.. _ImplicitArguments: Implicit arguments ------------------ An implicit argument of a function is an argument which can be inferred from contextual knowledge. There are different kinds of implicit arguments that can be considered implicit in different ways. There are also various commands to control the setting or the inference of implicit arguments. The different kinds of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Implicit arguments inferable from the knowledge of other arguments of a function ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ The first kind of implicit arguments covers the arguments that are inferable from the knowledge of the type of other arguments of the function, or of the type of the surrounding context of the application. Especially, such implicit arguments correspond to parameters dependent in the type of the function. Typical implicit arguments are the type arguments in polymorphic functions. There are several kinds of such implicit arguments. **Strict Implicit Arguments** An implicit argument can be either strict or non strict. An implicit argument is said to be *strict* if, whatever the other arguments of the function are, it is still inferable from the type of some other argument. Technically, an implicit argument is strict if it corresponds to a parameter which is not applied to a variable which itself is another parameter of the function (since this parameter may erase its arguments), not in the body of a match, and not itself applied or matched against patterns (since the original form of the argument can be lost by reduction). For instance, the first argument of :: cons: forall A:Set, A -> list A -> list A in module ``List.v`` is strict because :g:`list` is an inductive type and :g:`A` will always be inferable from the type :g:`list A` of the third argument of :g:`cons`. Also, the first argument of :g:`cons` is strict with respect to the second one, since the first argument is exactly the type of the second argument. On the contrary, the second argument of a term of type :: forall P:nat->Prop, forall n:nat, P n -> ex nat P is implicit but not strict, since it can only be inferred from the type :g:`P n` of the third argument and if :g:`P` is, e.g., :g:`fun _ => True`, it reduces to an expression where ``n`` does not occur any longer. The first argument :g:`P` is implicit but not strict either because it can only be inferred from :g:`P n` and :g:`P` is not canonically inferable from an arbitrary :g:`n` and the normal form of :g:`P n`. Consider, e.g., that :g:`n` is :math:`0` and the third argument has type :g:`True`, then any :g:`P` of the form :: fun n => match n with 0 => True | _ => anything end would be a solution of the inference problem. **Contextual Implicit Arguments** An implicit argument can be *contextual* or not. An implicit argument is said *contextual* if it can be inferred only from the knowledge of the type of the context of the current expression. For instance, the only argument of:: nil : forall A:Set, list A` is contextual. Similarly, both arguments of a term of type:: forall P:nat->Prop, forall n:nat, P n \/ n = 0 are contextual (moreover, :g:`n` is strict and :g:`P` is not). **Reversible-Pattern Implicit Arguments** There is another class of implicit arguments that can be reinferred unambiguously if all the types of the remaining arguments are known. This is the class of implicit arguments occurring in the type of another argument in position of reversible pattern, which means it is at the head of an application but applied only to uninstantiated distinct variables. Such an implicit argument is called *reversible- pattern implicit argument*. A typical example is the argument :g:`P` of nat_rec in :: nat_rec : forall P : nat -> Set, P 0 -> (forall n : nat, P n -> P (S n)) -> forall x : nat, P x (:g:`P` is reinferable by abstracting over :g:`n` in the type :g:`P n`). See :ref:`controlling-rev-pattern-implicit-args` for the automatic declaration of reversible-pattern implicit arguments. Implicit arguments inferable by resolution ++++++++++++++++++++++++++++++++++++++++++ This corresponds to a class of non-dependent implicit arguments that are solved based on the structure of their type only. Maximal or non maximal insertion of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case a function is partially applied, and the next argument to be applied is an implicit argument, two disciplines are applicable. In the first case, the function is considered to have no arguments furtherly: one says that the implicit argument is not maximally inserted. In the second case, the function is considered to be implicitly applied to the implicit arguments it is waiting for: one says that the implicit argument is maximally inserted. Each implicit argument can be declared to be inserted maximally or non