Syntax extensions and notation scopes — Coq 8.13.1 documentation

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# Syntax extensions and notation scopes

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 to provide custom symbolic notations for terms are `Notation` and `Infix`; they will be described in the next section. There is also a variant of `Notation` which does not modify the parser; this provides a form of abbreviation. It is sometimes expected that the same symbolic notation has different meanings in different contexts; to achieve this form of overloading, Coq offers a notion of notation scopes. The main command to provide custom notations for tactics is `Tactic Notation`.

Set Printing Depth 50.

## Notations

### Basic notations

Command `Notation string := one_term ( syntax_modifier+, )? : scope_name?`

  Defines a notation, an alternate syntax for entering or displaying a specific term or term pattern.

  This command supports the `local` attribute, which limits its effect to the current module. If the command is inside a section, its effect is limited to the section.

  Specifying `scope_name` associates the notation with that scope. Otherwise it is a lonely notation, that is, not associated with a scope.

For example, the following definition permits using the infix expression `A /\ B` to represent `(and A B)`:

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

`"A /\ B"` is a notation, which tells how to represent the abbreviated term `(and A B)`.

Notations must be in double quotes, except when the abbreviation has the form of an ordinary applicative expression; see Abbreviations. The notation consists of tokens separated by spaces. Tokens which are identifiers (such as `A`, `x0'`, etc.) are the parameters of the notation. Each of them must occur at least once in the abbreviated term. The other elements of the string (such as `/\`) are the symbols.

Identifiers enclosed in single quotes are treated as symbols and thus lose their role of parameters. In the same vein, every symbol of at least 3 characters and starting with a simple quote must be quoted (then it starts with two single quotes). Here is an example.

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 (such as through `Reserved Notation` or for notations that contain only literals), explicit precedences and associativity rules have to be given.

Note

The right-hand side of a notation is interpreted at the time the notation is given. In particular, disambiguation of constants, implicit arguments and other notations are resolved at the time of the declaration of the notation. The right-hand side is currently typed only at use time but this may change in the future.

### 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

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 conjunction 1.

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 do not know of a special convention for the associativity of disjunction and conjunction, so let us apply right associativity (which is the choice of Coq).

Precedence levels and associativity rules of notations are specified with a list of parenthesized `syntax_modifier`s. Here is how the previous examples refine:

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

By default, a notation is considered nonassociative, 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. Some associativities are predefined in the `Notations` module.

### Complex notations

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

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

One can also define notations for incomplete terms, with the hole expected to be inferred during type checking.

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 the inner sub-expression is 200, and the default level for the notation itself is 0.

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

One can also define notations for binders.

Notation "{ x : A | P }" := (sig A (fun x => P)).

In the last case though, there is a conflict with the notation for type casts. The notation for type casts, 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 the following:

Reset Initial.

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

More generally, it is required that notations are explicitly factorized on the left. See the next section for more about factorization.

### Simple factorization rules

Coq extensible parsing is performed by Camlp5 which is essentially a LL1 parser: it decides which notation to parse by looking at tokens from left to right. 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.

Notation "x < y" := (lt x y) (at level 70).
Fail Notation "x < y < z" := (x < y /\ y < z) (at level 70).
  Toplevel input, characters 0-60: > Fail Notation "x < y < z" := (x < y /\ y < z) (at level 70). > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: Notation "_ < _ < _" was already defined with a different format. [notation-incompatible-format,parsing] The command has indeed failed with message: Notation "_ < _ < _" is already defined at level 70 with arguments constr at next level, constr at next level, constr at next level while it is now required to be at level 70 with arguments constr at next level, constr, constr at next level.

In order to factorize the left part of the rules, the subexpression referred to 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 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.

Notation "x < y" := (lt x y) (at level 70).
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 the chapter on The Coq library.

### Use of notations for printing

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

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

However, printing, especially pretty-printing, also requires some care. We may want specific indentations, line breaks, alignment if on several lines, etc. For pretty-printing, Coq relies on OCaml formatting library, which provides indentation and automatic line breaks depending on page width by means of formatting boxes.

The default printing of notations is 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 next to the curly braces.

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

Check (sig (fun x : nat => x=x)).
  {{ x : nat | x = x }} : Set

The second, more powerful control on printing is by using `syntax_modifier`s. Here is an example

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

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

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

- tokens of the form `'/ '` are translated into breaking points. If there is a line break, indents the number of spaces appearing after the “`/`” (no indentation in the example)

- tokens of the form `'//'` force writing on a new line

- well-bracketed pairs of tokens of the form `'[ '` and `']'` are translated into printing boxes; if there is a line break, an extra indentation of the number of spaces after the “`[`” is applied

- well-bracketed pairs of tokens of the form `'[hv '` and `']'` are translated into horizontal-or-else-vertical printing boxes; if the content of the box does not fit on a single line, then every breaking point forces a new line and an extra indentation of the number of spaces after the “`[hv`” is applied at the beginning of each new line

- well-bracketed pairs of tokens of the form `'[v '` and `']'` are translated into vertical printing boxes; every breaking point forces a new line, even if the line is large enough to display the whole content of the box, and an extra indentation of the number of spaces after the “`[v`” is applied at the beginning of each new line (3 spaces in the example)

- extra spaces in other tokens are preserved in the output

Notations disappear when a section is closed. No typing of the denoted expression is performed at definition time. Type checking is done only at the time of use of the notation.

Note

The default for a notation is to be used both for parsing and printing. It is possible to declare a notation only for parsing by adding the option `only parsing` to the list of `syntax_modifier`s of `Notation`. Symmetrically, the `only printing` `syntax_modifier` can be used to declare that a notation should only be used for printing.

If a notation to be used both for parsing and printing is overridden, both the parsing and printing are invalided, even if the overriding rule is only parsing.

If a given notation string occurs only in `only printing` rules, the parser is not modified at all.

To a given notation string and scope can be attached at most one notation with both parsing and printing or with only parsing. Contrastingly, an arbitrary number of `only printing` notations differing in their right-hand sides but only a unique right-hand side can be attached to a given string and scope. Obviously, expressions printed by means of such extra printing rules will not be reparsed to the same form.

Note

When several notations can be used to print a given term, the notations which capture the largest subterm of the term are used preferentially. Here is an example:

Notation "x < y" := (lt x y) (at level 70).
Notation "x < y < z" := (lt x y /\ lt y z) (at level 70, y at next level).
Check (0 < 1 /\ 1 < 2).
  0 < 1 < 2 : Prop

When several notations match the same subterm, or incomparable subterms of the term to print, the notation declared most recently is selected. Moreover, reimporting a library or module declares the notations of this library or module again. If the notation is in a scope (see Notation scopes), either the scope has to be opened or a delimiter has to exist in the scope for the notation to be usable.

### The Infix command

The `Infix` command is a shortcut for declaring notations for infix symbols.

Command `Infix string := one_term ( syntax_modifier+, )? : scope_name?`

  This command is equivalent to

  > `Notation "x string y" := (one_term x y) ( syntax_modifier+, )? : scope_name?`

  where `x` and `y` are fresh names and omitting the quotes around `string`. Here is an example:

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

### Reserving notations

Command `Reserved Notation string ( syntax_modifier+, )?`

  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, this command declares a parsing rule (`string`) in advance without giving its interpretation. Here is an example from the initial state of 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.

  Note

  The notations mentioned in the module Notations are reserved. Hence their precedence and associativity cannot be changed.

  Command `Reserved Infix string ( syntax_modifier+, )?`

    This command declares an infix parsing rule without giving its interpretation.

