Chapter 12 Syntax extensions and interpretation scopes

# Coq Reference Manual

# Chapter 12 Syntax extensions and interpretation scopes

- 12.1 Notations
- 12.2 Interpretation scopes
- 12.3 Abbreviations
- 12.4 Tactic Notations

In this chapter, we introduce advanced commands to modify the way Coq parses and prints objects, i.e. the translations between the concrete and internal representations of terms and commands. The main commands are Notation and Infix which are described in section 12.1. It also happens that the same symbolic notation is expected in different contexts. To achieve this form of overloading, Coq offers a notion of interpretation scope. This is described in Section 12.2.

Remark: The commands Grammar, Syntax and Distfix which were present for a while in Coq are no longer available from Coq version 8.0. The underlying AST structure is also no longer available. The functionalities of the command Syntactic Definition are still available; see Section 12.3.

## 12.1 Notations

### 12.1.1 Basic notations

A notation is a symbolic abbreviation denoting some term or term pattern.

A typical notation is the use of the infix symbol `/\` to denote the logical conjunction (and). Such a notation is declared by

Coq < Notation "A /\ B" := (and A B).

The expression (and A B) is the abbreviated term and the string `"A /\ B"` (called a notation) tells how it is symbolically written.

A notation is always surrounded by double quotes (except when the abbreviation is a single identifier; see 12.3). The notation is composed of tokens separated by spaces. Identifiers in the string (such as A and B) are the parameters of the notation. They must occur at least once each in the denoted term. The other elements of the string (such as `/\`) are the symbols.

An identifier can be used as a symbol but it must be surrounded by simple quotes to avoid the confusion with a parameter. Similarly, every symbol of at least 3 characters and starting with a simple quote must be quoted (then it starts by two single quotes). Here is an example.

Coq < Notation "'IF' c1 'then' c2 'else' c3" := (IF_then_else c1 c2 c3).

A notation binds a syntactic expression to a term. Unless the parser and pretty-printer of Coq already know how to deal with the syntactic expression (see 12.1.7), explicit precedences and associativity rules have to be given.

Remark: The right-hand side of a notation is interpreted at the time the notation is given. In particular, implicit arguments (see Section 2.7), coercions (see Section 2.8), etc. are resolved at the time of the declaration of the notation.

### 12.1.2 Precedences and associativity

Mixing different symbolic notations in the same text may cause serious parsing ambiguity. To deal with the ambiguity of notations, Coq uses precedence levels ranging from 0 to 100 (plus one extra level numbered 200) and associativity rules.

Consider for example the new notation

Coq < Notation "A \/ B" := (or A B).

Clearly, an expression such as forall A:Prop, True `/\` A `\/` A `\/` False is ambiguous. To tell the Coq parser how to interpret the expression, a priority between the symbols `/\` and `\/` has to be given. Assume for instance that we want conjunction to bind more than disjunction. This is expressed by assigning a precedence level to each notation, knowing that a lower level binds more than a higher level. Hence the level for disjunction must be higher than the level for conjunction.

Since connectives are not tight articulation points of a text, it is reasonable to choose levels not so far from the highest level which is 100, for example 85 for disjunction and 80 for conjunction1.

Similarly, an associativity is needed to decide whether True `/\` False `/\` False defaults to True `/\` (False `/\` False) (right associativity) or to (True `/\` False) `/\` False (left associativity). We may even consider that the expression is not well-formed and that parentheses are mandatory (this is a “no associativity”)2. We don’t know of a special convention of the associativity of disjunction and conjunction, so let’s apply for instance a right associativity (which is the choice of Coq).

Precedence levels and associativity rules of notations have to be given between parentheses in a list of modifiers that the Notation command understands. Here is how the previous examples refine.

Coq < Notation "A /\ B" := (and A B) (at level 80, right associativity).
Coq < Notation "A \/ B" := (or A B) (at level 85, right associativity).

By default, a notation is considered non associative, but the precedence level is mandatory (except for special cases whose level is canonical). The level is either a number or the phrase next level whose meaning is obvious. The list of levels already assigned is on Figure 3.1.

### 12.1.3 Complex notations

Notations can be made from arbitrarily complex symbols. One can for instance define prefix notations.

Coq < Notation "~ x" := (not x) (at level 75, right associativity).

One can also define notations for incomplete terms, with the hole expected to be inferred at typing time.

Coq < Notation "x = y" := (@eq _ x y) (at level 70, no associativity).

One can define closed notations whose both sides are symbols. In this case, the default precedence level for inner subexpression is 200.