maximally. In Coq, maximally-inserted implicit arguments are written between curly braces "{ }" and non-maximally-inserted implicit arguments are written in square brackets "[ ]". .. seealso:: :flag:`Maximal Implicit Insertion` Trailing Implicit Arguments +++++++++++++++++++++++++++ An implicit argument is considered trailing when all following arguments are declared implicit. Trailing implicit arguments cannot be declared non maximally inserted, otherwise they would never be inserted. .. exn:: Argument @name is a trailing implicit, so it can't be declared non maximal. Please use %{ %} instead of [ ]. For instance: .. coqtop:: all fail Fail Definition double [n] := n + n. Casual use of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In a given expression, if it is clear that some argument of a function can be inferred from the type of the other arguments, the user can force the given argument to be guessed by replacing it by “_”. If possible, the correct argument will be automatically generated. .. exn:: Cannot infer a term for this placeholder. :name: Cannot infer a term for this placeholder. (Casual use of implicit arguments) |Coq| was not able to deduce an instantiation of a “_”. .. _declare-implicit-args: Declaration of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case one wants that some arguments of a given object (constant, inductive types, constructors, assumptions, local or not) are always inferred by |Coq|, one may declare once and for all which are the expected implicit arguments of this object. There are two ways to do this, *a priori* and *a posteriori*. Implicit Argument Binders +++++++++++++++++++++++++ .. insertprodn implicit_binders implicit_binders .. prodn:: implicit_binders ::= %{ {+ @name } {? : @type } %} | [ {+ @name } {? : @type } ] In the first setting, one wants to explicitly give the implicit arguments of a declared object as part of its definition. To do this, one has to surround the bindings of implicit arguments by curly braces or square braces: .. coqtop:: all Definition id {A : Type} (x : A) : A := x. This automatically declares the argument A of id as a maximally inserted implicit argument. One can then do as-if the argument was absent in every situation but still be able to specify it if needed: .. coqtop:: all Definition compose {A B C} (g : B -> C) (f : A -> B) := fun x => g (f x). Goal forall A, compose id id = id (A:=A). For non maximally inserted implicit arguments, use square brackets: .. coqtop:: all Fixpoint map [A B : Type] (f : A -> B) (l : list A) : list B := match l with | nil => nil | cons a t => cons (f a) (map f t) end. Print Implicit map. The syntax is supported in all top-level definitions: :cmd:`Definition`, :cmd:`Fixpoint`, :cmd:`Lemma` and so on. For (co-)inductive datatype declarations, the semantics are the following: an inductive parameter declared as an implicit argument need not be repeated in the inductive definition and will become implicit for the inductive type and the constructors. For example: .. coqtop:: all Inductive list {A : Type} : Type := | nil : list | cons : A -> list -> list. Print list. One can always specify the parameter if it is not uniform using the usual implicit arguments disambiguation syntax. The syntax is also supported in internal binders. For instance, in the following kinds of expressions, the type of each declaration present in :token:`binders` can be bracketed to mark the declaration as implicit: :n:`fun (@ident:forall {* @binder }, @type) => @term`, :n:`forall (@ident:forall {* @binder }, @type), @type`, :n:`let @ident {* @binder } := @term in @term`, :n:`fix @ident {* @binder } := @term in @term` and :n:`cofix @ident {* @binder } := @term in @term`. Here is an example: .. coqtop:: all Axiom Ax : forall (f:forall {A} (a:A), A * A), let g {A} (x y:A) := (x,y) in f 0 = g 0 0. .. warn:: Ignoring implicit binder declaration in unexpected position This is triggered when setting an argument implicit in an expression which does not correspond to the type of an assumption or to the body of a definition. Here is an example: .. coqtop:: all warn Definition f := forall {y}, y = 0. .. warn:: Making shadowed name of implicit argument accessible by position This is triggered when two variables of same name are set implicit in the same block of binders, in which case the first occurrence is considered to be unnamed. Here is an example: .. coqtop:: all warn Check let g {x:nat} (H:x=x) {x} (H:x=x) := x in 0. Declaring Implicit Arguments ++++++++++++++++++++++++++++ .. cmd:: Arguments @smart_qualid {* @argument_spec_block } {* , {* @more_implicits_block } } {? : {+, @arguments_modifier } } :name: Arguments .. insertprodn smart_qualid arguments_modifier .. prodn:: smart_qualid ::= @qualid | @by_notation by_notation ::= @string {? % @scope } argument_spec_block ::= @argument_spec | / | & | ( {+ @argument_spec } ) {? % @scope } | [ {+ @argument_spec } ] {? % @scope } | %{ {+ @argument_spec } %} {? % @scope } argument_spec ::= {? ! } @name {? % @scope } more_implicits_block ::= @name | [ {+ @name } ] | %{ {+ @name } %} arguments_modifier ::= simpl nomatch | simpl never | default implicits | clear bidirectionality hint | clear implicits | clear scopes | clear scopes and implicits | clear implicits and scopes | rename | assert | extra scopes This command sets implicit arguments *a posteriori*, where the list of :n:`@name`\s is a prefix of the list of arguments of :n:`@smart_qualid`. Arguments in square brackets are declared as implicit and arguments in curly brackets are declared as maximally inserted. After the command is issued, implicit arguments can and must be omitted in any expression that applies :token:`qualid`. This command supports the :attr:`local` and :attr:`global` attributes. Default behavior is to limit the effect to the current section but also to extend their effect outside the current module or library file. Applying :attr:`local` limits the effect of the command to the current module if it's not in a section. Applying :attr:`global` within a section extends the effect outside the current sections and current module if the command occurs. A command containing :n:`@argument_spec_block & @argument_spec_block` provides :ref:`bidirectionality_hints`. Use the :n:`@more_implicits_block` to specify multiple implicit arguments declarations for names of constants, inductive types, constructors and lemmas that can only be applied to a fixed number of arguments (excluding, for instance, constants whose type is polymorphic). The longest applicable list of implicit arguments will be used to select which implicit arguments are inserted. For printing, the omitted arguments are the ones of the longest list of implicit arguments of the sequence. See the example :ref:`here`. The :n:`@arguments_modifier` values have various effects: * :n:`clear implicits` - clears implicit arguments * :n:`default implicits` - automatically determine the implicit arguments of the object. See :ref:`auto_decl_implicit_args`. * :n:`rename` - rename implicit arguments for the object * :n:`assert` - assert that the object has the expected number of arguments with the expected names. See the example here: :ref:`renaming_implicit_arguments`. .. exn:: The / modifier may only occur once. :undocumented: .. exn:: The & modifier may only occur once. :undocumented: .. example:: .. coqtop:: reset all Inductive list (A : Type) : Type := | nil : list A | cons : A -> list A -> list A. Check (cons nat 3 (nil nat)). Arguments cons [A] _ _. Arguments nil {A}. Check (cons 3 nil). Fixpoint map (A B : Type) (f : A -> B) (l : list A) : list B := match l with nil => nil | cons a t => cons (f a) (map A B f t) end. Fixpoint length (A : Type) (l : list A) : nat := match l with nil => 0 | cons _ m => S (length A m) end. Arguments map [A B] f l. Arguments length {A} l. (* A has to be maximally inserted *) Check (fun l:list (list nat) => map length l). .. _example_more_implicits: .. example:: Multiple implicit arguments with :n:`@more_implicits_block` .. coqtop:: all Arguments map [A B] f l, [A] B f l, A B f l. Check (fun l => map length l = map (list nat) nat length l). .. note:: Use the :cmd:`Print Implicit` command to see the implicit arguments of an object (see :ref:`displaying-implicit-args`). .. _auto_decl_implicit_args: Automatic declaration of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The :n:`default implicits @arguments_modifier` clause tells |Coq| to automatically determine the implicit arguments of the object. Auto-detection is governed by flags specifying whether strict, contextual, or reversible-pattern implicit arguments must be considered or not (see :ref:`controlling-strict-implicit-args`, :ref:`controlling-contextual-implicit-args`, :ref:`controlling-rev-pattern-implicit-args` and also :ref:`controlling-insertion-implicit-args`). .. example:: Default implicits .. coqtop:: reset all Inductive list (A:Set) : Set := | nil : list A | cons : A -> list A -> list A. Arguments cons : default implicits. Print Implicit cons. Arguments nil : default implicits. Print Implicit nil. Set Contextual Implicit. Arguments nil : default implicits. Print Implicit nil. The computation of implicit arguments takes account of the unfolding of constants. For instance, the variable ``p`` below has type ``(Transitivity R)`` which is reducible to ``forall x,y:U, R x y -> forall z:U, R y z -> R x z``. As the variables ``x``, ``y`` and ``z`` appear strictly in the body of the type, they are implicit. .. coqtop:: all Parameter X : Type. Definition Relation := X -> X -> Prop. Definition Transitivity (R:Relation) := forall x y:X, R x y -> forall z:X, R y z -> R x z. Parameters (R : Relation) (p : Transitivity R). Arguments p : default implicits. Print p. Print Implicit p. Parameters (a b c : X) (r1 : R a b) (r2 : R b c). Check (p r1 r2). Mode for automatic declaration of implicit arguments ++++++++++++++++++++++++++++++++++++++++++++++++++++ .. flag:: Implicit Arguments This flag (off by default) allows to systematically declare implicit the arguments detectable as such. Auto-detection of implicit arguments is governed by flags controlling whether strict and contextual implicit arguments have to be considered or not. .. _controlling-strict-implicit-args: Controlling strict implicit arguments +++++++++++++++++++++++++++++++++++++ .. flag:: Strict Implicit When the mode for automatic declaration of implicit arguments is on, the default is to automatically set implicit only the strict implicit arguments plus, for historical reasons, a small subset of the non-strict implicit arguments. To relax this constraint and to set implicit all non strict implicit arguments by default, you can turn this flag off. .. flag:: Strongly Strict Implicit Use this flag (off by default) to capture exactly the strict implicit arguments and no more than the strict implicit arguments. .. _controlling-contextual-implicit-args: Controlling contextual implicit arguments +++++++++++++++++++++++++++++++++++++++++ .. flag:: Contextual Implicit By default, |Coq| does not automatically set implicit the contextual implicit arguments. You can turn this flag on to tell |Coq| to also infer contextual implicit argument. .. _controlling-rev-pattern-implicit-args: Controlling reversible-pattern implicit arguments +++++++++++++++++++++++++++++++++++++++++++++++++ .. flag:: Reversible Pattern Implicit By default, |Coq| does not automatically set implicit the reversible-pattern implicit arguments. You can turn this flag on to tell |Coq| to also infer reversible-pattern implicit argument. .. _controlling-insertion-implicit-args: Controlling the insertion of implicit arguments not followed by explicit arguments ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ .. flag:: Maximal Implicit Insertion Assuming the implicit argument mode is on, this flag (off by default) declares implicit arguments to be automatically inserted when a function is partially applied and the next argument of the function is an implicit one. Combining manual declaration and automatic declaration ++++++++++++++++++++++++++++++++++++++++++++++++++++++ When some arguments are manually specified implicit with binders in a definition and the automatic declaration mode in on, the manual implicit arguments are added to the automatically declared ones. In that case, and when the flag :flag:`Maximal Implicit Insertion` is set to off, some trailing implicit arguments can be inferred to be non maximally inserted. In this case, they are converted to maximally inserted ones. .. example:: .. coqtop:: all Set Implicit Arguments. Axiom eq0_le0 : forall (n : nat) (x : n = 0), n <= 0. Print Implicit eq0_le0. Axiom eq0_le0' : forall (n : nat) {x : n = 0}, n <= 0. Print Implicit eq0_le0'. .. _explicit-applications: Explicit applications ~~~~~~~~~~~~~~~~~~~~~ In presence of non-strict or contextual arguments, or in presence of partial applications, the synthesis of implicit arguments may fail, so one may have to explicitly give certain implicit arguments of an application. Use the :n:`(@ident := @term)` form of :token:`arg` to do so, where :token:`ident` is the name of the implicit argument and :token:`term` is its corresponding explicit term. Alternatively, one can deactivate the hiding of implicit arguments for a single function application using the :n:`@ @qualid {? @univ_annot } {* @term1 }` form of :token:`term10`. .. example:: Syntax for explicitly giving implicit arguments (continued) .. coqtop:: all Check (p r1 (z:=c)). Check (p (x:=a) (y:=b) r1 (z:=c) r2). .. _renaming_implicit_arguments: Renaming implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. example:: (continued) Renaming implicit arguments .. coqtop:: all Arguments p [s t] _ [u] _: rename. Check (p r1 (u:=c)). Check (p (s:=a) (t:=b) r1 (u:=c) r2). Fail Arguments p [s t] _ [w] _ : assert. .. _displaying-implicit-args: Displaying implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. cmd:: Print Implicit @smart_qualid Displays the implicit arguments associated with an object, identifying which arguments are applied maximally or not. Displaying implicit arguments when pretty-printing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. flag:: Printing Implicit By default, the basic pretty-printing rules hide the inferrable implicit arguments of an application. Turn this flag on to force printing all implicit arguments. .. flag:: Printing Implicit Defensive By default, the basic pretty-printing rules display implicit arguments that are