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Definition Classes: root

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). 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

Definition Classes: induction

ConstructorPatternMap

ConstructorPatternMapper

IndexedConstructorPatternMap

IndexedConstructorPatternMapper

IndexedConstructorSeqDom

IndexedConstructorSeqMap

IndexedConstructorSeqMapper

IndexedConstructorShape

IndexedInductiveDefinition

IndexedIterFuncPtnMap

IndexedIterFuncPtnMapper

IndexedIterFuncShape

IndexedRecursiveDefinition

provingground.induction

ConstructorShape

Companion trait ConstructorShape

object ConstructorShape

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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
case class CnstFuncConsShape[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], head: ConstructorShape[HShape, H, HC]) extends ConstructorShape[::[CnstFuncConsShape.type, HShape], H, Func[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
case class FuncConsShape[HS <: HList, H <: Term with Subs[H], HC <: Term with Subs[HC], F <: Term with Subs[F]](tail: IterFuncShape[H, F], head: ConstructorShape[HS, H, HC]) extends ConstructorShape[::[FuncConsShape.type, HS], H, Func[F, HC]] with Product with Serializable
ConstructorShape corresponding to an introduction rule of the form (.. -> W) -> head
case class IdShape[H <: Term with Subs[H]]() extends ConstructorShape[HNil, H, H] with Product with Serializable
ConstructorShape corresponding to the introduction rule W; all constructor patterns are constructed from this.

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object IdShape extends Serializable

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Packages

Inductive Type Definitions

Recursion and Induction functions

Indexed Versions

ConstructorShape

Companion trait ConstructorShape

object ConstructorShape

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Inductive Type Definitions

Recursion and Induction functions

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ConstructorShape

Companion trait ConstructorShape

object ConstructorShape

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ConstructorShape