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 IdentityTyp extends Serializable
    Definition Classes
    HoTT
  • InducFn
  • RecFn

case class InducFn[U <: Term with Subs[U], V <: Term with Subs[V]](domain: Typ[U], targetFmly: FuncLike[U, FuncLike[U, FuncLike[Equality[U], Typ[V]]]], data: FuncLike[U, V], start: U, end: U) extends IndInducFuncLike[Equality[U], V, Func[U, Func[U, Typ[Term]]], FuncLike[U, FuncLike[U, FuncLike[Equality[U], Typ[V]]]]] with Product with Serializable

inductive definition for identity type family.

Self Type
InducFn[U, V]
Linear Supertypes
Serializable, Product, Equals, IndInducFuncLike[Equality[U], V, Func[U, Func[U, Typ[Term]]], FuncLike[U, FuncLike[U, FuncLike[Equality[U], Typ[V]]]]], FuncLike[Equality[U], V], (Equality[U]) => V, Term, Subs[FuncLike[Equality[U], V]], AnyRef, Any
Type Hierarchy
Ordering
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  2. By Inheritance
Inherited
  1. InducFn
  2. Serializable
  3. Product
  4. Equals
  5. IndInducFuncLike
  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
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Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new InducFn(domain: Typ[U], targetFmly: FuncLike[U, FuncLike[U, FuncLike[Equality[U], Typ[V]]]], data: FuncLike[U, V], start: U, end: U)

Type Members

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

Value Members

  1. def act(t: Equality[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
  2. def andThen[A](g: (V) => A): (Equality[U]) => A
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  3. def apply(arg: Equality[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
  4. def applyUnchecked(arg: Equality[U]): V
    Definition Classes
    FuncLike
  5. def canApply(arg: Equality[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
  6. val codXs: FuncLike[U, FuncLike[U, FuncLike[Equality[U], Typ[V]]]]

    the dependent codomain on the family.

    the dependent codomain on the family.

    Definition Classes
    InducFn → IndInducFuncLike
  7. def compose[A](g: (A) => Equality[U]): (A) => V
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  8. val data: FuncLike[U, V]
  9. val defnData: Vector[FuncLike[U, V]]

    the definition data for all the introduction rules

    the definition data for all the introduction rules

    Definition Classes
    InducFn → IndInducFuncLike
  10. lazy val depcodom: Func[Equality[U], Typ[V]]
    Definition Classes
    InducFn → FuncLike
  11. def dependsOn(that: Term): Boolean

    returns whether this depends on that

    returns whether this depends on that

    Definition Classes
    Term
  12. lazy val dom: IdentityTyp[U]
    Definition Classes
    InducFn → FuncLike
  13. lazy val domW: Func[U, Func[U, IdentityTyp[U]]]

    the domain family, e.g.

    the domain family, e.g. Vec

    Definition Classes
    InducFn → IndInducFuncLike
  14. val domain: Typ[U]
  15. val end: U
  16. def equals(that: Any): Boolean
    Definition Classes
    IndInducFuncLike → AnyRef → Any
  17. def fromData(data: Vector[Term]): InducFn[U, V]
  18. lazy val fullIndData: (Func[U, Func[U, Typ[Term]]], Vector[Term], FuncLike[U, FuncLike[U, FuncLike[Equality[U], Typ[V]]]], Vector[Term])
    Definition Classes
    IndInducFuncLike
  19. def hashCode(): Int
    Definition Classes
    IndInducFuncLike → AnyRef → Any
  20. def indepOf(that: Term): Boolean

    returns whether this is independent of that.

    returns whether this is independent of that.

    Definition Classes
    Term
  21. val index: Vector[U]

    indices of the introduction rules.

    indices of the introduction rules.

    Definition Classes
    InducFn → IndInducFuncLike
  22. val intros: Vector[Term]
    Definition Classes
    InducFn → IndInducFuncLike
  23. 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
  24. lazy val p: Equality[U]
  25. def productElementNames: Iterator[String]
    Definition Classes
    Product
  26. def replace(x: Term, y: Term): FuncLike[Equality[U], V] with Subs[FuncLike[Equality[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
  27. val start: U
  28. def subs(x: Term, y: Term): InducFn[U with Subs[U], V]

    substitute x by y recursively in this.

    substitute x by y recursively in this.

    Definition Classes
    InducFn → FuncLike → Subs
  29. val targetFmly: FuncLike[U, FuncLike[U, FuncLike[Equality[U], Typ[V]]]]
  30. def toString(): String
    Definition Classes
    IndInducFuncLike → Function1 → AnyRef → Any
  31. lazy val typ: GenFuncTyp[Equality[U], V]

    the HoTT-type of the term

    the HoTT-type of the term

    Definition Classes
    InducFn → FuncLike → Term
  32. def unlift: PartialFunction[Equality[U], B]
    Implicit
    This member is added by an implicit conversion from InducFn[U, V] toUnliftOps[Equality[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
  33. 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