Coq < Notation "( x , y )" := (@pair _ _ x y) (at level 0).

One can also define notations for binders.

Coq < Notation "{ x : A | P }" := (sig A (fun x => P)) (at level 0).

In the last case though, there is a conflict with the notation for type casts. This last notation, as shown by the command Print Grammar constr is at level 100. To avoid `x : A` being parsed as a type cast, it is necessary to put x at a level below 100, typically 99. Hence, a correct definition is

Coq < Notation "{ x : A | P }" := (sig A (fun x => P)) (at level 0, x at level 99).

See the next section for more about factorization.

### 12.1.4 Simple factorization rules

Coq extensible parsing is performed by Camlp5 which is essentially a LL1 parser. Hence, some care has to be taken not to hide already existing rules by new rules. Some simple left factorization work has to be done. Here is an example.

Coq < Notation "x < y" := (lt x y) (at level 70).
Coq < Notation "x < y < z" := (x < y /\ y < z) (at level 70).

In order to factorize the left part of the rules, the subexpression referred by y has to be at the same level in both rules. However the default behavior puts y at the next level below 70 in the first rule (no associativity is the default), and at the level 200 in the second rule (level 200 is the default for inner expressions). To fix this, we need to force the parsing level of y, as follows.

Coq < Notation "x < y" := (lt x y) (at level 70).
Coq < Notation "x < y < z" := (x < y /\ y < z) (at level 70, y at next level).

For the sake of factorization with Coq predefined rules, simple rules have to be observed for notations starting with a symbol: e.g. rules starting with “{” or “(” should be put at level 0. The list of Coq predefined notations can be found in Chapter 3.

The command to display the current state of the Coq term parser is

> Print Grammar constr.

Variant:

Print Grammar pattern.
This displays the state of the subparser of patterns (the parser used in the grammar of the match with constructions).

### 12.1.5 Displaying symbolic notations

The command Notation has an effect both on the Coq parser and on the Coq printer. For example:

Coq < Check (and True True).
True /\ True
: Prop

However, printing, especially pretty-printing, requires more care than parsing. We may want specific indentations, line breaks, alignment if on several lines, etc.

The default printing of notations is very rudimentary. For printing a notation, a formatting box is opened in such a way that if the notation and its arguments cannot fit on a single line, a line break is inserted before the symbols of the notation and the arguments on the next lines are aligned with the argument on the first line.

A first, simple control that a user can have on the printing of a notation is the insertion of spaces at some places of the notation. This is performed by adding extra spaces between the symbols and parameters: each extra space (other than the single space needed to separate the components) is interpreted as a space to be inserted by the printer. Here is an example showing how to add spaces around the bar of the notation.

Coq < Notation "{{ x : A | P }}" := (sig (fun x : A => P))
(at level 0, x at level 99).
Coq < Check (sig (fun x : nat => x=x)).
{{x : nat | x = x}}
: Set

The second, more powerful control on printing is by using the format modifier. Here is an example

Coq < Notation "'If' c1 'then' c2 'else' c3" := (IF_then_else c1 c2 c3)
(at level 200, right associativity, format
"'[v ' 'If' c1 '/' '[' 'then' c2 ']' '/' '[' 'else' c3 ']' ']'").
Identifier 'If' now a keyword

A format is an extension of the string denoting the notation with the possible following elements delimited by single quotes:

- extra spaces are translated into simple spaces
- tokens of the form `'/ '` are translated into breaking point, in case a line break occurs, an indentation of the number of spaces after the “`/`” is applied (2 spaces in the given example)
- token of the form `'//'` force writing on a new line
- well-bracketed pairs of tokens of the form `'[ '` and `']'` are translated into printing boxes; in case a line break occurs, an extra indentation of the number of spaces given after the “`[`” is applied (4 spaces in the example)
- well-bracketed pairs of tokens of the form `'[hv '` and `']'` are translated into horizontal-orelse-vertical printing boxes; if the content of the box does not fit on a single line, then every breaking point forces a newline and an extra indentation of the number of spaces given after the “`[`” is applied at the beginning of each newline (3 spaces in the example)
- well-bracketed pairs of tokens of the form `'[v '` and `']'` are translated into vertical printing boxes; every breaking point forces a newline, even if the line is large enough to display the whole content of the box, and an extra indentation of the number of spaces given after the “`[`” is applied at the beginning of each newline

Notations do not survive the end of sections. No typing of the denoted expression is performed at definition time. Type-checking is done only at the time of use of the notation.

