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 ConstructorSeqMap
    Definition Classes
    induction
  • Cons
  • ConsIndException
  • Empty
  • InducDataSym
  • RecDataSym

case class Cons[C <: Term with Subs[C], H <: Term with Subs[H], Cod <: Term with Subs[Cod], RD <: Term with Subs[RD], ID <: Term with Subs[ID], TR <: Term with Subs[TR], TI <: Term with Subs[TI], TIntros <: HList](cons: C, pattern: ConstructorPatternMap[Cod, C, H, RD, ID], tail: ConstructorSeqMap[Cod, H, TR, TI, TIntros]) extends ConstructorSeqMap[Cod, H, Func[RD, TR], Func[ID, TI], ::[C, TIntros]] with Product with Serializable

prepending an introduction rule to ConstructorSeqMap

cons

the introduction rule

pattern

the mapped version of the introduction rule for the given codomain

tail

the tail ConstructorSeqMap

Linear Supertypes
Serializable, Product, Equals, ConstructorSeqMap[Cod, H, Func[RD, TR], Func[ID, TI], ::[C, TIntros]], AnyRef, Any
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. Cons
  2. Serializable
  3. Product
  4. Equals
  5. ConstructorSeqMap
  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 Cons(cons: C, pattern: ConstructorPatternMap[Cod, C, H, RD, ID], tail: ConstructorSeqMap[Cod, H, TR, TI, TIntros])

    cons

    the introduction rule

    pattern

    the mapped version of the introduction rule for the given codomain

    tail

    the tail ConstructorSeqMap

Value Members

  1. val W: Typ[H]

    the inductive type being defined

    the inductive type being defined

    Definition Classes
    Cons → ConstructorSeqMap
  2. val cons: C
  3. def data(X: Typ[Cod]): RD
  4. def dataCons(X: Typ[Cod]): DataCons[H, Cod, RD]
  5. val defn: (RD) => (Func[H, Cod]) => (H) => Option[Cod]
  6. def indDataCons(fibre: Func[H, Typ[Cod]]): DataCons[H, Cod, ID]
  7. def induc(fibre: Func[H, Typ[Cod]]): Func[ID, TI]

    inductive definition function ind_W, Xs

    inductive definition function ind_W, Xs

    Definition Classes
    ConstructorSeqMap
  8. def inducData(fibre: Func[H, Typ[Cod]]): ID with Subs[ID]
  9. def inducDataLambda(fibre: Func[H, Typ[Cod]]): (FuncLike[H, Cod]) => Func[ID with Subs[ID], TI]

    converts an inductive definition built by InductiveDefinition to a lambda

    converts an inductive definition built by InductiveDefinition to a lambda

    Definition Classes
    Cons → ConstructorSeqMap
  10. def inducDefn(fibre: Func[H, Typ[Cod]]): DataCons[H, Cod, ID]
    Definition Classes
    Cons → ConstructorSeqMap
  11. val inducDefn: (ID) => (FuncLike[H, Cod]) => (H) => Option[Cod]
  12. val pattern: ConstructorPatternMap[Cod, C, H, RD, ID]
  13. def productElementNames: Iterator[String]
    Definition Classes
    Product
  14. def rec(X: Typ[Cod]): Func[RD, TR]

    the recursion function rec_W,X

    the recursion function rec_W,X

    Definition Classes
    ConstructorSeqMap
  15. def recDataLambda(X: Typ[Cod]): (Func[H, Cod]) => Func[RD, TR]

    converts a recursive definition built by RecursiveDefinition to a lambda

    converts a recursive definition built by RecursiveDefinition to a lambda

    Definition Classes
    Cons → ConstructorSeqMap
  16. def recDefn(X: Typ[Cod]): DataCons[H, Cod, RD]

    recursive definition function rec_W, X in raw form

    recursive definition function rec_W, X in raw form

    Definition Classes
    Cons → ConstructorSeqMap
  17. def subs(x: Term, y: Term): Cons[C with Subs[C], H, Cod, RD, ID, TR, TI, TIntros]
    Definition Classes
    Cons → ConstructorSeqMap
  18. val tail: ConstructorSeqMap[Cod, H, TR, TI, TIntros]