diff options
| author |  Maxime Dénès <mail@maximedenes.fr> | 2018-03-13 09:02:54 +0100 |
|---|---|---|
| committer | Maxime Dénès <mail@maximedenes.fr> | 2018-03-15 14:11:05 +0100 |
| commit | d28b991e14ff909a00cf2153d69a57493ac361f0 (patch) | |
| tree | bfb0c4501942e98cf6a196fe40321c5fca41a7b4 | |
| parent | 2f30bb4855cc4f3f96aa6904b9b8c48414100bb4 (diff) | |
[Sphinx] Add chapter 2
Thanks to Paul Steckler for porting this chapter.
| -rw-r--r-- | Makefile.doc | 2 | ||||
| -rw-r--r-- | doc/sphinx/index.rst | 2 | ||||
| -rw-r--r-- | doc/sphinx/language/gallina-extensions.rst | 3894 |
3 files changed, 2052 insertions, 1846 deletions
diff --git a/Makefile.doc b/Makefile.doc index 0f99f0530..0f424af09 100644 --- a/Makefile.doc +++ b/Makefile.doc @@ -58,7 +58,7 @@ DOCCOMMON:=doc/common/version.tex doc/common/title.tex doc/common/macros.tex REFMANCOQTEXFILES:=$(addprefix doc/refman/, \ RefMan-gal.v.tex \ - RefMan-mod.v.tex RefMan-tac.v.tex \ + RefMan-tac.v.tex \ RefMan-cic.v.tex RefMan-lib.v.tex \ RefMan-tacex.v.tex RefMan-syn.v.tex \ RefMan-oth.v.tex RefMan-ltac.v.tex \ diff --git a/doc/sphinx/index.rst b/doc/sphinx/index.rst index f5a57b9dd..f2e444c0c 100644 --- a/doc/sphinx/index.rst +++ b/doc/sphinx/index.rst @@ -16,6 +16,8 @@ Table of contents .. toctree:: :caption: The language + language/gallina-extensions + .. toctree:: :caption: The proof engine diff --git a/doc/sphinx/language/gallina-extensions.rst b/doc/sphinx/language/gallina-extensions.rst index a1950d136..d618d90ad 100644 --- a/doc/sphinx/language/gallina-extensions.rst +++ b/doc/sphinx/language/gallina-extensions.rst @@ -1,1148 +1,1351 @@ -\chapter[Extensions of \Gallina{}]{Extensions of \Gallina{}\label{Gallina-extension}\index{Gallina}} -%HEVEA\cutname{gallina-ext.html} - -{\gallina} is the kernel language of {\Coq}. We describe here extensions of -the Gallina's syntax. - -\section{Record types -\comindex{Record} -\comindex{Inductive} -\comindex{CoInductive} -\label{Record}} - -The \verb+Record+ construction is a macro allowing the definition of -records as is done in many programming languages. Its syntax is -described on Figure~\ref{record-syntax}. In fact, the \verb+Record+ -macro is more general than the usual record types, since it allows -also for ``manifest'' expressions. In this sense, the \verb+Record+ -construction allows defining ``signatures''. - -\begin{figure}[h] -\begin{centerframe} -\begin{tabular}{lcl} -{\sentence} & ++= & {\record}\\ - & & \\ -{\record} & ::= & - {\recordkw} {\ident} \zeroone{\binders} \zeroone{{\tt :} {\sort}} \verb.:=. \\ -&& ~~~~\zeroone{\ident} - \verb!{! \zeroone{\nelist{\field}{;}} \verb!}! \verb:.:\\ - & & \\ -{\recordkw} & ::= & - {\tt Record} $|$ {\tt Inductive} $|$ {\tt CoInductive}\\ - & & \\ -{\field} & ::= & {\name} \zeroone{\binders} : {\type} \zeroone{{\tt where} {\it notation}} \\ - & $|$ & {\name} \zeroone{\binders} {\typecstrtype} := {\term}\\ -\end{tabular} -\end{centerframe} -\caption{Syntax for the definition of {\tt Record}} -\label{record-syntax} -\end{figure} - -\noindent In the expression -\begin{quote} -{\tt Record {\ident} {\params} : {\sort} := {\ident$_0$} \{ \\ - {\ident$_1$} \binders$_1$ : {\term$_1$} ; ... ; \\ - {\ident$_n$} \binders$_n$ : {\term$_n$} \}.} -\end{quote} -\noindent the identifier {\ident} is the name of the defined record -and {\sort} is its type. The identifier {\ident$_0$} is the name of -its constructor. If {\ident$_0$} is omitted, the default name {\tt -Build\_{\ident}} is used. -If {\sort} is omitted, the default sort is {\Type}. -The identifiers {\ident$_1$}, \dots, {\ident$_n$} are the names of -fields and {\tt forall {\binders$_1$}, {\term$_1$}}, \dots, -{\tt forall {\binders$_n$}, {\term$_n$}} -their respective types. Remark that the type of {\ident$_i$} may -depend on the previous {\ident$_j$} (for $j<i$). Thus the order of the -fields is important. Finally, {\params} are the parameters of the -record. - -More generally, a record may have explicitly defined (a.k.a. -manifest) fields. For instance, {\tt Record} {\ident} {\tt [} -{\params} {\tt ]} \texttt{:} {\sort} := \verb+{+ {\ident$_1$} -\texttt{:} {\type$_1$} \verb+;+ {\ident$_2$} \texttt{:=} {\term$_2$} -\verb+;+ {\ident$_3$} \texttt{:} {\type$_3$} \verb+}+ in which case -the correctness of {\type$_3$} may rely on the instance {\term$_2$} of -{\ident$_2$} and {\term$_2$} in turn may depend on {\ident$_1$}. - - -\Example -The set of rational numbers may be defined as: -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Record Rat : Set := mkRat - {sign : bool; - top : nat; - bottom : nat; - Rat_bottom_cond : 0 <> bottom; - Rat_irred_cond : - forall x y z:nat, (x * y) = top /\ (x * z) = bottom -> x = 1}. -\end{coq_example} - -Remark here that the field \verb+Rat_bottom_cond+ depends -on the field \verb+bottom+ and \verb+Rat_irred_cond+ depends -on both \verb+top+ and \verb+bottom+. - -Let us now see the work done by the {\tt Record} macro. First the -macro generates a variant type definition with just one constructor: -\begin{quote} -{\tt Variant {\ident} {\params} : {\sort} := \\ - {\ident$_0$} ({\ident$_1$} : {\term$_1$}) ... ({\ident$_n$} : {\term$_n$}).} -\end{quote} -To build an object of type {\ident}, one should provide the -constructor {\ident$_0$} with $n$ terms filling the fields of -the record. - -As an example, let us define the rational $1/2$: -\begin{coq_example*} -Theorem one_two_irred : - forall x y z:nat, x * y = 1 /\ x * z = 2 -> x = 1. -Admitted. -\end{coq_example*} -\begin{coq_example} -Definition half := mkRat true 1 2 (O_S 1) one_two_irred. -\end{coq_example} -\begin{coq_example} -Check half. -\end{coq_example} - -\begin{figure}[t] -\begin{centerframe} -\begin{tabular}{lcl} -{\term} & ++= & - \verb!{|! \zeroone{\nelist{\fielddef}{;}} \verb!|}! \\ - & & \\ -{\fielddef} & ::= & {\name} \zeroone{\binders} := {\term} \\ -\end{tabular} -\end{centerframe} -\caption{Syntax for constructing elements of a \texttt{Record} using named fields} -\label{fig:fieldsyntax} -\end{figure} +.. include:: ../replaces.rst + +.. _extensionsofgallina: + +Extensions of |Gallina| +======================= + +|Gallina| is the kernel language of |Coq|. We describe here extensions of +|Gallina|’s syntax. + +.. _record-types: + +Record types +---------------- + +The ``Record`` construction is a macro allowing the definition of +records as is done in many programming languages. Its syntax is +described in the grammar below. In fact, the ``Record`` macro is more general +than the usual record types, since it allows also for “manifest” +expressions. In this sense, the ``Record`` construction allows defining +“signatures”. + +.. _record_grammar: + + .. productionlist:: `sentence` + record : `record_keyword` ident [binders] [: sort] := [ident] { [`field` ; … ; `field`] }. + record_keyword : Record | Inductive | CoInductive + field : name [binders] : type [ where notation ] + : | name [binders] [: term] := term + +In the expression: + +.. cmd:: Record @ident {* @param } {? : @sort} := {? @ident} { {*; @ident {* @binder } : @term } }. + +the first identifier `ident` is the name of the defined record and `sort` is its +type. The optional identifier following ``:=`` is the name of its constructor. If it is omitted, +the default name ``Build_``\ `ident`, where `ident` is the record name, is used. If `sort` is +omitted, the default sort is `\Type`. The identifiers inside the brackets are the names of +fields. For a given field `ident`, its type is :g:`forall binder …, term`. +Remark that the type of a particular identifier may depend on a previously-given identifier. Thus the +order of the fields is important. Finally, each `param` is a parameter of the record. + +More generally, a record may have explicitly defined (a.k.a. manifest) +fields. For instance, we might have:: + + Record ident param : sort := { ident₁ : type₁ ; ident₂ := term₂ ; ident₃ : type₃ }. + +in which case the correctness of |type_3| may rely on the instance |term_2| of |ident_2| and |term_2| in turn +may depend on |ident_1|. + +.. example:: + + .. coqtop:: reset all + + Record Rat : Set := mkRat + {sign : bool; + top : nat; + bottom : nat; + Rat_bottom_cond : 0 <> bottom; + Rat_irred_cond : + forall x y z:nat, (x * y) = top /\ (x * z) = bottom -> x = 1}. + +Remark here that the fields ``Rat_bottom_cond`` depends on the field ``bottom`` and ``Rat_irred_cond`` +depends on both ``top`` and ``bottom``. + +Let us now see the work done by the ``Record`` macro. First the macro +generates a variant type definition with just one constructor: + +.. cmd:: Variant @ident {* @params} : @sort := @ident {* (@ident : @term_1)}. + +To build an object of type `ident`, one should provide the constructor +|ident_0| with the appropriate number of terms filling the fields of the record. + +.. example:: Let us define the rational :math:`1/2`: + + .. coqtop:: in + + Theorem one_two_irred : forall x y z:nat, x * y = 1 /\ x * z = 2 -> x = 1. + Admitted. + + Definition half := mkRat true 1 2 (O_S 1) one_two_irred. + Check half. + +.. _record-named-fields-grammar: + + .. productionlist:: + term : {| [`field_def` ; … ; `field_def`] |} + field_def : name [binders] := `term` Alternatively, the following syntax allows creating objects by using named fields, as -shown on Figure~\ref{fig:fieldsyntax}. The -fields do not have to be in any particular order, nor do they have to be all -present if the missing ones can be inferred or prompted for (see -Section~\ref{Program}). - -\begin{coq_example} -Definition half' := - {| sign := true; - Rat_bottom_cond := O_S 1; - Rat_irred_cond := one_two_irred |}. -\end{coq_example} +shown in this grammar. The fields do not have to be in any particular order, nor do they have +to be all present if the missing ones can be inferred or prompted for +(see :ref:`programs`). + +.. coqtop:: all + + Definition half' := + {| sign := true; + Rat_bottom_cond := O_S 1; + Rat_irred_cond := one_two_irred |}. This syntax can be disabled globally for printing by -\begin{quote} -{\tt Unset Printing Records.} -\optindex{Printing Records} -\end{quote} + +.. cmd:: Unset Printing Records. + For a given type, one can override this using either -\begin{quote} -{\tt Add Printing Record {\ident}.} -\end{quote} + +.. cmd:: Add Printing Record @ident. + to get record syntax or -\begin{quote} -{\tt Add Printing Constructor {\ident}.} -\end{quote} + +.. cmd:: Add Printing Constructor @ident. + to get constructor syntax. This syntax can also be used for pattern matching. -\begin{coq_example} -Eval compute in ( - match half with - | {| sign := true; top := n |} => n - | _ => 0 - end). -\end{coq_example} +.. coqtop:: all + + Eval compute in ( + match half with + | {| sign := true; top := n |} => n + | _ => 0 + end). The macro generates also, when it is possible, the projection -functions for destructuring an object of type {\ident}. These +functions for destructuring an object of type `\ident`. These projection functions are given the names of the corresponding -fields. If a field is named ``\verb=_='' then no projection is built +fields. If a field is named `_` then no projection is built for it. In our example: -\begin{coq_example} -Eval compute in top half. -Eval compute in bottom half. -Eval compute in Rat_bottom_cond half. -\end{coq_example} +.. coqtop:: all + + Eval compute in top half. + Eval compute in bottom half. + Eval compute in Rat_bottom_cond half. An alternative syntax for projections based on a dot notation is available: -\begin{coq_example} -Eval compute in half.(top). -\end{coq_example} - -It can be activated for printing with the command -\optindex{Printing Projections} -\begin{quote} -{\tt Set Printing Projections.} -\end{quote} - -\begin{coq_example} -Set Printing Projections. -Check top half. -\end{coq_example} - -The corresponding grammar rules are given in Figure~\ref{fig:projsyntax}. -When {\qualid} denotes a projection, the syntax {\tt - {\term}.({\qualid})} is equivalent to {\qualid~\term}, the syntax -{\term}{\tt .(}{\qualid}~{\termarg}$_1$ {\ldots} {\termarg}$_n${\tt )} to -{\qualid~{\termarg}$_1$ {\ldots} {\termarg}$_n$~\term}, and the syntax -{\term}{\tt .(@}{\qualid}~{\term}$_1$~\ldots~{\term}$_n${\tt )} to -{@\qualid~{\term}$_1$ {\ldots} {\term}$_n$~\term}. In each case, {\term} -is the object projected and the other arguments are the parameters of -the inductive type. - -\begin{figure}[t] -\begin{centerframe} -\begin{tabular}{lcl} -{\term} & ++= & {\term} {\tt .(} {\qualid} {\tt )}\\ - & $|$ & {\term} {\tt .(} {\qualid} \nelist{\termarg}{} {\tt )}\\ - & $|$ & {\term} {\tt .(} {@}{\qualid} \nelist{\term}{} {\tt )} -\end{tabular} -\end{centerframe} -\caption{Syntax for \texttt{Record} projections} -\label{fig:projsyntax} -\end{figure} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{Remarks} - -\item Records defined with the {\tt Record} keyword are not allowed to be -recursive (references to the record's name in the type of its field -raises an error). To define recursive records, one can use the {\tt -Inductive} and {\tt CoInductive} keywords, resulting in an inductive -or co-inductive record. -A \emph{caveat}, however, is that records -cannot appear in mutually inductive (or co-inductive) definitions. - -\item Induction schemes are automatically generated for inductive records. -Automatic generation of induction schemes for non-recursive records -defined with the {\tt Record} keyword can be activated with the -{\tt Nonrecursive Elimination Schemes} option -(see~\ref{set-nonrecursive-elimination-schemes}). - -\item {\tt Structure} is a synonym of the keyword {\tt Record}. - -\end{Remarks} - -\begin{Warnings} -\item {\tt {\ident$_i$} cannot be defined.} +.. coqtop:: all + + Eval compute in half.(top). + +It can be activated for printing with + +.. cmd:: Set Printing Projections. + +.. example:: + + .. coqtop:: all + + Set Printing Projections. + Check top half. + +.. _record_projections_grammar: + + .. productionlist:: terms + term : term `.` ( qualid ) + : | term `.` ( qualid arg … arg ) + : | term `.` ( @`qualid` `term` … `term` ) + + Syntax of Record projections + +The corresponding grammar rules are given in the preceding grammar. When `qualid` +denotes a projection, the syntax `term.(qualid)` is equivalent to `qualid term`, +the syntax `term.(qualid` |arg_1| |arg_n| `)` to `qualid` |arg_1| `…` |arg_n| `term`, +and the syntax `term.(@qualid` |term_1| |term_n| `)` to `@qualid` |term_1| `…` |term_n| `term`. +In each case, `term` is the object projected and the +other arguments are the parameters of the inductive type. + +.. note::. Records defined with the ``Record`` keyword are not allowed to be + recursive (references to the record's name in the type of its field + raises an error). To define recursive records, one can use the ``Inductive`` + and ``CoInductive`` keywords, resulting in an inductive or co-inductive record. + A *caveat*, however, is that records cannot appear in mutually inductive + (or co-inductive) definitions. + +.. note:: Induction schemes are automatically generated for inductive records. + Automatic generation of induction schemes for non-recursive records + defined with the ``Record`` keyword can be activated with the + ``Nonrecursive Elimination Schemes`` option (see :ref:`TODO-13.1.1-nonrecursive-elimination-schemes`). + +.. note::``Structure`` is a synonym of the keyword ``Record``. + +.. warn:: @ident cannot be defined. It can happen that the definition of a projection is impossible. This message is followed by an explanation of this impossibility. There may be three reasons: - \begin{enumerate} - \item The name {\ident$_i$} already exists in the environment (see - Section~\ref{Axiom}). - \item The body of {\ident$_i$} uses an incorrect elimination for - {\ident} (see Sections~\ref{Fixpoint} and~\ref{Caseexpr}). - \item The type of the projections {\ident$_i$} depends on previous - projections which themselves could not be defined. - \end{enumerate} -\end{Warnings} - -\begin{ErrMsgs} - -\item \errindex{Records declared with the keyword Record or Structure cannot be recursive.