not detected as strict implicit arguments. This “defensive” mode can quickly make the display cumbersome so this can be deactivated by turning this flag off. .. seealso:: :flag:`Printing All`. Interaction with subtyping ~~~~~~~~~~~~~~~~~~~~~~~~~~ When an implicit argument can be inferred from the type of more than one of the other arguments, then only the type of the first of these arguments is taken into account, and not an upper type of all of them. As a consequence, the inference of the implicit argument of “=” fails in .. coqtop:: all Fail Check nat = Prop. but succeeds in .. coqtop:: all Check Prop = nat. Deactivation of implicit arguments for parsing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. flag:: Parsing Explicit Turning this flag on (it is off by default) deactivates the use of implicit arguments. In this case, all arguments of constants, inductive types, constructors, etc, including the arguments declared as implicit, have to be given as if no arguments were implicit. By symmetry, this also affects printing. .. _canonical-structure-declaration: Canonical structures ~~~~~~~~~~~~~~~~~~~~ A canonical structure is an instance of a record/structure type that can be used to solve unification problems involving a projection applied to an unknown structure instance (an implicit argument) and a value. The complete documentation of canonical structures can be found in :ref:`canonicalstructures`; here only a simple example is given. .. cmd:: Canonical {? Structure } @smart_qualid Canonical {? Structure } @ident_decl @def_body :name: Canonical Structure; _ The first form of this command declares an existing :n:`@smart_qualid` as a canonical instance of a structure (a record). The second form defines a new constant as if the :cmd:`Definition` command had been used, then declares it as a canonical instance as if the first form had been used on the defined object. This command supports the :attr:`local` attribute. When used, the structure is canonical only within the :cmd:`Section` containing it. Assume that :token:`qualid` denotes an object ``(Build_struct`` |c_1| … |c_n| ``)`` in the structure :g:`struct` of which the fields are |x_1|, …, |x_n|. Then, each time an equation of the form ``(``\ |x_i| ``_)`` |eq_beta_delta_iota_zeta| |c_i| has to be solved during the type checking process, :token:`qualid` is used as a solution. Otherwise said, :token:`qualid` is canonically used to extend the field |c_i| into a complete structure built on |c_i|. Canonical structures are particularly useful when mixed with coercions and strict implicit arguments. .. example:: Here is an example. .. coqtop:: all Require Import Relations. Require Import EqNat. Set Implicit Arguments. Unset Strict Implicit. Structure Setoid : Type := {Carrier :> Set; Equal : relation Carrier; Prf_equiv : equivalence Carrier Equal}. Definition is_law (A B:Setoid) (f:A -> B) := forall x y:A, Equal x y -> Equal (f x) (f y). Axiom eq_nat_equiv : equivalence nat eq_nat. Definition nat_setoid : Setoid := Build_Setoid eq_nat_equiv. Canonical nat_setoid. Thanks to :g:`nat_setoid` declared as canonical, the implicit arguments :g:`A` and :g:`B` can be synthesized in the next statement. .. coqtop:: all abort Lemma is_law_S : is_law S. .. note:: If a same field occurs in several canonical structures, then only the structure declared first as canonical is considered. .. attr:: canonical(false) To prevent a field from being involved in the inference of canonical instances, its declaration can be annotated with the :attr:`canonical(false)` attribute (cf. the syntax of :n:`@record_field`). .. example:: For instance, when declaring the :g:`Setoid` structure above, the :g:`Prf_equiv` field declaration could be written as follows. .. coqdoc:: #[canonical(false)] Prf_equiv : equivalence Carrier Equal See :ref:`canonicalstructures` for a more realistic example. .. attr:: canonical This attribute can decorate a :cmd:`Definition` or :cmd:`Let` command. It is equivalent to having a :cmd:`Canonical Structure` declaration just after the command. .. cmd:: Print Canonical Projections {* @smart_qualid } This displays the list of global names that are components of some canonical structure. For each of them, the canonical structure of which it is a projection is indicated. If constants are given as its arguments, only the unification rules that involve or are synthesized from simultaneously all given constants will be shown. .. example:: For instance, the above example gives the following output: .. coqtop:: all Print Canonical Projections. .. coqtop:: all Print Canonical Projections nat. .. note:: The last line in the first example would not show up if the corresponding projection (namely :g:`Prf_equiv`) were annotated as not canonical, as described above. Implicit types of variables ~~~~~~~~~~~~~~~~~~~~~~~~~~~ It is possible to bind variable names to a given type (e.g. in a development using arithmetic, it may be convenient to bind the names :g:`n` or :g:`m` to the type :g:`nat` of natural numbers). .. cmd:: Implicit {| Type | Types } @reserv_list :name: Implicit Type; Implicit Types .. insertprodn reserv_list simple_reserv .. prodn:: reserv_list ::= {+ ( @simple_reserv ) } | @simple_reserv simple_reserv ::= {+ @ident } : @type Sets the type of bound variables starting with :token:`ident` (either :token:`ident` itself or :token:`ident` followed by one or more single quotes, underscore or digits) to :token:`type` (unless the bound variable is already declared with an explicit type, in which case, that type will be used). .. example:: .. coqtop:: all Require Import List. Implicit Types m n : nat. Lemma cons_inj_nat : forall m n l, n :: l = m :: l -> n = m. Proof. intros m n. Abort. Lemma cons_inj_bool : forall (m n:bool) l, n :: l = m :: l -> n = m. Abort. .. flag:: Printing Use Implicit Types By default, the type of bound variables is not printed when the variable name is associated to an implicit type which matches the actual type of the variable. This feature can be deactivated by turning this flag off. .. _implicit-generalization: Implicit generalization ~~~~~~~~~~~~~~~~~~~~~~~ .. index:: `{ } .. index:: `[ ] .. index:: `( ) .. index:: `{! } .. index:: `[! ] .. index:: `(! ) .. insertprodn generalizing_binder typeclass_constraint .. prodn:: generalizing_binder ::= `( {+, @typeclass_constraint } ) | `%{ {+, @typeclass_constraint } %} | `[ {+, @typeclass_constraint } ] typeclass_constraint ::= {? ! } @term | %{ @name %} : {? ! } @term | @name : {? ! } @term Implicit generalization is an automatic elaboration of a statement with free variables into a closed statement where these variables are quantified explicitly. Use the :cmd:`Generalizable` command to designate which variables should be generalized. It is activated for a binder by prefixing a \`, and for terms by surrounding it with \`{ }, or \`[ ] or \`( ). Terms surrounded by \`{ } introduce their free variables as maximally inserted implicit arguments, terms surrounded by \`[ ] introduce them as non maximally inserted implicit arguments and terms surrounded by \`( ) introduce them as explicit arguments. Generalizing binders always introduce their free variables as maximally inserted implicit arguments. The binder itself introduces its argument as usual. In the following statement, ``A`` and ``y`` are automatically generalized, ``A`` is implicit and ``x``, ``y`` and the anonymous equality argument are explicit. .. coqtop:: all reset Generalizable All Variables. Definition sym `(x:A) : `(x = y -> y = x) := fun _ p => eq_sym p. Print sym. Dually to normal binders, the name is optional but the type is required: .. coqtop:: all Check (forall `{x = y :> A}, y = x). When generalizing a binder whose type is a typeclass, its own class arguments are omitted from the syntax and are generalized using automatic names, without instance search. Other arguments are also generalized unless provided. This produces a fully general statement. this behaviour may be disabled by prefixing the type with a ``!`` or by forcing the typeclass name to be an explicit application using ``@`` (however the later ignores implicit argument information). .. coqtop:: all Class Op (A:Type) := op : A -> A -> A. Class Commutative (A:Type) `(Op A) := commutative : forall x y, op x y = op y x. Instance nat_op : Op nat := plus. Set Printing Implicit. Check (forall `{Commutative }, True). Check (forall `{Commutative nat}, True). Fail Check (forall `{Commutative nat _}, True). Fail Check (forall `{!Commutative nat}, True). Arguments Commutative _ {_}. Check (forall `{!Commutative nat}, True). Check (forall `{@Commutative nat plus}, True). Multiple binders can be merged using ``,`` as a separator: .. coqtop:: all Check (forall `{Commutative A, Hnat : !Commutative nat}, True). .. cmd:: Generalizable {| {| Variable | Variables } {+ @ident } | All Variables | No Variables } Controls the set of generalizable identifiers. By default, no variables are generalizable. This command supports the :attr:`global` attribute. The :n:`{| Variable | Variables } {+ @ident }` form allows generalization of only the given :n:`@ident`\s. Using this command multiple times adds to the allowed identifiers. The other forms clear the list of :n:`@ident`\s. The :n:`All Variables` form generalizes all free variables in the context that appear under a generalization delimiter. This may result in confusing errors in case of typos. In such cases, the context will probably contain some unexpected generalized variables. The :n:`No Variables` form disables implicit generalization entirely. This is the default behavior (before any :cmd:`Generalizable` command has been entered).