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
  • package induction

    Much of the richness of HoTT is in the definitions of Inductive types (and their indexed counterparts) and of (dependent) functions on these by recursion and induction These are implemented using several layers of recursive definitions and diagonals (i.e., fixed points).

    Much of the richness of HoTT is in the definitions of Inductive types (and their indexed counterparts) and of (dependent) functions on these by recursion and induction These are implemented using several layers of recursive definitions and diagonals (i.e., fixed points). In HoTT, recursion and induction are applications of (dependent) functions rec_W,X and ind_W, Xs to the definition data.

    It is useful to capture information regarding inductive types and the recursion and induction functions in scala types. Our implementation is designed to do this.

    Inductive Type Definitions

    Inductive types are specified by introduction rules. Each introduction rule is specified in ConstructorShape (without specifying the type) and ConstructorTL including the specific type. The full definition is in ConstructorSeqTL.

    Recursion and Induction functions

    These are defined recursively, first for each introduction rule and then for the inductive type as a whole. A subtlety is that the scala type of the rec_W,X and induc_W, Xs functions depends on the scala type of the codomain X (or family Xs). To make these types visible, some type level calculations using implicits are done, giving traits ConstructorPatternMap and ConstructorSeqMap that have recursive definition of the recursion and induction functions, the former for the case of a single introduction rule. Traits ConstructorSeqMapper and ConstructorPatternMapper provide the lifts.

    Indexed Versions

    There are indexed versions of all these definitions, to work with indexed inductive type families.

    Definition Classes
    provingground
  • object InductiveDefinition
    Definition Classes
    induction
  • DataCons
  • Empty

case class Empty[H <: Term with Subs[H], C <: Term with Subs[C]](fibre: Func[H, Typ[C]]) extends InductiveDefinition[H, C] with Product with Serializable

empty InductiveDefinition, always a formal application.

Linear Supertypes
Serializable, Product, Equals, InductiveDefinition[H, C], InducFuncLike[H, C], FuncLike[H, C], (H) => C, Term, Subs[FuncLike[H, C]], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. Empty
  2. Serializable
  3. Product
  4. Equals
  5. InductiveDefinition
  6. InducFuncLike
  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
  1. Hide All
  2. Show All
Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new Empty(fibre: Func[H, Typ[C]])

Type Members

  1. abstract type Obj <: FuncLike[H, C]
    Definition Classes
    FuncLike

Value Members

  1. def act(arg: H): C

    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
    InductiveDefinition → FuncLike
  2. def andThen[A](g: (C) => A): (H) => A
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  3. def apply(arg: H): C

    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: H): C
    Definition Classes
    FuncLike
  5. def baseFunction: ExstFunc
    Definition Classes
    InducFuncLike
  6. def canApply(arg: H): 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 caseFn(f: => FuncLike[H, C])(arg: H): Option[C]
    Definition Classes
    Empty → InductiveDefinition
  8. def compose[A](g: (A) => H): (A) => C
    Definition Classes
    Function1
    Annotations
    @unspecialized()
  9. def dataSubs(that: InductiveDefinition[H, C], x: Term, y: Term): InductiveDefinition[H, C]
    Definition Classes
    Empty → InductiveDefinition
  10. val defnData: Vector[Nothing]

    the definition data for all the introduction rules

    the definition data for all the introduction rules

    Definition Classes
    Empty → InducFuncLike
  11. val depcodom: Func[H, Typ[C]]
    Definition Classes
    Empty → FuncLike
  12. def dependsOn(that: Term): Boolean

    returns whether this depends on that

    returns whether this depends on that

    Definition Classes
    Term
  13. val dom: Typ[H]
    Definition Classes
    Empty → FuncLike
  14. def equals(that: Any): Boolean
    Definition Classes
    InducFuncLike → AnyRef → Any
  15. val fibre: Func[H, Typ[C]]
    Definition Classes
    Empty → InductiveDefinition
  16. def fromData(data: Vector[Term]): Empty[H, C]
    Definition Classes
    Empty → InductiveDefinition → InducFuncLike
  17. lazy val fullData: (Typ[H], (H) => Typ[C], Vector[Term])
    Definition Classes
    InducFuncLike
  18. def hashCode(): Int
    Definition Classes
    InducFuncLike → AnyRef → Any
  19. def indepOf(that: Term): Boolean

    returns whether this is independent of that.

    returns whether this is independent of that.

    Definition Classes
    Term
  20. val intros: Vector[Term]
    Definition Classes
    Empty → InducFuncLike
  21. def newobj: Empty[H, C]

    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
    Empty → Subs
  22. def productElementNames: Iterator[String]
    Definition Classes
    Product
  23. def replace(x: Term, y: Term): FuncLike[H, C] with Subs[FuncLike[H, C]]

    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
  24. def subs(x: Term, y: Term): Empty[H, C]

    substitute x by y recursively in this.

    substitute x by y recursively in this.

    Definition Classes
    Empty → InductiveDefinition → FuncLike → Subs
  25. def toString(): String
    Definition Classes
    InducFuncLike → Function1 → AnyRef → Any
  26. val typ: PiDefn[H, C]

    the HoTT-type of the term

    the HoTT-type of the term

    Definition Classes
    Empty → FuncLike → Term
  27. def unlift: PartialFunction[H, B]
    Implicit
    This member is added by an implicit conversion from Empty[H, C] toUnliftOps[H, B] performed by method UnliftOps in scala.Function1.This conversion will take place only if C is a subclass of Option[B] (C <: Option[B]).
    Definition Classes
    UnliftOps
  28. 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