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

Syntax extensions and interpretation 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 interpretation scopes. The main command to provide custom notations for tactics is Tactic Notation.

Set Printing Depth 50.

Notations

Basic notations

Command Notation

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

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

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

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

A notation is always surrounded by double quotes (except when the abbreviation has the form of an ordinary applicative expression; see Abbreviations). The notation is composed of tokens separated by spaces. Identifiers in the string (such as A and B) are the parameters of the notation. Each of them must occur at least once in the denoted term. The other elements of the string (such as /\) are the symbols.

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

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

A notation binds a syntactic expression to a term. Unless the parser and pretty-printer of Coq already know how to deal with the syntactic expression (see Reserving notations), 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.

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 of the associativity of disjunction and conjunction, so let us apply for instance a right associativity (which is the choice of Coq).

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

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 types 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).

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).

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 at level 200, 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.

Command Print Grammar constr.

This command displays the current state of the Coq term parser.

Command Print Grammar pattern.

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

Displaying symbolic notations

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 around the bar of the notation.

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 the format modifier. 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

Sometimes, a notation is expected only for the parser. To do so, the option only parsing is allowed in the list of modifiers of Notation. Conversely, the only printing modifier can be used to declare that a notation should only be used for printing and should not declare a parsing rule. In particular, such notations do not modify the parser.

The Infix command

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

Command Infix "symbol" := term (modifiers)?.

This command is equivalent to

Notation "x symbol y" := (term x y) (modifiers)?.

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

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

Reserving notations

Command Reserved Notation string (modifiers)?

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.

Variant Reserved Infix "symbol" modifiers*

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

Simultaneous definition of terms and notations

Thanks to reserved notations, the inductive, co-inductive, record, recursive and corecursive definitions can benefit from customized notations. To do this, insert a where notation clause after the definition of the (co)inductive type or (co)recursive term (or after the definition of each of them in case of mutual definitions). The exact syntax is given by decl_notation for inductive, co-inductive, recursive and corecursive definitions and in Record types for records. 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).

plus is defined plus is recursively defined (decreasing on 1st argument)

Displaying information about notations

Flag Printing Notations

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

See also

Printing All

To disable other elements in addition to notations.

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) "'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)

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 ident, 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

Notice the modifier x ident in the declaration of the notation. It tells to parse x as a single identifier.

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 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 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 ident).

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 ident).

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 ident is replaced by x at level 99 as ident 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 an identifier.

The notation { x | P } is already defined in the standard library with the as ident 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 an identifier, one could have said p at level 99 as strict pattern.

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

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 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 are other examples:

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

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.

Notations with recursive patterns can be reserved like standard notations, they can also be declared within interpretation 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 modifiers 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 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.:

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

Custom entries

Command Declare Custom Entry ident

This command allows to define new grammar entries, called custom entries, that can later be referred to using the entry name custom ident.

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 + {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 num, where num 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 to 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 to the custom entry expr. The level can be omitted, as in

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

in which case Coq tries to infer it.

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.

Command Print Custom Grammar ident.

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

Summary

Syntax of notations

The different syntactic forms taken by the commands declaring notations are given below. The optional scope is described in Interpretation scopes.

notation          ::=  [Local] Notation string := term [(modifiers)] [: scope].
                       [Local] Infix string := qualid [(modifiers)] [: scope].
                       [Local] Reserved Notation string [(modifiers)] .
                       Inductive ind_body [decl_notation] with … with ind_body [decl_notation].
                       CoInductive ind_body [decl_notation] with … with ind_body [decl_notation].
                       Fixpoint fix_body [decl_notation] with … with fix_body [decl_notation].
                       CoFixpoint cofix_body [decl_notation] with … with cofix_body [decl_notation].
                       [Local] Declare Custom Entry ident.
decl_notation     ::=  [where string := term [: scope] and … and string := term [: scope]].
modifiers         ::=  modifier, … , modifier
modifier          ::=  at level num
                       in custom ident
                       in custom ident at level num
                       ident , … , ident at level num [binderinterp]
                       ident , … , ident at next level [binderinterp]
                       ident explicit_subentry
                       left associativity
                       right associativity
                       no associativity
                       only parsing
                       only printing
                       format string
explicit_subentry ::=  ident
                       global
                       bigint
                       [strict] pattern [at level num]
                       binder
                       closed binder
                       constr [binderinterp]
                       constr at level num [binderinterp]
                       constr at next level [binderinterp]
                       custom [binderinterp]
                       custom at level num [binderinterp]
                       custom at next level [binderinterp]
binderinterp      ::=  as ident
                       as pattern
                       as strict pattern

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.

Persistence of notations

Notations disappear when a section is closed.

Command Local Notation notation

Notations survive modules unless the command Local Notation is used instead of Notation.

Command Local Declare Custom Entry ident

Custom entries survive modules unless the command Local Declare Custom Entry is used instead of Declare Custom Entry.

Interpretation scopes

An interpretation scope is a set of notations for terms with their interpretations. Interpretation scopes provide a weak, purely syntactical form of notation overloading: the same notation, for instance the infix symbol +, can be used to denote distinct definitions of the additive operator. Depending on which interpretation scopes are currently open, the interpretation is different. Interpretation scopes can include an interpretation for numerals and strings, either at the OCaml level or using Numeral Notation or String Notation.

