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
  • object PlusTyp extends Serializable
    Definition Classes
    HoTT
  • FirstIncl
  • RecFn
  • ScndIncl

case class FirstIncl[U <: Term with Subs[U], V <: Term with Subs[V]](typ: PlusTyp[U, V], value: U) extends Term with Subs[FirstIncl[U, V]] with Product with Serializable

A -> A + B

Linear Supertypes
Serializable, Product, Equals, Term, Subs[FirstIncl[U, V]], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. FirstIncl
  2. Serializable
  3. Product
  4. Equals
  5. Term
  6. Subs
  7. AnyRef
  8. Any
Implicitly
  1. by RichTerm
  2. by any2stringadd
  3. by StringFormat
  4. by Ensuring
  5. by ArrowAssoc
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Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new FirstIncl(typ: PlusTyp[U, V], value: U)

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 FirstIncl[U, V] toany2stringadd[FirstIncl[U, V]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (FirstIncl[U, V], B)
    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toArrowAssoc[FirstIncl[U, V]] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  5. def :->[V <: Term with Subs[V]](that: V): Func[FirstIncl[U, V], V]

    constructor for (pure) lambda functions, see lmbda

    constructor for (pure) lambda functions, see lmbda

    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toRichTerm[FirstIncl[U, V]] performed by method RichTerm in provingground.HoTT.
    Definition Classes
    RichTerm
  6. def :~>[V <: Term with Subs[V]](that: V): FuncLike[FirstIncl[U, V], V]

    constructor for (in general dependent) lambda functions, see lambda

    constructor for (in general dependent) lambda functions, see lambda

    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toRichTerm[FirstIncl[U, V]] performed by method RichTerm in provingground.HoTT.
    Definition Classes
    RichTerm
  7. def =:=(rhs: FirstIncl[U, V]): IdentityTyp[FirstIncl[U, V]]

    equality type 'term = rhs'

    equality type 'term = rhs'

    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toRichTerm[FirstIncl[U, V]] performed by method RichTerm in provingground.HoTT.
    Definition Classes
    RichTerm
  8. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  9. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  10. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
  11. def dependsOn(that: Term): Boolean

    returns whether this depends on that

    returns whether this depends on that

    Definition Classes
    Term
  12. def ensuring(cond: (FirstIncl[U, V]) => Boolean, msg: => Any): FirstIncl[U, V]
    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toEnsuring[FirstIncl[U, V]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  13. def ensuring(cond: (FirstIncl[U, V]) => Boolean): FirstIncl[U, V]
    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toEnsuring[FirstIncl[U, V]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  14. def ensuring(cond: Boolean, msg: => Any): FirstIncl[U, V]
    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toEnsuring[FirstIncl[U, V]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  15. def ensuring(cond: Boolean): FirstIncl[U, V]
    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toEnsuring[FirstIncl[U, V]] 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 FirstIncl[U, V] toStringFormat[FirstIncl[U, V]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  18. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  19. def indepOf(that: Term): Boolean

    returns whether this is independent of that.

    returns whether this is independent of that.

    Definition Classes
    Term
  20. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  21. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  22. 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
    FirstIncl → Subs
  23. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  24. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  25. def productElementNames: Iterator[String]
    Definition Classes
    Product
  26. def refl: Refl[FirstIncl[U, V]]

    reflexivity term refl : term = term

    reflexivity term refl : term = term

    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toRichTerm[FirstIncl[U, V]] performed by method RichTerm in provingground.HoTT.
    Definition Classes
    RichTerm
  27. def replace(x: Term, y: Term): FirstIncl[U, V] with Subs[FirstIncl[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
  28. def subs(x: Term, y: Term): FirstIncl[U with Subs[U], V]

    substitute x by y recursively in this.

    substitute x by y recursively in this.

    Definition Classes
    FirstIncl → Subs
  29. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  30. val typ: PlusTyp[U, V]

    the HoTT-type of the term

    the HoTT-type of the term

    Definition Classes
    FirstIncl → Term
  31. 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
    Term
  32. val value: U
  33. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  34. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  35. 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): (FirstIncl[U, V], B)
    Implicit
    This member is added by an implicit conversion from FirstIncl[U, V] toArrowAssoc[FirstIncl[U, V]] 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 Term

Inherited from Subs[FirstIncl[U, V]]

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion RichTerm fromFirstIncl[U, V] to RichTerm[FirstIncl[U, V]]

Inherited by implicit conversion any2stringadd fromFirstIncl[U, V] to any2stringadd[FirstIncl[U, V]]

Inherited by implicit conversion StringFormat fromFirstIncl[U, V] to StringFormat[FirstIncl[U, V]]

Inherited by implicit conversion Ensuring fromFirstIncl[U, V] to Ensuring[FirstIncl[U, V]]

Inherited by implicit conversion ArrowAssoc fromFirstIncl[U, V] to ArrowAssoc[FirstIncl[U, V]]

Ungrouped