConstructorPattern[FuncLike[TF, U]], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. DepFuncPtn
  2. Serializable
  3. Product
  4. Equals
  5. RecursiveiConstructorPattern
  6. iConstructorPattern
  7. AnyRef
  8. Any
Implicitly
  1. by any2stringadd
  2. by StringFormat
  3. by Ensuring
  4. by ArrowAssoc
  1. Hide All
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Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new DepFuncPtn(tail: IterFuncPtn[H, C, TF], tailIndex: Ind, index: Ind, headfibre: (TF) => iConstructorPattern[U] { ... /* 2 definitions in type refinement */ })

Type Members

  1. type ArgType = TF

    scala type of argument to constructor A -> ...

    scala type of argument to constructor A -> ... (or A ~> ...)

    Definition Classes
    RecursiveiConstructorPattern
  2. type Cod = C
  3. type HeadInducDataType = VV
    Definition Classes
    DepFuncPtn → RecursiveiConstructorPattern
  4. type HeadRecDataType = V

    (scala) type of recursive data for head.

    (scala) type of recursive data for head.

    Definition Classes
    DepFuncPtn → RecursiveiConstructorPattern
  5. type HeadType = U

    scala type of the head T for constructor A -> T for Pi-Types, the head may have varying HoTT type but must have fixed scala type.

    scala type of the head T for constructor A -> T for Pi-Types, the head may have varying HoTT type but must have fixed scala type.

    Definition Classes
    RecursiveiConstructorPattern
  6. type InducDataType = FuncLike[IterFuncPattern.IterFuncPtn.Family, Func[DepTargetType, VV]]
    Definition Classes
    DepFuncPtn → iConstructorPattern
  7. type RecDataType = FuncLike[IterFuncPattern.IterFuncPtn.Family, Func[TargetType, V]]

    (scala) type of data for recursion corresponding to the single constructor

    (scala) type of data for recursion corresponding to the single constructor

    Definition Classes
    DepFuncPtn → iConstructorPattern
  8. type iConstructorType = FuncLike[TF, U]

    type of a constructor for this pattern.

    type of a constructor for this pattern.

    Definition Classes
    iConstructorPattern

Value Members

  1. final def !=(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  2. final def ##: Int
    Definition Classes
    AnyRef → Any
  3. def +(other: String): String
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toany2stringadd[DepFuncPtn[U, V, VV, W, TF]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def -->:(thatInd: (IterFuncPtn[H, IndexedConstructorPatterns.Cod, F], Ind)): FuncPtn[F, FuncLike[TF, U]]
    Definition Classes
    iConstructorPattern
  5. def ->[B](y: B): (DepFuncPtn[U, V, VV, W, TF], B)
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toArrowAssoc[DepFuncPtn[U, V, VV, W, TF]] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  6. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  7. def apply(tps: F): PiDefn[ArgType, U]
    Definition Classes
    DepFuncPtn → iConstructorPattern
  8. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  9. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
  10. def codClass[CC <: Term with Subs[CC]](w: Typ[H]): IndexedConstructorPatterns[CC, H, F]
    Definition Classes
    iConstructorPattern
  11. def constructor(tp: => F, name: AnySym): iConstructorDefn[iConstructorType with Subs[iConstructorType]]

    constructor for this pattern given inductive type and name.

    constructor for this pattern given inductive type and name.

    Definition Classes
    iConstructorPattern
  12. def ensuring(cond: (DepFuncPtn[U, V, VV, W, TF]) => Boolean, msg: => Any): DepFuncPtn[U, V, VV, W, TF]
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toEnsuring[DepFuncPtn[U, V, VV, W, TF]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  13. def ensuring(cond: (DepFuncPtn[U, V, VV, W, TF]) => Boolean): DepFuncPtn[U, V, VV, W, TF]
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toEnsuring[DepFuncPtn[U, V, VV, W, TF]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  14. def ensuring(cond: Boolean, msg: => Any): DepFuncPtn[U, V, VV, W, TF]
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toEnsuring[DepFuncPtn[U, V, VV, W, TF]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  15. def ensuring(cond: Boolean): DepFuncPtn[U, V, VV, W, TF]
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toEnsuring[DepFuncPtn[U, V, VV, W, TF]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  16. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  17. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toStringFormat[DepFuncPtn[U, V, VV, W, TF]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  18. def funcFrom[T <: Term with Subs[T]](tail: Typ[T], fmly: FamilyType): CnstFncPtn[T, FuncLike[TF, U]]
    Definition Classes
    iConstructorPattern
  19. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  20. def headData(data: RecDataType, arg: ArgType, f: => I): HeadRecDataType

    returns data for recursion to be passed on to the head given an argument (when matching with the construtor).

    returns data for recursion to be passed on to the head given an argument (when matching with the construtor).

