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 IndexedConstructorPatternMap
    Definition Classes
    induction
  • IndexedCnstDepFncPtnMap
  • IndexedCnstFncPtnMap
  • IndexedFuncPtnMap
  • IndexedIdMap
  • IndexedIndexedFuncPtnMap
  • IndexedRecursiveConstructorPatternMap
c

provingground.induction.IndexedConstructorPatternMap

IndexedRecursiveConstructorPatternMap

sealed abstract class IndexedRecursiveConstructorPatternMap[Cod <: Term with Subs[Cod], ArgType <: Term with Subs[ArgType], HeadConstructorType <: Term with Subs[HeadConstructorType], CT <: FuncLike[ArgType, HeadConstructorType] with Subs[CT], H <: Term with Subs[H], RecDataType <: Term with Subs[RecDataType], InducDataType <: Term with Subs[InducDataType], HeadRecDataType <: Term with Subs[HeadRecDataType], HeadInducDataType <: Term with Subs[HeadInducDataType], Fb <: Term with Subs[Fb], Index <: HList, IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], IDFT <: Term with Subs[IDFT]] extends IndexedConstructorPatternMap[Cod, CT, H, RecDataType, InducDataType, Fb, Index, IF, IDF, IDFT]

Self Type
IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]
Linear Supertypes
IndexedConstructorPatternMap[Cod, CT, H, RecDataType, InducDataType, Fb, Index, IF, IDF, IDFT], AnyRef, Any
Known Subclasses
IndexedCnstDepFncPtnMap, IndexedCnstFncPtnMap, IndexedFuncPtnMap, IndexedIndexedFuncPtnMap
Type Hierarchy
Ordering
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Inherited
  1. IndexedRecursiveConstructorPatternMap
  2. IndexedConstructorPatternMap
  3. AnyRef
  4. Any
Implicitly
  1. by any2stringadd
  2. by StringFormat
  3. by Ensuring
  4. by ArrowAssoc
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Visibility
  1. Public
  2. Protected

Abstract Value Members

  1. abstract def apply(tp: Fb): Typ[CT]

    returns HoTT type corresponding to the pattern given the (inductive) type family W (to be the head).

    returns HoTT type corresponding to the pattern given the (inductive) type family W (to be the head).

    Definition Classes
    IndexedConstructorPatternMap
  2. abstract val family: TypFamilyMap[H, Fb, Cod, Index, IF, IDF, IDFT]

    the inductive type family

    the inductive type family

    Definition Classes
    IndexedConstructorPatternMap
  3. abstract def headData(data: RecDataType, arg: ArgType, f: => IF): HeadRecDataType

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

  4. abstract def headInducData(data: InducDataType, arg: ArgType, f: => IDF): HeadInducDataType
  5. abstract val headfibre: (ArgType) => IndexedConstructorPatternMap[Cod, HeadConstructorType, H, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]

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

  6. abstract def inducDataTyp(w: Fb, xs: IDFT)(cons: CT): Typ[InducDataType]

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

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

    Definition Classes
    IndexedConstructorPatternMap
  7. abstract def recDataTyp(wb: Fb, 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
    IndexedConstructorPatternMap
  8. abstract def subs(x: Term, y: Term): IndexedConstructorPatternMap[Cod, CT, H, RecDataType, InducDataType, Fb, Index, IF, IDF, IDFT]
    Definition Classes
    IndexedConstructorPatternMap
  9. abstract val univLevel: Int
    Definition Classes
    IndexedConstructorPatternMap

Concrete 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 IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toany2stringadd[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT], B)
    Implicit
    This member is added by an implicit conversion from IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toArrowAssoc[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] 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 ensuring(cond: (IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]) => Boolean, msg: => Any): IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]
    Implicit
    This member is added by an implicit conversion from IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toEnsuring[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  9. def ensuring(cond: (IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]) => Boolean): IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]
    Implicit
    This member is added by an implicit conversion from IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toEnsuring[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  10. def ensuring(cond: Boolean, msg: => Any): IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]
    Implicit
    This member is added by an implicit conversion from IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toEnsuring[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  11. def ensuring(cond: Boolean): IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]
    Implicit
    This member is added by an implicit conversion from IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toEnsuring[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  12. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  13. def equals(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef → Any
  14. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toStringFormat[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  15. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  16. def hashCode(): Int
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  17. def inducDefCase(cons: CT, data: InducDataType, f: => IDF): (H) => Option[Cod]

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

    given a term, matches to see if this is the image of a given (quasi)-constructor, with this constructor pattern. optionally returns simplification (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 inductively, to be used recursively in definition.

    Definition Classes
    IndexedRecursiveConstructorPatternMap → IndexedConstructorPatternMap
  18. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  19. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  20. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  21. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  22. def recDefCase(cons: CT, data: RecDataType, f: => IF): (H) => Option[Cod]

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

    given a term, matches to see if this is the image of a given (quasi)-constructor, with this constructor pattern. optionally returns simplification (if the term matches), determined by the recursion data.

    cons

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

    data

    definition data for the image of the constructor.

    f

    the function being defined recursively, to be used recursively in definition.

    Definition Classes
    IndexedRecursiveConstructorPatternMap → IndexedConstructorPatternMap
  23. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  24. def toString(): String
    Definition Classes
    AnyRef → Any
  25. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  26. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  27. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])

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): (IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT], B)
    Implicit
    This member is added by an implicit conversion from IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] toArrowAssoc[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]] 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 IndexedConstructorPatternMap[Cod, CT, H, RecDataType, InducDataType, Fb, Index, IF, IDF, IDFT]

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd fromIndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] to any2stringadd[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]]

Inherited by implicit conversion StringFormat fromIndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] to StringFormat[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]]

Inherited by implicit conversion Ensuring fromIndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] to Ensuring[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]]

Inherited by implicit conversion ArrowAssoc fromIndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT] to ArrowAssoc[IndexedRecursiveConstructorPatternMap[Cod, ArgType, HeadConstructorType, CT, H, RecDataType, InducDataType, HeadRecDataType, HeadInducDataType, Fb, Index, IF, IDF, IDFT]]

Ungrouped