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
  • package coarse

    an earlier implementation of induction, without clean separation of codomain scala types; use induction instead

    an earlier implementation of induction, without clean separation of codomain scala types; use induction instead

    Definition Classes
    induction
  • BaseConstructorTypes
  • Constructor
  • ConstructorDefn
  • ConstructorPattern
  • ConstructorSeq
  • ConstructorTyp
  • Curry
  • FmlyPtn
  • Implicits
  • IndexedConstructorPatterns
  • InductiveTyp
  • InductiveTypDefinition
  • IterFuncPattern

sealed trait ConstructorPattern[Cod <: Term with Subs[Cod], CnstrctrType <: Term with Subs[CnstrctrType], H <: Term with Subs[H]] extends AnyRef

A composite pattern for inductive types. Typically (A -> B -> W)-> C -> W -> (D -> W) -> W as a function of W May have Pi-types instead of function types Assumed to have fixed type for codomain X.

Cod

scala type of objects in codomain for recursion and induction functions. The type of the codomain is needed as there are inner types for data for recursion and induction functions.

CnstrctrType

scala type of a constructor corresponding to this pattern.

H

scala type of the terms of the inductive type.

Self Type
ConstructorPattern[Cod, CnstrctrType, H]
Linear Supertypes
AnyRef, Any
Known Subclasses
CnstDepFuncPtn, CnstFncPtn, DepFuncPtn, FuncPtn, IdTarg, IdW, RecursiveConstructorPattern
Type Hierarchy
Ordering
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  2. By Inheritance
Inherited
  1. ConstructorPattern
  2. AnyRef
  3. Any
Implicitly
  1. by any2stringadd
  2. by StringFormat
  3. by Ensuring
  4. by ArrowAssoc
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Visibility
  1. Public
  2. Protected

Type Members

  1. type ConstructorType = CnstrctrType
  2. type DomainType = H
  3. abstract type InducDataType <: Term with Subs[InducDataType]

    (scala) type of data for recursion corresponding to the single constructor

  4. abstract type RecDataType <: Term with Subs[RecDataType]

    (scala) type of data for recursion corresponding to the single constructor

Abstract Value Members

  1. abstract def apply(tp: Typ[H]): Typ[ConstructorType]

    returns HoTT type corresponding to the pattern given the (inductive) type W (to be the head).

  2. abstract def inducDataTyp(w: Typ[H], xs: Func[H, Typ[Cod]])(cons: ConstructorType): Typ[InducDataType]

    domain containing the induction data for the constructor, i.e., the HoTT type of the induction data.

  3. abstract def inducDefCase(cons: ConstructorType, data: InducDataType, f: => FuncLike[H, Cod]): (H) => Option[Cod]

    given a term, matches to see if this is the image of a given (quasi)-constructor, with this constructor pattern.

    given a term, matches to see if this is the image of a given (quasi)-constructor, with this constructor pattern. optionally returns simplification (if the term matches).

    cons

    constructor, actually quasi-constructor, with which to match.

    data

    definition data for the image of the constructor.

    f

    the function being defined inductively, to be used recursively in definition.

  4. abstract def recDataTyp(w: Typ[H], x: Typ[Cod]): Typ[RecDataType]

    domain containing the recursion data for the constructor, i.e., the HoTT type of recursion data.

  5. abstract def recDefCase(cons: ConstructorType, data: RecDataType, f: => Func[H, Cod]): (H) => Option[Cod]

    given a term, matches to see if this is the image of a given (quasi)-constructor, with this constructor pattern.

    given a term, matches to see if this is the image of a given (quasi)-constructor, with this constructor pattern. optionally returns simplification (if the term matches), determined by the recursion data.

    cons

    constructor, actually quasi-constructor, with which to match.

    data

    definition data for the image of the constructor.

    f

    the function being defined recursively, to be used recursively in definition.

  6. abstract def subs(x: Term, y: Term): ConstructorPattern[Cod, CnstrctrType, H]
  7. abstract val univLevel: Int
  8. abstract def withCod[CC <: Term with Subs[CC]](w: Typ[H]): ConstructorPattern[CC, ConstructorType, H]

    changes codomain and propagates changes to other types.

