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 ProdTyp[U <: Term with Subs[U], V <: Term with Subs[V]](first: Typ[U], second: Typ[V]) extends Typ[PairTerm[U, V]] with AbsPair[Typ[U], Typ[V]] with Subs[ProdTyp[U, V]] with Product with Serializable

    The product type A times B

    The product type A times B

    first

    the first component

    second

    the second component

    Definition Classes
    HoTT
  • Elem
  • InducFn
  • RecFn

case class RecFn[W <: Term with Subs[W]](codom: Typ[W], data: Func[U, Func[V, W]]) extends RecFunc[PairTerm[U, V], W] with Product with Serializable

Recursion function from product type

codom

the codomain

data

the definition data

Self Type
RecFn[W]
Linear Supertypes
Serializable, Product, Equals, RecFunc[PairTerm[U, V], W], Func[PairTerm[U, V], W], FuncLike[PairTerm[U, V], W], (PairTerm[U, V]) => W, Term, Subs[Func[PairTerm[U, V], W]], AnyRef, Any
Type Hierarchy
Ordering
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Inherited
  1. RecFn
  2. Serializable
  3. Product
  4. Equals
  5. RecFunc
  6. Func
  7. FuncLike
  8. Function1
  9. Term
  10. Subs
  11. AnyRef
  12. 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 RecFn(codom: Typ[W], data: Func[U, Func[V, W]])

    codom

    the codomain

    data

    the definition data

Type Members

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

Value Members

  1. def act(w: PairTerm[U, V]): W

    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
    RecFn → Func → FuncLike
  2. def andThen[A](g: (W) => A): (PairTerm[U, V]) => A
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  3. def apply(arg: PairTerm[U, V]): W

    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: PairTerm[U, V]): W
    Definition Classes
    FuncLike
  5. def baseFunction: ExstFunc
    Definition Classes
    RecFunc
  6. def canApply(arg: PairTerm[U, V]): 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. val codom: Typ[W]

    codomain

    codomain

    Definition Classes
    RecFn → Func
  8. def compose[A](g: (A) => PairTerm[U, V]): (A) => W
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  9. val data: Func[U, Func[V, W]]
  10. val defnData: Vector[Func[U, Func[V, W]]]

    definition data for all introduction rules.

    definition data for all introduction rules.

    Definition Classes
    RecFn → RecFunc
  11. val depcodom: (PairTerm[U, V]) => Typ[W]
    Definition Classes
    Func → FuncLike
  12. def dependsOn(that: Term): Boolean

    returns whether this depends on that

    returns whether this depends on that

    Definition Classes
    Term
  13. lazy val dom: ProdTyp[U, V]

    domain

    domain

    Definition Classes
    RecFn → Func → FuncLike
  14. def equals(that: Any): Boolean
    Definition Classes
    RecFunc → AnyRef → Any
  15. def fromData(data: Vector[Term]): RecFn[W]
    Definition Classes
    RecFn → RecFunc
  16. lazy val fullData: (Typ[PairTerm[U, V]], Typ[W], Vector[Term])
    Definition Classes
    RecFunc
  17. def hashCode(): Int
    Definition Classes
    RecFunc → AnyRef → Any
  18. def indepOf(that: Term): Boolean

    returns whether this is independent of that.

    returns whether this is independent of that.

    Definition Classes
    Term
  19. lazy val intros: Vector[Term]
    Definition Classes
    RecFn → RecFunc
  20. 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
    RecFn → Subs
  21. def productElementNames: Iterator[String]
    Definition Classes
    Product
  22. def replace(x: Term, y: Term): Func[PairTerm[U, V], W] with Subs[Func[PairTerm[U, V], W]]

    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
  23. def subs(x: Term, y: Term): RecFn[W]

    substitute x by y recursively in this.

    substitute x by y recursively in this.

    Definition Classes
    RecFn → Func → FuncLike → Subs
  24. def toString(): String
    Definition Classes
    RecFunc → Function1 → AnyRef → Any
  25. lazy val typ: FuncTyp[PairTerm[U, V], W]

    the HoTT-type of the term

    the HoTT-type of the term

    Definition Classes
    RecFn → Func → FuncLike → Term
  26. def unlift: PartialFunction[PairTerm[U, V], B]
    Implicit
    This member is added by an implicit conversion from RecFn[W] toUnliftOps[PairTerm[U, V], B] performed by method UnliftOps in scala.Function1.This conversion will take place only if W is a subclass of Option[B] (W <: Option[B]).
    Definition Classes
    UnliftOps
  27. 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
    RecFunc → Term