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" := (c1 /\ c2 \/ ~ c1 /\ c3) (at level 200, right associativity).
 Identifier 'IF' now a keyword
A notation binds a syntactic expression to a term. Unless the parser
and prettyprinter 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 righthand 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 righthand 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 wellformed 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 subexpression is 200, and the default level for the notation itself is 0.
 Notation "( x , y )" := (@pair _ _ x y).
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:
 Notation "{ x : A  P }" := (sig A (fun x => P)) (x at level 99).
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).
 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 prettyprinting, also requires some care. We may want specific indentations, line breaks, alignment if on several lines, etc. For prettyprinting, 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
 Definition IF_then_else (P Q R:Prop) := P /\ Q \/ ~ P /\ R.
 IF_then_else is defined
 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 linewellbracketed 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 appliedwellbracketed pairs of tokens of the form
'[hv '
and']'
are translated into horizontalorelsevertical 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 linewellbracketed 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 righthand sides but only a unique
righthand 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
andy
are fresh names and omitting the quotes aroundstring
. 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, coinductive, 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,
coinductive, 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
 Arguments S _ : clear scopes.
 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).
 plus is defined plus is recursively defined (guarded on 1st argument)
Displaying information about notations¶

Flag
Printing Notations
¶ This flag controls whether to use notations for printing terms wherever possible. Default is on.

Flag
Printing Raw Literals
¶ This flag controls whether to use string and number notations for printing terms wherever possible (see String notations). Default is off.

