Version 2 (modified by 9 years ago) (diff) | ,
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# Normalising and Solving Type Equalities

The following is based on ideas for the new, post-ICFP'08 solving algorithm. Most of the code is in the module `TcTyFuns`

.

## Normal equalities

Central to the algorithm are **normal equalities**, which can be regarded as a set of rewrite rules. Normal equalities are carefully oriented and contain synonym families only as the head symbols of left-hand sides. They assume one of the following three forms:

`co :: F t1..tn ~ t`

,`co :: x ~ t`

, where`x`

is a flexible type variable, or`co :: a ~ t`

, where`a`

is a rigid type variable (skolem) and`t`

is*not*a flexible type variable.

The types `t`

, `t1`

, ..., `tn`

may not contain any occurrences of synonym families. Moreover, in Forms (2) & (3), the left- and right-hand side need to be different, and the left-hand side may not occur in the right-hand side.

Coercions `co`

are either wanteds (represented by a flexible type variable) or givens *aka* locals (represented by a type term of kind CO).

Important to note is the following:

- We explicitly permit equalities of the form
`x ~ y`

and`a ~ b`

, where both sides are either flexible or rigid type variables. - Rewrite rules of the forms described under Cases (2) & (3) require extra care during rewriting. They may only be applied if they are
*not*recursive.

In GHC, `TcTyFuns.RewriteInst`

represents normal equalities, emphasising their role as rewrite rules.

## Termination

The Note [skolemOccurs loop] in the old code explains that equalities of the form `x ~ t`

(where `x`

is a flexible type variable) may not be used as rewrite rules, but only be solved by applying Rule Unify. As Unify carefully avoids cycles, this prevents the use of equalities introduced by the Rule SkolemOccurs as rewrite rules. For this to work, SkolemOccurs also had to apply to equalities of the form `a ~ t[[a]]`

. This was a somewhat intricate set up that's being simplified in the new algorithm. Whether equalities of the form `x ~ t`

are used as rewrite rules or solved by Unify doesn't matter anymore. Instead, we are careful to not use equalities of Form (2) or (3) for rewriting if their left-hand side symbol occurs in the right-hand side.