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
  • object FmlyPtn
    Definition Classes
    coarse
  • DepFuncFmlyPtn
  • FuncFmlyPtn
  • IdFmlyPtn
  • RecFmlyPtn

case class FuncFmlyPtn[TT <: Term with Subs[TT], FV <: Term with Subs[FV], I <: Term with Subs[I], IT <: Term with Subs[IT], DI <: Term with Subs[DI], S <: Term with Subs[S], O <: Term with Subs[O], C <: Term with Subs[C], HTot <: Term with Subs[HTot]](tail: Typ[TT], head: FmlyPtn[O, C, FV] { ... /* 5 definitions in type refinement */ }) extends FmlyPtn[O, C, Func[TT, FV]] with Product with Serializable

Linear Supertypes
Serializable, Product, Equals, FmlyPtn[O, C, Func[TT, FV]], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. FuncFmlyPtn
  2. Serializable
  3. Product
  4. Equals
  5. FmlyPtn
  6. AnyRef
  7. Any
Implicitly
  1. by any2stringadd
  2. by StringFormat
  3. by Ensuring
  4. by ArrowAssoc
  1. Hide All
  2. Show All
Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new FuncFmlyPtn(tail: Typ[TT], head: FmlyPtn[O, C, FV] { ... /* 5 definitions in type refinement */ })

Type Members

  1. type ArgType = AbsPair[TT, S]

    type of total index

    type of total index

    Definition Classes
    FuncFmlyPtn → FmlyPtn
  2. type Cod = C
    Definition Classes
    FmlyPtn
  3. type FamilyType = Func[TT, FV]

    type of W, i.e., sections to a universe.

    type of W, i.e., sections to a universe.

    Definition Classes
    FmlyPtn
  4. type IterDepFunc = FuncLike[TT, (head)#IterDepFunc]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  5. type IterFunc = FuncLike[TT, (head)#IterFunc]

    Type of Curried function to X.

    Type of Curried function to X.

    Definition Classes
    FuncFmlyPtn → FmlyPtn
  6. type IterTypFunc = FuncLike[TT, (head)#IterTypFunc]

    Type of curried function to typ[X]

    Type of curried function to typ[X]

    Definition Classes
    FuncFmlyPtn → FmlyPtn
  7. type Total = AbsPair[TT, HTot]

    type of the total space, i.e., all terms in some W.

    type of the total space, i.e., all terms in some W.

    Definition Classes
    FuncFmlyPtn → FmlyPtn

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 FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toany2stringadd[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot], B)
    Implicit
    This member is added by an implicit conversion from FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toArrowAssoc[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  5. def ->:[TT <: Term with Subs[TT]](tail: Typ[TT]): FmlyPtn[O, C, Func[TT, Func[TT, FV]]]
    Definition Classes
    FmlyPtn
  6. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  7. def argOpt(l: List[Term]): Option[ArgType]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  8. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  9. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
  10. def contractType(w: FamilyType)(arg: ArgType): Typ[O]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  11. def curry(f: Func[Total, Cod]): IterFunc
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  12. def curryTyp(w: Func[Total, Typ[Cod]]): IterTypFunc
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  13. def depCurry(f: FuncLike[Total, Cod]): IterDepFunc
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  14. def depFill(g: IterDepFunc)(arg: ArgType): FuncLike[O, C]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  15. def depTotalDomain(g: IterDepFunc): ProdTyp[TT, (head)#Total]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  16. def depUncurry(g: IterDepFunc): FuncLike[Total, Cod]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  17. def domTotal(w: FamilyType): Typ[Total]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  18. def ensuring(cond: (FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]) => Boolean, msg: => Any): FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]
    Implicit
    This member is added by an implicit conversion from FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toEnsuring[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  19. def ensuring(cond: (FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]) => Boolean): FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]
    Implicit
    This member is added by an implicit conversion from FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toEnsuring[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  20. def ensuring(cond: Boolean, msg: => Any): FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]
    Implicit
    This member is added by an implicit conversion from FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toEnsuring[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  21. def ensuring(cond: Boolean): FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]
    Implicit
    This member is added by an implicit conversion from FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toEnsuring[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  22. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  23. def fill(g: IterFunc)(arg: ArgType): Func[O, C]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  24. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toStringFormat[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  25. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  26. val head: FmlyPtn[O, C, FV] { ... /* 5 definitions in type refinement */ }
  27. val headfibre: (Term) => FmlyPtn[O, C, FV] { ... /* 5 definitions in type refinement */ }
  28. def incl(term: O, arg: ArgType, w: FamilyType): Total
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  29. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  30. def iterDepFuncTyp(w: FamilyType, xs: IterTypFunc): Typ[IterDepFunc]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  31. def iterFuncTyp(w: FamilyType, x: Typ[Cod]): Typ[IterFunc]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  32. val me: FmlyPtn[O, C, Func[TT, FV]] { ... /* 5 definitions in type refinement */ }
    Definition Classes
    FmlyPtn
  33. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  34. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  35. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  36. def productElementNames: Iterator[String]
    Definition Classes
    Product
  37. def subs(x: Term, y: Term): FuncFmlyPtn[TT, FV, subs.newHead.type.IterFunc, subs.newHead.type.IterTypFunc, subs.newHead.type.IterDepFunc, subs.newHead.type.ArgType, O, C, subs.newHead.type.Total] forSome {val newHead: FmlyPtn[O, C, FV]}
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  38. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  39. val tail: Typ[TT]
  40. def totalDomain(g: IterFunc): SigmaTyp[TT, (head)#Total]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  41. def totalTypDomain(g: IterTypFunc): Typ[Total]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  42. def uncurry(g: IterFunc): Func[Total, Cod]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  43. def uncurryTyp(g: IterTypFunc): Func[Total, Typ[Cod]]
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  44. val univLevel: Int

    the universe containing the type

    the universe containing the type

    Definition Classes
    FuncFmlyPtn → FmlyPtn
  45. def value(x: Total): O

    project an element of the total type to its value, i.e., drop arguments

    project an element of the total type to its value, i.e., drop arguments

    Definition Classes
    FuncFmlyPtn → FmlyPtn
  46. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  47. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  48. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  49. def withCod[CC <: Term with Subs[CC]](w: Typ[O]): FuncFmlyPtn[TT, FV, withCod.newHead.type.IterFunc, withCod.newHead.type.IterTypFunc, withCod.newHead.type.IterDepFunc, withCod.newHead.type.ArgType, O, CC, withCod.newHead.type.Total] forSome {val newHead: FmlyPtn[O, CC, FV]}
    Definition Classes
    FuncFmlyPtn → FmlyPtn
  50. def ~>:[TT <: Term with Subs[TT]](tailVar: TT): FmlyPtn[O, C, FuncLike[TT, Func[TT, FV]]]
    Definition Classes
    FmlyPtn

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): (FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot], B)
    Implicit
    This member is added by an implicit conversion from FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] toArrowAssoc[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]] 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 Serializable

Inherited from Product

Inherited from Equals

Inherited from FmlyPtn[O, C, Func[TT, FV]]

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd fromFuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] to any2stringadd[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]]

Inherited by implicit conversion StringFormat fromFuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] to StringFormat[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]]

Inherited by implicit conversion Ensuring fromFuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] to Ensuring[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]]

Inherited by implicit conversion ArrowAssoc fromFuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot] to ArrowAssoc[FuncFmlyPtn[TT, FV, I, IT, DI, S, O, C, HTot]]

Ungrouped