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 IndexedConstructorPatternMap
    Definition Classes
    induction
  • IndexedCnstDepFncPtnMap
  • IndexedCnstFncPtnMap
  • IndexedFuncPtnMap
  • IndexedIdMap
  • IndexedIndexedFuncPtnMap
  • IndexedRecursiveConstructorPatternMap

case class IndexedCnstDepFncPtnMap[T <: Term with Subs[T], Cod <: Term with Subs[Cod], HC <: Term with Subs[HC], H <: Term with Subs[H], HR <: Term with Subs[HR], HI <: Term with Subs[HI], Fb <: Term with Subs[Fb], Index <: HList, IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], IDFT <: Term with Subs[IDFT]](tail: Typ[T], headfibre: (T) => IndexedConstructorPatternMap[Cod, HC, H, HR, HI, Fb, Index, IF, IDF, IDFT])(implicit evidence$7: TermList[Index]) extends IndexedRecursiveConstructorPatternMap[Cod, T, HC, FuncLike[T, HC], H, FuncLike[T, HR], FuncLike[T, HI], HR, HI, Fb, Index, IF, IDF, IDFT] with Product with Serializable

Self Type
IndexedCnstDepFncPtnMap[T, Cod, HC, H, HR, HI, Fb, Index, IF, IDF, IDFT]
Linear Supertypes
Serializable, Product, Equals, IndexedRecursiveConstructorPatternMap[Cod, T, HC, FuncLike[T, HC], H, FuncLike[T, HR], FuncLike[T, HI], HR, HI, Fb, Index, IF, IDF, IDFT], IndexedConstructorPatternMap[Cod, FuncLike[T, HC], H, FuncLike[T, HR], FuncLike[T, HI], Fb, Index, IF, IDF, IDFT], AnyRef, Any
Type Hierarchy
Ordering
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Inherited
  1. IndexedCnstDepFncPtnMap
  2. Serializable
  3. Product
  4. Equals
  5. IndexedRecursiveConstructorPatternMap
  6. IndexedConstructorPatternMap
  7. AnyRef
  8. 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 IndexedCnstDepFncPtnMap(tail: Typ[T], headfibre: (T) => IndexedConstructorPatternMap[Cod, HC, H, HR, HI, Fb, Index, IF, IDF, IDFT])(implicit arg0: TermList[Index])

Value Members

  1. def apply(W: Fb): PiDefn[T, HC]

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

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

    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedConstructorPatternMap
  2. val family: TypFamilyMap[H, Fb, Cod, Index, IF, IDF, IDFT]

    the inductive type family

    the inductive type family

    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedConstructorPatternMap
  3. def headData(data: FuncLike[T, HR], arg: T, f: => IF): HR

    returns data for recursion to be passed on to the head given an argument (when matching with the constructor).

    returns data for recursion to be passed on to the head given an argument (when matching with the constructor).

    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedRecursiveConstructorPatternMap
  4. def headInducData(data: FuncLike[T, HI], arg: T, f: => IDF): HI
    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedRecursiveConstructorPatternMap
  5. val headfibre: (T) => IndexedConstructorPatternMap[Cod, HC, H, HR, HI, Fb, Index, IF, IDF, IDFT]

    The head pattern, constant T for A -> T and T(a) for A ~> T(a)

    The head pattern, constant T for A -> T and T(a) for A ~> T(a)

    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedRecursiveConstructorPatternMap
  6. def inducDataTyp(w: Fb, xs: IDFT)(cons: FuncLike[T, HC]): PiDefn[T, HI]

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

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

    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedConstructorPatternMap
  7. def inducDefCase(cons: FuncLike[T, HC], data: FuncLike[T, HI], f: => IDF): (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.

    Definition Classes
    IndexedRecursiveConstructorPatternMap → IndexedConstructorPatternMap
  8. def productElementNames: Iterator[String]
    Definition Classes
    Product
  9. def recDataTyp(w: Fb, x: Typ[Cod]): PiDefn[T, HR]

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

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

    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedConstructorPatternMap
  10. def recDefCase(cons: FuncLike[T, HC], data: FuncLike[T, HR], f: => IF): (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.

    Definition Classes
    IndexedRecursiveConstructorPatternMap → IndexedConstructorPatternMap
  11. def subs(x: Term, y: Term): IndexedCnstDepFncPtnMap[T, Cod, HC, H, HR, HI, Fb, Index, IF, IDF, IDFT]
    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedConstructorPatternMap
  12. val tail: Typ[T]
  13. val univLevel: Int
    Definition Classes
    IndexedCnstDepFncPtnMap → IndexedConstructorPatternMap