Command Declare Scope scope

This adds a new scope named scope. Note that the initial state of Coq declares by default the following interpretation scopes: core_scope, type_scope, function_scope, nat_scope, bool_scope, list_scope, int_scope, uint_scope.

The syntax to associate a notation to a scope is given above. Here is a typical example which declares the notation for conjunction in the scope type_scope.

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

Note

A notation not defined in a scope is called a lonely notation. No example of lonely notations can be found in the initial state of Coq though.

Global interpretation rules for notations

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

Command Open Scope scope

The command to add a scope to the interpretation scope stack is Open Scope scope.

Command Close Scope scope

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

Notice that this command does not only cancel the last Open Scope scope but all its invocations.

Note

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

Command Local Open Scope scope.

Command Local Close Scope scope.

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

Command Global Open Scope scope.

Command Global Close Scope scope.

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

Local interpretation rules for notations

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

Local opening of an interpretation scope

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

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

Command Delimit Scope scope with ident

To bind a delimiting key to a scope, use the command Delimit Scope scope with ident

Command Undelimit Scope scope

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

Binding arguments of a constant to an interpretation scope

Command Arguments qualid name%ident+

It is possible to set in advance that some arguments of a given constant have to be interpreted in a given scope. The command is Arguments qualid name%ident+ where the list is a prefix of the arguments of qualid optionally annotated with their scope ident. Grouping round parentheses can be used to decorate multiple arguments with the same scope. ident can be either a scope name or its delimiting key. For example the following command puts the first two arguments of plus_fct in the scope delimited by the key F (Rfun_scope) and the last argument in the scope delimited by the key R (R_scope).

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

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

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

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

Variant Arguments qualid : clear scopes

This command can be used to clear argument scopes of qualid.

Variant Arguments qualid name%ident+ : extra scopes

Defines extra argument scopes, to be used in case of coercion to Funclass (see the Implicit Coercions chapter) or with a computed type.

Variant Global Arguments qualid name%ident+

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

Variant Local Arguments qualid name%ident+

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

See also

The command About can be used to show the scopes bound to the arguments of a function.

Note

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

Parameter g : bool -> bool.

g is declared

Declare Scope mybool_scope.

Notation "@@" := true (only parsing) : bool_scope.

Setting notation at level 0.

Notation "@@" := false (only parsing): mybool_scope.

Bind Scope bool_scope with bool.

Notation "# x #" := (g x) (at level 40).

Check # @@ #.

# true # : bool

Arguments g _%mybool_scope.

Check # @@ #.

# true # : bool

Delimit Scope mybool_scope with mybool.

Check # @@%mybool #.

# false # : bool

Binding types of arguments to an interpretation scope

Command Bind Scope scope with qualid

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

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

More generally, any coercion class (see the Implicit Coercions chapter) can be bound to an interpretation scope. The command to do it is Bind Scope scope with class

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 interpretation scope

The scope type_scope has a special status. It is a primitive interpretation scope which is temporarily activated each time a subterm of an expression is expected to be a type. It is delimited by the key type, and bound to the coercion class Sortclass. It is also used in certain situations where an expression is statically known to be a type, including the conclusion and the type of hypotheses within an Ltac goal match (see Pattern matching on goals), 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 interpretation scope

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

Interpretation scopes used in the standard library of Coq

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

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 numerals 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 numerals 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 numerals 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 numerals 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 numerals 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 numerals 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-digits number (possibly with leading 0's), or of the form """". Their respective denotations are the ASCII code of c, the decimal ASCII code nnn, or the ascii code of the character " (i.e. the ASCII code 34), all of them being represented in the type ascii.

Displaying information about scopes

Command Print Visibility

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

Variant Print Visibility scope

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

Command Print Scopes

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

Variant Print Scope scope

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

Abbreviations

Command Local? Notation ident ident+ := term (only parsing)?.

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

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.

Numeral notations

Command Numeral Notation ident1 ident2 ident3 : scope.

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

The token ident1 should be the name of an inductive type, while ident2 and ident3 should be the names of the parsing and printing functions, respectively. The parsing function ident2 should have one of the following types:

• Decimal.int -> ident1
• Decimal.int -> option ident1
• Decimal.uint -> ident1
• Decimal.uint -> option ident1
• Z -> ident1
• Z -> option ident1
• Decimal.decimal -> ident1
• Decimal.decimal -> option ident1

And the printing function ident3 should have one of the following types:

• ident1 -> Decimal.int
• ident1 -> option Decimal.int
• ident1 -> Decimal.uint
• ident1 -> option Decimal.uint
• ident1 -> Z
• ident1 -> option Z
• ident1 -> Decimal.decimal
• ident1 -> option Decimal.decimal

When parsing, the application of the parsing function ident2 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.

Variant Numeral Notation ident1 ident2 ident3 : scope (warning after num).

When a literal larger than num is parsed, a warning message about possible stack overflow, resulting from evaluating ident2, will be displayed.

