HaskHOL.Core.Kernel.Terms

This module exports a safe view of HOL terms for HaskHOL. It also defines the primitive functions related to terms. For clarity, these functions have been seperated based on their influential system: HOL Light, Stateless HOL, and HOL2P.

Note that, per the stateless approach, any stateful, but still primitive, functions related to terms have been relocated to the HaskHOL.Core.State module.

A View of HOL Terms

A Quick Note on View Patterns

The primitive data types of HaskHOL are implemented using view patterns in order to simulate private data types:

• Internal constructors are hidden to prevent manual construction of terms.
• View constructors (those of HOLTypeView, HOLTermView, and HOLThmView) are exposed to enable pattern matching.
• View patterns, as defined by instances of the view function from the Viewable class, provide a conversion between the two sets of constructors.

data HOLTerm

The HOLTerm data type defines the internal constructors for HOL terms in HaskHOL. For more details, see the documentation for its view pattern data type, HOLTermView.

Instances

Eq HOLTerm

Ord HOLTerm

Typeable HOLTerm

NFData HOLTerm

Lift HOLTerm

Protected HOLTerm

HOLTermRep HOLTerm a

ShowHOL HOLTerm thry

HOLTermRep (PTerm thry) thry

data HOLTermView

The view pattern data type for HOL terms.

Constructors

Var String HOLType

A term variable consisting of a name and type.

Const String HOLType ConstTag

A term constant consisting of a name, type, and tag. See ConstTag for more information.

Comb HOLTerm HOLTerm

A term application consisting of a function term and argument term.

Abs HOLTerm HOLTerm

A term abstraction consisting of a bound term and a body term. Note that the bound term must be a type variable.

TyComb HOLTerm HOLType

A term-level, type application consisting of a body term and an argument type. Note that the body term must have a universal type.

TyAbs HOLType HOLTerm

A term-level, type abstraction consisting of a bound type and a body term. Note that the bound type must be a small, type variable.

data ConstTag

The data type for constant tags, ConstTag, follows identically from the implementation in Stateless HOL. For more information regarding construction of the different tags, see the documentation of the following functions: newPrimConst, newDefinedConst, and newDefinedTypeOp.

Instances

Eq ConstTag

Ord ConstTag

Typeable ConstTag

NFData ConstTag

Lift ConstTag

type HOLTermEnv = [(HOLTerm, HOLTerm)]

Type synonym for the commonly used, list-based, term environment.

HOL Light Term Primitives

Alpha-Equivalence of Terms

alphaOrder :: HOLTerm -> HOLTerm -> Ordering

Provides an ordering for two terms modulo alpha-equivalence

aConv :: HOLTerm -> HOLTerm -> Bool

Tests if two terms are alpha-equivalent

Predicates, Constructors, and Destructors for Basic Terms

isVar :: HOLTerm -> Bool

Predicate for term variables.

isConst :: HOLTerm -> Bool

Predicate for term constants.

isAbs :: HOLTerm -> Bool

Predicate for term abstractions.

isComb :: HOLTerm -> Bool

Predicate for term combinations.

mkVar :: String -> HOLType -> HOLTerm

Constructs a term variable of a given name and type.

mkAbs :: HOLTerm -> HOLTerm -> Either String HOLTerm

Constructs a term abstraction of a given bound term and body term. Fails with Left if the bound term is not a variable.

mkComb :: HOLTerm -> HOLTerm -> Either String HOLTerm

Constructs a combination of two given terms. Fails with Left in the following cases:

• The first term does not have a function type.
• The types of the two terms does not agree.

destVar :: HOLTerm -> Maybe (String, HOLType)

Destructs a term variable, returning its name and type. Fails with Nothing if the provided term is not a variable.

destConst :: HOLTerm -> Maybe (String, HOLType)

Destructs a term constant, returning its name and type. Note that no constant tag information is returned. Fails with Nothing if the provided term is not a constant.

destComb :: HOLTerm -> Maybe (HOLTerm, HOLTerm)

Destructs a term combination, returning its function and argument terms. Fails with Nothing if the provided term is not a combination.

destAbs :: HOLTerm -> Maybe (HOLTerm, HOLTerm)

Destructs a term abstraction, returning its bound term and body term. Fails with Nothing if the provided term is not an abstraction.

Term and Type Variable Extractors

frees :: HOLTerm -> [HOLTerm]

Returns a list of all free, term variables in a term.

catFrees :: [HOLTerm] -> [HOLTerm]

Returns a list of all free, term variables in a list of terms.

freesIn :: [HOLTerm] -> HOLTerm -> Bool

Checks a list of term variables to see if they are all free in a give term.

varFreeIn :: HOLTerm -> HOLTerm -> Bool

Checks if a variable or constant term is free in a given term.

typeVarsInTerm :: HOLTerm -> [HOLType]

Returns a list of all free, type variables in a term, not including type operator variables.

typeVarsInTerms :: [HOLTerm] -> [HOLType]

Returns a list of all free, type variables in a list of terms, not including type operator variables.