} - - The record name {\ident} appears in the type of its fields, but uses - the keyword {\tt Record}. Use the keyword {\tt Inductive} or {\tt - CoInductive} instead. -\item \errindex{Cannot handle mutually (co)inductive records.} - - Records cannot be defined as part of mutually inductive (or - co-inductive) definitions, whether with records only or mixed with - standard definitions. -\item During the definition of the one-constructor inductive - definition, all the errors of inductive definitions, as described in - Section~\ref{gal-Inductive-Definitions}, may also occur. - -\end{ErrMsgs} - -\SeeAlso Coercions and records in Section~\ref{Coercions-and-records} -of the chapter devoted to coercions. - -\subsection{Primitive Projections} -\optindex{Primitive Projections} -\optindex{Printing Primitive Projection Parameters} -\optindex{Printing Primitive Projection Compatibility} -\index{Primitive projections} -\label{prim-proj} - -The option {\tt Set Primitive Projections} turns on the use of primitive -projections when defining subsequent records (even through the {\tt - Inductive} and {\tt CoInductive} commands). Primitive projections -extended the Calculus of Inductive Constructions with a new binary term -constructor {\tt r.(p)} representing a primitive projection p applied to -a record object {\tt r} (i.e., primitive projections are always -applied). Even if the record type has parameters, these do not appear at -applications of the projection, considerably reducing the sizes of terms -when manipulating parameterized records and typechecking time. On the -user level, primitive projections can be used as a replacement for the -usual defined ones, although there are a few notable differences. + + #. The name `ident` already exists in the environment (see Section :ref:`TODO-1.3.1-axioms`). + #. The body of `ident` uses an incorrect elimination for + `ident` (see Sections :ref:`TODO-1.3.4-fixpoint` and :ref:`TODO-4.5.3-case-expr`). + #. The type of the projections `ident` depends on previous + projections which themselves could not be defined. + +.. exn:: Records declared with the keyword Record or Structure cannot be recursive. + + The record name `ident` appears in the type of its fields, but uses + the keyword ``Record``. Use the keyword ``Inductive`` or ``CoInductive`` instead. + +.. exn:: Cannot handle mutually (co)inductive records. + + Records cannot be defined as part of mutually inductive (or + co-inductive) definitions, whether with records only or mixed with + standard definitions. + +During the definition of the one-constructor inductive definition, all +the errors of inductive definitions, as described in Section +:ref:`TODO-1.3.3-inductive-definitions`, may also occur. + +**See also** Coercions and records in Section :ref:`TODO-18.9-coercions-and-records` of the chapter devoted to coercions. + +.. _primitive_projections: + +Primitive Projections +~~~~~~~~~~~~~~~~~~~~~ + +The option ``Set Primitive Projections`` turns on the use of primitive +projections when defining subsequent records (even through the ``Inductive`` +and ``CoInductive`` commands). Primitive projections +extended the Calculus of Inductive Constructions with a new binary +term constructor `r.(p)` representing a primitive projection `p` applied +to a record object `r` (i.e., primitive projections are always applied). +Even if the record type has parameters, these do not appear at +applications of the projection, considerably reducing the sizes of +terms when manipulating parameterized records and typechecking time. +On the user level, primitive projections can be used as a replacement +for the usual defined ones, although there are a few notable differences. The internally omitted parameters can be reconstructed at printing time even though they are absent in the actual AST manipulated by the kernel. This -can be obtained by setting the {\tt Printing Primitive Projection Parameters} +can be obtained by setting the ``Printing Primitive Projection Parameters`` flag. Another compatibility printing can be activated thanks to the -{\tt Printing Primitive Projection Compatibility} option which governs the +``Printing Primitive Projection Compatibility`` option which governs the printing of pattern-matching over primitive records. -\subsubsection{Primitive Record Types} -When the {\tt Set Primitive Projections} option is on, definitions of +Primitive Record Types +++++++++++++++++++++++ + +When the ``Set Primitive Projections`` option is on, definitions of record types change meaning. When a type is declared with primitive -projections, its {\tt match} construct is disabled (see -\ref{primproj:compat} though). To eliminate the (co-)inductive type, one -must use its defined primitive projections. +projections, its :g:`match` construct is disabled (see :ref:`primitive_projections` though). +To eliminate the (co-)inductive type, one must use its defined primitive projections. + +For compatibility, the parameters still appear to the user when +printing terms even though they are absent in the actual AST +manipulated by the kernel. This can be changed by unsetting the +``Printing Primitive Projection Parameters`` flag. Further compatibility +printing can be deactivated thanks to the ``Printing Primitive Projection +Compatibility`` option which governs the printing of pattern-matching +over primitive records. There are currently two ways to introduce primitive records types: -\begin{itemize} -\item Through the {\tt Record} command, in which case the type has to be - non-recursive. The defined type enjoys eta-conversion definitionally, - that is the generalized form of surjective pairing for records: - {\tt $r$ = Build\_R ($r$.($p_1$) .. $r$.($p_n$))}. Eta-conversion allows to define - dependent elimination for these types as well. -\item Through the {\tt Inductive} and {\tt CoInductive} commands, when - the body of the definition is a record declaration of the form {\tt - Build\_R \{ $p_1$ : $t_1$; .. ; $p_n$ : $t_n$ \}}. In this case the types can be - recursive and eta-conversion is disallowed. These kind of record types - differ from their traditional versions in the sense that dependent - elimination is not available for them and only non-dependent case analysis - can be defined. -\end{itemize} - -\subsubsection{Reduction} - -The basic reduction rule of a primitive projection is {\tt $p_i$ - (Build\_R $t_1$ .. $t_n$) $\rightarrow_{\iota}$ $t_i$}. However, to take the $\delta$ flag into + +#. Through the ``Record`` command, in which case the type has to be + non-recursive. The defined type enjoys eta-conversion definitionally, + that is the generalized form of surjective pairing for records: + `r` ``= Build_``\ `R` ``(``\ `r`\ ``.(``\ |p_1|\ ``) …`` `r`\ ``.(``\ |p_n|\ ``))``. + Eta-conversion allows to define dependent elimination for these types as well. +#. Through the ``Inductive`` and ``CoInductive`` commands, when + the body of the definition is a record declaration of the form + ``Build_``\ `R` ``{`` |p_1| ``:`` |t_1|\ ``; … ;`` |p_n| ``:`` |t_n| ``}``. + In this case the types can be recursive and eta-conversion is disallowed. These kind of record types + differ from their traditional versions in the sense that dependent + elimination is not available for them and only non-dependent case analysis + can be defined. + +Reduction ++++++++++ + +The basic reduction rule of a primitive projection is +|p_i| ``(Build_``\ `R` |t_1| … |t_n|\ ``)`` :math:`{\rightarrow_{\iota}}` |t_i|. +However, to take the :math:`{\delta}` flag into account, projections can be in two states: folded or unfolded. An unfolded primitive projection application obeys the rule above, while the folded version delta-reduces to the unfolded version. This allows to precisely mimic the usual unfolding rules of constants. Projections -obey the usual {\tt simpl} flags of the {\tt Arguments} command in particular. - +obey the usual ``simpl`` flags of the ``Arguments`` command in particular. There is currently no way to input unfolded primitive projections at the -user-level, and one must use the {\tt Printing Primitive Projection - Compatibility} to display unfolded primitive projections as matches -and distinguish them from folded ones. +user-level, and one must use the ``Printing Primitive Projection Compatibility`` +to display unfolded primitive projections as matches and distinguish them from folded ones. + + +Compatibility Projections and :g:`match` +++++++++++++++++++++++++++++++++++++++++ -\subsubsection{Compatibility Projections and {\tt match}} -\label{primproj:compat} To ease compatibility with ordinary record types, each primitive projection is also defined as a ordinary constant taking parameters and an object of the record type as arguments, and whose body is an application of the unfolded primitive projection of the same name. These constants are used when elaborating partial applications of the projection. One can distinguish them from applications of the primitive -projection if the {\tt Printing Primitive Projection Parameters} option -is off: for a primitive projection application, parameters are printed +projection if the ``Printing Primitive Projection Parameters`` option +is off: For a primitive projection application, parameters are printed as underscores while for the compatibility projections they are printed as usual. -Additionally, user-written {\tt match} constructs on primitive records +Additionally, user-written :g:`match` constructs on primitive records are desugared into substitution of the projections, they cannot be -printed back as {\tt match} constructs. - - % - r.(p) and (p r) elaborate to native projection application, and - % the parameters cannot be mentioned. The following arguments are - % parsed according to the remaining implicit arguments declared for the - % projection (i.e. the implicit arguments after the record type - % argument). In dot notation, the record type argument is considered - % explicit no matter what its implicit status is. - % - r.(@p params) and @p args are parsed as regular applications of the - % projection with explicit parameters. - % - [simpl p] is forbidden, but [simpl @p] will simplify both the projection - % and its explicit [@p] version. - % - [unfold p] has no effect on projection applications unless it is applied - % to a constructor. If the explicit version appears it reduces to the - % projection application. - % - [pattern x at n], [rewrite x at n] and in general abstraction and selection - % of occurrences may fail due to the disappearance of parameters. - -\section{Variants and extensions of {\mbox{\tt match}} -\label{Extensions-of-match} -\index{match@{\tt match\ldots with\ldots end}}} - -\subsection{Multiple and nested pattern-matching -\index{ML-like patterns} -\label{Mult-match}} - -The basic version of \verb+match+ allows pattern-matching on simple +printed back as :g:`match` constructs. + +Variants and extensions of :g:`match` +------------------------------------- + +.. _extended pattern-matching: + +Multiple and nested pattern-matching +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The basic version of :g:`match` allows pattern-matching on simple patterns. As an extension, multiple nested patterns or disjunction of patterns are allowed, as in ML-like languages. The extension just acts as a macro that is expanded during parsing -into a sequence of {\tt match} on simple patterns. Especially, a -construction defined using the extended {\tt match} is generally -printed under its expanded form (see~\texttt{Set Printing Matching} in -section~\ref{SetPrintingMatching}). - -\SeeAlso Chapter~\ref{Mult-match-full}. - -\subsection{Pattern-matching on boolean values: the {\tt if} expression -\label{if-then-else} -\index{if@{\tt if ... then ... else}}} - -For inductive types with exactly two constructors and for -pattern-matchings expressions which do not depend on the arguments of -the constructors, it is possible to use a {\tt if ... then ... else} -notation. For instance, the definition - -\begin{coq_example} -Definition not (b:bool) := - match b with - | true => false - | false => true - end. -\end{coq_example} - -\noindent can be alternatively written - -\begin{coq_eval} -Reset not. -\end{coq_eval} -\begin{coq_example} -Definition not (b:bool) := if b then false else true. -\end{coq_example} - -More generally, for an inductive type with constructors {\tt C$_1$} -and {\tt C$_2$}, we have the following equivalence - -\smallskip - -{\tt if {\term} \zeroone{\ifitem} then {\term}$_1$ else {\term}$_2$} $\equiv$ -\begin{tabular}[c]{l} -{\tt match {\term} \zeroone{\ifitem} with}\\ -{\tt \verb!|! C$_1$ \_ {\ldots} \_ \verb!=>! {\term}$_1$} \\ -{\tt \verb!|! C$_2$ \_ {\ldots} \_ \verb!=>! {\term}$_2$} \\ -{\tt end} -\end{tabular} - -Here is an example. - -\begin{coq_example} -Check (fun x (H:{x=0}+{x<>0}) => - match H with - | left _ => true - | right _ => false - end). -\end{coq_example} - -Notice that the printing uses the {\tt if} syntax because {\tt sumbool} is -declared as such (see Section~\ref{printing-options}). - -\subsection{Irrefutable patterns: the destructuring {\tt let} variants -\index{let in@{\tt let ... in}} -\label{Letin}} +into a sequence of match on simple patterns. Especially, a +construction defined using the extended match is generally printed +under its expanded form (see ``Set Printing Matching`` in :ref:`controlling-match-pp`). + +See also: :ref:`extended pattern-matching`. + + +Pattern-matching on boolean values: the if expression +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For inductive types with exactly two constructors and for pattern-matching +expressions that do not depend on the arguments of the constructors, it is possible +to use a ``if … then … else`` notation. For instance, the definition + +.. coqtop:: all + + Definition not (b:bool) := + match b with + | true => false + | false => true + end. + +can be alternatively written + +.. coqtop:: reset all + + Definition not (b:bool) := if b then false else true. + +More generally, for an inductive type with constructors |C_1| and |C_2|, +we have the following equivalence + +:: + + if term [dep_ret_type] then term₁ else term₂ ≡ + match term [dep_ret_type] with + | C₁ _ … _ => term₁ + | C₂ _ … _ => term₂ + end + +.. example:: + + .. coqtop:: all + + Check (fun x (H:{x=0}+{x<>0}) => + match H with + | left _ => true + | right _ => false + end). + +Notice that the printing uses the :g:`if` syntax because `sumbool` is +declared as such (see :ref:`controlling-match-pp`). + + +Irrefutable patterns: the destructuring let variants +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Pattern-matching on terms inhabiting inductive type having only one -constructor can be alternatively written using {\tt let ... in ...} +constructor can be alternatively written using :g:`let … in …` constructions. There are two variants of them. -\subsubsection{First destructuring {\tt let} syntax} -The expression {\tt let -(}~{\ident$_1$},\ldots,{\ident$_n$}~{\tt ) :=}~{\term$_0$}~{\tt -in}~{\term$_1$} performs case analysis on a {\term$_0$} which must be in -an inductive type with one constructor having itself $n$ arguments. Variables -{\ident$_1$}\ldots{\ident$_n$} are bound to the $n$ arguments of the -constructor in expression {\term$_1$}. For instance, the definition -\begin{coq_example} -Definition fst (A B:Set) (H:A * B) := match H with - | pair x y => x - end. -\end{coq_example} +First destructuring let syntax +++++++++++++++++++++++++++++++ + +The expression :g:`let (`\ |ident_1|:g:`, … ,` |ident_n|\ :g:`) :=` |term_0|\ :g:`in` |term_1| performs +case analysis on |term_0| which must be in an inductive type with one +constructor having itself :math:`n` arguments. Variables |ident_1| … |ident_n| are +bound to the :math:`n` arguments of the constructor in expression |term_1|. For +instance, the definition + +.. coqtop:: reset all + + Definition fst (A B:Set) (H:A * B) := match H with + | pair x y => x + end. + +can be alternatively written + +.. coqtop:: reset all -can be alternatively written + Definition fst (A B:Set) (p:A * B) := let (x, _) := p in x. -\begin{coq_eval} -Reset fst. -\end{coq_eval} -\begin{coq_example} -Definition fst (A B:Set) (p:A * B) := let (x, _) := p in x. -\end{coq_example} -Notice that reduction is different from regular {\tt let ... in ...