  When a format is attached to a reserved notation (with the `format` `syntax_modifier`), it is used by default by all subsequent interpretations of the corresponding notation. Individual interpretations can override the format.

### Simultaneous definition of terms and notations

Thanks to reserved notations, inductive, co-inductive, record, recursive and corecursive definitions can use customized notations. To do this, insert a `decl_notations` 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 by `decl_notation` for inductive, co-inductive, recursive and corecursive definitions and in Record types for records. Note that only syntax modifiers that do not require adding or changing a parsing rule are accepted.

> decl_notations
> ::=
> where decl_notation and decl_notation*
> decl_notation
> ::=
> string := one_term ( syntax_modifier+, )? : scope_name?

Here are examples:

Reserved Notation "A & B" (at level 80).

Inductive and' (A B : Prop) : Prop := conj' : A -> B -> A & B where "A & B" := (and' A B).
  and' is defined and'_rect is defined and'_ind is defined and'_rec is defined and'_sind is defined

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).
  Toplevel input, characters 0-125: > 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). > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Warning: Not a truly recursive fixpoint. [non-recursive,fixpoints] plus is defined plus is recursively defined (guarded on 1st argument)

### Displaying information about notations

Flag `Printing Notations`

  Controls whether to use notations for printing terms wherever possible. Default is on.

Flag `Printing Parentheses`

  If on, parentheses are printed even if implied by associativity and precedence Default is off.

See also

`Printing All` to disable other elements in addition to notations.

Command `Print Grammar ident`

  Shows the grammar for the nonterminal `ident`, which must be one of the following:

  - `constr` - for `term`s

  - `pattern` - for `pattern`s

  - `tactic` - for currently-defined tactic notations, `tactic`s and tacticals (corresponding to `ltac_expr` in the documentation).

  - `vernac` - for `command`s

  This command doesn't display all nonterminals of the grammar. For example, productions shown by `Print Grammar tactic` refer to nonterminals `tactic_then_locality` and `for_each_goal` which are not shown and can't be printed.

  Most of the grammar in the documentation was updated in 8.12 to make it accurate and readable. This was done using a new developer tool that extracts the grammar from the source code, edits it and inserts it into the documentation files. While the edited grammar is equivalent to the original, for readability some nonterminals have been renamed and others have been eliminated by substituting the nonterminal definition where the nonterminal was referenced. This command shows the original grammar, so it won't exactly match the documentation.

  The Coq parser is based on Camlp5. The documentation for Extensible grammars is the most relevant but it assumes considerable knowledge. Here are the essentials:

  Productions can contain the following elements:

  - nonterminal names - identifiers in the form `[a-zA-Z0-9_]*`

  - `"…"` - a literal string that becomes a keyword and cannot be used as an `ident`. The string doesn't have to be a valid identifier; frequently the string will contain only punctuation characters.

  - `IDENT "…"` - a literal string that has the form of an `ident`

  - `OPT element` - optionally include `element` (e.g. a nonterminal, IDENT "…" or "…")

  - `LIST1 element` - a list of one or more `element`s

  - `LIST0 element` - an optional list of `element`s

  - `LIST1 element SEP sep` - a list of `element`s separated by `sep`

  - `LIST0 element SEP sep` - an optional list of `element`s separated by `sep`

  - `[ elements1 | elements2 | … ]` - alternatives (either `elements1` or `elements2` or …)

  Nonterminals can have multiple levels to specify precedence and associativity of its productions. This feature of grammars makes it simple to parse input such as `1+2*3` in the usual way as `1+(2*3)`. However, most nonterminals have a single level.

  For example, this output from `Print Grammar tactic` shows the first 3 levels for `ltac_expr`, designated as "5", "4" and "3". Level 3 is right-associative, which applies to the productions within it, such as the `try` construct:

      Entry ltac_expr is
      [ "5" RIGHTA
        [ binder_tactic ]
      | "4" LEFTA
        [ SELF; ";"; binder_tactic
        | SELF; ";"; SELF
        | SELF; ";"; tactic_then_locality; for_each_goal; "]" ]
      | "3" RIGHTA
        [ IDENT "try"; SELF
        :

  The interpretation of `SELF` depends on its position in the production and the associativity of the level:

  - At the beginning of a production, `SELF` means the next level. In the fragment shown above, the next level for `try` is "2". (This is defined by the order of appearance in the grammar or output; the levels could just as well be named "foo" and "bar".)

  - In the middle of a production, `SELF` means the top level ("5" in the fragment)

  - At the end of a production, `SELF` means the next level within `LEFTA` levels and the current level within `RIGHTA` levels.

  `NEXT` always means the next level. `nonterminal LEVEL "…"` is a reference to the specified level for `nonterminal`.

  Associativity explains `SELF` and `NEXT` in somewhat more detail.

  The output for `Print Grammar constr` includes `Notation` definitions, which are dynamically added to the grammar at run time. For example, in the definition for `term`, the production on the second line shown here is defined by a `Reserved Notation` command in `Notations.v`:

      | "50" LEFTA
        [ SELF; "||"; NEXT

  Similarly, `Print Grammar tactic` includes `Tactic Notation`s, such as `dintuition`.

  The file doc/tools/docgram/fullGrammar in the source tree extracts the full grammar for Coq (not including notations and tactic notations defined in `*.v` files nor some optionally-loaded plugins) in a single file with minor changes to handle nonterminals using multiple levels (described in doc/tools/docgram/README.md). This is complete and much easier to read than the grammar source files. doc/tools/docgram/orderedGrammar has the edited grammar that's used in the documentation.

  Developer documentation for parsing is in dev/doc/parsing.md.

### Locating notations

To know to which notations a given symbol belongs to, use the `Locate` command. You can call it on 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.

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

### Inheritance of the properties of arguments of constants bound to a notation

If the right-hand side of a notation is a partially applied constant, the notation inherits the implicit arguments (see Implicit arguments) and notation scopes (see Notation scopes) of the constant. For instance:

Record R := {dom : Type; op : forall {A}, A -> dom}.
  R is defined dom is defined op is defined
Notation "# x" := (@op x) (at level 8).

Check fun x: R => # x 3.
  fun x : R => # x 3 : forall x : R, dom x

As an exception, if the right-hand side is just of the form `@qualid`, this conventionally stops the inheritance of implicit arguments (but not of notation scopes).

### Notations and binders

Notations can include binders. This section lists different ways to deal with binders. For further examples, see also Notations with recursive patterns involving binders.

#### Binders bound in the notation and parsed as identifiers

Here is the basic example of a notation using a binder:

Notation "'sigma' x : A , B" := (sigT (fun x : A => B)) (at level 200, x name, A at level 200, right associativity).
  Identifier 'sigma' now a keyword

The binding variables in the right-hand side that occur as a parameter of the notation (here `x`) dynamically bind all the occurrences in their respective binding scope after instantiation of the parameters of the notation. This means that the term bound to `B` can refer to the variable name bound to `x` as shown in the following application of the notation:

Check sigma z : nat, z = 0.
  sigma z : nat, z = 0 : Set

Note the `syntax_modifier x name` in the declaration of the notation. It tells to parse `x` as a single identifier (or as the unnamed variable `_`).

