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 ConstructorShape
    Definition Classes
    induction
  • CnstDepFuncConsShape
  • CnstFuncConsShape
  • FuncConsShape
  • IdShape

case class CnstDepFuncConsShape[HShape <: HList, H <: Term with Subs[H], HC <: Term with Subs[HC], T <: Term with Subs[T], HS <: Term with Subs[HS]](tail: Typ[T], headfibre: (T) => ConstructorShape[HShape, H, HC]) extends ConstructorShape[::[CnstDepFuncConsShape.type, HShape], H, FuncLike[T, HC]] with Product with Serializable

ConstructorShape corresponding to an introduction rule of the form A ~> head with A not dependent on the inductive type W being constructed

Linear Supertypes
Serializable, Product, Equals, ConstructorShape[::[CnstDepFuncConsShape.type, HShape], H, FuncLike[T, HC]], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. CnstDepFuncConsShape
  2. Serializable
  3. Product
  4. Equals
  5. ConstructorShape
  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 CnstDepFuncConsShape(tail: Typ[T], headfibre: (T) => ConstructorShape[HShape, H, HC])

Value Members

  1. def -->:(that: IdShape.type): FuncConsShape[::[CnstDepFuncConsShape.type, HShape], H, FuncLike[T, HC], H]

    returns shape that -> this' where that must be the shape for inductive type W

    returns shape that -> this' where that must be the shape for inductive type W

    Definition Classes
    ConstructorShape
  2. def -->:[F <: Term with Subs[F]](that: IterFuncShape[H, F]): FuncConsShape[::[CnstDepFuncConsShape.type, HShape], H, FuncLike[T, HC], F]

    returns shape that -> this' where that is of the form W, A -> W etc; invoking this is an error if we that is independent of W

    returns shape that -> this' where that is of the form W, A -> W etc; invoking this is an error if we that is independent of W

    Definition Classes
    ConstructorShape
  3. def ->:[T <: Term with Subs[T]](tail: Typ[T]): CnstFuncConsShape[::[CnstDepFuncConsShape.type, HShape], H, FuncLike[T, HC], T, Nothing]

    returns shape tail ->: this where tail must be independent of the inductive type W being defined.

    returns shape tail ->: this where tail must be independent of the inductive type W being defined.

    Definition Classes
    ConstructorShape
  4. def apply(W: Typ[H]): Typ[FuncLike[T, HC]]

    returns HoTT type of the introduction rule given the (inductive) type W (to be the head).

    returns HoTT type of the introduction rule given the (inductive) type W (to be the head).

    Definition Classes
    CnstDepFuncConsShape → ConstructorShape
  5. val headfibre: (T) => ConstructorShape[HShape, H, HC]
  6. lazy val introArgs: Int
    Definition Classes
    CnstDepFuncConsShape → ConstructorShape
  7. def lift[Cod <: Term with Subs[Cod], RecDataType <: Term with Subs[RecDataType], InducDataType <: Term with Subs[InducDataType]](implicit mp: ConstructorPatternMapper[::[CnstDepFuncConsShape.type, HShape], Cod, FuncLike[T, HC], H, RecDataType, InducDataType]): ConstructorPatternMap[Cod, FuncLike[T, HC], H, RecDataType, InducDataType]

    lift to ConstructorPatternMap using an implicit ConstructorPatternMapper, which could from the mapper method.

    lift to ConstructorPatternMap using an implicit ConstructorPatternMapper, which could from the mapper method.

    Definition Classes
    ConstructorShape
  8. def mapped[Cod <: Term with Subs[Cod]]: ConstructorPatternMap[Cod, FuncLike[T, HC], H, RecDataType, InducDataType] forSome {type RecDataType <: Term with Subs[RecDataType], type InducDataType <: Term with Subs[InducDataType]}

    lift to ConstructorPatternMap using the result of the mapper method.

    lift to ConstructorPatternMap using the result of the mapper method.

    Definition Classes
    ConstructorShape
  9. def mapper[Cod <: Term with Subs[Cod]]: ConstructorPatternMapper[::[CnstDepFuncConsShape.type, HShape], Cod, FuncLike[T, HC], H, FuncLike[T, RecDataType], FuncLike[T, InducDataType]] forSome {type RecDataType <: Term with Subs[RecDataType], type InducDataType <: Term with Subs[InducDataType]}

    helper to give ConstructorPatternMap when scala type of codomain is specified.

    helper to give ConstructorPatternMap when scala type of codomain is specified.

    Definition Classes
    CnstDepFuncConsShape → ConstructorShape
  10. def productElementNames: Iterator[String]
    Definition Classes
    Product
  11. def subs(x: Term, y: Term): CnstDepFuncConsShape[HShape, H, HC, T, Nothing]
    Definition Classes
    CnstDepFuncConsShape → ConstructorShape
  12. def symbcons(name: AnySym, tp: Typ[H]): ConstructorShape.symbcons._1.type.Obj with Subs[ConstructorShape.symbcons._1.type.Obj] forSome {val _1: Typ[FuncLike[T, HC]]}

    returns term giving introduction rule given inductive type and name

    returns term giving introduction rule given inductive type and name

    Definition Classes
    ConstructorShape
  13. val tail: Typ[T]
  14. def ~>:[T <: Term with Subs[T]](tailVar: T): CnstDepFuncConsShape[::[CnstDepFuncConsShape.type, HShape], H, FuncLike[T, HC], T, Nothing]

    returns dependent shape tail ~>: this where tail must be independent of the inductive type W being defined.

    returns dependent shape tail ~>: this where tail must be independent of the inductive type W being defined.

    Definition Classes
    ConstructorShape