} -construction since it happens only if {\term$_0$} is in constructor -form. Otherwise, the reduction is blocked. +Notice that reduction is different from regular :g:`let … in …` +construction since it happens only if |term_0| is in constructor form. +Otherwise, the reduction is blocked. -The pretty-printing of a definition by matching on a -irrefutable pattern can either be done using {\tt match} or the {\tt -let} construction (see Section~\ref{printing-options}). +The pretty-printing of a definition by matching on a irrefutable +pattern can either be done using :g:`match` or the :g:`let` construction +(see Section :ref:`controlling-match-pp`). -If {\term} inhabits an inductive type with one constructor {\tt C}, -we have an equivalence between +If term inhabits an inductive type with one constructor `C`, we have an +equivalence between -{\tt let ({\ident}$_1$,\ldots,{\ident}$_n$) \zeroone{\ifitem} := {\term} in {\term}'} +:: -\noindent and + let (ident₁, …, identₙ) [dep_ret_type] := term in term' -{\tt match {\term} \zeroone{\ifitem} with C {\ident}$_1$ {\ldots} {\ident}$_n$ \verb!=>! {\term}' end} +and +:: -\subsubsection{Second destructuring {\tt let} syntax\index{let '... in@\texttt{let '... in}}} + match term [dep_ret_type] with + C ident₁ … identₙ => term' + end -Another destructuring {\tt let} syntax is available for inductive types with + +Second destructuring let syntax ++++++++++++++++++++++++++++++++ + +Another destructuring let syntax is available for inductive types with one constructor by giving an arbitrary pattern instead of just a tuple -for all the arguments. For example, the preceding example can be written: -\begin{coq_eval} -Reset fst. -\end{coq_eval} -\begin{coq_example} -Definition fst (A B:Set) (p:A*B) := let 'pair x _ := p in x. -\end{coq_example} +for all the arguments. For example, the preceding example can be +written: + +.. coqtop:: reset all + + Definition fst (A B:Set) (p:A*B) := let 'pair x _ := p in x. This is useful to match deeper inside tuples and also to use notations -for the pattern, as the syntax {\tt let 'p := t in b} allows arbitrary +for the pattern, as the syntax :g:`let ’p := t in b` allows arbitrary patterns to do the deconstruction. For example: -\begin{coq_example} -Definition deep_tuple (A:Set) (x:(A*A)*(A*A)) : A*A*A*A := - let '((a,b), (c, d)) := x in (a,b,c,d). -Notation " x 'With' p " := (exist _ x p) (at level 20). -Definition proj1_sig' (A:Set) (P:A->Prop) (t:{ x:A | P x }) : A := - let 'x With p := t in x. -\end{coq_example} +.. coqtop:: all + + Definition deep_tuple (A:Set) (x:(A*A)*(A*A)) : A*A*A*A := + let '((a,b), (c, d)) := x in (a,b,c,d). + + Notation " x 'With' p " := (exist _ x p) (at level 20). + + Definition proj1_sig' (A:Set) (P:A->Prop) (t:{ x:A | P x }) : A := + let 'x With p := t in x. When printing definitions which are written using this construct it -takes precedence over {\tt let} printing directives for the datatype -under consideration (see Section~\ref{printing-options}). +takes precedence over let printing directives for the datatype under +consideration (see Section :ref:`controlling-match-pp`). + -\subsection{Controlling pretty-printing of {\tt match} expressions -\label{printing-options}} +.. _controlling-match-pp: -The following commands give some control over the pretty-printing of -{\tt match} expressions. +Controlling pretty-printing of match expressions +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -\subsubsection{Printing nested patterns -\label{SetPrintingMatching} -\optindex{Printing Matching}} +The following commands give some control over the pretty-printing +of :g:`match` expressions. + +Printing nested patterns ++++++++++++++++++++++++++ The Calculus of Inductive Constructions knows pattern-matching only over simple patterns. It is however convenient to re-factorize nested -pattern-matching into a single pattern-matching over a nested pattern. -{\Coq}'s printer try to do such limited re-factorization. +pattern-matching into a single pattern-matching over a nested +pattern. |Coq|’s printer tries to do such limited re-factorization. + +.. cmd:: Set Printing Matching. -\begin{quote} -{\tt Set Printing Matching.} -\end{quote} -This tells {\Coq} to try to use nested patterns. This is the default +This tells |Coq| to try to use nested patterns. This is the default behavior. -\begin{quote} -{\tt Unset Printing Matching.} -\end{quote} -This tells {\Coq} to print only simple pattern-matching problems in -the same way as the {\Coq} kernel handles them. +.. cmd:: Unset Printing Matching. + +This tells |Coq| to print only simple pattern-matching problems in the +same way as the |Coq| kernel handles them. + +.. cmd:: Test Printing Matching. -\begin{quote} -{\tt Test Printing Matching.} -\end{quote} This tells if the printing matching mode is on or off. The default is on. -\subsubsection{Factorization of clauses with same right-hand side} -\label{SetPrintingFactorizableMatchPatterns} -\optindex{Printing Factorizable Match Patterns} +Factorization of clauses with same right-hand side +++++++++++++++++++++++++++++++++++++++++++++++++++ -When several patterns share the same right-hand side, it is -additionally possible to share the clauses using disjunctive patterns. -Assuming that the printing matching mode is on, whether {\Coq}'s -printer shall try to do this kind of factorization is governed by the -following commands: +When several patterns share the same right-hand side, it is additionally +possible to share the clauses using disjunctive patterns. Assuming that the +printing matching mode is on, whether |Coq|'s printer shall try to do this kind +of factorization is governed by the following commands: -\begin{quote} -{\tt Set Printing Factorizable Match Patterns.} -\end{quote} -This tells {\Coq}'s printer to try to use disjunctive patterns. This is the default -behavior. +.. cmd:: Set Printing Factorizable Match Patterns. -\begin{quote} -{\tt Unset Printing Factorizable Match Patterns.} -\end{quote} -This tells {\Coq}'s printer not to try to use disjunctive patterns. - -\begin{quote} -{\tt Test Printing Factorizable Match Patterns.} -\end{quote} -This tells if the factorization of clauses with same right-hand side is -on or off. - -\subsubsection{Use of a default clause} -\label{SetPrintingAllowDefaultClause} -\optindex{Printing Allow Default Clause} - -When several patterns share the same right-hand side which do not -depend on the arguments of the patterns, yet an extra factorization is -possible: the disjunction of patterns can be replaced with a ``{\tt - \_}'' default clause. Assuming that the printing matching mode and -the factorization mode are on, whether {\Coq}'s printer shall try to -use a default clause is governed by the following commands: - -\begin{quote} -{\tt Set Printing Allow Default Clause.} -\end{quote} -This tells {\Coq}'s printer to use a default clause when relevant. This is the default -behavior. +This tells |Coq|'s printer to try to use disjunctive patterns. This is the +default behavior. + +.. cmd:: Unset Printing Factorizable Match Patterns. + +This tells |Coq|'s printer not to try to use disjunctive patterns. + +.. cmd:: Test Printing Factorizable Match Patterns. + +This tells if the factorization of clauses with same right-hand side is on or +off. + +Use of a default clause ++++++++++++++++++++++++ + +When several patterns share the same right-hand side which do not depend on the +arguments of the patterns, yet an extra factorization is possible: the +disjunction of patterns can be replaced with a `_` default clause. Assuming that +the printing matching mode and the factorization mode are on, whether |Coq|'s +printer shall try to use a default clause is governed by the following commands: + +.. cmd:: Set Printing Allow Default Clause. -\begin{quote} -{\tt Unset Printing Allow Default Clause.} -\end{quote} -This tells {\Coq}'s printer not to use a default clause. +This tells |Coq|'s printer to use a default clause when relevant. This is the +default behavior. + +.. cmd:: Unset Printing Allow Default Clause. + +This tells |Coq|'s printer not to use a default clause. + +.. cmd:: Test Printing Allow Default Clause. -\begin{quote} -{\tt Test Printing Allow Default Clause.} -\end{quote} This tells if the use of a default clause is allowed. -\subsubsection{Printing of wildcard pattern -\optindex{Printing Wildcard}} +Printing of wildcard patterns +++++++++++++++++++++++++++++++ Some variables in a pattern may not occur in the right-hand side of -the pattern-matching clause. There are options to control the -display of these variables. +the pattern-matching clause. There are options to control the display +of these variables. + +.. cmd:: Set Printing Wildcard. -\begin{quote} -{\tt Set Printing Wildcard.} -\end{quote} The variables having no occurrences in the right-hand side of the pattern-matching clause are just printed using the wildcard symbol -``{\tt \_}''. - -\begin{quote} -{\tt Unset Printing Wildcard.} -\end{quote} -The variables, even useless, are printed using their usual name. But some -non dependent variables have no name. These ones are still printed -using a ``{\tt \_}''. - -\begin{quote} -{\tt Test Printing Wildcard.} -\end{quote} +“_”. + +.. cmd:: Unset Printing Wildcard. + +The variables, even useless, are printed using their usual name. But +some non-dependent variables have no name. These ones are still +printed using a “_”. + +.. cmd:: Test Printing Wildcard. + This tells if the wildcard printing mode is on or off. The default is to print wildcard for useless variables. -\subsubsection{Printing of the elimination predicate -\optindex{Printing Synth}} + +Printing of the elimination predicate ++++++++++++++++++++++++++++++++++++++ In most of the cases, the type of the result of a matched term is mechanically synthesizable. Especially, if the result type does not depend of the matched term. -\begin{quote} -{\tt Set Printing Synth.} -\end{quote} -The result type is not printed when {\Coq} knows that it can -re-synthesize it. +.. cmd:: Set Printing Synth. + +The result type is not printed when |Coq| knows that it can re- +synthesize it. + +.. cmd:: Unset Printing Synth. -\begin{quote} -{\tt Unset Printing Synth.} -\end{quote} This forces the result type to be always printed. -\begin{quote} -{\tt Test Printing Synth.} -\end{quote} +.. cmd:: Test Printing Synth. + This tells if the non-printing of synthesizable types is on or off. The default is to not print synthesizable types. -\subsubsection{Printing matching on irrefutable pattern -\label{AddPrintingLet} -\comindex{Add Printing Let {\ident}} -\comindex{Remove Printing Let {\ident}} -\comindex{Test Printing Let for {\ident}} -\comindex{Print Table Printing Let}} - -If an inductive type has just one constructor, -pattern-matching can be written using the first destructuring let syntax. - -\begin{quote} -{\tt Add Printing Let {\ident}.} -\end{quote} -This adds {\ident} to the list of inductive types for which -pattern-matching is written using a {\tt let} expression. - -\begin{quote} -{\tt Remove Printing Let {\ident}.} -\end{quote} -This removes {\ident} from this list. Note that removing an inductive -type from this list has an impact only for pattern-matching written using -\texttt{match}. Pattern-matching explicitly written using a destructuring -let are not impacted. - -\begin{quote} -{\tt Test Printing Let for {\ident}.} -\end{quote} -This tells if {\ident} belongs to the list. - -\begin{quote} -{\tt Print Table Printing Let.} -\end{quote} -This prints the list of inductive types for which pattern-matching is -written using a {\tt let} expression. -The list of inductive types for which pattern-matching is written -using a {\tt let} expression is managed synchronously. This means that -it is sensible to the command {\tt Reset}. - -\subsubsection{Printing matching on booleans -\comindex{Add Printing If {\ident}} -\comindex{Remove Printing If {\ident}} -\comindex{Test Printing If for {\ident}} -\comindex{Print Table Printing If}} - -If an inductive type is isomorphic to the boolean type, -pattern-matching can be written using {\tt if} ... {\tt then} ... {\tt - else} ... - -\begin{quote} -{\tt Add Printing If {\ident}.} -\end{quote} -This adds {\ident} to the list of inductive types for which -pattern-matching is written using an {\tt if} expression. - -\begin{quote} -{\tt Remove Printing If {\ident}.} -\end{quote} -This removes {\ident} from this list. - -\begin{quote} -{\tt Test Printing If for {\ident}.} -\end{quote} -This tells if {\ident} belongs to the list. - -\begin{quote} -{\tt Print Table Printing If.} -\end{quote} -This prints the list of inductive types for which pattern-matching is -written using an {\tt if} expression. +Printing matching on irrefutable patterns +++++++++++++++++++++++++++++++++++++++++++ -The list of inductive types for which pattern-matching is written -using an {\tt if} expression is managed synchronously. This means that -it is sensible to the command {\tt Reset}. +If an inductive type has just one constructor, pattern-matching can be +written using the first destructuring let syntax. + +.. cmd:: Add Printing Let @ident. + + This adds `ident` to the list of inductive types for which pattern-matching + is written using a let expression. + +.. cmd:: Remove Printing Let @ident. + + This removes ident from this list. Note that removing an inductive + type from this list has an impact only for pattern-matching written + using :g:`match`. Pattern-matching explicitly written using a destructuring + :g:`let` are not impacted. + +.. cmd:: Test Printing Let for @ident. + + This tells if `ident` belongs to the list. + +.. cmd:: Print Table Printing Let. + + This prints the list of inductive types for which pattern-matching is + written using a let expression. + + The list of inductive types for which pattern-matching is written + using a :g:`let` expression is managed synchronously. This means that it is + sensitive to the command ``Reset``. + + +Printing matching on booleans ++++++++++++++++++++++++++++++ + +If an inductive type is isomorphic to the boolean type, pattern-matching +can be written using ``if`` … ``then`` … ``else`` …: + +.. cmd:: Add Printing If @ident. + + This adds ident to the list of inductive types for which pattern-matching is + written using an if expression. + +.. cmd:: Remove Printing If @ident. + + This removes ident from this list. + +.. cmd:: Test Printing If for @ident. + + This tells if ident belongs to the list. -\subsubsection{Example} +.. cmd:: Print Table Printing If. + + This prints the list of inductive types for which pattern-matching is + written using an if expression. + +The list of inductive types for which pattern-matching is written +using an ``if`` expression is managed synchronously. This means that it is +sensitive to the command ``Reset``. This example emphasizes what the printing options offer. -\begin{coq_example} -Definition snd (A B:Set) (H:A * B) := match H with - | pair x y => y - end. -Test Printing Let for prod. -Print snd. -Remove Printing Let prod. -Unset Printing Synth. -Unset Printing Wildcard. -Print snd. -\end{coq_example} -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\subsection{Printing \mbox{\tt match} templates} - -The {\tt Show Match} vernacular command prints a {\tt match} template for -a