Chapter 18  Type Classes

• Class and Instance declarations
• Binding classes
• Parameterized Instances
• Building hierarchies
• Summary of the commands

Matthieu Sozeau

This chapter presents a quick reference of the commands related to type classes. For an actual introduction to type classes, there is a description of the system [126] and the literature on type classes in Haskell which also applies.

18.1  Class and Instance declarations

The syntax for class and instance declarations is the same as record syntax of Coq:

The α_i : τ_i variables are called the parameters of the class and the f_k : type_k are called the methods. Each class definition gives rise to a corresponding record declaration and each instance is a regular definition whose name is given by ident and type is an instantiation of the record type.

We’ll use the following example class in the rest of the chapter:

This class implements a boolean equality test which is compatible with leibniz equality on some type. An example implementation is:

If one does not give all the members in the Instance declaration, Coq enters the proof-mode and the user is asked to build inhabitants of the remaining fields, e.g.:

One has to take care that the transparency of every field is determined by the transparency of the Instance proof. One can use alternatively the Program Instance variant which has richer facilities for dealing with obligations.

18.2  Binding classes

Once a type class is declared, one can use it in class binders:

When one calls a class method, a constraint is generated that is satisfied only in contexts where the appropriate instances can be found. In the example above, a constraint EqDec A is generated and satisfied by eqa : EqDec A. In case no satisfying constraint can be found, an error is raised:

The algorithm used to solve constraints is a variant of the eauto tactic that does proof search with a set of lemmas (the instances). It will use local hypotheses as well as declared lemmas in the typeclass_instances database. Hence the example can also be written:

However, the generalizing binders should be used instead as they have particular support for type classes:

• They automatically set the maximally implicit status for type class arguments, making derived functions as easy to use as class methods. In the example above, A and eqa should be set maximally implicit.
• They support implicit quantification on class arguments and partialy applied type classes (§18.2.1)
• They support implicit quantification on superclasses (§18.4.1)

18.2.1  Implicit quantification

Implicit quantification is an automatic elaboration of a statement with free variables into a closed statement where these variables are quantified explicitely. Implicit generalization is done only inside binders begining with a backquote ‘ and the codomain of Instance declarations.

Following the previous example, one can write:

Here A is implicitely generalized, and the resulting function is equivalent to the one above. One must be careful that all the free variables are generalized, which may result in confusing errors in case of typos. In such cases, the context will probably contain some unexpected generalized variable.

The generalizing binders `{ } and `( ) work similarly to their explicit counterparts, only binding the generalized variables implicitly, as maximally-inserted arguments. In these binders, the binding name for the bound object is optional, whereas the type is mandatory, dually to regular binders.

18.3  Parameterized Instances

One can declare parameterized instances as in Haskell simply by giving the constraints as a binding context before the instance, e.g.:

These instances are used just as well as lemmas in the instance hint database.

18.4  Building hierarchies

18.4.1  Superclasses

One can also parameterize classes by other classes, generating a hierarchy of classes and superclasses. In the same way, we give the superclasses as a binding context:

Contrary to Haskell, we have no special syntax for superclasses, but this declaration is morally equivalent to:

Class `(E : EqDec A) => Ord A :=
  { le : A -> A -> bool }.

This declaration means that any instance of the Ord class must have an instance of EqDec. The parameters of the subclass contain at least all the parameters of its superclasses in their order of appearance (here A is the only one). As we have seen, Ord is encoded as a record type with two parameters: a type A and an E of type EqDec A. However, one can still use it as if it had a single parameter inside generalizing binders: the generalization of superclasses will be done automatically.

In some cases, to be able to specify sharing of structures, one may want to give explicitely the superclasses. It is is possible to do it directly in regular binders, and using the ! modifier in class binders. For example:

The ! modifier switches the way a binder is parsed back to the regular interpretation of Coq. In particular, it uses the implicit arguments mechanism if available, as shown in the example.

18.4.2  Substructures

Substructures are components of a class which are instances of a class themselves. They often arise when using classes for logical properties, e.g.:

This declares singleton classes for reflexive and transitive relations, (see 1 for an explanation). These may be used as part of other classes:

The syntax :> indicates that each PreOrder can be seen as a Reflexive relation. So each time a reflexive relation is needed, a preorder can be used instead. This is very similar to the coercion mechanism of Structure declarations. The implementation simply declares each projection as an instance.

One can also declare existing objects or structure projections using the Existing Instance command to achieve the same effect.

18.5  Summary of the commands

18.5.1  Class ident binder₁ … binder_n : sort:= { field₁ ; …; field_k }.

The Class command is used to declare a type class with parameters binder₁ to binder_n and fields field₁ to field_k.

Variants:

1. Class ident binder₁ …binder_n : sort:= ident₁ : type₁. This variant declares a singleton class whose only method is ident₁. This singleton class is a so-called definitional class, represented simply as a definition identbinder₁ …binder_n := type₁ and whose instances are themselves objects of this type. Definitional classes are not wrapped inside records, and the trivial projection of an instance of such a class is convertible to the instance itself. This can be useful to make instances of existing objects easily and to reduce proof size by not inserting useless projections. The class constant itself is declared rigid during resolution so that the class abstraction is maintained.

18.5.2  Instance ident binder₁ …binder_n : Class t₁ …t_n [| priority] := { field₁ := b₁ ; …; field_i := b_i }

The Instance command is used to declare a type class instance named ident of the class Class with parameters t₁ to t_n and fields b₁ to b_i, where each field must be a declared field of the class. Missing fields must be filled in interactive proof mode.

An arbitrary context of the form binder₁ …binder_n can be put after the name of the instance and before the colon to declare a parameterized instance. An optional priority can be declared, 0 being the highest priority as for auto hints.

Variants:

1. Instance ident binder₁ …binder_n : Class t₁ …t_n [| priority] := term This syntax is used for declaration of singleton class instances. It does not include curly braces and one need not even mention the unique field name.
2. Global Instance One can use the Global modifier on instances declared in a section so that their generalization is automatically redeclared after the section is closed.
3. Program Instance Switches the type-checking to Program (chapter 22) and uses the obligation mechanism to manage missing fields.

Besides the Class and Instance vernacular commands, there are a few other commands related to type classes.

18.5.3  Existing Instance ident

This commands adds an arbitrary constant whose type ends with an applied type class to the instance database. It can be used for redeclaring instances at the end of sections, or declaring structure projections as instances. This is almost equivalent to Hint Resolve ident : typeclass_instances.

18.5.4  Typeclasses Transparent, Opaque ident₁ …ident_n

This commands defines the transparency of ident₁ …ident_n during type class resolution. It is useful when some constants prevent some unifications and make resolution fail. It is also useful to declare constants which should never be unfolded during proof-search, like fixpoints or anything which does not look like an abbreviation. This can additionally speed up proof search as the typeclass map can be indexed by such rigid constants (see 8.13.1). By default, all constants and local variables are considered transparent. One should take care not to make opaque any constant that is used to abbreviate a type, like relation A := A -> A -> Prop.

This is equivalent to Hint Transparent,Opaque ident : typeclass_instances.

18.5.5  Typeclasses eauto := [debug] [dfs | bfs] [depth]

This commands allows to customize the type class resolution tactic, based on a variant of eauto. The flags semantics are:

• debug In debug mode, the trace of successfully applied tactics is printed.
• dfs, bfs This sets the search strategy to depth-first search (the default) or breadth-first search.
• depth This sets the depth of the search (the default is 100).

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