wiki:TypeFunctions/TotalFamilies

Version 1 (modified by chak, 6 years ago) (diff)

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Total Families

To guarantee the termination and completeness of the solving of equality constraints, we need to impose rather draconian restrictions on the instances of type synonym families. More specifically, to achieve both termination and completeness, we need the Strong Termination Condition and if we settle for termination alone (accepting to be incomplete for some rather exotic programs), we need the Relaxed (Termination) Condition as defined in Type Checking with Open Type Functions. The Strong Termination Condition corresponds closely to the conditions imposed on functional dependencies. The Relaxed Condition is somewhat more liberal, but still does not permit, for example, nested applications of type families in the right-hand side of a type instance. For many practically attractive uses of type families, where the the system is actually terminating, this is still to restrictive.

An example

data Z; data S a;

type family x :+ y
type instance Z   :+ y = y
type instance S x :+ y = S (x :+ y)

type family x :* y
type instance Z     :* y = Z
type instance (S x) :* y = x :* y :+ y

The family (:+) meets the Relaxed Condition, but not the Strong Termination Condition. However, (:*) meets not even the Relaxed Condition.