#### Binders bound in the notation and parsed as patterns

In the same way as patterns can be used as binders, as in `fun '(x,y) => x+y` or `fun '(existT _ x _) => x`, notations can be defined so that any `pattern` can be used in place of the binder. Here is an example:

Notation "'subset' ' p , P " := (sig (fun p => P)) (at level 200, p pattern, format "'subset' ' p , P").
  Identifier 'subset' now a keyword

Check subset '(x,y), x+y=0.
  subset '(x, y), x + y = 0 : Set

The `syntax_modifier p pattern` in the declaration of the notation tells to parse `p` as a pattern. Note that a single variable is both an identifier and a pattern, so, e.g., the following also works:

Check subset 'x, x=0.
  subset 'x, x = 0 : Set

If one wants to prevent such a notation to be used for printing when the pattern is reduced to a single identifier, one has to use instead the `syntax_modifier p strict pattern`. For parsing, however, a `strict pattern` will continue to include the case of a variable. Here is an example showing the difference:

Notation "'subset_bis' ' p , P" := (sig (fun p => P)) (at level 200, p strict pattern).
  Identifier 'subset_bis' now a keyword
Notation "'subset_bis' p , P " := (sig (fun p => P)) (at level 200, p name).

Check subset_bis 'x, x=0.
  subset_bis x, x = 0 : Set

The default level for a `pattern` is 0. One can use a different level by using `pattern at level` \(n\) where the scale is the same as the one for terms (see Notations).

#### Binders bound in the notation and parsed as terms

Sometimes, for the sake of factorization of rules, a binder has to be parsed as a term. This is typically the case for a notation such as the following:

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

This is so because the grammar also contains rules starting with `{}` and followed by a term, such as the rule for the notation `{ A } + { B }` for the constant `sumbool` (see Specification).

Then, in the rule, `x name` is replaced by `x at level 99 as name` meaning that `x` is parsed as a term at level 99 (as done in the notation for `sumbool`), but that this term has actually to be a name, i.e. an identifier or `_`.

The notation `{ x | P }` is already defined in the standard library with the `as name` `syntax_modifier`. We cannot redefine it but one can define an alternative notation, say `{ p such that P }`, using instead `as pattern`.

Notation "{ p 'such' 'that' P }" := (sig (fun p => P)) (at level 0, p at level 99 as pattern).
  Identifier 'such' now a keyword

Then, the following works:

Check {(x,y) such that x+y=0}.
  {(x, y) such that x + y = 0} : Set

To enforce that the pattern should not be used for printing when it is just a name, one could have said `p at level 99 as strict pattern`.

Note also that in the absence of a `as name`, `as strict pattern` or `as pattern` `syntax_modifier`s, the default is to consider sub-expressions occurring in binding position and parsed as terms to be `as name`.

#### Binders bound in the notation and parsed as general binders

It is also possible to rely on Coq's syntax of binders using the `binder` modifier as follows:

Notation "'myforall' p , [ P , Q ] " := (forall p, P -> Q) (at level 200, p binder).
  Identifier 'myforall' now a keyword

In this case, all of `ident`, `{ident}`, `[ident]`, `ident:type`, `{ident:type}`, `[ident:type]`, `'pattern` can be used in place of the corresponding notation variable. In particular, the binder can declare implicit arguments:

Check fun (f : myforall {a}, [a=0, Prop]) => f eq_refl.
  fun f : myforall a, [a = 0, Prop] => f 0 eq_refl : (myforall a, [a = 0, Prop]) -> Prop
Check myforall '((x,y):nat*nat), [ x = y, True ].
  myforall '(x, y), [x = y, True] : Prop

By using instead `closed binder`, the same list of binders is allowed except that `ident:type` requires parentheses around.

#### Binders not bound in the notation

We can also have binders in the right-hand side of a notation which are not themselves bound in the notation. In this case, the binders are considered up to renaming of the internal binder. E.g., for the notation

Notation "'exists_different' n" := (exists p: nat, p<>n) (at level 200).
  Identifier 'exists_different' now a keyword

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

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

Notation "[> a , .. , b <]" := (cons a .. (cons b nil) .., cons b .. (cons a nil) ..).
  Setting notation at level 0.

#### Notations with expressions used both as binder and term

It is possible to use parameters of the notation both in term and binding position. Here is an example:

Definition force n (P: nat -> Prop) := forall n', n' >= n -> P n'.
  force is defined
Notation "▢_ n P" := (force n (fun n => P)) (at level 0, n name, P at level 9, format "▢_ n P").

Check exists p, ▢_p (p >= 1).
  exists p : nat, ▢_p (p >= 1) : Prop

More generally, the parameter can be a pattern, as in the following variant:

Definition force2 q (P: nat*nat -> Prop) := (forall n', n' >= fst q -> forall p', p' >= snd q -> P q).
  force2 is defined
Notation "▢_ p P" := (force2 p (fun p => P)) (at level 0, p pattern at level 0, P at level 9, format "▢_ p P").

Check exists x y, ▢_(x,y) (x >= 1 /\ y >= 2).
  exists x y : nat, ▢_(x, y) (x >= 1 /\ y >= 2) : Prop

This support is experimental. For instance, the notation is used for printing only if the occurrence of the parameter in term position comes in the right-hand side before the occurrence in binding position.

### Notations with recursive patterns

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

Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..).
  Setting notation at level 0.

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 times (but at least once) a sequence of expressions separated by the sequence of tokens `s` (in the example, `s` is just “`;`”).

The right-hand side must contain a subterm of the form either `φ(x, .. φ(y,t) ..)` or `φ(y, .. φ(x,t) ..)` where \(φ([~]_E , [~]_I)\), called the iterator of the recursive notation is an arbitrary expression with distinguished placeholders and where \(t\) is called the terminating expression of the recursive notation. In the example, we choose the names \(x\) and \(y\) but in practice they can of course be chosen arbitrarily. Note that the placeholder \([~]_I\) has to occur only once but \([~]_E\) can occur several times.

Parsing the notation produces a list of expressions which are used to fill the first placeholder of the iterating pattern which itself is repeatedly nested as many times as the length of the list, the second placeholder being the nesting point. In the innermost occurrence of the nested iterating pattern, the second placeholder 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 is another example with the pattern associating on the left:

Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) (at level 0).

Here is an example with more involved recursive patterns:

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).
  Setting notation at level 0.

To give a flavor of the extent and limits of the mechanism, here is an example showing a notation for a chain of equalities. It relies on an artificial expansion of the intended denotation so as to expose a `φ(x, .. φ(y,t) ..)` structure, with the drawback that if ever the beta-redexes are contracted, the notations stops to be used for printing. Support for notations defined in this way should be considered experimental.

Notation "x ⪯ y ⪯ .. ⪯ z ⪯ t" := ((fun b A a => a <= b /\ A b) y .. ((fun b A a => a <= b /\ A b) z (fun b => b <= t)) .. x) (at level 70, y at next level, z at next level, t at next level).

Note finally that notations with recursive patterns can be reserved like standard notations, they can also be declared within notation scopes.

### Notations with recursive patterns involving binders

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

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 Notations with recursive patterns except that in the iterator \(φ([~]_E , [~]_I)\), the placeholder \([~]_E\) can also occur in 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 `syntax_modifier`s of the notation. The binders of the parsed sequence are used to fill the occurrences of the first placeholder of the iterating pattern which is repeatedly nested as many times as the number of binders generated. If ever the generalization operator `'` (see Implicit generalization) is used in the binding list, the added binders are taken into account too.

There are two flavors of binder parsing. 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 valid.