Flag
Printing Parentheses
¶ When this flag is 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
 forterm
spattern
 forpattern
stactic
 for currentlydefined tactic notations,tactic
s and tacticals (corresponding toltac_expr
in the documentation).vernac
 forcommand
sltac2
 for Ltac2 notations (corresponding toltac2_expr
)
This command doesn't display all nonterminals of the grammar. For example, productions shown by
Print Grammar tactic
refer to nonterminalstactic_then_locality
andfor_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
[azAZ09_]*
"…"
 a literal string that becomes a keyword and cannot be used as anident
. 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 anident
OPT element
 optionally includeelement
(e.g. a nonterminal, IDENT "…" or "…")LIST1 element
 a list of one or moreelement
sLIST0 element
 an optional list ofelement
sLIST1 element SEP sep
 a list ofelement
s separated bysep
LIST0 element SEP sep
 an optional list ofelement
s separated bysep
[ elements1  elements2  … ]
 alternatives (eitherelements1
orelements2
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 as1+(2*3)
. However, most nonterminals have a single level.For example, this output from
Print Grammar tactic
shows the first 3 levels forltac_expr
, designated as "5", "4" and "3". Level 3 is rightassociative, which applies to the productions within it, such as thetry
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 fortry
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 withinLEFTA
levels and the current level withinRIGHTA
levels.
NEXT
always means the next level.nonterminal LEVEL "…"
is a reference to the specified level fornonterminal
.Associativity explains
SELF
andNEXT
in somewhat more detail.The output for
Print Grammar constr
includesNotation
definitions, which are dynamically added to the grammar at run time. For example, in the definition forterm
, the production on the second line shown here is defined by aReserved Notation
command inNotations.v
: "50" LEFTA [ SELF; ""; NEXT
Similarly,
Print Grammar tactic
includesTactic Notation
s, such asdintuition
.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 optionallyloaded 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 righthand 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 righthand 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 righthand 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:
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 subexpressions 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 righthand 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) ..).
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 righthand 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) ..).
On the righthand 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 threedots notation “…
” used in this manual to denote
a sequence of arbitrary size.
On the lefthand 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 righthand 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).
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
betaredexes 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 wellbracketed binders of the form (a b c:T)
or {a b c:T}
etc. are accepted.
With closed binders, the recursive sequence in the lefthand 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, subexpressions 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
In addition to global
, one can restrict the syntax of a
subexpression 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 righthand
side. E.g.:
 Set Warnings "deprecatedidententry".
 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
 Set Warnings "+deprecatedidententry".
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.Nonlocal custom entries survive module closing and are declared when a file is Required.
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).
 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).
 Notation "{ x }" := x (in custom expr, x constr).
 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 subentries. 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 subexpressions. 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 subexpressions of the main grammar, or from another custom entry as is the case in
 Notation "[ e ]" := e (e custom expr at level 2).
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
in which case Coq infer it. If the subexpression 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 subexpression 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
subexpression of a notation in a custom entry, the default is to
consider that this subexpression 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 subexpression 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 righthand 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 righthand 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:
 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 unqualified identifiers), use the following form:
 Declare Custom Entry expr.
 Notation "x" := x (in custom expr at level 0, x global).
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 levelin scope ident
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 patternlevel
::=
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.
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 "deprecatedidententry"
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 "deprecatedasidentkind"
.
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.
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. Specifyinglocal
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. Specifyingglobal
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. Specifyinglocal
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. Specifyingglobal
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.
The arguments of an abbreviation can be interpreted
in a scope stack locally extended with a given scope by using the modifier
ident+, in scope scope_name
.s
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 classclass
. 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 theArguments
command. The association may be convenient when a notation scope is naturally associated with atype
(e.g.nat
and the natural numbers).Whether the argument of a function has some type
type
is determined statically. For instance, iff
is a polymorphic function of typeforall X:Type, X > X
and typet
is bound to a scopescope
, thena
of typet
inf t a
is not recognized as an argument to be interpreted in scopescope
. 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 theArguments
command.Note
The scopes
type_scope
andfunction_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 classSortclass
, as described above.function_scope
This scope is delimited by the key
function
, and bound to the coercion classFunclass
, 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
andS
. The scope is delimited by the keynat
, and bound to the typenat
(see above).N_scope
This scope includes the standard arithmetical operators and relations on type
N
(binary natural numbers). It is delimited by the keyN
and comes with an interpretation for numbers as closed terms of typeN
.Z_scope
This scope includes the standard arithmetical operators and relations on type
Z
(binary integer numbers). It is delimited by the keyZ
and comes with an interpretation for numbers as closed terms of typeZ
.positive_scope
This scope includes the standard arithmetical operators and relations on type
positive
(binary strictly positive numbers). It is delimited by keypositive
and comes with an interpretation for numbers as closed terms of typepositive
.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 crossproduct) and comes with an interpretation for numbers as closed terms of typeQ
.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 keyR
and comes with an interpretation for numbers using theIZR
morphism from binary integer numbers toR
andZ.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 typebool
(see above).list_scope
This scope includes notations for the list operators. It is delimited by the key
list
, and bound to the typelist
(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"
wherec
is an ASCII character, or of the form"nnn"
where nnn is a threedigit number (possibly with leading 0s), or of the form""""
. Their respective denotations are the ASCII code ofc
, the decimal ASCII codennn
, or the ascii code of the character"
(i.e. the ASCII code 34), all of them being represented in the typeascii
.
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 scopescope_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 mostrecently 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 ident_{parm}* := one_term ( syntax_modifier+, )?
¶ Defines an abbreviation
ident
with the parametersident_{parm}
.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 righthand 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 righthand 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 qualid_{type} qualid_{parse} qualid_{print} ( number_modifier+, )? : scope_name
¶  number_modifier
::=
warning after bignat
abstract after bignat
number_string_vianumber_string_via
::=
via qualid mapping [ qualid => qualid[ qualid ] => qualid+, ]This command allows the user to customize the way number literals are parsed and printed.
qualid_{type}
the name of an inductive type, while
qualid_{parse}
andqualid_{print}
should be the names of the parsing and printing functions, respectively. The parsing functionqualid_{parse}
should have one of the following types:Number.int > qualid_{type}
Number.int > option qualid_{type}
Number.uint > qualid_{type}
Number.uint > option qualid_{type}
Z > qualid_{type}
Z > option qualid_{type}
PrimInt63.pos_neg_int63 > qualid_{type}
PrimInt63.pos_neg_int63 > option qualid_{type}
PrimFloat.float > qualid_{type}
PrimFloat.float > option qualid_{type}
Number.number > qualid_{type}
Number.number > option qualid_{type}
And the printing function
qualid_{print}
should have one of the following types:qualid_{type} > Number.int
qualid_{type} > option Number.int
qualid_{type} > Number.uint
qualid_{type} > option Number.uint
qualid_{type} > Z
qualid_{type} > option Z
qualid_{type} > PrimInt63.pos_neg_int63
qualid_{type} > option PrimInt63.pos_neg_int63
qualid_{type} > PrimFloat.float
qualid_{type} > option PrimFloat.float
qualid_{type} > Number.number
qualid_{type} > option Number.number
When parsing, the application of the parsing function
qualid_{parse}
to the number will be fully reduced, and universes of the resulting term will be refreshed.Note that only fullyreduced ground terms (terms containing only function application, constructors, inductive type families, sorts, primitive integers, primitive floats, primitive arrays and type constants for primitive types) will be considered for printing.
Note
Instead of an inductive type,
qualid_{type}
can bePrimInt63.int
orPrimFloat.float
, in which casequalid_{print}
takesPrimInt63.int_wrapper
orPrimFloat.float_wrapper
as input instead ofPrimInt63.int
orPrimFloat.float
. See below for an example.Note
When
PrimFloat.float
is used as input type ofqualid_{parse}
, only numerical values will be parsed this way, (no infinities nor NaN). Similarly, printersqualid_{print}
with output typePrimFloat.float
oroption PrimFloat.float
are ignored when they return non numerical values.
via qualid_{ind} mapping [ qualid_{constant} => qualid_{constructor}+, ]
When using this option,
qualid_{type}
no longer needs to be an inductive type and is instead mapped to the inductive typequalid_{ind}
according to the provided list of pairs, whose first componentqualid_{constant}
is a constant of typequalid_{type}
(or a function of type_ >* qualid_{type}
) and the second a constructor of typequalid_{ind}
. The typequalid_{type}
is then replaced byqualid_{ind}
in the above parser and printer types.When
qualid_{constant}
is surrounded by square brackets, all the implicit arguments ofqualid_{constant}
(whether maximally inserted or not) are ignored when translating toqualid_{constructor}
(i.e., before applyingqualid_{print}
) and replaced with implicit argument holes_
when translating fromqualid_{constructor}
toqualid_{constant}
(afterqualid_{parse}
). 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 typequalid_{type}
, use an abbreviation as in the example below.warning after bignat
displays a warning message about a possible stack overflow when calling
qualid_{parse}
to parse a literal larger thanbignat
.
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 tobignat
.