Coq < Check
(IF_then_else (IF_then_else True False True)
(IF_then_else True False True)
(IF_then_else True False True)).
If If True
then False
else True
then If True
then False
else True
else If True
then False
else True
: Prop

Remark: Sometimes, a notation is expected only for the parser. To do so, the option only parsing is allowed in the list of modifiers of Notation.

Conversely, the only printing can be used to declare that a notation should only be used for printing and should not declare a parsing rule. In particular, such notations do not modify the parser.

### 12.1.6 The Infix command

The Infix command is a shortening for declaring notations of infix symbols. Its syntax is

> Infix "symbol" := qualid ( modifier , … , modifier ).

and it is equivalent to

> Notation "x symbol y" := (qualid x y) ( modifier , … , modifier ).

where x and y are fresh names distinct from qualid. Here is an example.

Coq < Infix "/\" := and (at level 80, right associativity).

### 12.1.7 Reserving notations

A given notation may be used in different contexts. Coq expects all uses of the notation to be defined at the same precedence and with the same associativity. To avoid giving the precedence and associativity every time, it is possible to declare a parsing rule in advance without giving its interpretation. Here is an example from the initial state of Coq.

Coq < Reserved Notation "x = y" (at level 70, no associativity).

Reserving a notation is also useful for simultaneously defining an inductive type or a recursive constant and a notation for it.

Remark: The notations mentioned on Figure 3.1 are reserved. Hence their precedence and associativity cannot be changed.

### 12.1.8 Simultaneous definition of terms and notations

Thanks to reserved notations, the inductive, co-inductive, recursive and corecursive definitions can benefit of customized notations. To do this, insert a where notation clause after the definition of the (co)inductive type or (co)recursive term (or after the definition of each of them in case of mutual definitions). The exact syntax is given on Figure 12.1. Here are examples:

Coq < Inductive and (A B:Prop) : Prop := conj : A -> B -> A /\ B
where "A /\ B" := (and A B).
Coq < Fixpoint plus (n m:nat) {struct n} : nat :=
match n with
| O => m
| S p => S (p+m)
end
where "n + m" := (plus n m).

### 12.1.9 Displaying informations about notations

To deactivate the printing of all notations, use the command

> Unset Printing Notations.

To reactivate it, use the command

> Set Printing Notations.

The default is to use notations for printing terms wherever possible.

See also: Set Printing All in Section 2.9.

### 12.1.10 Locating notations

To know to which notations a given symbol belongs to, use the command

> Locate symbol

where symbol is any (composite) symbol surrounded by double quotes. To locate a particular notation, use a string where the variables of the notation are replaced by “_” and where possible single quotes inserted around identifiers or tokens starting with a single quote are dropped.

Example:

Coq < Locate "exists".
Notation
"'exists' x .. y , p" := ex (fun y => .. (ex (fun y => p)) ..)
: type_scope (default interpretation)
"'exists' ! x .. y , p" := ex
(unique
(fun y =>
.. (ex (unique (fun y => p))) ..))
: type_scope (default interpretation)
Coq < Locate "exists _ .. _ , _".
Notation
"'exists' x .. y , p" := ex (fun y => .. (ex (fun y => p)) ..)
: type_scope (default interpretation)

See also: Section 6.3.10.

> sentence | ::= | [Local] Notation string := term [modifiers] [:scope] .
> | | | [Local] Infix string := qualid [modifiers] [:scope] .
> | | | [Local] Reserved Notation string [modifiers] .
> | | | Inductive ind_body [decl_notation] with … with ind_body [decl_notation].
> | | | CoInductive ind_body [decl_notation] with … with ind_body [decl_notation].
> | | | Fixpoint fix_body [decl_notation] with … with fix_body [decl_notation] .
> | | | CoFixpoint cofix_body [decl_notation] with … with cofix_body [decl_notation] .
> decl_notation | ::= | [where string := term [:scope] and … and string := term [:scope]].
> modifiers | ::= | ident , … , ident at level natural
> | | | ident , … , ident at next level
> | | | at level natural
> | | | left associativity
> | | | right associativity
> | | | no associativity
> | | | ident ident
> | | | ident binder
> | | | ident closed binder
> | | | ident global
> | | | ident bigint
> | | | only parsing
> | | | only printing
> | | | format string

> Figure 12.1: Syntax of the variants of Notation

### 12.1.11 Notations and simple binders

Notations can be defined for binders as in the example:

Coq < Notation "{ x : A | P }" := (sig (fun x : A => P)) (at level 0).