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 more 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:

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).
  Identifier 'mylet' now a keyword

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

Notation "'FUNAPP' x .. y , f" := (fun x => .. (fun y => (.. (f x) ..) y ) ..) (at level 200, x binder, y binder, right associativity).
  Identifier 'FUNAPP' now a keyword

If an occurrence of the \([~]_E\) is not in position of a binding variable but of a term, it is the name used in the binding which is used. Here is an example:

Notation "'exists_non_null' x .. y , P" := (ex (fun x => x <> 0 /\ .. (ex (fun y => y <> 0 /\ P)) ..)) (at level 200, x binder).
  Identifier 'exists_non_null' now a keyword

### Predefined entries

By default, sub-expressions are parsed as terms and the corresponding grammar entry is called `constr`. However, one may sometimes want to restrict the syntax of terms in a notation. For instance, the following notation will accept to parse only global reference in position of `x`:

Notation "'apply' f a1 .. an" := (.. (f a1) .. an) (at level 10, f global, a1, an at level 9).
  Identifier 'apply' now a keyword Identifier 'apply' now a keyword

In addition to `global`, one can restrict the syntax of a sub-expression by using the entry names `ident`, `name` or `pattern` already seen in Binders not bound in the notation, even when the corresponding expression is not used as a binder in the right-hand side. E.g.:

Set Warnings "-deprecated-ident-entry".

Notation "'apply_id' f a1 .. an" := (.. (f a1) .. an) (at level 10, f ident, a1, an at level 9).
  Identifier 'apply_id' now a keyword Identifier 'apply_id' now a keyword

Set Warnings "+deprecated-ident-entry".

Note

As of version 8.13, the entry `ident` is a deprecated alias for `name`. In the future, it is planned to strictly parse an identifier (excluding `_`).

### Custom entries

Command `Declare Custom Entry ident`

  Defines new grammar entries, called custom entries, that can later be referred to using the entry name `custom ident`.

  This command supports the `local` attribute, which limits the entry to the current module.

Example

For instance, we may want to define an ad hoc parser for arithmetical operations and proceed as follows:

Inductive Expr := | One : Expr | Mul : Expr -> Expr -> Expr | Add : Expr -> Expr -> Expr.
  Expr is defined Expr_rect is defined Expr_ind is defined Expr_rec is defined Expr_sind is defined
Declare Custom Entry expr.
Notation "[ e ]" := e (e custom expr at level 2).
  Setting notation at level 0.
Notation "1" := One (in custom expr at level 0).
Notation "x y" := (Mul x y) (in custom expr at level 1, left associativity).
Notation "x + y" := (Add x y) (in custom expr at level 2, left associativity).
Notation "( x )" := x (in custom expr, x at level 2).
  Setting notation at level 0.
Notation "{ x }" := x (in custom expr, x constr).
  Setting notation at level 0.
Notation "x" := x (in custom expr at level 0, x ident).
Axiom f : nat -> Expr.
  f is declared
Check fun x y z => [1 + y z + {f x}].
  fun (x : nat) (y z : Expr) => [1 + y z + {apply f x}] : nat -> Expr -> Expr -> Expr
Unset Printing Notations.
Check fun x y z => [1 + y z + {f x}].
  fun (x : nat) (y z : Expr) => Add (Add One (Mul y z)) (f x) : forall (_ : nat) (_ : Expr) (_ : Expr), Expr
Set Printing Notations.
Check fun e => match e with | [1 + 1] => [1] | [x y + z] => [x + y z] | y => [y + e] end.
  fun e : Expr => match e with | [1 + 1] => [1] | [x y + z] => [x + y z] | _ => [e + e] end : Expr -> Expr

Custom entries have levels, like the main grammar of terms and grammar of patterns have. The lower level is 0 and this is the level used by default to put rules delimited with tokens on both ends. The level is left to be inferred by Coq when using `in custom ident`. The level is otherwise given explicitly by using the syntax `in custom ident at level natural`, where `natural` refers to the level.

Levels are cumulative: a notation at level `n` of which the left end is a term shall use rules at level less than `n` to parse this subterm. More precisely, it shall use rules at level strictly less than `n` if the rule is declared with `right associativity` and rules at level less or equal than `n` if the rule is declared with `left associativity`. Similarly, a notation at level `n` of which the right end is a term shall use by default rules at level strictly less than `n` to parse this subterm if the rule is declared left associative and rules at level less or equal than `n` if the rule is declared right associative. This is what happens for instance in the rule

Notation "x + y" := (Add x y) (in custom expr at level 2, left associativity).

where `x` is any expression parsed in entry `expr` at level less or equal than `2` (including, recursively, the given rule) and `y` is any expression parsed in entry `expr` at level strictly less than `2`.

Rules associated with an entry can refer different sub-entries. The grammar entry name `constr` can be used to refer to the main grammar of term as in the rule

Notation "{ x }" := x (in custom expr at level 0, x constr).

which indicates that the subterm `x` should be parsed using the main grammar. If not indicated, the level is computed as for notations in `constr`, e.g. using 200 as default level for inner sub-expressions. The level can otherwise be indicated explicitly by using `constr at level n` for some `n`, or `constr at next level`.

Conversely, custom entries can be used to parse sub-expressions of the main grammar, or from another custom entry as is the case in

Notation "[ e ]" := e (e custom expr at level 2).
  Setting notation at level 0.

to indicate that `e` has to be parsed at level `2` of the grammar associated with the custom entry `expr`. The level can be omitted, as in

Notation "[ e ]" := e (e custom expr).

in which case Coq infer it. If the sub-expression is at a border of the notation (as e.g. `x` and `y` in `x + y`), the level is determined by the associativity. If the sub-expression is not at the border of the notation (as e.g. `e` in `"[ e ]`), the level is inferred to be the highest level used for the entry. In particular, this level depends on the highest level existing in the entry at the time of use of the notation.

In the absence of an explicit entry for parsing or printing a sub-expression of a notation in a custom entry, the default is to consider that this sub-expression is parsed or printed in the same custom entry where the notation is defined. In particular, if `x at level n` is used for a sub-expression of a notation defined in custom entry `foo`, it shall be understood the same as `x custom foo at level n`.

In general, rules are required to be productive on the right-hand side, i.e. that they are bound to an expression which is not reduced to a single variable. If the rule is not productive on the right-hand side, as it is the case above for

Notation "( x )" := x (in custom expr at level 0, x at level 2).

and

Notation "{ x }" := x (in custom expr at level 0, x constr).

it is used as a grammar coercion which means that it is used to parse or print an expression which is not available in the current grammar at the current level of parsing or printing for this grammar but which is available in another grammar or in another level of the current grammar. For instance,

Notation "( x )" := x (in custom expr at level 0, x at level 2).

tells that parentheses can be inserted to parse or print an expression declared at level `2` of `expr` whenever this expression is expected to be used as a subterm at level 0 or 1. This allows for instance to parse and print `Add x y` as a subterm of `Mul (Add x y) z` using the syntax `(x + y) z`. Similarly,

Notation "{ x }" := x (in custom expr at level 0, x constr).

gives a way to let any arbitrary expression which is not handled by the custom entry `expr` be parsed or printed by the main grammar of term up to the insertion of a pair of curly brackets.

Another special situation is when parsing global references or identifiers. To indicate that a custom entry should parse identifiers, use the following form:

Reset Initial.
Declare Custom Entry expr.

Notation "x" := x (in custom expr at level 0, x ident).

Similarly, to indicate that a custom entry should parse global references (i.e. qualified or non qualified identifiers), use the following form:

Reset Initial.
Declare Custom Entry expr.

Notation "x" := x (in custom expr at level 0, x global).

Command `Print Custom Grammar ident`

  This displays the state of the grammar for terms associated with the custom entry `ident`.

### Syntax

Here are the syntax elements used by the various notation commands.

