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 andrewscurtis
    Definition Classes
    provingground
  • package examples
    Definition Classes
    provingground
  • 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 interface
    Definition Classes
    provingground
  • package learning
    Definition Classes
    provingground
  • package library
    Definition Classes
    provingground
  • package scalahott
    Definition Classes
    provingground
  • BigOps
  • Bool
  • BoolType
  • EnumFin
  • EnumFuncs
  • EnumType
  • FreeGroup
  • FunctionFeedback
  • IndexedVecTyp
  • InducPairs
  • IntRing
  • IntTypes
  • IntVector
  • LinNormBound
  • ListTyp
  • ListType
  • MatrixTypes
  • NatRing
  • NatTypLong
  • NatVecTyps
  • Norm
  • PlusTypInduc
  • QField
  • RecEnum
  • RepTerm
  • ScalaPolyRep
  • ScalaRep
  • ScalaTyp
  • ScalaTypUniv
  • ScalaUniv
  • ScalaVec
  • SigmaPiEnum
  • SymbScalaTyp
  • SymbolicCRing
  • SymbolicField
  • SymbolicGroup
  • VecTyp
  • VecTyps
  • package scratch
    Definition Classes
    provingground
  • package translation

    Translation primarily using a functorial framework - see Translator$, for natural language processing as well as serialization, formatted output, parsing, interface with formal languages etc.

    Translation primarily using a functorial framework - see Translator$, for natural language processing as well as serialization, formatted output, parsing, interface with formal languages etc.

    Besides the Translator framework and helper typeclasses is Functors, several structures for concrete languages including our implementation of HoTT are in this package.

    Definition Classes
    provingground

package scalahott

Content Hierarchy
Ordering
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Visibility
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  2. Protected

Type Members

  1. case class IndexedVecTyp[X, +U <: RepTerm[X] with Subs[U]](basetyp: Typ[U], dim: SafeLong)(implicit baserep: ScalaRep[U, X]) extends Typ[RepTerm[Vector[X]]] with Product with Serializable

  2. case class IntVector(dim: Int) extends ScalaTyp[Vector[Int]] with Product with Serializable
  3. sealed abstract class LinNormBound extends AnyRef
  4. case class ListTyp[U <: Term with Subs[U], X]()(implicit baserep: ScalaRep[U, X]) extends ScalaTyp[List[X]] with Product with Serializable

  5. trait RepTerm[A] extends Term with Subs[RepTerm[A]]
  6. trait ScalaPolyRep[U <: Term with Subs[U], W] extends AnyRef
  7. trait ScalaRep[+U <: Term with Subs[U], V] extends AnyRef

    Representation by a scala object of a HoTT term

    Representation by a scala object of a HoTT term

    It is assumed that there is a single HoTT type corresponding to the given scala type, when the scala rep is in scope. If one needs several HoTT types, eg. vectors of different lengths, one uses ScalaPolyRep.

    ScalaRep objects are recursively constructed from

    U

    the HoTT type represented

    V

    scala type representing the given object.

  8. class ScalaTyp[A] extends Typ[RepTerm[A]]
  9. case class ScalaTypUniv[A]() extends Typ[Typ[RepTerm[A]]] with BaseUniv with Product with Serializable
  10. case class ScalaUniv[U <: Term with Subs[U]](univ: Typ[Typ[U]]) extends Product with Serializable

    Wrapper for universe with refined scala type for objects (i.e., types) in it.

    Wrapper for universe with refined scala type for objects (i.e., types) in it. Refined scala types typically recursively built from (dependent) function types and types of already refined types.

  11. class ScalaVec[X] extends AnyRef

  12. case class SymbScalaTyp[A](name: AnySym) extends ScalaTyp[A] with Symbolic with Product with Serializable
  13. class SymbolicCRing[A] extends AnyRef

  14. class SymbolicField[A] extends SymbolicCRing[A]

  15. class SymbolicGroup[A] extends ScalaTyp[A]
  16. case class VecTyp[X, +U <: RepTerm[X] with Subs[U]](basetyp: Typ[U], dim: Long)(implicit _baserep: ScalaRep[U, X]) extends Typ[RepTerm[Vector[X]]] with Product with Serializable

  17. class VecTyps[X, U <: RepTerm[X] with Subs[U]] extends AnyRef

Value Members

  1. object BigOps
  2. object Bool extends ScalaTyp[Boolean]
  3. object BoolType
  4. object EnumFin
  5. object EnumFuncs
  6. object EnumType
  7. object FreeGroup extends SymbolicGroup[Word]
  8. object FunctionFeedback
  9. object IndexedVecTyp extends Serializable
  10. object InducPairs

    Recursion and induction for (dependent) pairs.

  11. object IntRing extends SymbolicCRing[SafeLong]
  12. object IntTypes
  13. object IntVector extends Serializable
  14. object LinNormBound
  15. object ListType
  16. object MatrixTypes
  17. object NatRing extends SymbolicCRing[SafeLong] with ExstInducStrucs
  18. object NatTypLong extends ScalaTyp[Long]
  19. object NatVecTyps extends VecTyps[SafeLong, Nat]
  20. object Norm
  21. object PlusTypInduc
  22. object QField extends SymbolicField[Rational]
  23. object RecEnum
  24. object ScalaPolyRep
  25. object ScalaRep
  26. object ScalaUniv extends Serializable
  27. object SigmaPiEnum
  28. object SymbolicCRing
  29. object VecTyp extends Serializable

Ungrouped