The binding variables in the left-hand-side that occur as a parameter of the notation naturally bind all their occurrences appearing in their respective scope after instantiation of the parameters of the notation.

Contrastingly, the binding variables that are not a parameter of the notation do not capture the variables of same name that could appear in their scope after instantiation of the notation. E.g., for the notation

Coq < Notation "'exists_different' n" := (exists p:nat, p<>n) (at level 200).

the next command fails because p does not bind in the instance of n.

Coq < Fail Check (exists_different p).
The command has indeed failed with message:
The reference p was not found in the current environment.

Remark: Binding variables must not necessarily be parsed using the ident entry. For factorization purposes, they can be said to be parsed at another level (e.g. x in `"{ x : A | P }"` must be parsed at level 99 to be factorized with the notation `"{ A } + { B }"` for which A can be any term). However, even if parsed as a term, this term must at the end be effectively a single identifier.

### 12.1.12 Notations with recursive patterns

A mechanism is provided for declaring elementary notations with recursive patterns. The basic example is:

Coq < Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..).

On the right-hand side, an extra construction of the form .. t .. can be used. Notice that .. is part of the Coq syntax and it must not be confused with the three-dots notation … used in this manual to denote a sequence of arbitrary size.

On the left-hand side, the part “x s .. s y” of the notation parses any number of time (but at least one time) a sequence of expressions separated by the sequence of tokens s (in the example, s is just “;”).

In the right-hand side, the term enclosed within .. must be a pattern with two holes of the form φ([ ]E,[ ]I) where the first hole is occupied either by x or by y and the second hole is occupied by an arbitrary term t called the terminating expression of the recursive notation. The subterm .. φ(x,t) .. (or .. φ(y,t) ..) must itself occur at second position of the same pattern where the first hole is occupied by the other variable, y or x. Otherwise said, the right-hand side must contain a subterm of the form either φ(x,.. φ(y,t) ..) or φ(y,.. φ(x,t) ..). The pattern φ is the iterator of the recursive notation and, of course, the name x and y can be chosen arbitrarily.

The parsing phase produces a list of expressions which are used to fill in order the first hole of the iterating pattern which is repeatedly nested as many times as the length of the list, the second hole being the nesting point. In the innermost occurrence of the nested iterating pattern, the second hole is finally filled with the terminating expression.

In the example above, the iterator φ([ ]E,[ ]I) is cons [ ]E [ ]I and the terminating expression is nil. Here are other examples:

Coq < Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) (at level 0).
Coq < Notation "[| t * ( x , y , .. , z ) ; ( a , b , .. , c ) * u |]" :=
(pair (pair .. (pair (pair t x) (pair t y)) .. (pair t z))
(pair .. (pair (pair a u) (pair b u)) .. (pair c u)))
(t at level 39).

Recursive patterns can occur several times on the right-hand side. Here is an example:

Coq < Notation "[> a , .. , b <]" :=
(cons a .. (cons b nil) .., cons b .. (cons a nil) ..).

Notations with recursive patterns can be reserved like standard notations, they can also be declared within interpretation scopes (see section 12.2).

### 12.1.13 Notations with recursive patterns involving binders

Recursive notations can also be used with binders. The basic example is:

Coq < Notation "'exists' x .. y , p" := (ex (fun x => .. (ex (fun y => p)) ..))
(at level 200, x binder, y binder, right associativity).

The principle is the same as in Section 12.1.12 except that in the iterator φ([ ]E,[ ]I), the first hole is a placeholder occurring at the position of the binding variable of a fun or a forall.

To specify that the part “x .. y” of the notation parses a sequence of binders, x and y must be marked as binder in the list of modifiers of the notation. Then, the list of binders produced at the parsing phase are used to fill in the first hole of the iterating pattern which is repeatedly nested as many times as the number of binders generated. If ever the generalization operator ‘ (see Section 2.7.19) is used in the binding list, the added binders are taken into account too.

Binders parsing exist in two flavors. If x and y are marked as binder, then a sequence such as a b c : T will be accepted and interpreted as the sequence of binders (a:T) (b:T) (c:T). For instance, in the notation above, the syntax exists a b : nat, a = b is provided.

The variables x and y can also be marked as closed binder in which case only well-bracketed binders of the form (a b c:T) or {a b c:T} etc. are accepted.