> syntax_modifier
> ::=
> at level natural
> |
> in custom ident at level natural?
> |
> ident+, at level
> |
> ident at level binder_interp?
> |
> ident explicit_subentry
> |
> ident binder_interp
> |
> left associativity
> |
> right associativity
> |
> no associativity
> |
> only parsing
> |
> only printing
> |
> format string string?
> explicit_subentry
> ::=
> ident
> |
> name
> |
> global
> |
> bigint
> |
> strict pattern at level natural?
> |
> binder
> |
> closed binder
> |
> constr at level? binder_interp?
> |
> custom ident at level? binder_interp?
> |
> pattern at level natural?
> binder_interp
> ::=
> as ident
> |
> as name
> |
> as pattern
> |
> as strict pattern
> level
> ::=
> level natural
> |
> next level

Note

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

Note

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

Note

The notation `"{ x }"` has a special status in the main grammars of terms and patterns so 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"`).

Note

Notations such as `"( p | q )"` (or starting with `"( x | "`, more generally) are deprecated as they conflict with the syntax for nested disjunctive patterns (see Extended pattern matching), and are not honored in pattern expressions.

Warning `Use of string Notation is deprecated as it is inconsistent with pattern syntax.`

  This warning is disabled by default to avoid spurious diagnostics due to legacy notation in the Coq standard library. It can be turned on with the `-w disj-pattern-notation` flag.

Note

As of version 8.13, the entry `ident` is a deprecated alias for `name`. In the future, it is planned to strictly parse an identifier (to the exclusion of `_`). If the intent was to use `ident` as an identifier (excluding `_`), just silence the warning with `Set Warnings "-deprecated-ident-entry"` and it should automatically get its intended meaning in version 8.15.

Similarly, `as ident` is a deprecated alias for `as name`, which will only accept an identifier in the future. If the intent was to use `as ident` as an identifier (excluding `_`), just silence the warning with `Set Warnings "-deprecated-as-ident-kind"`.

However, this deprecation does not apply to custom entries, where it already denotes an identifier, as expected.

## Notation scopes

A notation scope is a set of notations for terms with their interpretations. Notation scopes provide a weak, purely syntactic form of notation overloading: a symbol may refer to different definitions depending on which notation scopes are currently open. For instance, the infix symbol `+` can be used to refer to distinct definitions of the addition operator, such as for natural numbers, integers or reals. Notation scopes can include an interpretation for numbers and strings with the `Number Notation` and `String Notation` commands.

> scope
> ::=
> scope_name
> |
> scope_key
> scope_name
> ::=
> ident
> scope_key
> ::=
> ident

Each notation scope has a single `scope_name`, which by convention ends with the suffix "_scope", as in "nat_scope". One or more `scope_key`s (delimiting keys) may be associated with a notation scope with the `Delimit Scope` command. Most commands use `scope_name`; `scope_key`s are used within `term`s.

Command `Declare Scope scope_name`

  Declares a new notation scope. Note that the initial state of Coq declares the following notation scopes: `core_scope`, `type_scope`, `function_scope`, `nat_scope`, `bool_scope`, `list_scope`, `int_scope`, `uint_scope`.

  Use commands such as `Notation` to add notations to the scope.

### Global interpretation rules for notations

At any time, the interpretation of a notation for a term is done within a stack of notation scopes and lonely notations. If a notation is defined in multiple scopes, Coq uses the interpretation from the most recently opened notation scope or declared lonely notation.

Note that "stack" is a misleading name. Each scope or lonely notation can only appear in the stack once. New items are pushed onto the top of the stack, except that adding a item that's already in the stack moves it to the top of the stack instead. Scopes are removed by name (e.g. by `Close Scope`) wherever they are in the stack, rather than through "pop" operations.

Use the `Print Visibility` command to display the current notation scope stack.

Command `Open Scope scope`

  Adds a scope to the notation scope stack. If the scope is already present, the command moves it to the top of the stack.

  If the command appears in a section: By default, the scope is only added within the section. Specifying `global` marks the scope for export as part of the current module. Specifying `local` behaves like the default.

  If the command does not appear in a section: By default, the scope marks the scope for export as part of the current module. Specifying `local` prevents exporting the scope. Specifying `global` behaves like the default.

Command `Close Scope scope`

  Removes a scope from the notation scope stack.

  If the command appears in a section: By default, the scope is only removed within the section. Specifying `global` marks the scope removal for export as part of the current module. Specifying `local` behaves like the default.

  If the command does not appear in a section: By default, the scope marks the scope removal for export as part of the current module. Specifying `local` prevents exporting the removal. Specifying `global` behaves like the default.

### 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.

#### Opening a notation scope locally

term_scope
::=
term0 % scope_key

The notation scope stack can be locally extended within a `term` with the syntax `(term)%scope_key` (or simply `term0%scope_key` for atomic terms).

In this case, `term` is interpreted in the scope stack extended with the scope bound to `scope_key`.

Command `Delimit Scope scope_name with scope_key`

  Binds the delimiting key `scope_key` to a scope.

Command `Undelimit Scope scope_name`

  Removes the delimiting keys associated with a scope.

#### Binding types or coercion classes to a notation scope

Command `Bind Scope scope_name with class+`

  Binds the notation scope `scope_name` to the type or coercion class `class`. When bound, arguments of that type for any function will be interpreted in that scope by default. This default can be overridden for individual functions with the `Arguments` command. The association may be convenient when a notation scope is naturally associated with a `type` (e.g. `nat` and the natural numbers).

  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`.

  Parameter U : Set.
    U is declared
  Declare Scope U_scope.
  Bind Scope U_scope with U.
  Parameter Uplus : U -> U -> U.
    Uplus is declared
  Parameter P : forall T: Set, T -> U -> Prop.
    P is declared
  Parameter f : forall T: Set, T -> U.
    f is declared
  Infix "+" := Uplus : U_scope.
  Unset Printing Notations.
  Open Scope nat_scope.
  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

  Note

  When active, a bound scope has effect on all defined functions (even if they are defined after the `Bind Scope` directive), except if argument scopes were assigned explicitly using the `Arguments` command.

  Note

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

### The `type_scope` notation 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 Pattern matching on goals and hypotheses: match goal), 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.

### The `function_scope` notation 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`.

### Notation 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`.

`type_scope`

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

`function_scope`

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

`nat_scope`

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

`N_scope`

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

`Z_scope`

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

`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.

`R_scope`

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

`bool_scope`

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

`list_scope`

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

`core_scope`

  This scope includes the notation for pairs. It is delimited by the 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 Lexical conventions). 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-digit number (possibly with leading 0s), 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`.

### Displaying information about scopes

Command `Print Visibility scope_name?`

  Displays the current notation scope stack. The top of the stack is displayed last. Notations in scopes whose interpretation is hidden by the same notation in a more recently opened scope are not displayed. Hence each notation is displayed only once.

  If `scope_name` is specified, displays the current notation scope stack as if the scope `scope_name` is pushed on top of the stack. This is useful to see how a subterm occurring locally in the scope is interpreted.

Command `Print Scopes`

  Displays, for each existing notation scope, all accessible notations (whether or not currently in the notation scope stack), the most-recently defined delimiting key and the class the notation scope is bound to. The display also includes lonely notations.

  Use the `Print Visibility` command to display the current notation scope stack.

Command `Print Scope scope_name`

  Displays all notations defined in the notation scope `scope_name`. It also displays the delimiting key and the class to which the scope is bound, if any.

## Abbreviations

Command `Notation ident identparm* := one_term ( only parsing )?`

  Defines an abbreviation `ident` with the parameters `identparm`.

  This command supports the `local` attribute, which limits the notation to the current module.