given type. See Section~\ref{Show}. - -% \subsection{Still not dead old notations} - -% The following variant of {\tt match} is inherited from older version -% of {\Coq}. - -% \medskip -% \begin{tabular}{lcl} -% {\term} & ::= & {\annotation} {\tt Match} {\term} {\tt with} {\terms} {\tt end}\\ -% \end{tabular} -% \medskip - -% This syntax is a macro generating a combination of {\tt match} with {\tt -% Fix} implementing a combinator for primitive recursion equivalent to -% the {\tt Match} construction of \Coq\ V5.8. It is provided only for -% sake of compatibility with \Coq\ V5.8. It is recommended to avoid it. -% (see Section~\ref{Matchexpr}). - -% There is also a notation \texttt{Case} that is the -% ancestor of \texttt{match}. Again, it is still in the code for -% compatibility with old versions but the user should not use it. - -% Explained in RefMan-gal.tex -%% \section{Forced type} - -%% In some cases, one may wish to assign a particular type to a term. The -%% syntax to force the type of a term is the following: - -%% \medskip -%% \begin{tabular}{lcl} -%% {\term} & ++= & {\term} {\tt :} {\term}\\ -%% \end{tabular} -%% \medskip - -%% It forces the first term to be of type the second term. The -%% type must be compatible with -%% the term. More precisely it must be either a type convertible to -%% the automatically inferred type (see Chapter~\ref{Cic}) or a type -%% coercible to it, (see \ref{Coercions}). When the type of a -%% whole expression is forced, it is usually not necessary to give the types of -%% the variables involved in the term. - -%% Example: - -%% \begin{coq_example} -%% Definition ID := forall X:Set, X -> X. -%% Definition id := (fun X x => x):ID. -%% Check id. -%% \end{coq_example} - -\section{Advanced recursive functions} - -The following \emph{experimental} command is available -when the {\tt FunInd} library has been loaded via {\tt Require Import FunInd}: -\begin{center} - \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - \{decrease\_annot\} : type$_0$ := \term$_0$} - \comindex{Function} - \label{Function} -\end{center} -This command can be seen as a generalization of {\tt Fixpoint}. It is actually -a wrapper for several ways of defining a function \emph{and other useful - related objects}, namely: an induction principle that reflects the -recursive structure of the function (see \ref{FunInduction}) and its -fixpoint equality. - The meaning of this -declaration is to define a function {\it ident}, similarly to {\tt - Fixpoint}. Like in {\tt Fixpoint}, the decreasing argument must be -given (unless the function is not recursive), but it might not -necessarily be \emph{structurally} decreasing. The point of the {\tt - \{\}} annotation is to name the decreasing argument \emph{and} to -describe which kind of decreasing criteria must be used to ensure -termination of recursive calls. - -The {\tt Function} construction also enjoys the {\tt with} extension -to define mutually recursive definitions. However, this feature does -not work for non structurally recursive functions. % VRAI?? - -See the documentation of {\tt functional induction} -(see Section~\ref{FunInduction}) and {\tt Functional Scheme} -(see Section~\ref{FunScheme} and \ref{FunScheme-examples}) for how to use the -induction principle to easily reason about the function. - -\noindent {\bf Remark: } To obtain the right principle, it is better -to put rigid parameters of the function as first arguments. For -example it is better to define plus like this: - -\begin{coq_example*} -Function plus (m n : nat) {struct n} : nat := - match n with - | 0 => m - | S p => S (plus m p) - end. -\end{coq_example*} -\noindent than like this: -\begin{coq_eval} -Reset plus. -\end{coq_eval} -\begin{coq_example*} -Function plus (n m : nat) {struct n} : nat := - match n with - | 0 => m - | S p => S (plus p m) - end. -\end{coq_example*} - -\paragraph[Limitations]{Limitations\label{sec:Function-limitations}} -\term$_0$ must be built as a \emph{pure pattern-matching tree} -(\texttt{match...with}) with applications only \emph{at the end} of -each branch. - -Function does not support partial application of the function being defined. Thus, the following example cannot be accepted due to the presence of partial application of \ident{wrong} into the body of \ident{wrong}~: -\begin{coq_eval} -Require List. -\end{coq_eval} -\begin{coq_example*} -Fail Function wrong (C:nat) : nat := - List.hd 0 (List.map wrong (C::nil)). -\end{coq_example*} - -For now dependent cases are not treated for non structurally terminating functions. - - - -\begin{ErrMsgs} -\item \errindex{The recursive argument must be specified} -\item \errindex{No argument name \ident} -\item \errindex{Cannot use mutual definition with well-founded - recursion or measure} - -\item \errindex{Cannot define graph for \ident\dots} (warning) - - The generation of the graph relation \texttt{(R\_\ident)} used to - compute the induction scheme of \ident\ raised a typing error. Only - the ident is defined; the induction scheme will not be generated. - - This error happens generally when: - - \begin{itemize} - \item the definition uses pattern matching on dependent types, which - \texttt{Function} cannot deal with yet. - \item the definition is not a \emph{pattern-matching tree} as - explained above. - \end{itemize} - -\item \errindex{Cannot define principle(s) for \ident\dots} (warning) - - The generation of the graph relation \texttt{(R\_\ident)} succeeded - but the induction principle could not be built. Only the ident is - defined. Please report. - -\item \errindex{Cannot build functional inversion principle} (warning) - - \texttt{functional inversion} will not be available for the - function. -\end{ErrMsgs} - - -\SeeAlso{\ref{FunScheme}, \ref{FunScheme-examples}, \ref{FunInduction}} - -Depending on the {\tt \{$\ldots$\}} annotation, different definition -mechanisms are used by {\tt Function}. More precise description -given below. - -\begin{Variants} -\item \texttt{ Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - : type$_0$ := \term$_0$} - - Defines the not recursive function \ident\ as if declared with - \texttt{Definition}. Moreover the following are defined: - - \begin{itemize} - \item {\tt\ident\_rect}, {\tt\ident\_rec} and {\tt\ident\_ind}, - which reflect the pattern matching structure of \term$_0$ (see the - documentation of {\tt Inductive} \ref{Inductive}); - \item The inductive \texttt{R\_\ident} corresponding to the graph of - \ident\ (silently); - \item \texttt{\ident\_complete} and \texttt{\ident\_correct} which are - inversion information linking the function and its graph. - \end{itemize} -\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - {\tt \{}{\tt struct} \ident$_0${\tt\}} : type$_0$ := \term$_0$} - - Defines the structural recursive function \ident\ as if declared - with \texttt{Fixpoint}. Moreover the following are defined: - - \begin{itemize} - \item The same objects as above; - \item The fixpoint equation of \ident: \texttt{\ident\_equation}. - \end{itemize} - -\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} {\tt - \{}{\tt measure \term$_1$} \ident$_0${\tt\}} : type$_0$ := - \term$_0$} -\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - {\tt \{}{\tt wf \term$_1$} \ident$_0${\tt\}} : type$_0$ := \term$_0$} - -Defines a recursive function by well founded recursion. \textbf{The -module \texttt{Recdef} of the standard library must be loaded for this -feature}. The {\tt \{\}} annotation is mandatory and must be one of -the following: -\begin{itemize} -\item {\tt \{measure} \term$_1$ \ident$_0${\tt\}} with \ident$_0$ - being the decreasing argument and \term$_1$ being a function - from type of \ident$_0$ to \texttt{nat} for which value on the - decreasing argument decreases (for the {\tt lt} order on {\tt - nat}) at each recursive call of \term$_0$. Parameters of the - function are bound in \term$_0$; -\item {\tt \{wf} \term$_1$ \ident$_0${\tt\}} with \ident$_0$ being - the decreasing argument and \term$_1$ an ordering relation on - the type of \ident$_0$ (i.e. of type T$_{\ident_0}$ - $\to$ T$_{\ident_0}$ $\to$ {\tt Prop}) for which - the decreasing argument decreases at each recursive call of - \term$_0$. The order must be well founded. Parameters of the - function are bound in \term$_0$. -\end{itemize} +.. example:: + + .. coqtop:: all + + Definition snd (A B:Set) (H:A * B) := match H with + | pair x y => y + end. + + Test Printing Let for prod. + + Print snd. + + Remove Printing Let prod. + + Unset Printing Synth. + + Unset Printing Wildcard. + + Print snd. + +.. _advanced-recursive-functions: + +Advanced recursive functions +---------------------------- + +The following experimental command is available when the ``FunInd`` library has been loaded via ``Require Import FunInd``: + +.. cmd:: Function @ident {* @binder} { @decrease_annot } : @type := @term. + +This command can be seen as a generalization of ``Fixpoint``. It is actually a wrapper +for several ways of defining a function *and other useful related +objects*, namely: an induction principle that reflects the recursive +structure of the function (see Section :ref:`TODO-8.5.5-functional-induction`) and its fixpoint equality. +The meaning of this declaration is to define a function ident, +similarly to ``Fixpoint`. Like in ``Fixpoint``, the decreasing argument must +be given (unless the function is not recursive), but it might not +necessarily be *structurally* decreasing. The point of the {} annotation +is to name the decreasing argument *and* to describe which kind of +decreasing criteria must be used to ensure termination of recursive +calls. + +The ``Function`` construction also enjoys the ``with`` extension to define +mutually recursive definitions. However, this feature does not work +for non structurally recursive functions. + +See the documentation of functional induction (:ref:`TODO-8.5.5-functional-induction`) +and ``Functional Scheme`` (:ref:`TODO-13.2-functional-scheme`) for how to use +the induction principle to easily reason about the function. + +Remark: To obtain the right principle, it is better to put rigid +parameters of the function as first arguments. For example it is +better to define plus like this: + +.. coqtop:: reset none + + Require Import FunInd. -Depending on the annotation, the user is left with some proof -obligations that will be used to define the function. These proofs -are: proofs that each recursive call is actually decreasing with -respect to the given criteria, and (if the criteria is \texttt{wf}) a -proof that the ordering relation is well founded. +.. coqtop:: all -%Completer sur measure et wf - -Once proof obligations are discharged, the following objects are -defined: - -\begin{itemize} -\item The same objects as with the \texttt{struct}; -\item The lemma \texttt{\ident\_tcc} which collects all proof - obligations in one property; -\item The lemmas \texttt{\ident\_terminate} and \texttt{\ident\_F} - which is needed to be inlined during extraction of \ident. -\end{itemize} - - - -%Complete!! -The way this recursive function is defined is the subject of several -papers by Yves Bertot and Antonia Balaa on the one hand, and Gilles Barthe, -Julien Forest, David Pichardie, and Vlad Rusu on the other hand. - -%Exemples ok ici + Function plus (m n : nat) {struct n} : nat := + match n with + | 0 => m + | S p => S (plus m p) + end. -\bigskip - -\noindent {\bf Remark: } Proof obligations are presented as several -subgoals belonging to a Lemma {\ident}{\tt\_tcc}. % These subgoals are independent which means that in order to -% abort them you will have to abort each separately. - - - -%The decreasing argument cannot be dependent of another?? - -%Exemples faux ici -\end{Variants} - - -\section{Section mechanism -\index{Sections} -\label{Section}} +than like this: + +.. coqtop:: reset all + + Function plus (n m : nat) {struct n} : nat := + match n with + | 0 => m + | S p => S (plus p m) + end. + + +*Limitations* + +|term_0| must be built as a *pure pattern-matching tree* (:g:`match … with`) +with applications only *at the end* of each branch. + +Function does not support partial application of the function being +defined. Thus, the following example cannot be accepted due to the +presence of partial application of `wrong` in the body of +`wrong` : + +.. coqtop:: all + + Fail Function wrong (C:nat) : nat := + List.hd 0 (List.map wrong (C::nil)). + +For now, dependent cases are not treated for non structurally +terminating functions. + +.. exn:: The recursive argument must be specified +.. exn:: No argument name @ident +.. exn:: Cannot use mutual definition with well-founded recursion or measure + +.. warn:: Cannot define graph for @ident + + The generation of the graph relation (`R_ident`) used to compute the induction scheme of ident + raised a typing error. Only `ident` is defined; the induction scheme + will not be generated. This error happens generally when: + + - the definition uses pattern matching on dependent types, + which ``Function`` cannot deal with yet. + - the definition is not a *pattern-matching tree* as explained above. + +.. warn:: Cannot define principle(s) for @ident + + The generation of the graph relation (`R_ident`) succeeded but the induction principle + could not be built. Only `ident` is defined. Please report. + +.. warn:: Cannot build functional inversion principle + + `functional inversion` will not be available for the function. + +See also: :ref:`TODO-13.2-generating-ind-principles` and ref:`TODO-8.5.5-functional-induction` + +Depending on the ``{…}`` annotation, different definition mechanisms are +used by ``Function``. A more precise description is given below. + +.. cmdv:: Function @ident {* @binder } : @type := @term + + Defines the not recursive function `ident` as if declared with `Definition`. Moreover + the following are defined: + + + `ident_rect`, `ident_rec` and `ident_ind`, which reflect the pattern + matching structure of `term` (see the documentation of :ref:`TODO-1.3.3-Inductive`); + + The inductive `R_ident` corresponding to the graph of `ident` (silently); + + `ident_complete` and `ident_correct` which are inversion information + linking the function and its graph. + +.. cmdv:: Function @ident {* @binder } { struct @ident } : @type := @term + + Defines the structural recursive function `ident` as if declared with ``Fixpoint``. Moreover the following are defined: + + + The same objects as above; + + The fixpoint equation of `ident`: `ident_equation`. + +.. cmdv:: Function @ident {* @binder } { measure @term @ident } : @type := @term +.. cmdv:: Function @ident {* @binder } { wf @term @ident } : @type := @term + + Defines a recursive function by well-founded recursion. The module ``Recdef`` + of the standard library must be loaded for this feature. The ``{}`` + annotation is mandatory and must be one of the following: + + + ``{measure`` `term` `ident` ``}`` with `ident` being the decreasing argument + and `term` being a function from type of `ident` to ``nat`` for which + value on the decreasing argument decreases (for the ``lt`` order on ``nat``) + at each recursive call of `term`. Parameters of the function are + bound in `term`\ ; + + ``{wf`` `term` `ident` ``}`` with `ident` being the decreasing argument and + `term` an ordering relation on the type of `ident` (i.e. of type + `T`\ :math:`_{\sf ident}` → `T`\ :math:`_{\sf ident}` → ``Prop``) for which the decreasing argument + decreases at each recursive call of `term`. The order must be well-founded. + Parameters of the function are bound in `term`. + + Depending on the annotation, the user is left with some proof + obligations that will be used to define the function. These proofs + are: proofs that each recursive call is actually decreasing with + respect to the given criteria, and (if the criteria is `wf`) a proof + that the ordering relation is well-founded. Once proof obligations are + discharged, the following objects are defined: + + + The same objects as with the struct; + + The lemma `ident`\ :math:`_{\sf tcc}` which collects all proof obligations in one + property; + + The lemmas `ident`\ :math:`_{\sf terminate}` and `ident`\ :math:`_{\sf F}` which is needed to be inlined + during extraction of ident. + + The way this recursive function is defined is the subject of several + papers by Yves Bertot and Antonia Balaa on the one hand, and Gilles + Barthe, Julien Forest, David Pichardie, and Vlad Rusu on the other + hand. Remark: Proof obligations are presented as several subgoals + belonging to a Lemma `ident`\ :math:`_{\sf tcc}`. + + +Section mechanism +----------------- The sectioning mechanism can be used to to organize a proof in structured sections. Then local declarations become available (see -Section~\ref{Basic-definitions}). - -\subsection{\tt Section {\ident}\comindex{Section}} - -This command is used to open a section named {\ident}. - -%% Discontinued ? -%% \begin{Variants} -%% \comindex{Chapter} -%% \item{\tt Chapter {\ident}}\\ -%% Same as {\tt Section {\ident}} -%% \end{Variants} - -\subsection{\tt End {\ident} -\comindex{End}} - -This command closes the section named {\ident}. After closing of the -section, the local declarations (variables and local definitions) get -{\em discharged}, meaning that they stop being visible and that all -global objects defined in the section are generalized with respect to -the variables and local definitions they each depended on in the -section. - - -Here is an example : -\begin{coq_example} -Section s1. -Variables x y : nat. -Let y' := y. -Definition x' := S x. -Definition x'' := x' + y'. -Print x'. -End s1. -Print x'. -Print x''. -\end{coq_example} -Notice the difference between the value of {\tt x'} and {\tt x''} -inside section {\tt s1} and outside. - -\begin{ErrMsgs} -\item \errindex{This is not the last opened section} -\end{ErrMsgs} - -\begin{Remarks} -\item Most commands, like {\tt Hint}, {\tt Notation}, option management, ... -which appear inside a section are canceled when the -section is closed. -% see Section~\ref{LongNames} -%\item Usually all identifiers must be distinct. -%However, a name already used in a closed section (see \ref{Section}) -%can be reused. In this case, the old name is no longer accessible. - -% Obsolète -%\item A module implicitly open a section. Be careful not to name a -%module with an identifier already used in the module (see \ref{compiled}). -\end{Remarks} - -\input{RefMan-mod.v} - -\section{Libraries and qualified names} - -\subsection{Names of libraries -\label{Libraries} -\index{Libraries}} - -The theories developed in {\Coq} are stored in {\em library files} -which are hierarchically classified into {\em libraries} and {\em - sublibraries}. To express this hierarchy, library names are -represented by qualified identifiers {\qualid}, i.e. as list of -identifiers separated by dots (see Section~\ref{qualid}). For -instance, the library file {\tt Mult} of the standard {\Coq} library -{\tt Arith} is named {\tt Coq.Arith.Mult}. The identifier that starts -the name of a library is called a {\em library root}. All library -files of the standard library of {\Coq} have the reserved root {\tt Coq} -but library file names based on other roots can be obtained by using -{\Coq} commands ({\tt coqc}, {\tt coqtop}, {\tt coqdep}, \dots) options -{\tt -Q} or {\tt -R} (see Section~\ref{coqoptions}). Also, when an -interactive {\Coq} session starts, a library of root {\tt Top} is -started, unless option {\tt -top} is set (see -Section~\ref{coqoptions}). - -\subsection{Qualified names -\label{LongNames} -\index{Qualified identifiers} -\index{Absolute names}} +Section :ref:`TODO-1.3.2-Definitions`). + + +.. cmd:: Section @ident. + + This command is used to open a section named `ident`. + + +.. cmd:: End @ident. + + This command closes the section named `ident`. After closing of the + section, the local declarations (variables and local definitions) get + *discharged*, meaning that they stop being visible and that all global + objects defined in the section are generalized with respect to the + variables and local definitions they each depended on in the section. + + .. example:: + + .. coqtop:: all + + Section s1. + + Variables x y : nat. + + Let y' := y. + + Definition x' := S x. + + Definition x'' := x' + y'. + + Print x'. + + End s1. + + Print x'. + + Print x''. + + Notice the difference between the value of `x’` and `x’’` inside section + `s1` and outside. + + .. exn:: This is not the last opened section + +**Remarks:** + +#. Most commands, like ``Hint``, ``Notation``, option management, … which + appear inside a section are canceled when the section is closed. + + +Module system +------------- + +The module system provides a way of packaging related elements +together, as well as a means of massive abstraction. + + .. productionlist:: modules + module_type : qualid + : | `module_type` with Definition qualid := term + : | `module_type` with Module qualid := qualid + : | qualid qualid … qualid + : | !qualid qualid … qualid + module_binding : ( [Import|Export] ident … ident : module_type ) + module_bindings : `module_binding` … `module_binding` + module_expression : qualid … qualid + : | !qualid … qualid + + Syntax of modules + +In the syntax of module application, the ! prefix indicates that any +`Inline` directive in the type of the functor arguments will be ignored +(see :ref:`named_module_type` below). + + +.. cmd:: Module @ident. + + This command is used to start an interactive module named `ident`. + +.. cmdv:: Module @ident {* @module_binding}. + + Starts an interactive functor with + parameters given by module_bindings. + +.. cmdv:: Module @ident : @module_type. + + Starts an interactive module specifying its module type. + +.. cmdv:: Module @ident {* @module_binding} : @module_type. + + Starts an interactive functor with parameters given by the list of `module binding`, and output module + type `module_type`. + +.. cmdv:: Module @ident <: {+<: @module_type }. + + Starts an interactive module satisfying each `module_type`. + + .. cmdv:: Module @ident {* @module_binding} <: {+<; @module_type }. + + Starts an interactive functor with parameters given by the list of `module_binding`. The output module type + is verified against each `module_type`. + +.. cmdv:: Module [ Import | Export ]. + + Behaves like ``Module``, but automatically imports or exports the module. + +Reserved commands inside an interactive module +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. cmd:: Include @module. + + Includes the content of module in the current + interactive module. Here module can be a module expression or a module + type expression. If module is a high-order module or module type + expression then the system tries to instantiate module by the current + interactive module. + +.. cmd:: Include {+<+ @module}. + + is a shortcut for the commands ``Include`` `module` for each `module`. + +.. cmd:: End @ident. + + This command closes the interactive module `ident`. If the module type + was given the content of the module is matched against it and an error + is signaled if the matching fails. If the module is basic (is not a + functor) its components (constants, inductive types, submodules etc.) + are now available through the dot notation. + + .. exn:: No such label @ident + + .. exn:: Signature components for label @ident do not match + + .. exn:: This is not the last opened module + +.. cmd:: Module @ident := @module_expression. + + This command defines the module identifier `ident` to be equal + to `module_expression`. + + .. cmdv:: Module @ident {* @module_binding} := @module_expression. + + Defines a functor with parameters given by the list of `module_binding` and body `module_expression`. + + .. cmdv:: Module @ident {* @module_binding} : @module_type := @module_expression. + + Defines a functor with parameters given by the list of `module_binding` (possibly none), and output module type `module_type`, + with body `module_expression`. + + .. cmdv:: Module @ident {* @module_binding} <: {+<: @module_type} := @module_expression. + + Defines a functor with parameters given by module_bindings (possibly none) with body `module_expression`. + The body is checked against each |module_type_i|. + + .. cmdv:: Module @ident {* @module_binding} := {+<+ @module_expression}. + + is equivalent to an interactive module where each `module_expression` is included. + +.. _named_module_type: + +.. cmd:: Module Type @ident. + +This command is used to start an interactive module type `ident`. + + .. cmdv:: Module Type @ident {* @module_binding}. + + Starts an interactive functor type with parameters given by `module_bindings`. + + +Reserved commands inside an interactive module type: +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. cmd:: Include @module. + + Same as ``Include`` inside a module. + +.. cmd:: Include {+<+ @module}. + + is a shortcut for the command ``Include`` `module` for each `module`. + +.. cmd:: @assumption_keyword Inline @assums. + + The instance of this assumption will be automatically expanded at functor application, except when + this functor application is prefixed by a ``!`` annotation. + +.. cmd:: End @ident. + + This command closes the interactive module type `ident`. + + .. exn:: This is not the last opened module type + +.. cmd:: Module Type @ident := @module_type. + + Defines a module type `ident` equal to `module_type`. + + .. cmdv:: Module Type @ident {* @module_binding} := @module_type. + + Defines a functor type `ident` specifying functors taking arguments `module_bindings` and + returning `module_type`. + + .. cmdv:: Module Type @ident {* @module_binding} := {+<+ @module_type }. + + is equivalent to an interactive module type were each `module_type` is included. + +.. cmd:: Declare Module @ident : @module_type. + + Declares a module `ident` of type `module_type`. + + .. cmdv:: Declare Module @ident {* @module_binding} : @module_type. + + Declares a functor with parameters given by the list of `module_binding` and output module type + `module_type`. + +.. example:: + + Let us define a simple module. + + .. coqtop:: all + + Module M. + + Definition T := nat. + + Definition x := 0. + + Definition y : bool. + + exact true. + + Defined. + + End M. + +Inside a module one can define constants, prove theorems and do any +other things that can be done in the toplevel. Components of a closed +module can be accessed using the dot notation: + +.. coqtop:: all + + Print M.x. + +A simple module type: + +.. coqtop:: all + + Module Type SIG. + + Parameter T : Set. + + Parameter x : T. + + End SIG. + +Now we can create a new module from M, giving it a less precise +specification: the y component is dropped as well as the body of x. + +.. coqtop:: all + + Module N : SIG with Definition T := nat := M. + + Print N.T. + + Print N.x. + + Fail Print N.y. + +.. reset to remove N (undo in last coqtop block doesn't seem to do that), invisibly redefine M, SIG +.. coqtop:: none reset + + Module M. + + Definition T := nat. + + Definition x := 0. + + Definition y : bool. + + exact true. + + Defined. + + End M. + + Module Type SIG. + + Parameter T : Set. + + Parameter x : T. + + End SIG. + +The definition of ``N`` using the module type expression ``SIG`` with +``Definition T := nat`` is equivalent to the following one: + +.. coqtop:: all + + Module Type SIG'. + + Definition T : Set := nat. + + Parameter x : T. + + End SIG'. + + Module N : SIG' := M. + +If we just want to be sure that the our implementation satisfies a +given module type without restricting the interface, we can use a +transparent constraint + +.. coqtop:: all + + Module P <: SIG := M. + + Print P.y. + +Now let us create a functor, i.e. a parametric module + +.. coqtop:: all + + Module Two (X Y: SIG). + + Definition T := (X.T * Y.T)%type. + + Definition x := (X.x, Y.x). + + End Two. + +and apply it to our modules and do some computations: + +.. coqtop:: all + + Module Q := Two M N. + + Eval compute in (fst Q.x + snd Q.x). + +In the end, let us define a module type with two sub-modules, sharing +some of the fields and give one of its possible implementations: + +.. coqtop:: all + + Module Type SIG2. + + Declare Module M1 : SIG. + + Module M2 <: SIG. + + Definition T := M1.T. + + Parameter x : T. + + End M2. + + End SIG2. + + Module Mod <: SIG2. + + Module M1. + + Definition T := nat. + + Definition x := 1. + + End M1. + + Module M2 := M. + + End Mod. + +Notice that ``M`` is a correct body for the component ``M2`` since its ``T`` +component is equal ``nat`` and hence ``M1.T`` as specified. + +**Remarks:** + +#. Modules and module types can be nested components of each other. +#. One can have sections inside a module or a module type, but not a + module or a module type inside a section. +#. Commands like ``Hint`` or ``Notation`` can also appear inside modules and + module types. Note that in case of a module definition like: + +:: + + Module N : SIG := M. + +or:: + + Module N : SIG. … End N. + +hints and the like valid for ``N`` are not those defined in ``M`` (or the module body) but the ones defined +in ``SIG``. + + +.. _import_qualid: + +.. cmd:: Import @qualid. + + If `qualid` denotes a valid basic module (i.e. its module type is a + signature), makes its components available by their short names. + +.. example:: + + .. coqtop:: reset all + + Module Mod. + + Definition T:=nat. + + Check T. + + End Mod. + + Check Mod.T. + + Fail Check T. + + Import Mod. + + Check T. + +Some features defined in modules are activated only when a module is +imported. This is for instance the case of notations (see :ref:`TODO-12.1-Notations`). + +Declarations made with the Local flag are never imported by theImport +command. Such declarations are only accessible through their fully +qualified name. + +.. example:: + + .. coqtop:: all + + Module A. + + Module B. + + Local Definition T := nat. + + End B. + + End A. + + Import A. + + Fail Check B.T. + + .. cmdv:: Export @qualid + + When the module containing the command Export qualid + is imported, qualid is imported as well. + + .. exn:: @qualid is not a module + + .. warn:: Trying to mask the absolute name @qualid! + +.. cmd:: Print Module @ident. + + Prints the module type and (optionally) the body of the module `ident`. + +.. cmd:: Print Module Type @ident. + + Prints the module type corresponding to `ident`. + +.. opt:: Short Module Printing + + This option (off by default) disables the printing of the types of fields, + leaving only their names, for the commands ``Print Module`` and ``Print Module Type``. + +.. cmd:: Locate Module @qualid. + + Prints the full name of the module `qualid`. + +Libraries and qualified names +--------------------------------- + +Names of libraries +~~~~~~~~~~~~~~~~~~ + +The theories developed in |Coq| are stored in *library files* which are +hierarchically classified into *libraries* and *sublibraries*. To +express this hierarchy, library names are represented by qualified +identifiers qualid, i.e. as list of identifiers separated by dots (see +:ref:`TODO-1.2.3-identifiers`). For instance, the library file ``Mult`` of the standard +|Coq| library ``Arith`` is named ``Coq.Arith.Mult``. The identifier that starts +the name of a library is called a *library root*. All library files of +the standard library of |Coq| have the reserved root |Coq| but library +file names based on other roots can be obtained by using |Coq| commands +(coqc, coqtop, coqdep, …) options ``-Q`` or ``-R`` (see :ref:`TODO-14.3.3-command-line-options`). +Also, when an interactive |Coq| session starts, a library of root ``Top`` is +started, unless option ``-top`` or ``-notop`` is set (see :ref:`TODO-14.3.3-command-line-options`). + + +Qualified names +~~~~~~~~~~~~~~~ Library files are modules which possibly contain submodules which eventually contain