\[\begin{split}\newcommand{\alors}{\textsf{then}} \newcommand{\alter}{\textsf{alter}} \newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\bool}{\textsf{bool}} \newcommand{\case}{\kw{case}} \newcommand{\conc}{\textsf{conc}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\conshl}{\textsf{cons\_hl}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\EqSt}{\textsf{EqSt}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\false}{\textsf{false}} \newcommand{\filter}{\textsf{filter}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\from}{\textsf{from}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\haslength}{\textsf{has\_length}} \newcommand{\hd}{\textsf{hd}} \newcommand{\ident}{\textsf{ident}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\lb}{\lambda} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\Nat}{\mathbb{N}} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\Prod}{\textsf{prod}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\si}{\textsf{if}} \newcommand{\sinon}{\textsf{else}} \newcommand{\Sort}{\cal S} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\true}{\textsf{true}} \newcommand{\Type}{\textsf{Type}} \newcommand{\unfold}{\textsf{unfold}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\cal W\!F}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\cal W\!F}(#2)} \newcommand{\WFTWOLINES}[2]{{\cal W\!F}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \newcommand{\zeros}{\textsf{zeros}} \end{split}\]

Type Classes

This chapter presents a quick reference of the commands related to type classes. For an actual introduction to typeclasses, there is a description of the system [SO08] and the literature on type classes in Haskell which also applies.

Class and Instance declarations

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

Class Id ( p\(_{1}\) : t\(_{1}\) ) ( p\(_{n}\) : t\(_{n}\) ) [: sort] := { f\(_{1}\) : u\(_{1}\) ; f\(_{m}\) : u\(_{m}\) }.

Instance ident : Id p\(_{1}\) p\(_{n}\) := { f\(_{1}\) := t\(_{1}\) ; f\(_{m}\) := t\(_{m}\) }.

The p\(_{i}\) : t\(_{i}\) variables are called the parameters of the class and the f\(_{i}\) : t\(_{i}\) 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:

Class EqDec (A : Type) := {   eqb : A -> A -> bool ;   eqb_leibniz : forall x y, eqb x y = true -> x = y }.
EqDec is defined eqb is defined eqb_leibniz is defined

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

Instance unit_EqDec : EqDec unit :=  { eqb x y := true ;    eqb_leibniz x y H :=          match x, y return x = y with tt, tt => eq_refl tt end }.
unit_EqDec is defined

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.:

Instance eq_bool : EqDec bool := { eqb x y := if x then y else negb y }.
1 subgoal ============================ forall x y : bool, (if x then y else negb y) = true -> x = y
Proof.
intros x y H.
1 subgoal x, y : bool H : (if x then y else negb y) = true ============================ x = y
destruct x ; destruct y ; (discriminate || reflexivity).
No more subgoals.
Defined.
eq_bool is defined

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.

Binding classes

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

Definition neqb {A} {eqa : EqDec A} (x y : A) := negb (eqb x y).
neqb is defined

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:

Fail Definition neqb' (A : Type) (x y : A) := negb (eqb x y).
The command has indeed failed with message: Unable to satisfy the following constraints: In environment: A : Type x, y : A ?EqDec : "EqDec A"

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:

Definition neqb' A (eqa : EqDec A) (x y : A) := negb (eqb x y).
neqb' is defined

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

  • They automatically set the maximally implicit status for typeclass 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 partially applied type classes (Implicit generalization). Any argument not given as part of a typeclass binder will be automatically generalized.
  • They also support implicit quantification on Superclasses.

Following the previous example, one can write:

Generalizable Variables A B C.
Definition neqb_implicit `{eqa : EqDec A} (x y : A) := negb (eqb x y).
neqb_implicit is defined

Here A is implicitly generalized, and the resulting function is equivalent to the one above.

Parameterized Instances

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

Instance prod_eqb `(EA : EqDec A, EB : EqDec B) : EqDec (A * B) := { eqb x y := match x, y with              | (la, ra), (lb, rb) => andb (eqb la lb) (eqb ra rb)              end }.
1 subgoal A : Type EA : EqDec A B : Type EB : EqDec B ============================ forall x y : A * B, (let (la, ra) := x in let (lb, rb) := y in (eqb la lb && eqb ra rb)%bool) = true -> x = y
Abort.