Warning
abstract after bignat
returns
(qualid_{parse} m)
when parsing a literalm
that's greater thanbignat
rather than reducing it to a normal form. Herem
will be aNumber.int
,Number.uint
,Z
orNumber.number
, depending on the type of the parsing functionqualid_{parse}
. This allows for a more compact representation of literals in types such asnat
, and limits parse failures due to stack overflow. Note that a warning will be emitted when an integer larger thanbignat
is parsed. Note that(abstract after bignat)
has no effect whenqualid_{parse}
lands in anoption
type.
Warning
To avoid stack overflow, large numbers in type are interpreted as applications of qualid_{parse}.
¶ 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 tobignat
. Typically, this indicates that the fully computed representation of numbers can be so large that nontailrecursive OCaml functions run out of stack space when trying to walk them.

Warning
The 'abstract after' directive has no effect when the parsing function (qualid_{parse}) targets an option type.
¶ As noted above, the
(abstract after natural)
directive has no effect whenqualid_{parse}
lands in anoption
type.

Error
'via' and 'abstract' cannot be used together.
¶ With the
abstract after
option, the parser functionqualid_{parse}
does not reduce large numbers to a normal form, which prevents doing the translation given in themapping
list.

Warning

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 returnsNone
, or if the interpretation is registered only for integers or nonnegative integers, and the given number has a fractional or exponent part or is negative.

Error
overflow in int63 literal bigint
¶ The constant's absolute value is too big to fit into a 63bit integer
PrimInt63.int
.