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

  Require Import List.
  Require Import Relations.
  Set Printing Notations.
  Notation Nlist := (list nat).
  Check 1 :: 2 :: 3 :: nil.
    1 :: 2 :: 3 :: nil : Nlist
  Notation reflexive R := (forall x, R x x).
  Check forall A: Prop, A <-> A.
    reflexive iff : Prop
  Check reflexive iff.
    reflexive iff : Prop
  Notation Plus1 B := (Nat.add B 1).
  Compute (Plus1 3).
    = 4 : nat

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

  An abbreviation is bound to an absolute name as an ordinary definition is and it also can be referred to by a qualified name.

  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 they are used. Especially, abbreviations can be bound to terms with holes (i.e. with “`_`”). For example:

  Set Strict Implicit.
  Set Printing Depth 50.
  Definition explicit_id (A: Set) (a: A) := a.
    explicit_id is defined
  Notation id := (explicit_id _).
  Check (id 0).
    id 0 : nat

  Abbreviations disappear when a section is closed. No typing of the denoted expression is performed at definition time. Type checking is done only at the time of use of the abbreviation.

  Like for notations, if the right-hand side of an abbreviation is a partially applied constant, the abbreviation inherits the implicit arguments and notation scopes of the constant. As an exception, if the right-hand side is just of the form `@qualid`, this conventionally stops the inheritance of implicit arguments.

  Like for notations, it is possible to bind binders in abbreviations. Here is an example:

  Definition force2 q (P: nat*nat -> Prop) := (forall n', n' >= fst q -> forall p', p' >= snd q -> P q).
    force2 is defined
  Notation F p P := (force2 p (fun p => P)).
  Check exists x y, F (x,y) (x >= 1 /\ y >= 2).
    exists x y : nat, F (x, y) (x >= 1 /\ y >= 2) : Prop

## Numbers and strings

primitive_notations
::=
number
|
string

Numbers and strings have no predefined semantics in the calculus. They are merely notations that can be bound to objects through the notation mechanism. Initially, numbers are bound to Peano’s representation of natural numbers (see Datatypes).

Note

Negative integers are not at the same level as `natural`, for this would make precedence unnatural.

### Number notations

Command `Number Notation qualidtype qualidparse qualidprint ( number_modifier+, )? : scope_name`
  number_modifier
  ::=
  warning after bignat
  |
  abstract after bignat
  |
  number_string_via
  number_string_via
  ::=
  via qualid mapping [ qualid => qualid​[ qualid ] => qualid+, ]

  This command allows the user to customize the way number literals are parsed and printed.

  > `qualidtype`

  >   the name of an inductive type, while `qualidparse` and `qualidprint` should be the names of the parsing and printing functions, respectively. The parsing function `qualidparse` should have one of the following types:

  >   > - `Number.int -> qualidtype`

  >   > - `Number.int -> option qualidtype`

  >   > - `Number.uint -> qualidtype`

  >   > - `Number.uint -> option qualidtype`

  >   > - `Z -> qualidtype`

  >   > - `Z -> option qualidtype`

  >   > - `Int63.int -> qualidtype`

  >   > - `Int63.int -> option qualidtype`

  >   > - `Number.number -> qualidtype`

  >   > - `Number.number -> option qualidtype`

  >   And the printing function `qualidprint` should have one of the following types:

  >   > - `qualidtype -> Number.int`

  >   > - `qualidtype -> option Number.int`

  >   > - `qualidtype -> Number.uint`

  >   > - `qualidtype -> option Number.uint`

  >   > - `qualidtype -> Z`

  >   > - `qualidtype -> option Z`

  >   > - `qualidtype -> Int63.int`

  >   > - `qualidtype -> option Int63.int`

  >   > - `qualidtype -> Number.number`

  >   > - `qualidtype -> option Number.number`

  >   Deprecated since version 8.12: Number notations on `Decimal.uint`, `Decimal.int` and `Decimal.decimal` are replaced respectively by number notations on `Number.uint`, `Number.int` and `Number.number`.

  >   When parsing, the application of the parsing function `qualidparse` to the number will be fully reduced, and universes of the resulting term will be refreshed.

  >   Note that only fully-reduced ground terms (terms containing only function application, constructors, inductive type families, sorts, and primitive integers) will be considered for printing.

  > `via qualidind mapping [ qualidconstant => qualidconstructor+, ]`

  >   When using this option, `qualidtype` no longer needs to be an inductive type and is instead mapped to the inductive type `qualidind` according to the provided list of pairs, whose first component `qualidconstant` is a constant of type `qualidtype` (or a function of type `_ ->* qualidtype`) and the second a constructor of type `qualidind`. The type `qualidtype` is then replaced by `qualidind` in the above parser and printer types.

  >   When `qualidconstant` is surrounded by square brackets, all the implicit arguments of `qualidconstant` (whether maximally inserted or not) are ignored when translating to `qualidconstructor` (i.e., before applying `qualidprint`) and replaced with implicit argument holes `_` when translating from `qualidconstructor` to `qualidconstant` (after `qualidparse`). See below for an example.

  >   Note

  >   The implicit status of the arguments is considered only at notation declaration time, any further modification of this status has no impact on the previously declared notations.

  >   Note

  >   In case of multiple implicit options (for instance `Arguments eq_refl {A}%type_scope {x}, [_] _`), an argument is considered implicit when it is implicit in any of the options.

  >   Note

  >   To use a `sort` as the target type `qualidtype`, use an abbreviation as in the example below.

  > `warning after bignat`

  >   displays a warning message about a possible stack overflow when calling `qualidparse` to parse a literal larger than `bignat`.

  >   Warning `Stack overflow or segmentation fault happens when working with large numbers in type (threshold may vary depending on your system limits and on the command executed).`

  >     When a `Number Notation` is registered in the current scope with `(warning after bignat)`, this warning is emitted when parsing a number greater than or equal to `bignat`.

  > `abstract after bignat`

  >   returns `(qualidparse m)` when parsing a literal `m` that's greater than `bignat` rather than reducing it to a normal form. Here `m` will be a `Number.int`, `Number.uint`, `Z` or `Number.number`, depending on the type of the parsing function `qualidparse`. This allows for a more compact representation of literals in types such as `nat`, and limits parse failures due to stack overflow. Note that a warning will be emitted when an integer larger than `bignat` is parsed. Note that `(abstract after bignat)` has no effect when `qualidparse` lands in an `option` type.

  >   Warning `To avoid stack overflow, large numbers in type are interpreted as applications of qualidparse.`

  >     When a `Number Notation` is registered in the current scope with `(abstract after bignat)`, this warning is emitted when parsing a number greater than or equal to `bignat`. Typically, this indicates that the fully computed representation of numbers can be so large that non-tail-recursive OCaml functions run out of stack space when trying to walk them.

  >   Warning `The 'abstract after' directive has no effect when the parsing function (qualidparse) targets an option type.`

  >     As noted above, the `(abstract after natural)` directive has no effect when `qualidparse` lands in an `option` type.

  >   Error `'via' and 'abstract' cannot be used together.`

  >     With the `abstract after` option, the parser function `qualidparse` does not reduce large numbers to a normal form, which prevents doing the translation given in the `mapping` list.

  Error `Cannot interpret this number as a value of type type`

    The number notation registered for `type` does not support the given number. This error is given when the interpretation function returns `None`, or if the interpretation is registered only for integers or non-negative integers, and the given number has a fractional or exponent part or is negative.

  Error `int63 are only non-negative numbers.`

    `Int63.int` are unsigned integers.

  Error `overflow in int63 literal bigint`

    The constant is too big to fit into an unsigned 63-bit integer `Int63.int`.