With closed binders, the recursive sequence in the left-hand side can be of the general form x s .. s y where s is an arbitrary sequence of tokens. With open binders though, s has to be empty. Here is an example of recursive notation with closed binders:

Coq < Notation "'mylet' f x .. y := t 'in' u":=
(let f := fun x => .. (fun y => t) .. in u)
(at level 200, x closed binder, y closed binder, right associativity).

A recursive pattern for binders can be used in position of a recursive pattern for terms. Here is an example:

Coq < Notation "'FUNAPP' x .. y , f" :=
(fun x => .. (fun y => (.. (f x) ..) y ) ..)
(at level 200, x binder, y binder, right associativity).

### 12.1.14 Summary

##### Syntax of notations

The different syntactic variants of the command Notation are given on Figure 12.1. The optional :scope is described in the Section 12.2.

Remark: No typing of the denoted expression is performed at definition time. Type-checking is done only at the time of use of the notation.

Remark: Many examples of Notation may be found in the files composing the initial state of Coq (see directory $COQLIB/theories/Init).

Remark: The notation `"{ x }"` has a special status in such a way that complex notations of the form `"x + { y }"` or `"x * { y }"` can be nested with correct precedences. Especially, every notation involving a pattern of the form `"{ x }"` is parsed as a notation where the pattern `"{ x }"` has been simply replaced by `"x"` and the curly brackets are parsed separately. E.g. `"y + { z }"` is not parsed as a term of the given form but as a term of the form `"y + z"` where `z` has been parsed using the rule parsing `"{ x }"`. Especially, level and precedences for a rule including patterns of the form `"{ x }"` are relative not to the textual notation but to the notation where the curly brackets have been removed (e.g. the level and the associativity given to some notation, say `"{ y } & { z }"` in fact applies to the underlying `"{ x }"`-free rule which is `"y & z"`).

##### Persistence of notations

Notations do not survive the end of sections. They survive modules unless the command Local Notation is used instead of Notation.

## 12.2 Interpretation scopes

An interpretation scope is a set of notations for terms with their interpretation. Interpretation scopes provide a weak, purely syntactical form of notation overloading: the same notation, for instance the infix symbol `+`, can be used to denote distinct definitions of the additive operator. Depending on which interpretation scope is currently open, the interpretation is different. Interpretation scopes can include an interpretation for numerals and strings. However, this is only made possible at the Objective Caml level.

See Figure 12.1 for the syntax of notations including the possibility to declare them in a given scope. Here is a typical example which declares the notation for conjunction in the scope type_scope.

    Notation "A /\ B" := (and A B) : type_scope.

Remark: A notation not defined in a scope is called a lonely notation.

### 12.2.1 Global interpretation rules for notations

At any time, the interpretation of a notation for term is done within a stack of interpretation scopes and lonely notations. In case a notation has several interpretations, the actual interpretation is the one defined by (or in) the more recently declared (or open) lonely notation (or interpretation scope) which defines this notation. Typically if a given notation is defined in some scope scope but has also an interpretation not assigned to a scope, then, if scope is open before the lonely interpretation is declared, then the lonely interpretation is used (and this is the case even if the interpretation of the notation in scope is given after the lonely interpretation: otherwise said, only the order of lonely interpretations and opening of scopes matters, and not the declaration of interpretations within a scope).

The initial state of Coq declares three interpretation scopes and no lonely notations. These scopes, in opening order, are core_scope, type_scope and nat_scope.

The command to add a scope to the interpretation scope stack is

> Open Scope scope.

It is also possible to remove a scope from the interpretation scope stack by using the command

> Close Scope scope.

Notice that this command does not only cancel the last Open Scope scope but all the invocation of it.

Remark: Open Scope and Close Scope do not survive the end of sections where they occur. When defined outside of a section, they are exported to the modules that import the module where they occur.

Variants:

1. Local Open Scope scope.
2. Local Close Scope scope.

   These variants are not exported to the modules that import the module where they occur, even if outside a section.

3. Global Open Scope scope.
4. Global Close Scope scope.

   These variants survive sections. They behave as if Global were absent when not inside a section.

### 12.2.2 Local interpretation rules for notations

In addition to the global rules of interpretation of notations, some ways to change the interpretation of subterms are available.

#### Local opening of an interpretation scope

It is possible to locally extend the interpretation scope stack using the syntax (term)%key (or simply term%key for atomic terms), where key is a special identifier called delimiting key and bound to a given scope.

In such a situation, the term term, and all its subterms, are interpreted in the scope stack extended with the scope bound to key.

