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
  • object HoTT

    Core of Homotopy Type Theory (HoTT) implementation.

    Core of Homotopy Type Theory (HoTT) implementation. Includes: - terms : Term, - types : Typ - universes - functions and dependent functions (see [FuncLike], [Func]) - function types FuncTyp and pi-types PiDefn, - lambda definitions LambdaLike, - pairs PairTerm and dependent pairs DepPair - product types ProdTyp and sigma types SigmaTyp - Coproduct types PlusTyp, the Unit type Unit and the empty type Zero - recursion and induction functions for products, coproducts

    General inductive types are not implemented here, but in the induction package.

    Definition Classes
    provingground
  • case class SigmaTyp[W <: Term with Subs[W], U <: Term with Subs[U]](fibers: TypFamily[W, U]) extends Typ[AbsPair[W, U]] with Subs[SigmaTyp[W, U]] with Product with Serializable

    Sigma Type, i.e., dependent pair type.

    Sigma Type, i.e., dependent pair type.

    Definition Classes
    HoTT
  • Elem
  • InducFn
  • RecFn

case class InducFn[V <: Term with Subs[V]](targetFmly: FuncLike[W, Func[U, Typ[V]]], data: FuncLike[W, FuncLike[U, V]]) extends InducFuncLike[AbsPair[W, U], V] with Product with Serializable

inductive definition on the Sigma-type

Self Type
InducFn[V]
Linear Supertypes
Serializable, Product, Equals, InducFuncLike[AbsPair[W, U], V], FuncLike[AbsPair[W, U], V], (AbsPair[W, U]) => V, Term, Subs[FuncLike[AbsPair[W, U], V]], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. InducFn
  2. Serializable
  3. Product
  4. Equals
  5. InducFuncLike
  6. FuncLike
  7. Function1
  8. Term
  9. Subs
  10. AnyRef
  11. Any
Implicitly
  1. by UnliftOps
  2. by any2stringadd
  3. by StringFormat
  4. by Ensuring
  5. by ArrowAssoc
  1. Hide All
  2. Show All
Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new InducFn(targetFmly: FuncLike[W, Func[U, Typ[V]]], data: FuncLike[W, FuncLike[U, V]])

Type Members

  1. abstract type Obj <: FuncLike[AbsPair[W, U], V]
    Definition Classes
    FuncLike

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 InducFn[V] toany2stringadd[InducFn[V]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (InducFn[V], B)
    Implicit
    This member is added by an implicit conversion from InducFn[V] toArrowAssoc[InducFn[V]] performed by method ArrowAssoc in scala.Predef.This conversion will take place only if V is a subclass of Option[Nothing] (V <: Option[Nothing]).
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  5. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  6. def act(w: AbsPair[W, U]): V

    the action of the function to define: define this method, but use apply.

    the action of the function to define: define this method, but use apply.

    Definition Classes
    InducFn → FuncLike
  7. def andThen[A](g: (V) => A): (AbsPair[W, U]) => A
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  8. def apply(arg: AbsPair[W, U]): V

    application of the function: use this but define the act method; checks HoTT-type of argument is in the domain and throws exception if it fails.

    application of the function: use this but define the act method; checks HoTT-type of argument is in the domain and throws exception if it fails.

    Definition Classes
    FuncLike → Function1
  9. def applyUnchecked(arg: AbsPair[W, U]): V
    Definition Classes
    FuncLike
  10. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  11. def baseFunction: ExstFunc
    Definition Classes
    InducFuncLike
  12. def canApply(arg: AbsPair[W, U]): Boolean

    checks if application is valid; can override to allow for example resizing universes

    checks if application is valid; can override to allow for example resizing universes

    arg

    the argument

    returns

    whether the argument has the correct type.

    Definition Classes
    FuncLike
  13. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
  14. def compose[A](g: (A) => AbsPair[W, U]): (A) => V
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  15. val data: FuncLike[W, FuncLike[U, V]]
  16. val defnData: Vector[FuncLike[W, FuncLike[U, V]]]

    the definition data for all the introduction rules

    the definition data for all the introduction rules

    Definition Classes
    InducFn → InducFuncLike
  17. lazy val depcodom: Func[AbsPair[W, U], Typ[V]]
    Definition Classes
    InducFn → FuncLike
  18. def dependsOn(that: Term): Boolean