  Error `qualidparse should go from Number.int to type or (option type). Instead of Number.int, the types Number.uint or Z or Int63.int or Number.number could be used (you may need to require BinNums or Number or Int63 first).`

    The parsing function given to the `Number Notation` command is not of the right type.

  Error `qualidprint should go from type to Number.int or (option Number.int). Instead of Number.int, the types Number.uint or Z or Int63.int or Number.number could be used (you may need to require BinNums or Number or Int63 first).`

    The printing function given to the `Number Notation` command is not of the right type.

  Error `Unexpected term term while parsing a number notation.`

    Parsing functions must always return ground terms, made up of function application, constructors, inductive type families, sorts and primitive integers. Parsing functions may not return terms containing axioms, bare (co)fixpoints, lambdas, etc.

  Error `Unexpected non-option term term while parsing a number notation.`

    Parsing functions expected to return an `option` must always return a concrete `Some` or `None` when applied to a concrete number expressed as a (hexa)decimal. They may not return opaque constants.

  Error `Multiple 'via' options.`

    At most one `via` option can be given.

  Error `Multiple 'warning after' or 'abstract after' options.`

    At most one `warning after` or `abstract after` option can be given.

### String notations

Command `String Notation qualidtype qualidparse qualidprint ( number_string_via )? : scope_name`

  > Allows the user to customize how strings are parsed and printed.

  > > `qualidtype`

  > >   the name of an inductive type, while `qualidparse` and `qualidprint` should be the names of the parsing and printing functions, respectively. The parsing function `qualidparse` should have one of the following types:

  > >   > - `Byte.byte -> qualidtype`

  > >   > - `Byte.byte -> option qualidtype`

  > >   > - `list Byte.byte -> qualidtype`

  > >   > - `list Byte.byte -> option qualidtype`

  > >   The printing function `qualidprint` should have one of the following types:

  > >   > - `qualidtype -> Byte.byte`

  > >   > - `qualidtype -> option Byte.byte`

  > >   > - `qualidtype -> list Byte.byte`

  > >   > - `qualidtype -> option (list Byte.byte)`

  > >   When parsing, the application of the parsing function `qualidparse` to the string will be fully reduced, and universes of the resulting term will be refreshed.

  > >   Note that only fully-reduced ground terms (terms containing only function application, constructors, inductive type families, sorts, and primitive integers) will be considered for printing.

  > > `via qualidind mapping [ qualidconstant => qualidconstructor+, ]`

  > >   works as for number notations above.

  Error `Cannot interpret this string as a value of type type`

    > The string notation registered for `type` does not support the given string. This error is given when the interpretation function returns `None`.

    Error `qualidparse should go from Byte.byte or (list Byte.byte) to type or (option type).`

      The parsing function given to the `String Notation` command is not of the right type.

    Error `qualidprint should go from type to Byte.byte or (option Byte.byte) or (list Byte.byte) or (option (list Byte.byte)).`

      The printing function given to the `String Notation` command is not of the right type.

    Error `Unexpected term term while parsing a string notation.`

      Parsing functions must always return ground terms, made up of function application, constructors, inductive type families, sorts and primitive integers. Parsing functions may not return terms containing axioms, bare (co)fixpoints, lambdas, etc.

    Error `Unexpected non-option term term while parsing a string notation.`

      Parsing functions expected to return an `option` must always return a concrete `Some` or `None` when applied to a concrete string expressed as a decimal. They may not return opaque constants.

Note

Number or string notations for parameterized inductive types can be added by declaring an abbreviation for the inductive which instantiates all parameters. See example below.

The following errors apply to both string and number notations:

> Error `type is not an inductive type.`

>   String and number notations can only be declared for inductive types. Declare string or numeral notations for non-inductive types using `number_string_via`.

> Error `qualid was already mapped to qualid and cannot be remapped to qualid`

>   Duplicates are not allowed in the `mapping` list.

> Error `Missing mapping for constructor qualid`

>   A mapping should be provided for `qualid` in the `mapping` list.

> Warning `type was already mapped to type, mapping it also to type might yield ill typed terms when using the notation.`

>   Two pairs in the `mapping` list associate types that might be incompatible.

> Warning `Type of qualid seems incompatible with the type of qualid. Expected type is: type instead of type. This might yield ill typed terms when using the notation.`

>   A mapping given in the `mapping` list associates a constant with a seemingly incompatible constructor.

> Error `Cannot interpret in scope_name because qualid could not be found in the current environment.`

>   The inductive type used to register the string or number notation is no longer available in the environment. Most likely, this is because the notation was declared inside a functor for an inductive type inside the functor. This use case is not currently supported.

>   Alternatively, you might be trying to use a primitive token notation from a plugin which forgot to specify which module you must `Require` for access to that notation.

> Error `Syntax error: [prim:reference] expected after 'Notation' (in [vernac:command]).`

>   The type passed to `String Notation` or `Number Notation` must be a single qualified identifier.

> Error `Syntax error: [prim:reference] expected after [prim:reference] (in [vernac:command]).`

>   Both functions passed to `String Notation` or `Number Notation` must be single qualified identifiers.

> Error `qualid is bound to a notation that does not denote a reference.`

>   Identifiers passed to `String Notation` or `Number Notation` must be global references, or notations which evaluate to single qualified identifiers.

Example: Number Notation for radix 3

The following example parses and prints natural numbers whose digits are `0`, `1` or `2` as terms of the following inductive type encoding radix 3 numbers.

Inductive radix3 : Set := | x0 : radix3 | x3 : radix3 -> radix3 | x3p1 : radix3 -> radix3 | x3p2 : radix3 -> radix3.
  radix3 is defined radix3_rect is defined radix3_ind is defined radix3_rec is defined radix3_sind is defined

We first define a parsing function

Definition of_uint_dec (u : Decimal.uint) : option radix3 := let fix f u := match u with | Decimal.Nil => Some x0 | Decimal.D0 u => match f u with Some u => Some (x3 u) | None => None end | Decimal.D1 u => match f u with Some u => Some (x3p1 u) | None => None end | Decimal.D2 u => match f u with Some u => Some (x3p2 u) | None => None end | _ => None end in f (Decimal.rev u).
  of_uint_dec is defined
Definition of_uint (u : Number.uint) : option radix3 := match u with Number.UIntDecimal u => of_uint_dec u | Number.UIntHexadecimal _ => None end.
  of_uint is defined

and a printing function

Definition to_uint_dec (x : radix3) : Decimal.uint := let fix f x := match x with | x0 => Decimal.Nil | x3 x => Decimal.D0 (f x) | x3p1 x => Decimal.D1 (f x) | x3p2 x => Decimal.D2 (f x) end in Decimal.rev (f x).
  to_uint_dec is defined
Definition to_uint (x : radix3) : Number.uint := Number.UIntDecimal (to_uint_dec x).
  to_uint is defined

before declaring the notation

Declare Scope radix3_scope.
Open Scope radix3_scope.
Number Notation radix3 of_uint to_uint : radix3_scope.

We can check the printer

Check x3p2 (x3p1 x0).
  12 : radix3

and the parser

Set Printing All.
Check 120.
  x3 (x3p2 (x3p1 x0)) : radix3

Digits other than `0`, `1` and `2` are rejected.

Check 3.
  Toplevel input, characters 6-7: > Check 3. > ^ Error: Cannot interpret this number as a value of type radix3

Example: Number Notation for a non inductive type

The following example encodes the terms in the form `sum unit ( ... (sum unit unit) ... )` as the number of units in the term. For instance `sum unit (sum unit unit)` is encoded as `3` while `unit` is `1` and `0` stands for `Empty_set`. The inductive `I` will be used as `qualidind`.

