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 IndexedConstructorSeqDom
    Definition Classes
    induction
  • Cons
  • Empty

case class Empty[H <: Term with Subs[H], F <: Term with Subs[F], Index <: HList](W: F, family: TypFamilyPtn[H, F, Index])(implicit evidence$6: TermList[Index]) extends IndexedConstructorSeqDom[HNil, H, F, Index, HNil] with Product with Serializable

Linear Supertypes
Serializable, Product, Equals, IndexedConstructorSeqDom[HNil, H, F, Index, HNil], AnyRef, Any
Type Hierarchy
Ordering
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Inherited
  1. Empty
  2. Serializable
  3. Product
  4. Equals
  5. IndexedConstructorSeqDom
  6. AnyRef
  7. Any
Implicitly
  1. by any2stringadd
  2. by StringFormat
  3. by Ensuring
  4. by ArrowAssoc
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Visibility
  1. Public
  2. Protected

Instance Constructors

  1. new Empty(W: F, family: TypFamilyPtn[H, F, Index])(implicit arg0: TermList[Index])

Value Members

  1. val W: F

    the inductive type family

    the inductive type family

    Definition Classes
    Empty → IndexedConstructorSeqDom
  2. val family: TypFamilyPtn[H, F, Index]

    the index family

    the index family

    Definition Classes
    Empty → IndexedConstructorSeqDom
  3. def getInduc[C <: Term with Subs[C], IDFT <: Term with Subs[IDFT], IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], RecType <: Term with Subs[RecType], InducType <: Term with Subs[InducType]](X: Typ[C])(implicit mapper: IndexedConstructorSeqMapper[HNil, C, H, RecType, InducType, HNil, F, Index, IF, IDF, IDFT]): (IDFT) => InducType
    Definition Classes
    IndexedConstructorSeqDom
  4. def getMapper[C <: Term with Subs[C], IDFT <: Term with Subs[IDFT], IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], RecType <: Term with Subs[RecType], InducType <: Term with Subs[InducType]](X: Typ[C])(implicit mapper: IndexedConstructorSeqMapper[HNil, C, H, RecType, InducType, HNil, F, Index, IF, IDF, IDFT]): IndexedConstructorSeqMap[C, H, RecType, InducType, HNil, F, Index, IF, IDF, IDFT]
    Definition Classes
    IndexedConstructorSeqDom
  5. def induc[IDFT <: Term with Subs[IDFT], C <: Term with Subs[C], IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], RecType <: Term with Subs[RecType], InducType <: Term with Subs[InducType]](Xs: IDFT)(implicit mapper: IndexedConstructorSeqMapper[HNil, C, H, RecType, InducType, HNil, F, Index, IF, IDF, IDFT]): InducType

    induction function based on implicits

    induction function based on implicits

    Definition Classes
    IndexedConstructorSeqDom
  6. def inducE[IDFT <: Term with Subs[IDFT]](Xs: IDFT): InducType forSome {type InducType <: Term with Subs[InducType]}

    existentitally typed induction function

    existentitally typed induction function

    Definition Classes
    IndexedConstructorSeqDom
  7. val intros: HNil.type

    the introduction rules

    the introduction rules

    Definition Classes
    Empty → IndexedConstructorSeqDom
  8. def mapped[C <: Term with Subs[C], IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], IDFT <: Term with Subs[IDFT]](implicit fmlyMapper: TypFamilyMapper[H, F, C, Index, IF, IDF, IDFT]): IndexedConstructorSeqMap.Empty[C, H, F, Index, IF, IDF, IDFT]

    existentitally typed lift given codomain

    existentitally typed lift given codomain

    Definition Classes
    Empty → IndexedConstructorSeqDom
  9. def productElementNames: Iterator[String]
    Definition Classes
    Product
  10. def rec[C <: Term with Subs[C], IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], IDFT <: Term with Subs[IDFT], RecType <: Term with Subs[RecType], InducType <: Term with Subs[InducType]](X: Typ[C])(implicit mapper: IndexedConstructorSeqMapper[HNil, C, H, RecType, InducType, HNil, F, Index, IF, IDF, IDFT]): RecType

    recursion function definition based on implcits

    recursion function definition based on implcits

    Definition Classes
    IndexedConstructorSeqDom
  11. def recE[C <: Term with Subs[C]](x: Typ[C]): RecType forSome {type RecType <: Term with Subs[RecType]}

    existentitally typed recursion function definition

    existentitally typed recursion function definition

    Definition Classes
    IndexedConstructorSeqDom
  12. def subs(x: Term, y: Term): Empty[H, F with Subs[F], Index]
    Definition Classes
    Empty → IndexedConstructorSeqDom
  13. def |:[HShape <: HList, HC <: Term with Subs[HC]](head: IndexedConstructor[HShape, H, F, HC, Index]): IndexedConstructorSeqDom[::[HShape, HNil], H, F, Index, ::[HC, HNil]]

    prepend introduction rule

    prepend introduction rule

    Definition Classes
    IndexedConstructorSeqDom