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

Sections and contexts

To ease the parametrization of developments by typeclasses, we provide a new way to introduce variables into section contexts, compatible with the implicit argument mechanism. The new command works similarly to the Variables vernacular, except it accepts any binding context as argument. For example:

Section EqDec_defs.
Context `{EA : EqDec A}.
A is declared EA is declared
Global Instance option_eqb : EqDec (option A) :=   { eqb x y := match x, y with          | Some x, Some y => eqb x y          | None, None => true          | _, _ => false          end }.
1 subgoal A : Type EA : EqDec A ============================ forall x y : option A, match x with | Some x0 => match y with | Some y0 => eqb x0 y0 | None => false end | None => match y with | Some _ => false | None => true end end = true -> x = y
Admitted.
option_eqb is declared
End EqDec_defs.
About option_eqb.
option_eqb : forall A : Type, EqDec A -> EqDec (option A) Arguments A, EA are implicit and maximally inserted Argument scopes are [type_scope _] Expands to: Constant Top.option_eqb

Here the Global modifier redeclares the instance at the end of the section, once it has been generalized by the context variables it uses.

Building hierarchies

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:

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

Contrary to Haskell, we have no special syntax for superclasses, but this declaration is 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.

Definition le_eqb `{Ord A} (x y : A) := andb (le x y) (le y x).
le_eqb is defined

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

Definition lt `{eqa : EqDec A, ! Ord eqa} (x y : A) := andb (le x y) (neqb x y).
lt is defined

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.

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.:

Require Import Relation_Definitions.
Class Reflexive (A : Type) (R : relation A) :=   reflexivity : forall x, R x x.
Class Transitive (A : Type) (R : relation A) :=   transitivity : forall x y z, R x y -> R y z -> R x z.

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

Class PreOrder (A : Type) (R : relation A) := { PreOrder_Reflexive :> Reflexive A R ;   PreOrder_Transitive :> Transitive A R }.
PreOrder is defined PreOrder_Reflexive is defined PreOrder_Transitive is defined

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.

Summary of the commands

Command Class ident binders? : sort? := ident? { ident :>? term+; }

The Class command is used to declare a typeclass with parameters binders and fields the declared record fields.

Variants:

Command Class ident binders? : sort? := ident : term

This variant declares a singleton class with a single method. This singleton class is a so-called definitional class, represented simply as a definition ident binders := term 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.

Command Existing Class ident

This variant declares a class a posteriori from a constant or inductive definition. No methods or instances are defined.

Warning ident is already declared as a typeclass

This command has no effect when used on a typeclass.

Command Instance ident binders? : class t1 tn [| priority] := { field1 := b1 ; …; fieldi := bi }

This command is used to declare a typeclass instance named ident of the class class with parameters t1 to tn and fields b1 to bi, where each field must be a declared field of the class. Missing fields must be filled in interactive proof mode.

An arbitrary context of binders 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. If the priority is not specified, it defaults to the number of non-dependent binders of the instance.

Variant Instance ident binders? : forall binders?, class term1 termn [| priority] := term

This syntax is used for declaration of singleton class instances or for directly giving an explicit term of type forall binders, class term1 termn. One need not even mention the unique field name for singleton classes.

Variant 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.

Variant Program Instance

Switches the type checking to Program (chapter Program) and uses the obligation mechanism to manage missing fields.

Variant Declare Instance

In a Module Type, this command states that a corresponding concrete instance should exist in any implementation of this Module Type. This is similar to the distinction between Parameter vs. Definition, or between Declare Module and Module.

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

Command Existing Instance ident+ [| priority]

This command adds an arbitrary list of constants whose type ends with an applied typeclass to the instance database with an optional priority. It can be used for redeclaring instances at the end of sections, or declaring structure projections as instances. This is equivalent to Hint Resolve ident : typeclass_instances, except it registers instances for Print Instances.

Command Context binders

Declares variables according to the given binding context, which might use Implicit generalization.

typeclasses eauto

This tactic uses a different resolution engine than eauto and auto. The main differences are the following:

  • Contrary to eauto and auto, the resolution is done entirely in the new proof engine (as of Coq 8.6), meaning that backtracking is available among dependent subgoals, and shelving goals is supported. typeclasses eauto is a multi-goal tactic. It analyses the dependencies between subgoals to avoid backtracking on subgoals that are entirely independent.

  • When called with no arguments, typeclasses eauto uses the typeclass_instances database by default (instead of core). Dependent subgoals are automatically shelved, and shelved goals can remain after resolution ends (following the behavior of Coq 8.5).

    Note

    As of Coq 8.6, all:once (typeclasses eauto) faithfully mimicks what happens during typeclass resolution when it is called during refinement/type inference, except that only declared class subgoals are considered at the start of resolution during type inference, while all can select non-class subgoals as well. It might move to all:typeclasses eauto in future versions when the refinement engine will be able to backtrack.