Inductive I := Iempty : I | Iunit : I | Isum : I -> I -> I.
  I is defined I_rect is defined I_ind is defined I_rec is defined I_sind is defined

We then define `qualidparse` and `qualidprint`

Definition of_uint (x : Number.uint) : I := let fix f n := match n with | O => Iempty | S O => Iunit | S n => Isum Iunit (f n) end in f (Nat.of_num_uint x).
  of_uint is defined
Definition to_uint (x : I) : Number.uint := let fix f i := match i with | Iempty => O | Iunit => 1 | Isum i1 i2 => f i1 + f i2 end in Nat.to_num_uint (f x).
  to_uint is defined
Inductive sum (A : Set) (B : Set) : Set := pair : A -> B -> sum A B.
  sum is defined sum_rect is defined sum_ind is defined sum_rec is defined sum_sind is defined

the number notation itself

Notation nSet := Set (only parsing).
Number Notation nSet of_uint to_uint (via I mapping [Empty_set => Iempty, unit => Iunit, sum => Isum]) : type_scope.

and check the printer

Local Open Scope type_scope.
Check sum unit (sum unit unit).
  3 : Set

and the parser

Set Printing All.
Check 3.
  sum unit (sum unit unit) : Set

Example: Number Notation with implicit arguments

The following example parses and prints natural numbers between `0` and `n-1` as terms of type `Fin.t n`.

Require Import Vector.
Print Fin.t.
  Inductive t : nat -> Set := F1 : forall n : nat, Fin.t (S n) | FS : forall n : nat, Fin.t n -> Fin.t (S n) Arguments Fin.t _%nat_scope Arguments Fin.F1 {n}%nat_scope Arguments Fin.FS {n}%nat_scope _

Note the implicit arguments of `Fin.F1` and `Fin.FS`, which won't appear in the corresponding inductive type.

Inductive I := I1 : I | IS : I -> I.
  I is defined I_rect is defined I_ind is defined I_rec is defined I_sind is defined
Definition of_uint (x : Number.uint) : I := let fix f n := match n with O => I1 | S n => IS (f n) end in f (Nat.of_num_uint x).
  of_uint is defined
Definition to_uint (x : I) : Number.uint := let fix f i := match i with I1 => O | IS n => S (f n) end in Nat.to_num_uint (f x).
  to_uint is defined
Declare Scope fin_scope.
Delimit Scope fin_scope with fin.
Local Open Scope fin_scope.
Number Notation Fin.t of_uint to_uint (via I mapping [[Fin.F1] => I1, [Fin.FS] => IS]) : fin_scope.

Now `2` is parsed as `Fin.FS (Fin.FS Fin.F1)`, that is `@Fin.FS _ (@Fin.FS _ (@Fin.F1 _))`.

Check 2.
  2 : Fin.t (S (S (S ?n))) where ?n : [ |- nat]

which can be of type `Fin.t 3` (numbers `0`, `1` and `2`)

Check 2 : Fin.t 3.
  2 : Fin.t 3 : Fin.t 3

but cannot be of type `Fin.t 2` (only `0` and `1`)

Check 2 : Fin.t 2.
  Toplevel input, characters 6-7: > Check 2 : Fin.t 2. > ^ Error: The term "2" has type "Fin.t (S (S (S ?n)))" while it is expected to have type "Fin.t 2".

Example: String Notation with a parameterized inductive type

The parameter `Byte.byte` for the parameterized inductive type `list` is given through an abbreviation.

Notation string := (list Byte.byte) (only parsing).
Definition id_string := @id string.
  id_string is defined
String Notation string id_string id_string : list_scope.

Check "abc"%list.
  "abc"%list : list Byte.byte

## Tactic Notations

Tactic notations allow customizing the syntax of tactics.

Command `Tactic Notation ( at level natural )? ltac_production_item+ := ltac_expr`
  ltac_production_item
  ::=
  string
  |
  ident ( ident , string? )?

  Defines a tactic notation, which extends the parsing and pretty-printing of tactics.

  This command supports the `local` attribute, which limits the notation to the current module.

  > `natural`

  >   The parsing precedence to assign to the notation. This information is particularly relevant for notations for tacticals. Levels can be in the range 0 .. 5 (default is 5).

  > `ltac_production_item+`

  >   The notation syntax. Notations for simple tactics should begin with a `string`. Note that `Tactic Notation foo := idtac` is not valid; it should be `Tactic Notation "foo" := idtac`.

  > `string`

  >   represents a literal value in the notation

  > `ident`

  >   is the name of a grammar nonterminal listed in the table below. In a few cases, to maintain backward compatibility, the name differs from the nonterminal name used elsewhere in the documentation.

  > `( identparm , strings? )`

  >   `identparm` is the parameter name associated with `ident`. The `strings` is the separator string to use when `ident` specifies a list with separators (i.e. `ident` ends with `_list_sep`).

  > `ltac_expr`

  >   The tactic expression to substitute for the notation. `identparm` tokens appearing in `ltac_expr` are substituted with the associated nonterminal value.

  For example, the following command defines a notation with a single parameter `x`.

  Tactic Notation "destruct_with_eqn" constr(x) := destruct x eqn:?.

  For a complex example, examine the 16 `Tactic Notation "setoid_replace"`s defined in `$COQLIB/theories/Classes/SetoidTactics.v`, which are designed to accept any subset of 4 optional parameters.

  The nonterminals that can specified in the tactic notation are:

  Specified `ident` | Parsed as | Interpreted as | as in tactic
  --- | --- | --- | ---
  `ident` | `ident` | a user-given name | `intro`
  `simple_intropattern` | `simple_intropattern` | an introduction pattern | `assert` `as`
  `hyp` | `ident` | a hypothesis defined in context | `clear`
  `reference` | `qualid` | a qualified identifier | name of an `L`tac-defined tactic
  `smart_global` | `reference` | a global reference of term | `unfold`, `with_strategy`
  `constr` | `one_term` | a term | `exact`
  `uconstr` | `one_term` | an untyped term | `refine`
  `integer` | `integer` | an integer |
  `int_or_var` | `int_or_var` | an integer | `do`
  `strategy_level` | `strategy_level` | a strategy level |
  `strategy_level_or_var` | `strategy_level_or_var` | a strategy level | `with_strategy`
  `tactic` | `ltac_expr` | a tactic |
  `tactic`n (n in 0..5) | `ltac_expr` n | a tactic at level n |
  entry`_list` | `entry*` | a list of how entry is interpreted |
  `ne_`entry`_list` | `entry+` | a list of how entry is interpreted |
  entry`_list_sep` | `entry*s` | a list of how entry is interpreted |
  `ne_`entry`_list_sep` | `entry+s` | a list of how entry is interpreted |

  Note

  In order to be bound in tactic definitions, each syntactic entry for argument type must include the case of a simple `L`tac identifier as part of what it parses. This is naturally the case for `ident`, `simple_intropattern`, `reference`, `constr`, ... but not for `integer` nor for `strategy_level`. This is the reason for introducing special entries `int_or_var` and `strategy_level_or_var` which evaluate to integers or strategy levels only, respectively, but which syntactically includes identifiers in order to be usable in tactic definitions.

  Note

  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`, `reference`, `strategy_level`, `strategy_level_or_var`, or `int_or_var`.

Footnotes

1

  which are the levels effectively chosen in the current implementation of Coq

2

  Coq accepts notations declared as nonassociative but the parser on which Coq is built, namely Camlp5, currently does not implement `no associativity` and replaces it with `left associativity`; hence it is the same for Coq: `no associativity` is in fact `left associativity` for the purposes of parsing

Other versions v: 8.13.1

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