    Definition Classes
    DepFuncPtn → RecursiveiConstructorPattern
  21. def headInducData(data: InducDataType, arg: ArgType, f: => DI): HeadInducDataType
    Definition Classes
    DepFuncPtn → RecursiveiConstructorPattern
  22. val headfibre: (TF) => iConstructorPattern[U] { ... /* 2 definitions in type refinement */ }

    The head pattern, constant T for A -> T and T(a) for A ~> T(a)

    The head pattern, constant T for A -> T and T(a) for A ~> T(a)

    Definition Classes
    DepFuncPtn → RecursiveiConstructorPattern
  23. def inClass[CC <: Term with Subs[CC]](w: Typ[H])(that: IndexedConstructorPatterns[CC, H, F]): iConstructorPattern[iConstructorType]
    Definition Classes
    DepFuncPtn → iConstructorPattern
  24. val index: Ind

    argument for the final image

    argument for the final image

    Definition Classes
    DepFuncPtn → iConstructorPattern
  25. def inducDataTyp(tps: F, xs: Func[Total, Typ[Cod]])(cons: iConstructorType): Typ[InducDataType]
    Definition Classes
    DepFuncPtn → iConstructorPattern
  26. def inducDefCase(cons: iConstructorType, data: InducDataType, f: => DI): (Total) => Option[IndexedConstructorPatterns.Cod]
    Definition Classes
    RecursiveiConstructorPattern → iConstructorPattern
  27. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  28. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  29. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  30. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  31. def piOf[T <: Term with Subs[T]](tailVar: T, fmly: FamilyType): CnstDepFuncPtn[U, FuncLike[TF, U], RecDataType, InducDataType, Nothing] forSome {type U >: tailVar.type <: Term with Subs[U]}
    Definition Classes
    iConstructorPattern
  32. def productElementNames: Iterator[String]
    Definition Classes
    Product
  33. def recDataTyp(tps: F, x: Typ[Cod]): Typ[RecDataType]

    domain containing the recursion data for the constructor, i.e., the HoTT type of recursion data.

    domain containing the recursion data for the constructor, i.e., the HoTT type of recursion data.

    Definition Classes
    DepFuncPtn → iConstructorPattern
  34. def recDefCase(cons: iConstructorType, data: RecDataType, f: => I): (Total) => Option[IndexedConstructorPatterns.Cod]

    given a term, matches to see if this is the image of a given (quasi)-constructor.

    given a term, matches to see if this is the image of a given (quasi)-constructor. returns simplification (wrapped in Some) if the term matches.

    cons

    constructor, actually quasi-constructor, with which to match.

    data

    definition data for the image of the constructor.

    f

    the function being defined, to be applied recursively.

    Definition Classes
    RecursiveiConstructorPattern → iConstructorPattern
  35. def subs(x: Term, y: Term): DepFuncPtn[U, (<refinement>.this)#RecDataType, (<refinement>.this)#InducDataType, Nothing, IterFuncPattern.IterFuncPtn.Family]
    Definition Classes
    DepFuncPtn → iConstructorPattern
  36. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  37. val tail: IterFuncPtn[H, C, TF]
  38. val tailIndex: Ind
  39. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  40. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  41. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  42. def withCod[CC <: Term with Subs[CC]](w: Typ[H]): iConstructorPattern[FuncLike[TF, U]]
    Definition Classes
    iConstructorPattern

Deprecated Value Members

  1. def finalize(): Unit
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.Throwable]) @Deprecated
    Deprecated
  2. def →[B](y: B): (DepFuncPtn[U, V, VV, W, TF], B)
    Implicit
    This member is added by an implicit conversion from DepFuncPtn[U, V, VV, W, TF] toArrowAssoc[DepFuncPtn[U, V, VV, W, TF]] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @deprecated
    Deprecated

    (Since version 2.13.0) Use -> instead. If you still wish to display it as one character, consider using a font with programming ligatures such as Fira Code.

Inherited from Serializable

Inherited from Product

Inherited from Equals

Inherited from RecursiveiConstructorPattern[TF, U, FuncLike[TF, U]]

Inherited from iConstructorPattern[FuncLike[TF, U]]

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd fromDepFuncPtn[U, V, VV, W, TF] to any2stringadd[DepFuncPtn[U, V, VV, W, TF]]

Inherited by implicit conversion StringFormat fromDepFuncPtn[U, V, VV, W, TF] to StringFormat[DepFuncPtn[U, V, VV, W, TF]]

Inherited by implicit conversion Ensuring fromDepFuncPtn[U, V, VV, W, TF] to Ensuring[DepFuncPtn[U, V, VV, W, TF]]

Inherited by implicit conversion ArrowAssoc fromDepFuncPtn[U, V, VV, W, TF] to ArrowAssoc[DepFuncPtn[U, V, VV, W, TF]]

Ungrouped