Variant Numeral Notation ident1 ident2 ident3 : scope (abstract after num).

When a literal m larger than num is parsed, the result will be (ident2 m), without reduction of this application to a normal form. Here m will be a Decimal.int or Decimal.uint or Z, depending on the type of the parsing function ident2. 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 num is parsed. Note that (abstract after num) has no effect when ident2 lands in an option type.

Error Cannot interpret this number as a value of type type

The numeral notation registered for type does not support the given numeral. 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 numeral has a fractional or exponent part or is negative.

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

The parsing function given to the Numeral Notation vernacular is not of the right type.

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

The printing function given to the Numeral Notation vernacular is not of the right type.

Error type is not an inductive type.

Numeral notations can only be declared for inductive types with no arguments.

Error Unexpected term term while parsing a numeral notation.

Parsing functions must always return ground terms, made up of applications of constructors, inductive types, 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 numeral notation.

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

Error Cannot interpret in scope because ident could not be found in the current environment.

The inductive type used to register the numeral notation is no longer available in the environment. Most likely, this is because the numeral 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 Numeral Notation must be a single identifier.

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

Both functions passed to Numeral Notation must be single identifiers.

Error The reference ident was not found in the current environment.

Identifiers passed to Numeral Notation must exist in the global environment.

Error ident is bound to a notation that does not denote a reference.

Identifiers passed to Numeral Notation must be global references, or notations which denote to single identifiers.

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 Numeral Notation is registered in the current scope with (warning after num), this warning is emitted when parsing a numeral greater than or equal to num.

Warning To avoid stack overflow, large numbers in type are interpreted as applications of ident2.

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

For example

Check 90000.

Toplevel input, characters 0-12: > Check 90000. > ^^^^^^^^^^^^ Warning: To avoid stack overflow, large numbers in nat are interpreted as applications of Nat.of_uint. [abstract-large-number,numbers] Nat.of_uint 90000 : nat

Warning The 'abstract after' directive has no effect when the parsing function (ident2) targets an option type.

As noted above, the (abstract after num) directive has no effect when ident2 lands in an option type.

String notations

Command String Notation ident1 ident2 ident3 : scope.

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

The token ident1 should be the name of an inductive type, while ident2 and ident3 should be the names of the parsing and printing functions, respectively. The parsing function ident2 should have one of the following types:

• Byte.byte -> ident1
• Byte.byte -> option ident1
• list Byte.byte -> ident1
• list Byte.byte -> option ident1

And the printing function ident3 should have one of the following types:

• ident1 -> Byte.byte
• ident1 -> option Byte.byte
• ident1 -> list Byte.byte
• ident1 -> option (list Byte.byte)

When parsing, the application of the parsing function ident2 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.

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 ident should go from Byte.byte or (list Byte.byte) to type or (option type).

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

Error ident 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 vernacular is not of the right type.

Error type is not an inductive type.

String notations can only be declared for inductive types with no arguments.

Error Unexpected term term while parsing a string notation.

Parsing functions must always return ground terms, made up of applications of constructors, inductive types, 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.

Error Cannot interpret in scope because ident could not be found in the current environment.

The inductive type used to register the string notation is no longer available in the environment. Most likely, this is because the string 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 must be a single identifier.

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

Both functions passed to String Notation must be single identifiers.

Error The reference ident was not found in the current environment.

Identifiers passed to String Notation must exist in the global environment.

Error ident is bound to a notation that does not denote a reference.

Identifiers passed to String Notation must be global references, or notations which denote to single identifiers.

Tactic Notations

Tactic notations allow to customize the syntax of tactics. They have the following syntax:

tacn                 ::=  Tactic Notation [tactic_level] [prod_item … prod_item] := tactic.
prod_item            ::=  string | tactic_argument_type(ident)
tactic_level         ::=  (at level num)
tactic_argument_type ::=  ident | simple_intropattern | reference
                          hyp | hyp_list | ne_hyp_list
                          constr | uconstr | constr_list | ne_constr_list
                          integer | integer_list | ne_integer_list
                          int_or_var | int_or_var_list | ne_int_or_var_list
                          tactic | tactic0 | tactic1 | tactic2 | tactic3
                          tactic4 | tactic5

Command Tactic Notation (at level num)? prod_item+ := tactic.

A tactic notation extends the parser and pretty-printer of tactics with a new rule made of the list of production items. It then evaluates into the tactic expression tactic. For simple tactics, it is recommended to use a terminal symbol, i.e. a string, for the first production item. The tactic level indicates the parsing precedence of the tactic notation. This information is particularly relevant for notations of tacticals. Levels 0 to 5 are available (default is 5).

Command Print Grammar tactic

To know the parsing precedences of the existing tacticals, use the command Print Grammar tactic.

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

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

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. This is the reason for introducing a special entry int_or_var which evaluates to integers only but which syntactically includes identifiers in order to be usable in tactic definitions.

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 or int_or_var.

Variant Local Tactic Notation

Tactic notations disappear when a section is closed. They survive when a module is closed unless the command Local Tactic Notation is used instead of Tactic Notation.

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