  • When called with specific databases (e.g. with), typeclasses eauto allows shelved goals to remain at any point during search and treat typeclass goals like any other.

  • The transparency information of databases is used consistently for all hints declared in them. It is always used when calling the unifier. When considering local hypotheses, we use the transparent state of the first hint database given. Using an empty database (created with Create HintDb for example) with unfoldable variables and constants as the first argument of typeclasses eauto hence makes resolution with the local hypotheses use full conversion during unification.

Variant typeclasses eauto num

Warning

The semantics for the limit num is different than for auto. By default, if no limit is given, the search is unbounded. Contrary to auto, introduction steps are counted, which might result in larger limits being necessary when searching with typeclasses eauto than with auto.

Variant typeclasses eauto with ident+

This variant runs resolution with the given hint databases. It treats typeclass subgoals the same as other subgoals (no shelving of non-typeclass goals in particular).

autoapply term with ident

The tactic autoapply applies a term using the transparency information of the hint database ident, and does no typeclass resolution. This can be used in Hint Extern’s for typeclass instances (in the hint database typeclass_instances) to allow backtracking on the typeclass subgoals created by the lemma application, rather than doing typeclass resolution locally at the hint application time.

Typeclasses Transparent, Typclasses Opaque

Command Typeclasses Transparent ident+

This command makes the identifiers transparent during typeclass resolution.

Command Typeclasses Opaque ident+

Make the identifiers opaque for typeclass search. 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 The hints databases for auto and eauto).

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.

Options

Flag Typeclasses Dependency Order

This option (on by default since 8.6) respects the dependency order between subgoals, meaning that subgoals on which other subgoals depend come first, while the non-dependent subgoals were put before the dependent ones previously (Coq 8.5 and below). This can result in quite different performance behaviors of proof search.

Flag Typeclasses Filtered Unification

This option, available since Coq 8.6 and off by default, switches the hint application procedure to a filter-then-unify strategy. To apply a hint, we first check that the goal matches syntactically the inferred or specified pattern of the hint, and only then try to unify the goal with the conclusion of the hint. This can drastically improve performance by calling unification less often, matching syntactic patterns being very quick. This also provides more control on the triggering of instances. For example, forcing a constant to explicitely appear in the pattern will make it never apply on a goal where there is a hole in that place.

Flag Typeclasses Limit Intros

This option (on by default) controls the ability to apply hints while avoiding (functional) eta-expansions in the generated proof term. It does so by allowing hints that conclude in a product to apply to a goal with a matching product directly, avoiding an introduction.

Warning

This can be expensive as it requires rebuilding hint clauses dynamically, and does not benefit from the invertibility status of the product introduction rule, resulting in potentially more expensive proof-search (i.e. more useless backtracking).

Flag Typeclass Resolution For Conversion

This option (on by default) controls the use of typeclass resolution when a unification problem cannot be solved during elaboration/type inference. With this option on, when a unification fails, typeclass resolution is tried before launching unification once again.

Flag Typeclasses Strict Resolution

Typeclass declarations introduced when this option is set have a stricter resolution behavior (the option is off by default). When looking for unifications of a goal with an instance of this class, we “freeze” all the existentials appearing in the goals, meaning that they are considered rigid during unification and cannot be instantiated.

Flag Typeclasses Unique Solutions

When a typeclass resolution is launched we ensure that it has a single solution or fail. This ensures that the resolution is canonical, but can make proof search much more expensive.

Flag Typeclasses Unique Instances

Typeclass declarations introduced when this option is set have a more efficient resolution behavior (the option is off by default). When a solution to the typeclass goal of this class is found, we never backtrack on it, assuming that it is canonical.

Flag Typeclasses Debug

Controls whether typeclass resolution steps are shown during search. Setting this flag also sets Typeclasses Debug Verbosity to 1.

Option Typeclasses Debug Verbosity num

Determines how much information is shown for typeclass resolution steps during search. 1 is the default level. 2 shows additional information such as tried tactics and shelving of goals. Setting this option also sets Typeclasses Debug.

Flag Refine Instance Mode

This option allows to switch the behavior of instance declarations made through the Instance command.

  • When it is on (the default), instances that have unsolved holes in their proof-term silently open the proof mode with the remaining obligations to prove.
  • When it is off, they fail with an error instead.

Typeclasses eauto :=

Command Typeclasses eauto := debug? {dfs | bfs}? depth

This command allows more global customization of the typeclass resolution tactic. The semantics of the options 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 limit of the search.