The type of terms generated by structures t over variables v. The structure type should implement Unifiable and the variable type should implement Variable.

The Show instance doesn't show the constructors, in order to improve legibility for large terms.

All the category theoretic instances (Functor, Foldable, Traversable,...) are provided because they are often useful; however, beware that since the implementations must be pure, they cannot read variables bound in the current context and therefore can create incoherent results. Therefore, you should apply the current bindings before using any of the functions provided by those classes.

O(n). Extract a pure term from a mutable term, or return Nothing if the mutable term actually contains variables. N.B., this function is pure, so you should manually apply bindings before calling it.

O(n). Embed a pure term as a mutable term.

The possible failure modes that could be encountered in unification and related functions. While many of the functions could be given more accurate types if we used ad-hoc combinations of these constructors (i.e., because they can only throw one of the errors), the extra complexity is not considered worth it.

This is a finally-tagless encoding of the UFailure data type so that we can abstract over clients adding additional domain-specific failure modes, introducing monoid instances, etc.

Since: 0.10.0

An implementation of syntactically unifiable structure. The Traversable constraint is there because we also require terms to be functors and require the distributivity of sequence or mapM.

Updated: 0.11 This class can now be derived so long as you have a Generic1 instance.

An implementation of unification variables. The Eq requirement is to determine whether two variables are equal as variables, without considering what they are bound to. We use Eq rather than having our own eqVar method so that clients can make use of library functions which commonly assume Eq.

The basic class for generating, reading, and writing to bindings stored in a monad. These three functionalities could be split apart, but are combined in order to simplify contexts. Also, because most functions reading bindings will also perform path compression, there's no way to distinguish "true" mutation from mere path compression.

The superclass constraints are there to simplify contexts, since we make the same assumptions everywhere we use BindingMonad.

Walk a term and determine which variables are still free. N.B., this function does not detect cyclic terms (i.e., throw errors), but it will return the correct answer for them in finite time.

Apply the current bindings from the monad so that any remaining variables in the result must, therefore, be free. N.B., this expensively clones term structure and should only be performed when a pure term is needed, or when occursFailure exceptions must be forced. This function does preserve sharing, however that sharing is no longer observed by the monad.

If any cyclic bindings are detected, then an occursFailure exception will be thrown.

Freshen all variables in a term, both bound and free. This ensures that the observability of sharing is maintained, while freshening the free variables. N.B., this expensively clones term structure and should only be performed when necessary.

If any cyclic bindings are detected, then an occursFailure exception will be thrown.

equals

equiv

unify

subsumes

Determine if two terms are structurally equal. This is essentially equivalent to (==) except that it does not require applying bindings before comparing, so it is more efficient. N.B., this function does not consider alpha-variance, and thus variables with different names are considered unequal. Cf., equiv.

Determine if two terms are structurally equivalent; that is, structurally equal modulo renaming of free variables. Returns a mapping from variable IDs of the left term to variable IDs of the right term, indicating the renaming used.

Unify two terms, or throw an error with an explanation of why unification failed. Since bindings are stored in the monad, the two input terms and the output term are all equivalent if unification succeeds. However, the returned value makes use of aggressive opportunistic observable sharing, so it will be more efficient to use it in future calculations than either argument.

A variant of unify which uses occursIn instead of visited-sets. This should only be used when eager throwing of occursFailure errors is absolutely essential (or for testing the correctness of unify). Performing the occurs-check is expensive. Not only is it slow, it's asymptotically slow since it can cause the same subterm to be traversed multiple times.

Determine whether the left term subsumes the right term. That is, whereas (tl =:= tr) will compute the most general substitution s such that (s tl === s tr), (tl <:= tr) computes the most general substitution s such that (s tl === tr). This means that tl is less defined than and consistent with tr.

N.B., this function updates the monadic bindings just like unify does. However, while the use cases for unification often want to keep the bindings around, the use cases for subsumption usually do not. Thus, you'll probably want to use a binding monad which supports backtracking in order to undo the changes. Unfortunately, leaving the monadic bindings unaltered and returning the necessary substitution directly imposes a performance penalty or else requires specifying too much about the implementation of variables.

Same as getFreeVars, but works on several terms simultaneously. This is more efficient than getting the free variables for each of the terms separately because we can make use of sharing across the whole collection. That is, each time we move to the next term, we still remember the bound variables we've already looked at (and therefore do not need to traverse, since we've already seen whatever free variables there are down there); whereas we would forget between each call to getFreeVars.

Since: 0.7.0

Same as applyBindings, but works on several terms simultaneously. This function preserves sharing across the entire collection of terms, whereas applying the bindings to each term separately would only preserve sharing within each term.

Since: 0.7.0

Same as freshen, but works on several terms simultaneously. This is different from freshening each term separately, because freshenAll preserves the relationship between the terms. For instance, the result of

mapM freshen [UVar 1, UVar 1]

would be [UVar 2, UVar 3] or something alpha-equivalent, whereas the result of

freshenAll [UVar 1, UVar 1]

would be [UVar 2, UVar 2] or something alpha-equivalent.

Since: 0.7.0

Canonicalize a chain of variables so they all point directly to the term at the end of the chain (or the free variable, if the chain is unbound), and return that end.

N.B., this is almost never the function you want. Cf., semiprune.

Canonicalize a chain of variables so they all point directly to the last variable in the chain, regardless of whether it is bound or not. This allows detecting many cases where multiple variables point to the same term, thereby allowing us to avoid re-unifying the term they point to.

Determine if a variable appears free somewhere inside a term. Since occurs checks only make sense when we're about to bind the variable to the term, we do not bother checking for the possibility of the variable occuring bound in the term.