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
  • package coarse

    an earlier implementation of induction, without clean separation of codomain scala types; use induction instead

    an earlier implementation of induction, without clean separation of codomain scala types; use induction instead

    Definition Classes
    induction
  • BaseConstructorTypes
  • Constructor
  • ConstructorDefn
  • ConstructorPattern
  • ConstructorSeq
  • ConstructorTyp
  • Curry
  • FmlyPtn
  • Implicits
  • IndexedConstructorPatterns
  • InductiveTyp
  • InductiveTypDefinition
  • IterFuncPattern

package coarse

an earlier implementation of induction, without clean separation of codomain scala types; use induction instead

Linear Supertypes
AnyRef, Any
Content Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. coarse
  2. AnyRef
  3. Any
  1. Hide All
  2. Show All
Visibility
  1. Public
  2. Protected

Type Members

  1. trait Constructor[Cod <: Term with Subs[Cod], H <: Term with Subs[H]] extends AnyRef

    Constructor for an inductive type, with given scala type and poly-pattern of this type.

    Constructor for an inductive type, with given scala type and poly-pattern of this type.

    abstraction of ConstructorDefn mainly to allow different type parameters.

  2. case class ConstructorDefn[U <: Term with Subs[U], C <: Term with Subs[C], H <: Term with Subs[H]](pattern: ConstructorPattern[C, U, H], cons: U, W: Typ[H]) extends Constructor[C, H] with Product with Serializable

    a constructor given by its parameters.

    a constructor given by its parameters.

    U

    scala type of polypattern.

    pattern

    poly-pattern for the constructor.

    cons

    constructor function.

  3. sealed trait ConstructorPattern[Cod <: Term with Subs[Cod], CnstrctrType <: Term with Subs[CnstrctrType], H <: Term with Subs[H]] extends AnyRef

    A composite pattern for inductive types.

    A composite pattern for inductive types. Typically (A -> B -> W)-> C -> W -> (D -> W) -> W as a function of W May have Pi-types instead of function types Assumed to have fixed type for codomain X.

    Cod

    scala type of objects in codomain for recursion and induction functions. The type of the codomain is needed as there are inner types for data for recursion and induction functions.

    CnstrctrType

    scala type of a constructor corresponding to this pattern.

    H

    scala type of the terms of the inductive type.

  4. trait ConstructorSeq[C <: Term with Subs[C], H <: Term with Subs[H]] extends AnyRef
  5. case class ConstructorTyp[C <: Term with Subs[C], F <: Term with Subs[F], H <: Term with Subs[H]](pattern: ConstructorPattern[C, F, H], typ: Typ[H]) extends Product with Serializable

    Constructor pattern with type, for convenient building.

  6. trait Curry[Iter <: Term with Subs[Term], Total <: Term with Subs[Total], Cod <: Term with Subs[Cod]] extends AnyRef
  7. sealed trait FmlyPtn[O <: Term with Subs[O], C <: Term with Subs[C], F <: Term with Subs[F]] extends AnyRef

    A pattern for families, e.g.

    A pattern for families, e.g. of inductive types to be defined for instance A -> B -> W, where W is the type to be defined; ends with the type with members.

    given a codomain C, or a family of codomains, we can lift functions W -> C to functions on families.

    O

    scala type of objects of W, i.e., members of the family.

    C

    scala type of the codomain, needed to deduce types for induced functions. This is used in more than one way, which perhaps should be separated: for constructor types and for inductive families. Hence we need two kinds of induced functions, e.g., (A -> B -> W) -> (A -> B -> X) and (A -> B -> W -> X).

    F

    scala type of sections, e.g. A -> B -> W

  8. class IndexedConstructorPatterns[C <: Term with Subs[C], H <: Term with Subs[H], F <: Term with Subs[F]] extends AnyRef

  9. trait InductiveTyp[C <: Term with Subs[C], H <: Term with Subs[H]] extends Typ[H]
  10. case class InductiveTypDefinition[C <: Term with Subs[C]](constructorDefs: List[(Typ[Term]) => Constructor[C, Term]]) extends InductiveTyp[C, Term] with Product with Serializable

Value Members

  1. object BaseConstructorTypes
  2. object Constructor
  3. object ConstructorPattern
  4. object ConstructorSeq
  5. object ConstructorTyp extends Serializable
  6. object Curry
  7. object FmlyPtn
  8. object Implicits
  9. object IndexedConstructorPatterns
  10. object InductiveTyp
  11. object IterFuncPattern

Inherited from AnyRef

Inherited from Any

Ungrouped