constructions (axioms, parameters, definitions, -lemmas, theorems, remarks or facts). The {\em absolute name}, or {\em -full name}, of a construction in some library file is a qualified +lemmas, theorems, remarks or facts). The *absolute name*, or *full +name*, of a construction in some library file is a qualified identifier starting with the logical name of the library file, followed by the sequence of submodules names encapsulating the construction and ended by the proper name of the construction. -Typically, the absolute name {\tt Coq.Init.Logic.eq} denotes Leibniz' -equality defined in the module {\tt Logic} in the sublibrary {\tt -Init} of the standard library of \Coq. - -The proper name that ends the name of a construction is the {\it short -name} (or sometimes {\it base name}) of the construction (for -instance, the short name of {\tt Coq.Init.Logic.eq} is {\tt eq}). Any -partial suffix of the absolute name is a {\em partially qualified name} -(e.g. {\tt Logic.eq} is a partially qualified name for {\tt -Coq.Init.Logic.eq}). Especially, the short name of a construction is -its shortest partially qualified name. - -{\Coq} does not accept two constructions (definition, theorem, ...) -with the same absolute name but different constructions can have the -same short name (or even same partially qualified names as soon as the -full names are different). - -Notice that the notion of absolute, partially qualified and -short names also applies to library file names. - -\paragraph{Visibility} - -{\Coq} maintains a table called {\it name table} which maps partially -qualified names of constructions to absolute names. This table is -updated by the commands {\tt Require} (see \ref{Require}), {\tt -Import} and {\tt Export} (see \ref{Import}) and also each time a new -declaration is added to the context. An absolute name is called {\it -visible} from a given short or partially qualified name when this -latter name is enough to denote it. This means that the short or -partially qualified name is mapped to the absolute name in {\Coq} name -table. Definitions flagged as {\tt Local} are only accessible with their -fully qualified name (see \ref{Definition}). +Typically, the absolute name ``Coq.Init.Logic.eq`` denotes Leibniz’ +equality defined in the module Logic in the sublibrary ``Init`` of the +standard library of |Coq|. + +The proper name that ends the name of a construction is the short name +(or sometimes base name) of the construction (for instance, the short +name of ``Coq.Init.Logic.eq`` is ``eq``). Any partial suffix of the absolute +name is a *partially qualified name* (e.g. ``Logic.eq`` is a partially +qualified name for ``Coq.Init.Logic.eq``). Especially, the short name of a +construction is its shortest partially qualified name. + +|Coq| does not accept two constructions (definition, theorem, …) with +the same absolute name but different constructions can have the same +short name (or even same partially qualified names as soon as the full +names are different). + +Notice that the notion of absolute, partially qualified and short +names also applies to library file names. + +**Visibility** + +|Coq| maintains a table called the name table which maps partially qualified +names of constructions to absolute names. This table is updated by the +commands ``Require`` (see :ref:`TODO-6.5.1-Require`), Import and Export (see :ref:`import_qualid`) and +also each time a new declaration is added to the context. An absolute +name is called visible from a given short or partially qualified name +when this latter name is enough to denote it. This means that the +short or partially qualified name is mapped to the absolute name in +|Coq| name table. Definitions flagged as Local are only accessible with +their fully qualified name (see :ref:`TODO-1.3.2-definitions`). It may happen that a visible name is hidden by the short name or a qualified name of another construction. In this case, the name that @@ -1150,1003 +1353,1004 @@ has been hidden must be referred to using one more level of qualification. To ensure that a construction always remains accessible, absolute names can never be hidden. -Examples: -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Check 0. -Definition nat := bool. -Check 0. -Check Datatypes.nat. -Locate nat. -\end{coq_example} +.. example:: + + .. coqtop:: all + + Check 0. + + Definition nat := bool. -\SeeAlso Command {\tt Locate} in Section~\ref{Locate} and {\tt Locate -Library} in Section~\ref{Locate Library}. + Check 0. -\subsection{Libraries and filesystem\label{loadpath}\index{Loadpath} -\index{Physical paths} \index{Logical paths}} + Check Datatypes.nat. + + Locate nat. + +See also: Command Locate in :ref:`TODO-6.3.10-locate-qualid` and Locate Library in +:ref:`TODO-6.6.11-locate-library`. + + +Libraries and filesystem +~~~~~~~~~~~~~~~~~~~~~~~~ Please note that the questions described here have been subject to -redesign in Coq v8.5. Former versions of Coq use the same terminology +redesign in |Coq| v8.5. Former versions of |Coq| use the same terminology to describe slightly different things. -Compiled files (\texttt{.vo} and \texttt{.vio}) store sub-libraries. In -order to refer to them inside {\Coq}, a translation from file-system -names to {\Coq} names is needed. In this translation, names in the -file system are called {\em physical} paths while {\Coq} names are -contrastingly called {\em logical} names. +Compiled files (``.vo`` and ``.vio``) store sub-libraries. In order to refer +to them inside |Coq|, a translation from file-system names to |Coq| names +is needed. In this translation, names in the file system are called +*physical* paths while |Coq| names are contrastingly called *logical* +names. -A logical prefix {\tt Lib} can be associated to a physical path -\textrm{\textsl{path}} using the command line option {\tt -Q} -\textrm{\textsl{path}} {\tt Lib}. All subfolders of {\textsl{path}} are -recursively associated to the logical path {\tt Lib} extended with the +A logical prefix Lib can be associated to a physical pathpath using +the command line option ``-Q`` `path` ``Lib``. All subfolders of path are +recursively associated to the logical path ``Lib`` extended with the corresponding suffix coming from the physical path. For instance, the -folder {\tt path/fOO/Bar} maps to {\tt Lib.fOO.Bar}. Subdirectories -corresponding to invalid {\Coq} identifiers are skipped, and, by -convention, subdirectories named {\tt CVS} or {\tt \_darcs} are -skipped too. - -Thanks to this mechanism, {\texttt{.vo}} files are made available through the -logical name of the folder they are in, extended with their own basename. For -example, the name associated to the file {\tt path/fOO/Bar/File.vo} is -{\tt Lib.fOO.Bar.File}. The same caveat applies for invalid identifiers. -When compiling a source file, the {\texttt{.vo}} file stores its logical name, -so that an error is issued if it is loaded with the wrong loadpath afterwards. - -Some folders have a special status and are automatically put in the path. -{\Coq} commands associate automatically a logical path to files -in the repository trees rooted at the directory from where the command -is launched, \textit{coqlib}\texttt{/user-contrib/}, the directories -listed in the \verb:$COQPATH:, \verb:${XDG_DATA_HOME}/coq/: and -\verb:${XDG_DATA_DIRS}/coq/: environment variables (see -\url{http://standards.freedesktop.org/basedir-spec/basedir-spec-latest.html}) -with the same physical-to-logical translation and with an empty logical prefix. - -The command line option \texttt{-R} is a variant of \texttt{-Q} which has the -strictly same behavior regarding loadpaths, but which also makes the -corresponding \texttt{.vo} files available through their short names in a -way not unlike the {\tt Import} command (see~{\ref{Import}}). For instance, -\texttt{-R} \textrm{\textsl{path}} \texttt{Lib} associates to the file -\texttt{path/fOO/Bar/File.vo} the logical name \texttt{Lib.fOO.Bar.File}, but -allows this file to be accessed through the short names \texttt{fOO.Bar.File}, -\texttt{Bar.File} and \texttt{File}. If several files with identical base name -are present in different subdirectories of a recursive loadpath, which of -these files is found first may be system-dependent and explicit -qualification is recommended. The {\tt From} argument of the {\tt Require} -command can be used to bypass the implicit shortening by providing an absolute -root to the required file (see~\ref{Require}). - -There also exists another independent loadpath mechanism attached to {\ocaml} -object files (\texttt{.cmo} or \texttt{.cmxs}) rather than {\Coq} object files -as described above. The {\ocaml} loadpath is managed using the option -\texttt{-I path} (in the {\ocaml} world, there is neither a notion of logical -name prefix nor a way to access files in subdirectories of \texttt{path}). -See the command \texttt{Declare ML Module} in Section~\ref{compiled} to -understand the need of the {\ocaml} loadpath. - -See Section~\ref{coqoptions} for a more general view over the {\Coq} -command line options. - -%% \paragraph{The special case of remarks and facts} -%% -%% In contrast with definitions, lemmas, theorems, axioms and parameters, -%% the absolute name of remarks includes the segment of sections in which -%% it is defined. Concretely, if a remark {\tt R} is defined in -%% subsection {\tt S2} of section {\tt S1} in module {\tt M}, then its -%% absolute name is {\tt M.S1.S2.R}. The same for facts, except that the -%% name of the innermost section is dropped from the full name. Then, if -%% a fact {\tt F} is defined in subsection {\tt S2} of section {\tt S1} -%% in module {\tt M}, then its absolute name is {\tt M.S1.F}. - -\section{Implicit arguments -\index{Implicit arguments} -\label{Implicit Arguments}} +folder ``path/fOO/Bar`` maps to ``Lib.fOO.Bar``. Subdirectories corresponding +to invalid |Coq| identifiers are skipped, and, by convention, +subdirectories named ``CVS`` or ``_darcs`` are skipped too. + +Thanks to this mechanism, .vo files are made available through the +logical name of the folder they are in, extended with their own +basename. For example, the name associated to the file +``path/fOO/Bar/File.vo`` is ``Lib.fOO.Bar.File``. The same caveat applies for +invalid identifiers. When compiling a source file, the ``.vo`` file stores +its logical name, so that an error is issued if it is loaded with the +wrong loadpath afterwards. + +Some folders have a special status and are automatically put in the +path. |Coq| commands associate automatically a logical path to files in +the repository trees rooted at the directory from where the command is +launched, coqlib/user-contrib/, the directories listed in the +`$COQPATH`, `${XDG_DATA_HOME}/coq/` and `${XDG_DATA_DIRS}/coq/` +environment variables (see`http://standards.freedesktop.org/basedir- +spec/basedir-spec-latest.html`_) with the same physical-to-logical +translation and with an empty logical prefix. + +The command line option ``-R`` is a variant of ``-Q`` which has the strictly +same behavior regarding loadpaths, but which also makes the +corresponding ``.vo`` files available through their short names in a way +not unlike the ``Import`` command (see :ref:`import_qualid`). For instance, ``-R`` `path` ``Lib`` +associates to the ``filepath/fOO/Bar/File.vo`` the logical name +``Lib.fOO.Bar.File``, but allows this file to be accessed through the +short names ``fOO.Bar.File,Bar.File`` and ``File``. If several files with +identical base name are present in different subdirectories of a +recursive loadpath, which of these files is found first may be system- +dependent and explicit qualification is recommended. The ``From`` argument +of the ``Require`` command can be used to bypass the implicit shortening +by providing an absolute root to the required file (see :ref:`TODO-6.5.1-require-qualid`). + +There also exists another independent loadpath mechanism attached to +OCaml object files (``.cmo`` or ``.cmxs``) rather than |Coq| object +files as described above. The OCaml loadpath is managed using +the option ``-I`` `path` (in the OCaml world, there is neither a +notion of logical name prefix nor a way to access files in +subdirectories of path). See the command ``Declare`` ``ML`` ``Module`` in +:ref:`TODO-6.5-compiled-files` to understand the need of the OCaml loadpath. + +See :ref:`TODO-14.3.3-command-line-options` for a more general view over the |Coq| command +line options. + + +Implicit arguments +------------------ An implicit argument of a function is an argument which can be inferred from contextual knowledge. There are different kinds of -implicit arguments that can be considered implicit in different -ways. There are also various commands to control the setting or the +implicit arguments that can be considered implicit in different ways. +There are also various commands to control the setting or the inference of implicit arguments. -\subsection{The different kinds of implicit arguments} -\subsubsection{Implicit arguments inferable from the knowledge of other -arguments of a function} +The different kinds of implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Implicit arguments inferable from the knowledge of other arguments of a function +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ The first kind of implicit arguments covers the arguments that are inferable from the knowledge of the type of other arguments of the function, or of the type of the surrounding context of the -application. Especially, such implicit arguments correspond to +application. Especially, such implicit arguments correspond to parameters dependent in the type of the function. Typical implicit -arguments are the type arguments in polymorphic functions. -There are several kinds of such implicit arguments. +arguments are the type arguments in polymorphic functions. There are +several kinds of such implicit arguments. + +**Strict Implicit Arguments** -\paragraph{Strict Implicit Arguments.} An implicit argument can be either strict or non strict. An implicit -argument is said {\em strict} if, whatever the other arguments of the +argument is said to be *strict* if, whatever the other arguments of the function are, it is still inferable from the type of some other argument. Technically, an implicit argument is strict if it corresponds to a parameter which is not applied to a variable which -itself is another parameter of the function (since this parameter -may erase its arguments), not in the body of a {\tt match}, and not -itself applied or matched against patterns (since the original -form of the argument can be lost by reduction). +itself is another parameter of the function (since this parameter may +erase its arguments), not in the body of a match, and not itself +applied or matched against patterns (since the original form of the +argument can be lost by reduction). For instance, the first argument of -\begin{quote} -\verb|cons: forall A:Set, A -> list A -> list A| -\end{quote} -in module {\tt List.v} is strict because {\tt list} is an inductive -type and {\tt A} will always be inferable from the type {\tt -list A} of the third argument of {\tt cons}. -On the contrary, the second argument of a term of type -\begin{quote} -\verb|forall P:nat->Prop, forall n:nat, P n -> ex nat P| -\end{quote} +:: + + cons: forall A:Set, A -> list A -> list A + +in module ``List.v`` is strict because :g:`list` is an inductive type and :g:`A` +will always be inferable from the type :g:`list A` of the third argument of +:g:`cons`. On the contrary, the second argument of a term of type +:: + + forall P:nat->Prop, forall n:nat, P n -> ex nat P + is implicit but not strict, since it can only be inferred from the -type {\tt P n} of the third argument and if {\tt P} is, e.g., {\tt -fun \_ => True}, it reduces to an expression where {\tt n} does not -occur any longer. The first argument {\tt P} is implicit but not -strict either because it can only be inferred from {\tt P n} and {\tt -P} is not canonically inferable from an arbitrary {\tt n} and the -normal form of {\tt P n} (consider e.g. that {\tt n} is {\tt 0} and -the third argument has type {\tt True}, then any {\tt P} of the form -{\tt fun n => match n with 0 => True | \_ => \mbox{\em anything} end} would -be a solution of the inference problem). - -\paragraph{Contextual Implicit Arguments.