To bind a delimiting key to a scope, use the command

> Delimit Scope scope with ident

To remove a delimiting key of a scope, use the command

> Undelimit Scope scope

#### Binding arguments of a constant to an interpretation scope

It is possible to set in advance that some arguments of a given constant have to be interpreted in a given scope. The command is

> Arguments qualid name%scope … name%scope

where the list is a prefix of the list of the arguments of qualid eventually annotated with their scope. Grouping round parentheses can be used to decorate multiple arguments with the same scope. scope can be either a scope name or its delimiting key. For example the following command puts the first two arguments of plus_fct in the scope delimited by the key F (Rfun_scope) and the last argument in the scope delimited by the key R (R_scope).

Coq < Arguments plus_fct (f1 f2)%F x%R.

The Arguments command accepts scopes decoration to all grouping parentheses. In the following example arguments A and B are marked as maximally inserted implicit arguments and are put into the type_scope scope.

Coq < Arguments respectful {A B}%type (R R')%signature _ _.

When interpreting a term, if some of the arguments of qualid are built from a notation, then this notation is interpreted in the scope stack extended by the scope bound (if any) to this argument. The effect of the scope is limited to the argument itself. It does not propagate to subterms but the subterms that, after interpretation of the notation, turn to be themselves arguments of a reference are interpreted accordingly to the arguments scopes bound to this reference.

Arguments scopes can be cleared with the following command:

> Arguments qualid : clear scopes

Variants:

1. Global Arguments qualid name%scope … name%scope

   This behaves like Arguments qualid name%scope … name%scope but survives when a section is closed instead of stopping working at section closing. Without the Global modifier, the effect of the command stops when the section it belongs to ends.

2. Local Arguments qualid name%scope … name%scope

   This behaves like Arguments qualid name%scope … name%scope but does not survive modules and files. Without the Local modifier, the effect of the command is visible from within other modules or files.

See also: The command to show the scopes bound to the arguments of a function is described in Section 2.

#### Binding types of arguments to an interpretation scope

When an interpretation scope is naturally associated to a type (e.g. the scope of operations on the natural numbers), it may be convenient to bind it to this type. When a scope scope is bound to a type type, any new function defined later on gets its arguments of type type interpreted by default in scope scope (this default behavior can however be overwritten by explicitly using the command Arguments).

Whether the argument of a function has some type type is determined statically. For instance, if f is a polymorphic function of type forall X:Type, X -> X and type t is bound to a scope scope, then a of type t in f t a is not recognized as an argument to be interpreted in scope scope.

More generally, any coercion class (see Chapter 18) can be bound to an interpretation scope. The command to do it is

> Bind Scope scope with class

Example:

Coq < Parameter U : Set.
U is declared
Coq < Bind Scope U_scope with U.
Coq < Parameter Uplus : U -> U -> U.
Uplus is declared
Coq < Parameter P : forall T:Set, T -> U -> Prop.
P is declared
Coq < Parameter f : forall T:Set, T -> U.
f is declared
Coq < Infix "+" := Uplus : U_scope.
Coq < Unset Printing Notations.
Coq < Open Scope nat_scope. (* Define + on the nat as the default for + *)
Coq < Check (fun x y1 y2 z t => P _ (x + t) ((f _ (y1 + y2) + z))).
fun (x y1 y2 : nat) (z : U) (t : nat) =>
P nat (Nat.add x t) (Uplus (f nat (Nat.add y1 y2)) z)
: forall (_ : nat) (_ : nat) (_ : nat) (_ : U) (_ : nat), Prop

Remark: The scopes type_scope and function_scope also have a local effect on interpretation. See the next section.

See also: The command to show the scopes bound to the arguments of a function is described in Section 2.

Remark: In notations, the subterms matching the identifiers of the notations are interpreted in the scope in which the identifiers occurred at the time of the declaration of the notation. Here is an example:

Coq < Parameter g : bool -> bool.
g is declared
Coq < Notation "@@" := true (only parsing) : bool_scope.
Setting notation at level 0.
Coq < Notation "@@" := false (only parsing): mybool_scope.
Coq < (* Defining a notation while the argument of g is bound to bool_scope *)
Bind Scope bool_scope with bool.
Coq < Notation "# x #" := (g x) (at level 40).
Coq < Check # @@ #.
g true
: bool
Coq < (* Rebinding the argument of g to mybool_scope has no effect on the notation *)
Arguments g _%mybool_scope.
Coq < Check # @@ #.
g true
: bool
Coq < (* But we can force the scope *)
Delimit Scope mybool_scope with mybool.
Coq < Check # @@%mybool #.
g false
: bool

### 12.2.3 The type_scope interpretation scope

The scope type_scope has a special status. It is a primitive interpretation scope which is temporarily activated each time a subterm of an expression is expected to be a type. It is delimited by the key type, and bound to the coercion class Sortclass. It is also used in certain situations where an expression is statically known to be a type, including the conclusion and the type of hypotheses within an Ltac goal match (see Section 9.2) the statement of a theorem, the type of a definition, the type of a binder, the domain and codomain of implication, the codomain of products, and more generally any type argument of a declared or defined constant.

