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 andrewscurtis
    Definition Classes
    provingground
  • package examples
    Definition Classes
    provingground
  • 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 interface
    Definition Classes
    provingground
  • package learning
    Definition Classes
    provingground
  • package library
    Definition Classes
    provingground
  • package scalahott
    Definition Classes
    provingground
  • package scratch
    Definition Classes
    provingground
  • package translation

    Translation primarily using a functorial framework - see Translator$, for natural language processing as well as serialization, formatted output, parsing, interface with formal languages etc.

    Translation primarily using a functorial framework - see Translator$, for natural language processing as well as serialization, formatted output, parsing, interface with formal languages etc.

    Besides the Translator framework and helper typeclasses is Functors, several structures for concrete languages including our implementation of HoTT are in this package.

    Definition Classes
    provingground
  • Base
  • Context
  • FiniteDistribution
  • Frankl
  • HoTT
  • JvmUtils
  • LinearStructure
  • MereProposition
  • PickledWeighted
  • ProbabilityDistribution
  • Subst
  • SubstImplicits
  • TermList
  • TermListImplicits
  • Utils
  • Weighted

sealed trait TermList[A] extends Subst[A]

allows substitution of a Term by another, as well as mapping to a vector of terms chiefly subtypes of Term and HLists of these;

Linear Supertypes
Subst[A], AnyRef, Any
Known Subclasses
HNilTermList
Type Hierarchy
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. TermList
  2. Subst
  3. AnyRef
  4. Any
Implicitly
  1. by TermListOp
  2. by SubstOp
  3. by any2stringadd
  4. by StringFormat
  5. by Ensuring
  6. by ArrowAssoc
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Visibility
  1. Public
  2. Protected

Abstract Value Members

  1. abstract def subst(a: A)(x: Term, y: Term): A
    Definition Classes
    Subst
  2. abstract def terms(a: A): Vector[Term]

Shadowed Implicit Value Members

  1. def subst(x: Term, y: Term): TermList[A]
    Implicit
    This member is added by an implicit conversion from TermList[A] toTermList.TermListOp[TermList[A]] performed by method TermListOp in provingground.TermListImplicits.This conversion will take place only if an implicit value of type TermList[TermList[A]] is in scope.
    Shadowing
    This implicitly inherited member is shadowed by one or more members in this class.
    To access this member you can use a type ascription:
    (termList: TermList.TermListOp[TermList[A]]).subst(x, y)
    Definition Classes
    SubstOp
  2. def subst(x: Term, y: Term): TermList[A]
    Implicit
    This member is added by an implicit conversion from TermList[A] toTermList.SubstOp[TermList[A]] performed by method SubstOp in provingground.SubstImplicits.This conversion will take place only if an implicit value of type Subst[TermList[A]] is in scope.
    Shadowing
    This implicitly inherited member is shadowed by one or more members in this class.
    To access this member you can use a type ascription:
    (termList: TermList.SubstOp[TermList[A]]).subst(x, y)
    Definition Classes
    SubstOp
  3. def terms: Vector[Term]
    Implicit
    This member is added by an implicit conversion from TermList[A] toTermList.TermListOp[TermList[A]] performed by method TermListOp in provingground.TermListImplicits.This conversion will take place only if an implicit value of type TermList[TermList[A]] is in scope.
    Shadowing
    This implicitly inherited member is shadowed by one or more members in this class.
    To access this member you can use a type ascription:
    (termList: TermList.TermListOp[TermList[A]]).terms
    Definition Classes
    TermListOp
  4. def ~->:(x: Term): Lambda[Term, TermList[A]]
    Implicit
    This member is added by an implicit conversion from TermList[A] toTermList.TermListOp[TermList[A]] performed by method TermListOp in provingground.TermListImplicits.This conversion will take place only if an implicit value of type TermList[TermList[A]] is in scope.
    Shadowing
    This implicitly inherited member is ambiguous. One or more implicitly inherited members have similar signatures, so calling this member may produce an ambiguous implicit conversion compiler error.
    To access this member you can use a type ascription:
    (termList: TermList.TermListOp[TermList[A]]).~->:(x)
    Definition Classes
    SubstOp
  5. def ~->:(x: Term): Lambda[Term, TermList[A]]
    Implicit
    This member is added by an implicit conversion from TermList[A] toTermList.SubstOp[TermList[A]] performed by method SubstOp in provingground.SubstImplicits.This conversion will take place only if an implicit value of type Subst[TermList[A]] is in scope.
    Shadowing
    This implicitly inherited member is ambiguous. One or more implicitly inherited members have similar signatures, so calling this member may produce an ambiguous implicit conversion compiler error.
    To access this member you can use a type ascription:
    (termList: TermList.SubstOp[TermList[A]]).~->:(x)
    Definition Classes
    SubstOp