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 IndexedIterFuncPtnMap
    Definition Classes
    induction
  • IdIterPtnMap
  • IndexedDepFuncIterPtnMap
  • IndexedFuncIterPtnMap

case class IndexedDepFuncIterPtnMap[TT <: Term with Subs[TT], V <: Term with Subs[V], T <: Term with Subs[T], D <: Term with Subs[D], H <: Term with Subs[H], Fb <: Term with Subs[Fb], Index <: HList, C <: Term with Subs[C], F <: Term with Subs[F], IF <: Term with Subs[IF], IDF <: Term with Subs[IDF], IDFT <: Term with Subs[IDFT]](tail: Typ[TT], headfibre: (TT) => IndexedIterFuncPtnMap[H, Fb, Index, C, V, T, D, IF, IDF, IDFT])(implicit evidence$4: TermList[Index]) extends IndexedIterFuncPtnMap[H, Fb, Index, C, FuncLike[TT, V], FuncLike[TT, T], FuncLike[TT, D], IF, IDF, IDFT] with Product with Serializable

Self Type
IndexedDepFuncIterPtnMap[TT, V, T, D, H, Fb, Index, C, F, IF, IDF, IDFT]
Linear Supertypes
Serializable, Product, Equals, IndexedIterFuncPtnMap[H, Fb, Index, C, FuncLike[TT, V], FuncLike[TT, T], FuncLike[TT, D], IF, IDF, IDFT], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. IndexedDepFuncIterPtnMap
  2. Serializable
  3. Product
  4. Equals
  5. IndexedIterFuncPtnMap
  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 IndexedDepFuncIterPtnMap(tail: Typ[TT], headfibre: (TT) => IndexedIterFuncPtnMap[H, Fb, Index, C, V, T, D, IF, IDF, IDFT])(implicit arg0: TermList[Index])

Type Members

  1. type DepTargetType = FuncLike[TT, D]
  2. type TargetType = Func[TT, T]

Value Members

  1. def apply(W: Fb): GenFuncTyp[TT, V]

    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
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap
  2. def depTarget(xs: IDFT): (FuncLike[TT, V]) => PiDefn[TT, D]

    dependent target scala type.

    dependent target scala type.

    Definition Classes
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap
  3. val family: TypFamilyMap[H, Fb, C, Index, IF, IDF, IDFT]
    Definition Classes
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap
  4. val headfibre: (TT) => IndexedIterFuncPtnMap[H, Fb, Index, C, V, T, D, IF, IDF, IDFT]
  5. def induced(f: IF): (FuncLike[TT, V]) => TargetType

    inductively defining the induced function.

    inductively defining the induced function. maps (g : tail --> head(W)) to func : tail --> head(X) given (head(W) --> head(X))

    f

    function from which to induce

    Definition Classes
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap
  6. def inducedDep(f: IDF): (FuncLike[TT, V]) => DepTargetType

    inductively defining the induced function.

    inductively defining the induced function. maps (g : tail --> head(W)) to func : (t : tail) ~> head(Xs(t)) given (head(W) --> (t: tail) ~> head(Xs(t)))

    f

    dependent function from which to induce

    Definition Classes
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap
  7. def productElementNames: Iterator[String]
    Definition Classes
    Product
  8. def subs(x: Term, y: Term): IndexedDepFuncIterPtnMap[TT, V, T, D, H, Fb, Index, C, Nothing, IF, IDF, IDFT]
    Definition Classes
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap
  9. val tail: Typ[TT]
  10. def target(x: Typ[C]): GenFuncTyp[TT, T]

    target scala type.

    target scala type.

    Definition Classes
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap
  11. val univLevel: Int

    the universe containing the type

    the universe containing the type

    Definition Classes
    IndexedDepFuncIterPtnMap → IndexedIterFuncPtnMap