### 12.2.4 The function_scope interpretation scope

The scope function_scope also has a special status. It is temporarily activated each time the argument of a global reference is recognized to be a Funclass instance, i.e., of type forall x:A, B or A -> B.

### 12.2.5 Interpretation scopes used in the standard library of Coq

We give an overview of the scopes used in the standard library of Coq. For a complete list of notations in each scope, use the commands Print Scopes or Print Scope scope.

#### type_scope

This scope includes infix * for product types and infix + for sum types. It is delimited by key type, and bound to the coercion class Sortclass, as described at 12.2.2.

#### nat_scope

This scope includes the standard arithmetical operators and relations on type nat. Positive numerals in this scope are mapped to their canonical representent built from O and S. The scope is delimited by key nat, and bound to the type nat (see 12.2.2).

#### N_scope

This scope includes the standard arithmetical operators and relations on type N (binary natural numbers). It is delimited by key N and comes with an interpretation for numerals as closed term of type N.

#### Z_scope

This scope includes the standard arithmetical operators and relations on type Z (binary integer numbers). It is delimited by key Z and comes with an interpretation for numerals as closed term of type Z.

#### positive_scope

This scope includes the standard arithmetical operators and relations on type positive (binary strictly positive numbers). It is delimited by key positive and comes with an interpretation for numerals as closed term of type positive.

#### Q_scope

This scope includes the standard arithmetical operators and relations on type Q (rational numbers defined as fractions of an integer and a strictly positive integer modulo the equality of the numerator-denominator cross-product). As for numerals, only 0 and 1 have an interpretation in scope Q_scope (their interpretations are 0/1 and 1/1 respectively).

#### Qc_scope

This scope includes the standard arithmetical operators and relations on the type Qc of rational numbers defined as the type of irreducible fractions of an integer and a strictly positive integer.

#### real_scope

This scope includes the standard arithmetical operators and relations on type R (axiomatic real numbers). It is delimited by key R and comes with an interpretation for numerals using the IZR morphism from binary integer numbers to R.

#### bool_scope

This scope includes notations for the boolean operators. It is delimited by key bool, and bound to the type bool (see 12.2.2).

#### list_scope

This scope includes notations for the list operators. It is delimited by key list, and bound to the type list (see 12.2.2).

#### function_scope

This scope is delimited by the key function, and bound to the coercion class Funclass, as described at 12.2.2.

#### core_scope

This scope includes the notation for pairs. It is delimited by key core.

#### string_scope

This scope includes notation for strings as elements of the type string. Special characters and escaping follow Coq conventions on strings (see Section 1.1). Especially, there is no convention to visualize non printable characters of a string. The file String.v shows an example that contains quotes, a newline and a beep (i.e. the ASCII character of code 7).

#### char_scope

This scope includes interpretation for all strings of the form `"`c`"` where c is an ASCII character, or of the form `"`nnn`"` where nnn is a three-digits number (possibly with leading 0’s), or of the form `""""`. Their respective denotations are the ASCII code of c, the decimal ASCII code nnn, or the ASCII code of the character `"` (i.e. the ASCII code 34), all of them being represented in the type ascii.

### 12.2.6 Displaying informations about scopes

#### Print Visibility

This displays the current stack of notations in scopes and lonely notations that is used to interpret a notation. The top of the stack is displayed last. Notations in scopes whose interpretation is hidden by the same notation in a more recently open scope are not displayed. Hence each notation is displayed only once.

Variant:

Print Visibility scope
This displays the current stack of notations in scopes and lonely notations assuming that scope is pushed on top of the stack. This is useful to know how a subterm locally occurring in the scope of scope is interpreted.

#### Print Scope scope

This displays all the notations defined in interpretation scope scope. It also displays the delimiting key if any and the class to which the scope is bound, if any.

#### Print Scopes

This displays all the notations, delimiting keys and corresponding class of all the existing interpretation scopes. It also displays the lonely notations.