Concrete 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 ConstructorPattern[Cod, CnstrctrType, H] toany2stringadd[ConstructorPattern[Cod, CnstrctrType, H]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def -->:(that: IdW[_]): FuncPtn[Cod, H, CnstrctrType, H]
  5. def -->:[F <: Term with Subs[F]](that: IterFuncPtn[H, Cod, F]): FuncPtn[Cod, F, CnstrctrType, H]

    function pattern.

  6. def ->[B](y: B): (ConstructorPattern[Cod, CnstrctrType, H], B)
    Implicit
    This member is added by an implicit conversion from ConstructorPattern[Cod, CnstrctrType, H] toArrowAssoc[ConstructorPattern[Cod, CnstrctrType, H]] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  7. def ->:[T <: Term with Subs[T]](tail: Typ[T]): CnstFncPtn[T, Cod, CnstrctrType, H]
  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 cons(tp: => Typ[H]): Constructor.ConstructorType
  12. def cons(tp: => Typ[H], name: AnySym): CnstrctrType
  13. def constructor(tp: => Typ[H], name: AnySym): ConstructorDefn[ConstructorType, Cod, H]

    constructor for this pattern given inductive type and name.

  14. def ensuring(cond: (ConstructorPattern[Cod, CnstrctrType, H]) => Boolean, msg: => Any): ConstructorPattern[Cod, CnstrctrType, H]
    Implicit
    This member is added by an implicit conversion from ConstructorPattern[Cod, CnstrctrType, H] toEnsuring[ConstructorPattern[Cod, CnstrctrType, H]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  15. def ensuring(cond: (ConstructorPattern[Cod, CnstrctrType, H]) => Boolean): ConstructorPattern[Cod, CnstrctrType, H]
    Implicit
    This member is added by an implicit conversion from ConstructorPattern[Cod, CnstrctrType, H] toEnsuring[ConstructorPattern[Cod, CnstrctrType, H]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  16. def ensuring(cond: Boolean, msg: => Any): ConstructorPattern[Cod, CnstrctrType, H]
    Implicit
    This member is added by an implicit conversion from ConstructorPattern[Cod, CnstrctrType, H] toEnsuring[ConstructorPattern[Cod, CnstrctrType, H]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  17. def ensuring(cond: Boolean): ConstructorPattern[Cod, CnstrctrType, H]
    Implicit
    This member is added by an implicit conversion from ConstructorPattern[Cod, CnstrctrType, H] toEnsuring[ConstructorPattern[Cod, CnstrctrType, H]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  18. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  19. def equals(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef → Any
  20. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from ConstructorPattern[Cod, CnstrctrType, H] toStringFormat[ConstructorPattern[Cod, CnstrctrType, H]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  21. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  22. def hashCode(): Int
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  23. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  24. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  25. def newconstructor(tp: Typ[H]): Constructor[Cod, H]

    constructor for this pattern given inductive type, with a name symbol generated.

  26. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  27. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  28. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  29. def toString(): String
    Definition Classes
    AnyRef → Any
  30. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  31. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  32. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  33. def ~>:[T <: Term with Subs[T]](tailVar: T): CnstDepFuncPtn[T, (<refinement>.this)#RecDataType, (<refinement>.this)#InducDataType, Cod, CnstrctrType, H]

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): (ConstructorPattern[Cod, CnstrctrType, H], B)
    Implicit
    This member is added by an implicit conversion from ConstructorPattern[Cod, CnstrctrType, H] toArrowAssoc[ConstructorPattern[Cod, CnstrctrType, H]] 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 AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd fromConstructorPattern[Cod, CnstrctrType, H] to any2stringadd[ConstructorPattern[Cod, CnstrctrType, H]]

Inherited by implicit conversion StringFormat fromConstructorPattern[Cod, CnstrctrType, H] to StringFormat[ConstructorPattern[Cod, CnstrctrType, H]]

Inherited by implicit conversion Ensuring fromConstructorPattern[Cod, CnstrctrType, H] to Ensuring[ConstructorPattern[Cod, CnstrctrType, H]]

Inherited by implicit conversion ArrowAssoc fromConstructorPattern[Cod, CnstrctrType, H] to ArrowAssoc[ConstructorPattern[Cod, CnstrctrType, H]]

Ungrouped