} -An implicit argument can be {\em contextual} or not. An implicit -argument is said {\em contextual} if it can be inferred only from the -knowledge of the type of the context of the current expression. For -instance, the only argument of -\begin{quote} -\verb|nil : forall A:Set, list A| -\end{quote} -is contextual. Similarly, both arguments of a term of type -\begin{quote} -\verb|forall P:nat->Prop, forall n:nat, P n \/ n = 0| -\end{quote} -are contextual (moreover, {\tt n} is strict and {\tt P} is not). - -\paragraph{Reversible-Pattern Implicit Arguments.} +type :g:`P n` of the third argument and if :g:`P` is, e.g., :g:`fun _ => True`, it +reduces to an expression where ``n`` does not occur any longer. The first +argument :g:`P` is implicit but not strict either because it can only be +inferred from :g:`P n` and :g:`P` is not canonically inferable from an arbitrary +:g:`n` and the normal form of :g:`P n`. Consider, e.g., that :g:`n` is :math:`0` and the third +argument has type :g:`True`, then any :g:`P` of the form +:: + + fun n => match n with 0 => True | _ => anything end + +would be a solution of the inference problem. + +**Contextual Implicit Arguments** + +An implicit argument can be *contextual* or not. An implicit argument +is said *contextual* if it can be inferred only from the knowledge of +the type of the context of the current expression. For instance, the +only argument of:: + + nil : forall A:Set, list A` + +is contextual. Similarly, both arguments of a term of type:: + + forall P:nat->Prop, forall n:nat, P n \/ n = 0 + +are contextual (moreover, :g:`n` is strict and :g:`P` is not). + +**Reversible-Pattern Implicit Arguments** + There is another class of implicit arguments that can be reinferred -unambiguously if all the types of the remaining arguments are -known. This is the class of implicit arguments occurring in the type -of another argument in position of reversible pattern, which means it -is at the head of an application but applied only to uninstantiated -distinct variables. Such an implicit argument is called {\em -reversible-pattern implicit argument}. A typical example is the -argument {\tt P} of {\tt nat\_rec} in -\begin{quote} -{\tt nat\_rec : forall P : nat -> Set, - P 0 -> (forall n : nat, P n -> P (S n)) -> forall x : nat, P x}. -\end{quote} -({\tt P} is reinferable by abstracting over {\tt n} in the type {\tt P n}). - -See Section~\ref{SetReversiblePatternImplicit} for the automatic declaration -of reversible-pattern implicit arguments. - -\subsubsection{Implicit arguments inferable by resolution} - -This corresponds to a class of non dependent implicit arguments that +unambiguously if all the types of the remaining arguments are known. +This is the class of implicit arguments occurring in the type of +another argument in position of reversible pattern, which means it is +at the head of an application but applied only to uninstantiated +distinct variables. Such an implicit argument is called *reversible- +pattern implicit argument*. A typical example is the argument :g:`P` of +nat_rec in +:: + + nat_rec : forall P : nat -> Set, P 0 -> + (forall n : nat, P n -> P (S n)) -> forall x : nat, P x + +(:g:`P` is reinferable by abstracting over :g:`n` in the type :g:`P n`). + +See :ref:`controlling-rev-pattern-implicit-args` for the automatic declaration of reversible-pattern +implicit arguments. + +Implicit arguments inferable by resolution +++++++++++++++++++++++++++++++++++++++++++ + +This corresponds to a class of non-dependent implicit arguments that are solved based on the structure of their type only. -\subsection{Maximal or non maximal insertion of implicit arguments} + +Maximal or non maximal insertion of implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case a function is partially applied, and the next argument to be -applied is an implicit argument, two disciplines are applicable. In the -first case, the function is considered to have no arguments furtherly: -one says that the implicit argument is not maximally inserted. In -the second case, the function is considered to be implicitly applied -to the implicit arguments it is waiting for: one says that the -implicit argument is maximally inserted. +applied is an implicit argument, two disciplines are applicable. In +the first case, the function is considered to have no arguments +furtherly: one says that the implicit argument is not maximally +inserted. In the second case, the function is considered to be +implicitly applied to the implicit arguments it is waiting for: one +says that the implicit argument is maximally inserted. Each implicit argument can be declared to have to be inserted maximally or non maximally. This can be governed argument per argument -by the command {\tt Implicit Arguments} (see~\ref{ImplicitArguments}) -or globally by the command {\tt Set Maximal Implicit Insertion} -(see~\ref{SetMaximalImplicitInsertion}). See also -Section~\ref{PrintImplicit}. +by the command ``Implicit Arguments`` (see Section :ref:`declare-implicit-args`) or globally by the +command ``Set Maximal Implicit Insertion`` (see Section :ref:`controlling-insertion-implicit-args`). +See also :ref:`displaying-implicit-args`. + -\subsection{Casual use of implicit arguments} +Casual use of implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In a given expression, if it is clear that some argument of a function can be inferred from the type of the other arguments, the user can -force the given argument to be guessed by replacing it by ``{\tt \_}''. If +force the given argument to be guessed by replacing it by “_”. If possible, the correct argument will be automatically generated. -\begin{ErrMsgs} +.. exn:: Cannot infer a term for this placeholder. -\item \errindex{Cannot infer a term for this placeholder} + |Coq| was not able to deduce an instantiation of a “_”. - {\Coq} was not able to deduce an instantiation of a ``{\tt \_}''. +.. _declare-implicit-args: -\end{ErrMsgs} - -\subsection{Declaration of implicit arguments -\comindex{Arguments}} -\label{ImplicitArguments} +Declaration of implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case one wants that some arguments of a given object (constant, inductive types, constructors, assumptions, local or not) are always -inferred by Coq, one may declare once and for all which are the expected -implicit arguments of this object. There are two ways to do this, -a priori and a posteriori. +inferred by |Coq|, one may declare once and for all which are the +expected implicit arguments of this object. There are two ways to do +this, *a priori* and *a posteriori*. + -\subsubsection{Implicit Argument Binders} +Implicit Argument Binders ++++++++++++++++++++++++++ In the first setting, one wants to explicitly give the implicit -arguments of a declared object as part of its definition. To do this, one has -to surround the bindings of implicit arguments by curly braces: -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Definition id {A : Type} (x : A) : A := x. -\end{coq_example} - -This automatically declares the argument {\tt A} of {\tt id} as a -maximally inserted implicit argument. One can then do as-if the argument -was absent in every situation but still be able to specify it if needed: -\begin{coq_example} -Definition compose {A B C} (g : B -> C) (f : A -> B) := - fun x => g (f x). -Goal forall A, compose id id = id (A:=A). -\end{coq_example} - -The syntax is supported in all top-level definitions: {\tt Definition}, -{\tt Fixpoint}, {\tt Lemma} and so on. For (co-)inductive datatype +arguments of a declared object as part of its definition. To do this, +one has to surround the bindings of implicit arguments by curly +braces: + +.. coqtop:: all + + Definition id {A : Type} (x : A) : A := x. + +This automatically declares the argument A of id as a maximally +inserted implicit argument. One can then do as-if the argument was +absent in every situation but still be able to specify it if needed: + +.. coqtop:: all + + Definition compose {A B C} (g : B -> C) (f : A -> B) := fun x => g (f x). + + Goal forall A, compose id id = id (A:=A). + + +The syntax is supported in all top-level definitions: +``Definition``, ``Fixpoint``, ``Lemma`` and so on. For (co-)inductive datatype declarations, the semantics are the following: an inductive parameter declared as an implicit argument need not be repeated in the inductive definition but will become implicit for the constructors of the inductive only, not the inductive type itself. For example: -\begin{coq_example} -Inductive list {A : Type} : Type := -| nil : list -| cons : A -> list -> list. -Print list. -\end{coq_example} +.. coqtop:: all + + Inductive list {A : Type} : Type := + | nil : list + | cons : A -> list -> list. + + Print list. One can always specify the parameter if it is not uniform using the usual implicit arguments disambiguation syntax. -\subsubsection{Declaring Implicit Arguments} -To set implicit arguments a posteriori, one can use the -command: -\begin{quote} -\tt Arguments {\qualid} \nelist{\possiblybracketedident}{} -\end{quote} -where the list of {\possiblybracketedident} is a prefix of the list of arguments -of {\qualid} where the ones to be declared implicit are surrounded by square -brackets and the ones to be declared as maximally inserted implicits are -surrounded by curly braces. +Declaring Implicit Arguments +++++++++++++++++++++++++++++ + +To set implicit arguments *a posteriori*, one can use the command: + +.. cmd:: Arguments @qualid {* @possibly_bracketed_ident }. + +where the list of `possibly_bracketed_ident` is a prefix of the list of +arguments of `qualid` where the ones to be declared implicit are +surrounded by square brackets and the ones to be declared as maximally +inserted implicits are surrounded by curly braces. -After the above declaration is issued, implicit arguments can just (and -have to) be skipped in any expression involving an application of -{\qualid}. +After the above declaration is issued, implicit arguments can just +(and have to) be skipped in any expression involving an application +of `qualid`. Implicit arguments can be cleared with the following syntax: -\begin{quote} -{\tt Arguments {\qualid} : clear implicits -\comindex{Arguments}} -\end{quote} - -\begin{Variants} -\item {\tt Global Arguments {\qualid} \nelist{\possiblybracketedident}{} -\comindex{Global Arguments}} - -Tell to recompute the implicit arguments of {\qualid} after ending of -the current section if any, enforcing the implicit arguments known -from inside the section to be the ones declared by the command. - -\item {\tt Local Arguments {\qualid} \nelist{\possiblybracketedident}{} -\comindex{Local Arguments}} - -When in a module, tell not to activate the implicit arguments of -{\qualid} declared by this command to contexts that require the -module. - -\item {\tt \zeroone{Global {\sl |} Local} Arguments {\qualid} \sequence{\nelist{\possiblybracketedident}{}}{,}} - -For names of constants, inductive types, constructors, lemmas which -can only be applied to a fixed number of arguments (this excludes for -instance constants whose type is polymorphic), multiple -implicit arguments decflarations can be given. -Depending on the number of arguments {\qualid} is applied -to in practice, the longest applicable list of implicit arguments is -used to select which implicit arguments are inserted. - -For printing, the omitted arguments are the ones of the longest list -of implicit arguments of the sequence. - -\end{Variants} - -\Example -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Inductive list (A:Type) : Type := - | nil : list A - | cons : A -> list A -> list A. -\end{coq_example*} -\begin{coq_example} -Check (cons nat 3 (nil nat)). -Arguments cons [A] _ _. -Arguments nil [A]. -Check (cons 3 nil). -Fixpoint map (A B:Type) (f:A->B) (l:list A) : list B := - match l with nil => nil | cons a t => cons (f a) (map A B f t) end. -Fixpoint length (A:Type) (l:list A) : nat := - match l with nil => 0 | cons _ m => S (length A m) end. -Arguments map [A B] f l. -Arguments length {A} l. (* A has to be maximally inserted *) -Check (fun l:list (list nat) => map length l). -Arguments map [A B] f l, [A] B f l, A B f l. -Check (fun l => map length l = map (list nat) nat length l). -\end{coq_example} - -\Rem To know which are the implicit arguments of an object, use the command -{\tt Print Implicit} (see \ref{PrintImplicit}). - -\subsection{Automatic declaration of implicit arguments} - -{\Coq} can also automatically detect what are the implicit arguments -of a defined object. The command is just -\begin{quote} -{\tt Arguments {\qualid} : default implicits -\comindex{Arguments}} -\end{quote} +.. cmd:: Arguments @qualid : clear implicits. + +.. cmdv:: Global Arguments @qualid {* @possibly_bracketed_ident } + + Says to recompute the implicit arguments of + `qualid` after ending of the current section if any, enforcing the + implicit arguments known from inside the section to be the ones + declared by the command. + +.. cmdv:: Local Arguments @qualid {* @possibly_bracketed_ident }. + + When in a module, tell not to activate the + implicit arguments ofqualid declared by this command to contexts that + require the module. + +.. cmdv:: {? Global | Local } Arguments @qualid {*, {+ @possibly_bracketed_ident } }. + + For names of constants, inductive types, + constructors, lemmas which can only be applied to a fixed number of + arguments (this excludes for instance constants whose type is + polymorphic), multiple implicit arguments declarations can be given. + Depending on the number of arguments qualid is applied to in practice, + the longest applicable list of implicit arguments is used to select + which implicit arguments are inserted. For printing, the omitted + arguments are the ones of the longest list of implicit arguments of + the sequence. + +.. example:: + + .. coqtop:: reset all + + Inductive list (A:Type) : Type := + | nil : list A + | cons : A -> list A -> list A. + + Check (cons nat 3 (nil nat)). + + Arguments cons [A] _ _. + + Arguments nil [A]. + + Check (cons 3 nil). + + Fixpoint map (A B:Type) (f:A->B) (l:list A) : list B := match l with nil => nil | cons a t => cons (f a) (map A B f t) end. + + Fixpoint length (A:Type) (l:list A) : nat := match l with nil => 0 | cons _ m => S (length A m) end. + + Arguments map [A B] f l. + + Arguments length {A} l. (* A has to be maximally inserted *) + + Check (fun l:list (list nat) => map length l). + + Arguments map [A B] f l, [A] B f l, A B f l. + + Check (fun l => map length l = map (list nat) nat length l). + +Remark: To know which are the implicit arguments of an object, use the +command ``Print Implicit`` (see :ref:`displaying-implicit-args`). + + +Automatic declaration of implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +|Coq| can also automatically detect what are the implicit arguments of a +defined object. The command is just + +.. cmd:: Arguments @qualid : default implicits. + The auto-detection is governed by options telling if strict, contextual, or reversible-pattern implicit arguments must be -considered or not (see -Sections~\ref{SetStrictImplicit},~\ref{SetContextualImplicit},~\ref{SetReversiblePatternImplicit} -and also~\ref{SetMaximalImplicitInsertion}). - -\begin{Variants} -\item {\tt Global Arguments {\qualid} : default implicits -\comindex{Global Arguments}} - -Tell to recompute the implicit arguments of {\qualid} after ending