This trait abstracts over the various ways how the laws of a type class
can depend on the laws of other type classes. An instance of this trait is
called a rule set.
For that matter, we divide type classes into kinds, where the classes
of one kind share the number of operations and meaning. For example,
Semigroup, Monoid and Group all belong to the same kind. On the
other hand, their additive variants also belong to a common kind, but to
a different one.
Users of this trait should extend the outer trait Laws and create
specialized subtypes for each kind of type class. (See
DefaultRuleSet for an example.)
They all define their own laws, as well as a couple of parent classes.
If we want to check the laws of AdditiveGroup, we want to avoid checking
properties twice, i.e. do not want to check Monoid laws via Group and
also via AdditiveMonoid.
To address this problem, we define the parent in the same kind as
parent, and other parents as bases. In this example, the parent of
AdditiveGroup is Group, and its only basis is Group. On the other
hand, the parent of Group is Monoid, and it does not have any bases.
The set of all properties of a certain class is now defined as union of
these sets:
the properties of the class itself
recursively, the properties of all its parents (ignoring their bases)
recursively, the set of all properties of its bases
Looking at our example, that means that AdditiveGroup includes the
Monoid law only once, because it is the parent of its basis. The
same laws are ignored by its parent AdditiveMonoid, hence no redundant
checks occur.
Of course, classes can have multiple parents and multiple (named) bases.
The only requirement here is that inside one kind, the identifier of
a property is unique, since duplicates are eliminated. To avoid name
clashes between different kinds, the names of properties pulled in
via a basis are prefixed with the name of the basis.
For better type-safety, parents are only allowed to come from the
same outer instance of Laws, whereas bases are allowed to come
from anywhere.
This trait abstracts over the various ways how the laws of a type class can depend on the laws of other type classes. An instance of this trait is called a rule set.
For that matter, we divide type classes into kinds, where the classes of one kind share the number of operations and meaning. For example,
Semigroup
,Monoid
andGroup
all belong to the same kind. On the other hand, their additive variants also belong to a common kind, but to a different one.Users of this trait should extend the outer trait Laws and create specialized subtypes for each kind of type class. (See DefaultRuleSet for an example.)
Consider this example hierarchy:
They all define their own laws, as well as a couple of parent classes. If we want to check the laws ofAdditiveGroup
, we want to avoid checking properties twice, i.e. do not want to checkMonoid
laws viaGroup
and also viaAdditiveMonoid
.To address this problem, we define the parent in the same kind as parent, and other parents as bases. In this example, the parent of
AdditiveGroup
isGroup
, and its only basis isGroup
. On the other hand, the parent ofGroup
isMonoid
, and it does not have any bases.The set of all properties of a certain class is now defined as union of these sets:
Looking at our example, that means that
AdditiveGroup
includes theMonoid
law only once, because it is the parent of its basis. The same laws are ignored by its parentAdditiveMonoid
, hence no redundant checks occur.Of course, classes can have multiple parents and multiple (named) bases. The only requirement here is that inside one kind, the identifier of a property is unique, since duplicates are eliminated. To avoid name clashes between different kinds, the names of properties pulled in via a basis are prefixed with the name of the basis.
For better type-safety, parents are only allowed to come from the same outer instance of Laws, whereas bases are allowed to come from anywhere.