    returns whether this depends on that

    returns whether this depends on that

    Definition Classes
    Term
  19. lazy val dom: SigmaTyp[W, U]
    Definition Classes
    InducFn → FuncLike
  20. def ensuring(cond: (InducFn[V]) => Boolean, msg: => Any): InducFn[V]
    Implicit
    This member is added by an implicit conversion from InducFn[V] toEnsuring[InducFn[V]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  21. def ensuring(cond: (InducFn[V]) => Boolean): InducFn[V]
    Implicit
    This member is added by an implicit conversion from InducFn[V] toEnsuring[InducFn[V]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  22. def ensuring(cond: Boolean, msg: => Any): InducFn[V]
    Implicit
    This member is added by an implicit conversion from InducFn[V] toEnsuring[InducFn[V]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  23. def ensuring(cond: Boolean): InducFn[V]
    Implicit
    This member is added by an implicit conversion from InducFn[V] toEnsuring[InducFn[V]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  24. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  25. def equals(that: Any): Boolean
    Definition Classes
    InducFuncLike → AnyRef → Any
  26. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from InducFn[V] toStringFormat[InducFn[V]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  27. def fromData(data: Vector[Term]): InducFn[V]
    Definition Classes
    InducFn → InducFuncLike
  28. lazy val fullData: (Typ[AbsPair[W, U]], (AbsPair[W, U]) => Typ[V], Vector[Term])
    Definition Classes
    InducFuncLike
  29. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  30. def hashCode(): Int
    Definition Classes
    InducFuncLike → AnyRef → Any
  31. def indepOf(that: Term): Boolean

    returns whether this is independent of that.

    returns whether this is independent of that.

    Definition Classes
    Term
  32. lazy val intros: Vector[Term]
    Definition Classes
    InducFn → InducFuncLike
  33. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  34. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  35. def newobj: Nothing

    A new object with the same type, to be used in place of a variable to avoid name clashes.

    A new object with the same type, to be used in place of a variable to avoid name clashes. Should throw exception when invoked for constants.

    Definition Classes
    InducFn → Subs
  36. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  37. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  38. def productElementNames: Iterator[String]
    Definition Classes
    Product
  39. def replace(x: Term, y: Term): FuncLike[AbsPair[W, U], V] with Subs[FuncLike[AbsPair[W, U], V]]

    refine substitution so if x and y are both of certain forms such as pairs or formal applications, components are substituted.

    refine substitution so if x and y are both of certain forms such as pairs or formal applications, components are substituted.

    Definition Classes
    Subs
  40. def subs(x: Term, y: Term): InducFn[V]

    substitute x by y recursively in this.

    substitute x by y recursively in this.

    Definition Classes
    InducFn → FuncLike → Subs
  41. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  42. val targetFmly: FuncLike[W, Func[U, Typ[V]]]
  43. def toString(): String
    Definition Classes
    InducFuncLike → Function1 → AnyRef → Any
  44. lazy val typ: GenFuncTyp[AbsPair[W, U], V]

    the HoTT-type of the term

    the HoTT-type of the term

    Definition Classes
    InducFn → FuncLike → Term
  45. def unlift: PartialFunction[AbsPair[W, U], B]
    Implicit
    This member is added by an implicit conversion from InducFn[V] toUnliftOps[AbsPair[W, U], B] performed by method UnliftOps in scala.Function1.This conversion will take place only if V is a subclass of Option[B] (V <: Option[B]).
    Definition Classes
    UnliftOps
  46. def usesVar(t: Term): Boolean

    returns whether the variable t is used as a variable in a lambda definition.

    returns whether the variable t is used as a variable in a lambda definition.

    Definition Classes
    InducFuncLike → Term
  47. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  48. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  49. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  50. val xy: AbsPair[W, U]

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): (InducFn[V], B)
    Implicit
    This member is added by an implicit conversion from InducFn[V] toArrowAssoc[InducFn[V]] performed by method ArrowAssoc in scala.Predef.This conversion will take place only if V is a subclass of Option[Nothing] (V <: Option[Nothing]).
    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 InducFuncLike[AbsPair[W, U], V]

Inherited from FuncLike[AbsPair[W, U], V]

Inherited from (AbsPair[W, U]) => V

Inherited from Term

Inherited from Subs[FuncLike[AbsPair[W, U], V]]

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion UnliftOps fromInducFn[V] to UnliftOps[AbsPair[W, U], B]

Inherited by implicit conversion any2stringadd fromInducFn[V] to any2stringadd[InducFn[V]]

Inherited by implicit conversion StringFormat fromInducFn[V] to StringFormat[InducFn[V]]

Inherited by implicit conversion Ensuring fromInducFn[V] to Ensuring[InducFn[V]]

Inherited by implicit conversion ArrowAssoc fromInducFn[V] to ArrowAssoc[InducFn[V]]

Ungrouped