Error
qualid_{parse} should go from Number.int to type or (option type). Instead of Number.int, the types Number.uint or Z or PrimInt63.pos_neg_int63 or PrimFloat.float or Number.number could be used (you may need to require BinNums or Number or PrimInt63 or PrimFloat first).
¶ The parsing function given to the
Number Notation
command is not of the right type.

Error
qualid_{print} should go from type to Number.int or (option Number.int). Instead of Number.int, the types Number.uint or Z or PrimInt63.pos_neg_int63 or Number.number could be used (you may need to require BinNums or Number or PrimInt63 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 nonoption term term while parsing a number notation.
¶ Parsing functions expected to return an
option
must always return a concreteSome
orNone
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
orabstract after
option can be given.
String notations¶

Command
String Notation qualid_{type} qualid_{parse} qualid_{print} ( number_string_via )? : scope_name
¶ Allows the user to customize how strings are parsed and printed.
qualid_{type}
the name of an inductive type, while
qualid_{parse}
andqualid_{print}
should be the names of the parsing and printing functions, respectively. The parsing functionqualid_{parse}
should have one of the following types:Byte.byte > qualid_{type}
Byte.byte > option qualid_{type}
list Byte.byte > qualid_{type}
list Byte.byte > option qualid_{type}
The printing function
qualid_{print}
should have one of the following types:qualid_{type} > Byte.byte
qualid_{type} > option Byte.byte
qualid_{type} > list Byte.byte
qualid_{type} > option (list Byte.byte)
When parsing, the application of the parsing function
qualid_{parse}
to the string will be fully reduced, and universes of the resulting term will be refreshed.Note that only fullyreduced ground terms (terms containing only function application, constructors, inductive type families, sorts, primitive integers, primitive floats, primitive arrays and type constants for primitive types) will be considered for printing.
via qualid_{ind} mapping [ qualid_{constant} => qualid_{constructor}+, ]
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 returnsNone
.
Error
qualid_{parse} 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
qualid_{print} 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
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 noninductive 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 themapping
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
orNumber 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
orNumber Notation
must be single qualified identifiers.
 Error
qualid is bound to a notation that does not denote a reference.
¶Identifiers passed to
String Notation
orNumber 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 67: > Check 3. > ^ Error: Cannot interpret this number as a value of type radix3
Example: Number Notation for primitive integers
This shows the use of the primitive
integers PrimInt63.int
as qualid_{type}
. It is the way
parsing and printing of primitive integers are actually implemented
in PrimInt63.v
.
 Require Import PrimInt63.
 Definition parser (x : pos_neg_int63) : option int := match x with Pos p => Some p  Neg _ => None end.
 parser is defined
 Definition printer (x : int_wrapper) : pos_neg_int63 := Pos (int_wrap x).
 printer is defined
 Number Notation int parser printer : uint63_scope.
Example: Number Notation for a noninductive 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 qualid_{ind}
.
 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 qualid_{parse}
and qualid_{print}
 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 n1
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 67: > 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 prettyprinting 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 thatTactic Notation foo := idtac
is not valid; it should beTactic 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.
( ident_{parm} , string_{s}? )
ident_{parm}
is the parameter name associated withident
. Thestring_{s}
is the separator string to use whenident
specifies a list with separators (i.e.ident
ends with_list_sep
).ltac_expr
The tactic expression to substitute for the notation.
ident_{parm}
tokens appearing inltac_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
a usergiven name
simple_intropattern
an introduction pattern
assert
as
hyp
a hypothesis defined in context
reference
a qualified identifier
name of an
L
_{tac}defined tacticsmart_global
a global reference of term
constr
a term
uconstr
an untyped term
integer
an integer
int_or_var
an integer
strategy_level
a strategy level
strategy_level_or_var
a strategy level
tactic
a tactic
tactic
n (n in 0..5)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 forident
,simple_intropattern
,reference
,constr
, ... but not forinteger
nor forstrategy_level
. This is the reason for introducing special entriesint_or_var
andstrategy_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*
andne_
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 eitherconstr
,hyp
,integer
,reference
,strategy_level
,strategy_level_or_var
, orint_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 withleft associativity
; hence it is the same for Coq:no associativity
is in factleft associativity
for the purposes of parsing