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
  • object TypFamilyPtn
    Definition Classes
    induction
  • DepFuncTypFamily
  • Exst
  • FuncTypFamily
  • IdTypFamily

case class DepFuncTypFamily[U <: Term with Subs[U], H <: Term with Subs[H], TF <: Term with Subs[TF], TI <: HList](head: Typ[U], tailfibre: (U) => TypFamilyPtn[H, TF, TI])(implicit evidence$4: TermList[TI]) extends TypFamilyPtn[H, FuncLike[U, TF], ::[U, TI]] with Product with Serializable

Linear Supertypes
Serializable, Product, Equals, TypFamilyPtn[H, FuncLike[U, TF], ::[U, TI]], AnyRef, Any
Type Hierarchy
Ordering
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  2. By Inheritance
Inherited
  1. DepFuncTypFamily
  2. Serializable
  3. Product
  4. Equals
  5. TypFamilyPtn
  6. AnyRef
  7. Any
Implicitly
  1. by any2stringadd
  2. by StringFormat
  3. by Ensuring
  4. by ArrowAssoc
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Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new DepFuncTypFamily(head: Typ[U], tailfibre: (U) => TypFamilyPtn[H, TF, TI])(implicit arg0: TermList[TI])

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 DepFuncTypFamily[U, H, TF, TI] toany2stringadd[DepFuncTypFamily[U, H, TF, TI]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (DepFuncTypFamily[U, H, TF, TI], B)
    Implicit
    This member is added by an implicit conversion from DepFuncTypFamily[U, H, TF, TI] toArrowAssoc[DepFuncTypFamily[U, H, TF, TI]] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  5. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  6. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  7. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
  8. def codFamily(typ: Typ[Term]): Term
    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  9. def codFromRecType(typ: Typ[Term]): Option[Typ[Term]]
    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  10. def constFinalCod[IDFT <: Term with Subs[IDFT]](depCod: IDFT): OptTyp
    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  11. def domFromRecType(typ: Typ[Term]): Option[Typ[Term]]
    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  12. def ensuring(cond: (DepFuncTypFamily[U, H, TF, TI]) => Boolean, msg: => Any): DepFuncTypFamily[U, H, TF, TI]
    Implicit
    This member is added by an implicit conversion from DepFuncTypFamily[U, H, TF, TI] toEnsuring[DepFuncTypFamily[U, H, TF, TI]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  13. def ensuring(cond: (DepFuncTypFamily[U, H, TF, TI]) => Boolean): DepFuncTypFamily[U, H, TF, TI]
    Implicit
    This member is added by an implicit conversion from DepFuncTypFamily[U, H, TF, TI] toEnsuring[DepFuncTypFamily[U, H, TF, TI]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  14. def ensuring(cond: Boolean, msg: => Any): DepFuncTypFamily[U, H, TF, TI]
    Implicit
    This member is added by an implicit conversion from DepFuncTypFamily[U, H, TF, TI] toEnsuring[DepFuncTypFamily[U, H, TF, TI]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  15. def ensuring(cond: Boolean): DepFuncTypFamily[U, H, TF, TI]
    Implicit
    This member is added by an implicit conversion from DepFuncTypFamily[U, H, TF, TI] toEnsuring[DepFuncTypFamily[U, H, TF, TI]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  16. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  17. def finalCod[IDFT <: Term with Subs[IDFT]](depCod: IDFT): Typ[_]
    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  18. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from DepFuncTypFamily[U, H, TF, TI] toStringFormat[DepFuncTypFamily[U, H, TF, TI]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  19. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  20. def getIndex(w: FuncLike[U, TF], typ: Typ[H]): Option[::[U, TI]]

    optional index a given family W and type W(a)

    optional index a given family W and type W(a)

    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  21. def getMapper[C <: Term with Subs[C], IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], IDFT <: Term with Subs[IDFT]](cod: Typ[C])(implicit mpr: TypFamilyMapper[H, FuncLike[U, TF], C, ::[U, TI], IF, IDF, IDFT]): TypFamilyMapper[H, FuncLike[U, TF], C, ::[U, TI], IF, IDF, IDFT]

    mappper to bridge to TypFamilyMap given scala type of codomain based on implicits

    mappper to bridge to TypFamilyMap given scala type of codomain based on implicits

    Definition Classes
    TypFamilyPtn
  22. val head: Typ[U]
  23. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  24. def mapped[C <: Term with Subs[C]]: TypFamilyMap[H, FuncLike[U, TF], C, ::[U, TI], IF, IDF, IDFT] forSome {type IF <: Term with Subs[IF], type IDF <: Term with Subs[IDF], type IDFT <: Term with Subs[IDFT]}

    lift to TypFamilyMap based on mapper

    lift to TypFamilyMap based on mapper

    Definition Classes
    TypFamilyPtn
  25. def mapper[C <: Term with Subs[C]]: TypFamilyMapper[H, FuncLike[U, TF], C, ::[U, TI], FuncLike[U, IF], FuncLike[U, IDF], FuncLike[U, IDFT]] forSome {type IF <: Term with Subs[IF], type IDF <: Term with Subs[IDF], type IDFT <: Term with Subs[IDFT]}

    existentital typed mappper to bridge to TypFamilyMap given scala type of codomain

    existentital typed mappper to bridge to TypFamilyMap given scala type of codomain

    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  26. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  27. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  28. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  29. def productElementNames: Iterator[String]
    Definition Classes
    Product
  30. def someTyp(w: FuncLike[U, TF]): Typ[H]

    A type for some index, used to infer H

    A type for some index, used to infer H

    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  31. def subs(x: Term, y: Term): DepFuncTypFamily[U, H, TF, TI]
    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  32. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  33. val tailfibre: (U) => TypFamilyPtn[H, TF, TI]
  34. val tlEvidence: TermList[::[U, TI]]
    Definition Classes
    TypFamilyPtn
  35. def typ(w: FuncLike[U, TF], index: ::[U, TI]): Typ[H]

    type W(a) given the family W and index a

    type W(a) given the family W and index a

    Definition Classes
    DepFuncTypFamily → TypFamilyPtn
  36. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  37. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  38. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  39. def ~>:[TT <: Term with Subs[TT]](variable: TT): DepFuncTypFamily[TT, H, FuncLike[U, TF], ::[U, TI]]
    Definition Classes
    TypFamilyPtn

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): (DepFuncTypFamily[U, H, TF, TI], B)
    Implicit
    This member is added by an implicit conversion from DepFuncTypFamily[U, H, TF, TI] toArrowAssoc[DepFuncTypFamily[U, H, TF, TI]] 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 TypFamilyPtn[H, FuncLike[U, TF], ::[U, TI]]

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd fromDepFuncTypFamily[U, H, TF, TI] to any2stringadd[DepFuncTypFamily[U, H, TF, TI]]

Inherited by implicit conversion StringFormat fromDepFuncTypFamily[U, H, TF, TI] to StringFormat[DepFuncTypFamily[U, H, TF, TI]]

Inherited by implicit conversion Ensuring fromDepFuncTypFamily[U, H, TF, TI] to Ensuring[DepFuncTypFamily[U, H, TF, TI]]

Inherited by implicit conversion ArrowAssoc fromDepFuncTypFamily[U, H, TF, TI] to ArrowAssoc[DepFuncTypFamily[U, H, TF, TI]]

Ungrouped