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 object Zero extends SmallTyp with Product with Serializable

    Empty type

    Empty type

    Definition Classes
    HoTT
  • Elem
  • InducFn
  • RecFn

case class InducFn[U <: Term with Subs[U]](depcodom: Func[Term, Typ[U]]) extends InducFuncLike[Term, U] with Product with Serializable

Linear Supertypes
Serializable, Product, Equals, InducFuncLike[Term, U], FuncLike[Term, U], (Term) => U, Term, Subs[FuncLike[Term, U]], 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
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Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new InducFn(depcodom: Func[Term, Typ[U]])

Type Members

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

Value Members

  1. def act(arg: Term): U

    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: (U) => A): (Term) => A
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  3. def apply(arg: Term): U

    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: Term): U
    Definition Classes
    FuncLike
  5. def baseFunction: ExstFunc
    Definition Classes
    InducFuncLike
  6. def canApply(arg: Term): 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
  7. def compose[A](g: (A) => Term): (A) => U
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  8. val defnData: Vector[Term]

    the definition data for all the introduction rules

    the definition data for all the introduction rules

    Definition Classes
    InducFn → InducFuncLike
  9. val depcodom: Func[Term, Typ[U]]
    Definition Classes
    InducFn → FuncLike
  10. def dependsOn(that: Term): Boolean

    returns whether this depends on that

    returns whether this depends on that

    Definition Classes
    Term
  11. val dom: Typ[Term]
    Definition Classes
    InducFn → FuncLike
  12. def equals(that: Any): Boolean
    Definition Classes
    InducFuncLike → AnyRef → Any
  13. def fromData(data: Vector[Term]): InducFuncLike[Term, U]
    Definition Classes
    InducFn → InducFuncLike
  14. lazy val fullData: (Typ[Term], (Term) => Typ[U], Vector[Term])
    Definition Classes
    InducFuncLike
  15. def hashCode(): Int
    Definition Classes
    InducFuncLike → AnyRef → Any
  16. def indepOf(that: Term): Boolean

    returns whether this is independent of that.

    returns whether this is independent of that.

    Definition Classes
    Term
  17. val intros: Vector[Term]
    Definition Classes
    InducFn → InducFuncLike
  18. def newobj: FuncLike[Term, U] with Subs[FuncLike[Term, U]]

    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
  19. def productElementNames: Iterator[String]
    Definition Classes
    Product
  20. def replace(x: Term, y: Term): FuncLike[Term, U] with Subs[FuncLike[Term, U]]

    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
  21. def subs(x: Term, y: Term): FuncLike[Term, U]

    substitute x by y recursively in this.

    substitute x by y recursively in this.

    Definition Classes
    InducFn → FuncLike → Subs
  22. def toString(): String
    Definition Classes
    InducFuncLike → Function1 → AnyRef → Any
  23. val typ: Typ[FuncLike[Term, U]]

    the HoTT-type of the term

    the HoTT-type of the term

    Definition Classes
    InducFn → FuncLike → Term
  24. def unlift: PartialFunction[Term, B]
    Implicit
    This member is added by an implicit conversion from InducFn[U] toUnliftOps[Term, B] performed by method UnliftOps in scala.Function1.This conversion will take place only if U is a subclass of Option[B] (U <: Option[B]).
    Definition Classes
    UnliftOps
  25. 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