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 IterFuncPattern
    Definition Classes
    coarse
  • DepFuncIterPtn
  • FuncIterPtn
  • IdIterPtn
  • IterFuncPtn
  • IterFuncTyp
  • RecIterPtn

trait RecIterPtn[TT <: Term with Subs[TT], V <: Term with Subs[V], T <: Term with Subs[T], D <: Term with Subs[D], O <: Term with Subs[O], C <: Term with Subs[C]] extends IterFuncPtn[O, C, FuncLike[TT, V]]

Linear Supertypes
IterFuncPtn[O, C, FuncLike[TT, V]], AnyRef, Any
Known Subclasses
DepFuncIterPtn
Type Hierarchy
Ordering
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  2. By Inheritance
Inherited
  1. RecIterPtn
  2. IterFuncPtn
  3. AnyRef
  4. 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 Cod = C
    Definition Classes
    IterFuncPtn
  2. type DepTargetType = FuncLike[TT, D]

    Note that the target may be a type family, not a type, so this is the actual type of objects.

    Note that the target may be a type family, not a type, so this is the actual type of objects.

    Definition Classes
    RecIterPtn → IterFuncPtn
  3. type Family = FuncLike[TT, V]

    scala type (upper bound) for a member of the family, i.e., sections

    scala type (upper bound) for a member of the family, i.e., sections

    Definition Classes
    IterFuncPtn
  4. abstract type TargetType <: FuncLike[TT, T] with Subs[TargetType]

    scala type of target for induced functions

    scala type of target for induced functions

    Definition Classes
    RecIterPtn → IterFuncPtn

Abstract Value Members

  1. abstract def apply(tp: Typ[O]): Typ[Family]

    returns the type corresponding to the pattern, such as A -> W, given the (inductive) type W, this is used mainly for constructor patterns, with the W being fixed.

    returns the type corresponding to the pattern, such as A -> W, given the (inductive) type W, this is used mainly for constructor patterns, with the W being fixed.

    Definition Classes
    IterFuncPtn
  2. abstract val headfibre: (TT) => IterFuncPtn[O, C, V] { ... /* 2 definitions in type refinement */ }
  3. abstract def induced(f: Func[O, Cod]): (Family) => TargetType

    function induced by f: W -> X of type (A -> W) -> (A -> X) etc

    function induced by f: W -> X of type (A -> W) -> (A -> X) etc

    f

    function from which to induce

    Definition Classes
    IterFuncPtn
  4. abstract def inducedDep(f: FuncLike[O, Cod]): (Family) => DepTargetType

    dependent function induced by dependent f: W -> X(s) of type (A -> W) -> ((a : A) ~> Xs(a)) etc

    dependent function induced by dependent f: W -> X(s) of type (A -> W) -> ((a : A) ~> Xs(a)) etc

    f

    dependent function from which to induce

    Definition Classes
    IterFuncPtn
  5. abstract def subs(x: Term, y: Term): IterFuncPtn[O, C, Family]
    Definition Classes
    IterFuncPtn
  6. abstract val tail: Typ[TT]
  7. abstract def target(x: Typ[Cod]): Typ[TargetType]

    target scala type.

    target scala type.

    Definition Classes
    IterFuncPtn
  8. abstract val univLevel: Int

    the universe containing the type

    the universe containing the type

    Definition Classes
    IterFuncPtn
  9. abstract def withCod[CC <: Term with Subs[CC]](w: Typ[O]): IterFuncPtn[O, CC, Family]
    Definition Classes
    IterFuncPtn

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 RecIterPtn[TT, V, T, D, O, C] toany2stringadd[RecIterPtn[TT, V, T, D, O, C]] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (RecIterPtn[TT, V, T, D, O, C], B)
    Implicit
    This member is added by an implicit conversion from RecIterPtn[TT, V, T, D, O, C] toArrowAssoc[RecIterPtn[TT, V, T, D, O, C]] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  5. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  6. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  7. def clone(): AnyRef
    Attributes
    protected[lang]
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.CloneNotSupportedException]) @native() @HotSpotIntrinsicCandidate()
  8. def depTarget(xs: Func[O, Typ[Cod]]): (Family) => PiDefn[TT, D]

    dependent target scala type.

    dependent target scala type.

    Definition Classes
    RecIterPtn → IterFuncPtn
  9. def ensuring(cond: (RecIterPtn[TT, V, T, D, O, C]) => Boolean, msg: => Any): RecIterPtn[TT, V, T, D, O, C]
    Implicit
    This member is added by an implicit conversion from RecIterPtn[TT, V, T, D, O, C] toEnsuring[RecIterPtn[TT, V, T, D, O, C]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  10. def ensuring(cond: (RecIterPtn[TT, V, T, D, O, C]) => Boolean): RecIterPtn[TT, V, T, D, O, C]
    Implicit
    This member is added by an implicit conversion from RecIterPtn[TT, V, T, D, O, C] toEnsuring[RecIterPtn[TT, V, T, D, O, C]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  11. def ensuring(cond: Boolean, msg: => Any): RecIterPtn[TT, V, T, D, O, C]
    Implicit
    This member is added by an implicit conversion from RecIterPtn[TT, V, T, D, O, C] toEnsuring[RecIterPtn[TT, V, T, D, O, C]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  12. def ensuring(cond: Boolean): RecIterPtn[TT, V, T, D, O, C]
    Implicit
    This member is added by an implicit conversion from RecIterPtn[TT, V, T, D, O, C] toEnsuring[RecIterPtn[TT, V, T, D, O, C]] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  13. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  14. def equals(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef → Any
  15. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from RecIterPtn[TT, V, T, D, O, C] toStringFormat[RecIterPtn[TT, V, T, D, O, C]] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  16. final def getClass(): Class[_ <: AnyRef]
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  17. def hashCode(): Int
    Definition Classes
    AnyRef → Any
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  18. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  19. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  20. final def notify(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  21. final def notifyAll(): Unit
    Definition Classes
    AnyRef
    Annotations
    @native() @HotSpotIntrinsicCandidate()
  22. final def synchronized[T0](arg0: => T0): T0
    Definition Classes
    AnyRef
  23. def toString(): String
    Definition Classes
    AnyRef → Any
  24. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])
  25. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException]) @native()
  26. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws(classOf[java.lang.InterruptedException])

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): (RecIterPtn[TT, V, T, D, O, C], B)
    Implicit
    This member is added by an implicit conversion from RecIterPtn[TT, V, T, D, O, C] toArrowAssoc[RecIterPtn[TT, V, T, D, O, C]] 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 IterFuncPtn[O, C, FuncLike[TT, V]]

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd fromRecIterPtn[TT, V, T, D, O, C] to any2stringadd[RecIterPtn[TT, V, T, D, O, C]]

Inherited by implicit conversion StringFormat fromRecIterPtn[TT, V, T, D, O, C] to StringFormat[RecIterPtn[TT, V, T, D, O, C]]

Inherited by implicit conversion Ensuring fromRecIterPtn[TT, V, T, D, O, C] to Ensuring[RecIterPtn[TT, V, T, D, O, C]]

Inherited by implicit conversion ArrowAssoc fromRecIterPtn[TT, V, T, D, O, C] to ArrowAssoc[RecIterPtn[TT, V, T, D, O, C]]

Ungrouped