Term Substitution and Instantiation

varSubst :: HOLTermEnv -> HOLTerm -> HOLTerm

Performs a basic term substitution using a substitution environment containing pairs consisting of a term variable and a term to be substituted for that variable. Note that the order of elements in a substitution pair follows the convention of most Haskell libraries, rather than the traditional HOL convention:

• The second element is substituted for the first, i.e. the substitution pair (A, \ x.x) indicates that the lambda term \x.x should be substituted for the term variable A.

class TypeSubst a b => Inst a b

The Inst class provides the framework for type instantiation in HaskHOL. Note that in the simplest cases, instantiation is simply a type substitution for the types of term variables and constants. Therefore, instantiation is constrained by the TypeSubst class.

The move to a polymorphic type system further complicates things as types can now be bound at the term level, requiring renaming for type instantiation. Since we have three different possible substitution environment types, we have three different possible instantiation environment types and, therefore, three different ways to handle renaming:

• For (x::HOLTerm, r::HOLTerm) substitution pairs we rename in the case where a type abstraction binds a type variable present in r and x is present in the body of the type abstraction.
• For (_::TypeOp, _::TypeOp) substitution pairs we can safely ignore renaming as our logic does not permit the binding of type operator variables.
• For (x::TypeOp, r::HOLTerm) substitution pairs we rename in the case where a type abstraction binds a type variable present in r and x is present in the body of the type abstraction.

Just as we did for the TypeSubst class, we hide the internals of Inst to prevent unsound re-definition. The correct functions to call for type instantiation are inst and instFull.

Instances

Inst TypeOp TypeOp

Inst TypeOp HOLType

Inst HOLType HOLType

inst :: Inst a b => [(a, b)] -> HOLTerm -> HOLTerm

Type instantiation for terms. Accepts the same types of substitution environments as discussed in the documentation for the TypeSubst class, with invalid substitution pairs being pruned internally by typeSubst as necessary.

For more information on why the Inst class constraint is necessary and how renaming of bound types is performed, see that classes documentation.

instFull :: SubstTrip -> HOLTerm -> HOLTerm

A version of inst that accepts a triplet of type substitution environments.

instConst :: TypeSubst a b => HOLTerm -> [(a, b)] -> Maybe HOLTerm

A simplified version of inst that works only for term constants. Fails with Nothing if the provided term is not a constant. Used internally by mkConst to guarantee that only constants are constructed.

instConstFull :: HOLTerm -> SubstTrip -> Maybe HOLTerm

A version of instConst that accepts a triplet of type substitition environments.

Commonly Used Terms and Functions

tmEq :: HOLType -> HOLTerm

Constructs an instance of the HOL equality constant, =, for a given type.

isEq :: HOLTerm -> Bool

Predicate for equations, i.e. terms of the form l = r.

primMkEq :: HOLTerm -> HOLTerm -> Either String HOLTerm

Constructs an equation term given the left and right hand side arguments. Fails with Left if the types of the terms are not alpha-equivalent.

destEq :: HOLTerm -> Maybe (HOLTerm, HOLTerm)

Destructs an equation term, returning the left and right hand side arguments. Fails with Nothing if the term is not an equation, i.e. of the form l = r.

variant :: [HOLTerm] -> HOLTerm -> HOLTerm

Renames a term variable to avoid sharing a name with any of a given list of term variables. Rreturns the original term if it's not a term variable.

variants :: [HOLTerm] -> [HOLTerm] -> [HOLTerm]

Renames a list of term variables to avoid sharing a name with any of a given list of term variables. As each term variable is processed it is added to the list of avoids such that the resultant list of term variables are all uniquely named.

Stateless HOL Term Primitives

Constructors for Constant Tags

newPrimConst :: String -> HOLType -> HOLTerm

Constructs a primitive constant given a name and type. Note that primitive constants are tagged with a Prim ConstTag indicating that they have no definition.

Type Operator Variable Extractors

typeOpVarsInTerm :: HOLTerm -> [TypeOp]

Returns the list of all type operator variables in a term.

typeOpVarsInTerms :: [HOLTerm] -> [TypeOp]

Returns the list of all type operator variables in a list of terms.

HOL2P Term Primitives

Predicates, Constructors, and Destructors for Term-Level Types

isTyAbs :: HOLTerm -> Bool

Predicate for type abstraction terms.

isTyComb :: HOLTerm -> Bool

Predicate for type combination terms.

mkTyAbs :: HOLType -> HOLTerm -> Either String HOLTerm

Constructs a type abstraction term given a bound type and a body term. Fails with Left in the following cases:

• The bound type is not a small type variable.
• The bound type variable occurs in the type of a free variable in the body term.

mkTyComb :: HOLTerm -> HOLType -> Either String HOLTerm

Constructs a type combination term given a body term and a type argument to apply. Fails with Left in the following cases:

• The type argument is not a small type.
• The type of the body term is not a universal type.

destTyAbs :: HOLTerm -> Maybe (HOLType, HOLTerm)

Destructs a type abstraction, returning its bound type and body term. Fails with Nothing if the provided term is not a type abstraction.

destTyComb :: HOLTerm -> Maybe (HOLTerm, HOLType)

Destructs a type combination, returning its body term and type argument. Fails with Nothing if the provided term is not a type combination.