## 12.3 Abbreviations

An abbreviation is a name, possibly applied to arguments, that denotes a (presumably) more complex expression. Here are examples:

Coq < Notation Nlist := (list nat).
Coq < Check 1 :: 2 :: 3 :: nil.
[1; 2; 3]
: Nlist
Coq < Notation reflexive R := (forall x, R x x).
Coq < Check forall A:Prop, A <-> A.
reflexive iff
: Prop
Coq < Check reflexive iff.
reflexive iff
: Prop

An abbreviation expects no precedence nor associativity, since it follows the usual syntax of application. Abbreviations are used as much as possible by the Coq printers unless the modifier `(only parsing)` is given.

Abbreviations are bound to an absolute name as an ordinary definition is, and they can be referred by qualified names too.

Abbreviations are syntactic in the sense that they are bound to expressions which are not typed at the time of the definition of the abbreviation but at the time it is used. Especially, abbreviations can be bound to terms with holes (i.e. with “_”). The general syntax for abbreviations is

> [Local] Notation ident [ident ident … ident ident] := term [(only parsing)] `.`

Example:

Coq < Definition explicit_id (A:Set) (a:A) := a.
explicit_id is defined
Coq < Notation id := (explicit_id _).
Coq < Check (id 0).
id 0
: nat

Abbreviations do not survive the end of sections. No typing of the denoted expression is performed at definition time. Type-checking is done only at the time of use of the abbreviation.

## 12.4 Tactic Notations

Tactic notations allow to customize the syntax of the tactics of the tactic language3. Tactic notations obey the following syntax

sentence | ::= | [Local] Tactic Notation [tactic_level] [prod_item … prod_item]
|  | := tactic .
prod_item | ::= | string | tactic_argument_type(ident)
tactic_level | ::= | (at level natural)
tactic_argument_type | ::= | ident | simple_intropattern | reference
| | | hyp | hyp_list | ne_hyp_list
| | | constr | uconstr | constr_list | ne_constr_list
| | | integer | integer_list | ne_integer_list
| | | int_or_var | int_or_var_list | ne_int_or_var_list
| | | tactic | tacticn (for 0≤ n≤ 5)

A tactic notation Tactic Notation tactic_level [prod_item … prod_item] := tactic extends the parser and pretty-printer of tactics with a new rule made of the list of production items. It then evaluates into the tactic expression tactic. For simple tactics, it is recommended to use a terminal symbol, i.e. a string, for the first production item. The tactic level indicates the parsing precedence of the tactic notation. This information is particularly relevant for notations of tacticals. Levels 0 to 5 are available (default is 0). To know the parsing precedences of the existing tacticals, use the command Print Grammar tactic.

Each type of tactic argument has a specific semantic regarding how it is parsed and how it is interpreted. The semantic is described in the following table. The last command gives examples of tactics which use the corresponding kind of argument.

Tactic argument type | parsed as | interpreted as | as in tactic
ident | identifier | a user-given name | intro
simple_intropattern | intro_pattern | an intro_pattern | intros
hyp | identifier | an hypothesis defined in context | clear
reference | qualified identifier | a global reference of term | unfold
constr | term | a term | exact
uconstr | term | an untyped term | refine
integer | integer | an integer |
int_or_var | identifier or integer | an integer | do
tactic | tactic at level 5 | a tactic |
tacticn | tactic at level n | a tactic |
entry_list | list of entry | a list of how entry is interpreted |
ne_entry_list | non-empty list of entry | a list of how entry is interpreted |

Remark: In order to be bound in tactic definitions, each syntactic entry for argument type must include the case of simple Ltac identifier as part of what it parses. This is naturally the case for ident, simple_intropattern, reference, constr, ... but not for integer. This is the reason for introducing a special entry int_or_var which evaluates to integers only but which syntactically includes identifiers in order to be usable in tactic definitions.

Remark: The entry_list and ne_entry_list entries can be used in primitive tactics or in other notations at places where a list of the underlying entry can be used: entry is either constr, hyp, integer or int_or_var.

Tactic notations do not survive the end of sections. They survive modules unless the command Local Tactic Notation is used instead of Tactic Notation.

1
  which are the levels effectively chosen in the current implementation of Coq
2
  Coq accepts notations declared as no associative but the parser on which Coq is built, namely Camlp4, currently does not implement the no-associativity and replaces it by a left associativity; hence it is the same for Coq: no-associativity is in fact left associativity
3
  Tactic notations are just a simplification of the Grammar tactic simple_tactic command that existed in versions prior to version 8.0.