of -the current section if any. - -\item {\tt Local Arguments {\qualid} : default implicits -\comindex{Local Arguments}} - -When in a module, tell not to activate the implicit arguments of -{\qualid} computed by this declaration to contexts that requires the -module. - -\end{Variants} - -\Example -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Inductive list (A:Set) : Set := - | nil : list A - | cons : A -> list A -> list A. -\end{coq_example*} -\begin{coq_example} -Arguments cons : default implicits. -Print Implicit cons. -Arguments nil : default implicits. -Print Implicit nil. -Set Contextual Implicit. -Arguments nil : default implicits. -Print Implicit nil. -\end{coq_example} - -The computation of implicit arguments takes account of the -unfolding of constants. For instance, the variable {\tt p} below has -type {\tt (Transitivity R)} which is reducible to {\tt forall x,y:U, R x -y -> forall z:U, R y z -> R x z}. As the variables {\tt x}, {\tt y} and -{\tt z} appear strictly in body of the type, they are implicit. - -\begin{coq_example*} -Variable X : Type. -Definition Relation := X -> X -> Prop. -Definition Transitivity (R:Relation) := - forall x y:X, R x y -> forall z:X, R y z -> R x z. -Variables (R : Relation) (p : Transitivity R). -Arguments p : default implicits. -\end{coq_example*} -\begin{coq_example} -Print p. -Print Implicit p. -\end{coq_example} -\begin{coq_example*} -Variables (a b c : X) (r1 : R a b) (r2 : R b c). -\end{coq_example*} -\begin{coq_example} -Check (p r1 r2). -\end{coq_example} - -\subsection{Mode for automatic declaration of implicit arguments -\label{Auto-implicit} -\optindex{Implicit Arguments}} +considered or not (see :ref:`controlling-strict-implicit-args`, :ref:`controlling-strict-implicit-args`, +:ref:`controlling-rev-pattern-implicit-args`, and also :ref:`controlling-insertion-implicit-args`). + +.. cmdv:: Global Arguments @qualid : default implicits + + Tell to recompute the + implicit arguments of qualid after ending of the current section if + any. + +.. cmdv:: Local Arguments @qualid : default implicits + + When in a module, tell not to activate the implicit arguments of `qualid` computed by this + declaration to contexts that requires the module. + +.. example:: + + .. coqtop:: reset all + + Inductive list (A:Set) : Set := + | nil : list A + | cons : A -> list A -> list A. + + Arguments cons : default implicits. + + Print Implicit cons. + + Arguments nil : default implicits. + + Print Implicit nil. + + Set Contextual Implicit. + + Arguments nil : default implicits. + + Print Implicit nil. + +The computation of implicit arguments takes account of the unfolding +of constants. For instance, the variable ``p`` below has type +``(Transitivity R)`` which is reducible to +``forall x,y:U, R x y -> forall z:U, R y z -> R x z``. As the variables ``x``, ``y`` and ``z`` +appear strictly in the body of the type, they are implicit. + +.. coqtop:: reset none + + Set Warnings "-local-declaration". + +.. coqtop:: all + + Variable X : Type. + + Definition Relation := X -> X -> Prop. + + Definition Transitivity (R:Relation) := forall x y:X, R x y -> forall z:X, R y z -> R x z. + + Variables (R : Relation) (p : Transitivity R). + + Arguments p : default implicits. + + Print p. + + Print Implicit p. + + Variables (a b c : X) (r1 : R a b) (r2 : R b c). + + Check (p r1 r2). + + +Mode for automatic declaration of implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case one wants to systematically declare implicit the arguments detectable as such, one may switch to the automatic declaration of -implicit arguments mode by using the command -\begin{quote} -\tt Set Implicit Arguments. -\end{quote} -Conversely, one may unset the mode by using {\tt Unset Implicit -Arguments}. The mode is off by default. Auto-detection of implicit -arguments is governed by options controlling whether strict and -contextual implicit arguments have to be considered or not. - -\subsection{Controlling strict implicit arguments -\optindex{Strict Implicit} -\label{SetStrictImplicit}} +implicit arguments mode by using the command: + +.. cmd:: Set Implicit Arguments. + +Conversely, one may unset the mode by using ``Unset Implicit Arguments``. +The mode is off by default. Auto-detection of implicit arguments is +governed by options controlling whether strict and contextual implicit +arguments have to be considered or not. + +.. _controlling-strict-implicit-args: + +Controlling strict implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ When the mode for automatic declaration of implicit arguments is on, the default is to automatically set implicit only the strict implicit -arguments plus, for historical reasons, a small subset of the non -strict implicit arguments. To relax this constraint and to -set implicit all non strict implicit arguments by default, use the command -\begin{quote} -\tt Unset Strict Implicit. -\end{quote} -Conversely, use the command {\tt Set Strict Implicit} to -restore the original mode that declares implicit only the strict implicit arguments plus a small subset of the non strict implicit arguments. - -In the other way round, to capture exactly the strict implicit arguments and no more than the strict implicit arguments, use the command: -\optindex{Strongly Strict Implicit} -\begin{quote} -\tt Set Strongly Strict Implicit. -\end{quote} -Conversely, use the command {\tt Unset Strongly Strict Implicit} to -let the option ``{\tt Strict Implicit}'' decide what to do. - -\Rem In versions of {\Coq} prior to version 8.0, the default was to +arguments plus, for historical reasons, a small subset of the non-strict +implicit arguments. To relax this constraint and to set +implicit all non strict implicit arguments by default, use the command: + +.. cmd:: Unset Strict Implicit. + +Conversely, use the command ``Set Strict Implicit`` to restore the +original mode that declares implicit only the strict implicit +arguments plus a small subset of the non strict implicit arguments. + +In the other way round, to capture exactly the strict implicit +arguments and no more than the strict implicit arguments, use the +command + +.. cmd:: Set Strongly Strict Implicit. + +Conversely, use the command ``Unset Strongly Strict Implicit`` to let the +option “Strict Implicit” decide what to do. + +Remark: In versions of |Coq| prior to version 8.0, the default was to declare the strict implicit arguments as implicit. -\subsection{Controlling contextual implicit arguments -\optindex{Contextual Implicit} -\label{SetContextualImplicit}} - -By default, {\Coq} does not automatically set implicit the contextual -implicit arguments. To tell {\Coq} to infer also contextual implicit -argument, use command -\begin{quote} -\tt Set Contextual Implicit. -\end{quote} -Conversely, use command {\tt Unset Contextual Implicit} to -unset the contextual implicit mode. - -\subsection{Controlling reversible-pattern implicit arguments -\optindex{Reversible Pattern Implicit} -\label{SetReversiblePatternImplicit}} - -By default, {\Coq} does not automatically set implicit the reversible-pattern -implicit arguments. To tell {\Coq} to infer also reversible-pattern implicit -argument, use command -\begin{quote} -\tt Set Reversible Pattern Implicit. -\end{quote} -Conversely, use command {\tt Unset Reversible Pattern Implicit} to -unset the reversible-pattern implicit mode. - -\subsection{Controlling the insertion of implicit arguments not followed by explicit arguments -\optindex{Maximal Implicit Insertion} -\label{SetMaximalImplicitInsertion}} +.. _controlling-contextual-implicit-args: + +Controlling contextual implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +By default, |Coq| does not automatically set implicit the contextual +implicit arguments. To tell |Coq| to infer also contextual implicit +argument, use command + +.. cmd:: Set Contextual Implicit. + +Conversely, use command ``Unset Contextual Implicit`` to unset the +contextual implicit mode. + +.. _controlling-rev-pattern-implicit-args: + +Controlling reversible-pattern implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +By default, |Coq| does not automatically set implicit the reversible-pattern +implicit arguments. To tell |Coq| to infer also reversible- +pattern implicit argument, use command + +.. cmd:: Set Reversible Pattern Implicit. + +Conversely, use command ``Unset Reversible Pattern Implicit`` to unset the +reversible-pattern implicit mode. + +.. _controlling-insertion-implicit-args: + +Controlling the insertion of implicit arguments not followed by explicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Implicit arguments can be declared to be automatically inserted when a function is partially applied and the next argument of the function is an implicit one. In case the implicit arguments are automatically -declared (with the command {\tt Set Implicit Arguments}), the command -\begin{quote} -\tt Set Maximal Implicit Insertion. -\end{quote} +declared (with the command ``Set Implicit Arguments``), the command + +.. cmd:: Set Maximal Implicit Insertion. + is used to tell to declare the implicit arguments with a maximal insertion status. By default, automatically declared implicit -arguments are not declared to be insertable maximally. To restore the -default mode for maximal insertion, use command {\tt Unset Maximal -Implicit Insertion}. +arguments are not declared to be insertable maximally. To restore the +default mode for maximal insertion, use the command -\subsection{Explicit applications -\index{Explicitly given implicit arguments} -\label{Implicits-explicitation} -\index{qualid@{\qualid}} \index{\symbol{64}}} +.. cmd:: Unset Maximal Implicit Insertion. -In presence of non strict or contextual argument, or in presence of +Explicit applications +~~~~~~~~~~~~~~~~~~~~~ + +In presence of non-strict or contextual argument, or in presence of partial applications, the synthesis of implicit arguments may fail, so one may have to give explicitly certain implicit arguments of an -application. The syntax for this is {\tt (\ident:=\term)} where {\ident} -is the name of the implicit argument and {\term} is its corresponding -explicit term. Alternatively, one can locally deactivate the hiding of -implicit arguments of a function by using the notation -{\tt @{\qualid}~{\term}$_1$..{\term}$_n$}. This syntax extension is -given Figure~\ref{fig:explicitations}. -\begin{figure} -\begin{centerframe} -\begin{tabular}{lcl} -{\term} & ++= & @ {\qualid} \nelist{\term}{}\\ -& $|$ & @ {\qualid}\\ -& $|$ & {\qualid} \nelist{\textrm{\textsl{argument}}}{}\\ -\\ -{\textrm{\textsl{argument}}} & ::= & {\term} \\ -& $|$ & {\tt ({\ident}:={\term})}\\ -\end{tabular} -\end{centerframe} -\caption{Syntax for explicitly giving implicit arguments} -\label{fig:explicitations} -\end{figure} - -\noindent {\bf Example (continued): } -\begin{coq_example} -Check (p r1 (z:=c)). -Check (p (x:=a) (y:=b) r1 (z:=c) r2). -\end{coq_example} - -\subsection{Renaming implicit arguments -\comindex{Arguments} -} +application. The syntax for this is ``(`` `ident` ``:=`` `term` ``)`` where `ident` is the +name of the implicit argument and term is its corresponding explicit +term. Alternatively, one can locally deactivate the hiding of implicit +arguments of a function by using the notation `@qualid` |term_1| … |term_n|. +This syntax extension is given in the following grammar: + +.. _explicit_app_grammar: + + .. productionlist:: explicit_apps + term : @ qualid term … `term` + : | @ qualid + : | qualid `argument` … `argument` + argument : `term` + : | (ident := `term`) + + Syntax for explicitly giving implicit arguments + +.. example:: (continued) + + .. coqtop:: all + + Check (p r1 (z:=c)). + + Check (p (x:=a) (y:=b) r1 (z:=c) r2). + + +Renaming implicit arguments +~~~~~~~~~~~~~~~~~~~~~~~~~~~ Implicit arguments names can be redefined using the following syntax: -\begin{quote} -{\tt Arguments {\qualid} \nelist{\name}{} : rename} -\end{quote} -With the {\tt assert} flag, {\tt Arguments} can be used to assert -that a given object has the expected number of arguments and that -these arguments are named as expected. +.. cmd:: Arguments @qualid {* @name} : @rename. + +With the assert flag, ``Arguments`` can be used to assert that a given +object has the expected number of arguments and that these arguments +are named as expected. + +.. example:: (continued) + +.. coqtop:: all + + Arguments p [s t] _ [u] _: rename. + + Check (p r1 (u:=c)). + + Check (p (s:=a) (t:=b) r1 (u:=c) r2). -\noindent {\bf Example (continued): } -\begin{coq_example} -Arguments p [s t] _ [u] _: rename. -Check (p r1 (u:=c)). -Check (p (s:=a) (t:=b) r1 (u:=c) r2). -Fail Arguments p [s t] _ [w] _ : assert. -\end{coq_example} + Fail Arguments p [s t] _ [w] _ : assert. +.. _displaying-implicit-args: -\subsection{Displaying what the implicit arguments are -\comindex{Print Implicit} -\label{PrintImplicit}} +Displaying what the implicit arguments are +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ To display the implicit arguments associated to an object, and to know if each of them is to be used maximally or not, use the command -\begin{quote} -\tt Print Implicit {\qualid}. -\end{quote} -\subsection{Explicit displaying of implicit arguments for pretty-printing -\optindex{Printing Implicit} -\optindex{Printing Implicit Defensive}} +.. cmd:: Print Implicit @qualid. + +Explicit displaying of implicit arguments for pretty-printing +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ By default the basic pretty-printing rules hide the inferable implicit arguments of an application. To force printing all implicit arguments, use command -\begin{quote} -{\tt Set Printing Implicit.} -\end{quote} + +.. cmd:: Set Printing Implicit. + Conversely, to restore the hiding of implicit arguments, use command -\begin{quote} -{\tt Unset Printing Implicit.} -\end{quote} - -By default the basic pretty-printing rules display the implicit arguments that are not detected as strict implicit arguments. This ``defensive'' mode can quickly make the display cumbersome so this can be deactivated by using the command -\begin{quote} -{\tt Unset Printing Implicit Defensive.} -\end{quote} + +.. cmd:: Unset Printing Implicit. + +By default the basic pretty-printing rules display the implicit +arguments that are not detected as strict implicit arguments. This +“defensive” mode can quickly make the display cumbersome so this can +be deactivated by using the command + +.. cmd:: Unset Printing Implicit Defensive. + Conversely, to force the display of non strict arguments, use command -\begin{quote} -{\tt Set Printing Implicit Defensive.} -\end{quote} -\SeeAlso {\tt Set Printing All} in Section~\ref{SetPrintingAll}. +.. cmd:: Set Printing Implicit Defensive. + +See also: ``Set Printing All`` in :ref:`printing_constructions_full`. -\subsection{Interaction with subtyping} +Interaction with subtyping +~~~~~~~~~~~~~~~~~~~~~~~~~~ When an implicit argument can be inferred from the type of more than one of the other arguments, then only the type of the first of these -arguments is taken into account, and not an upper type of all of -them. As a consequence, the inference of the implicit argument of -``='' fails in -\begin{coq_example*} -Fail Check nat = Prop. -\end{coq_example*} +arguments is taken into account, and not an upper type of all of them. +As a consequence, the inference of the implicit argument of “=” fails +in + +.. coqtop:: all + + Fail Check nat = Prop. but succeeds in -\begin{coq_example*} -Check Prop = nat. -\end{coq_example*} -\subsection{Deactivation of implicit arguments for parsing} -\optindex{Parsing Explicit} +.. coqtop:: all + + Check Prop = nat. + + +Deactivation of implicit arguments for parsing +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Use of implicit arguments can be deactivated by issuing the command: -\begin{quote} -{\tt Set Parsing Explicit.} -\end{quote} + +.. cmd:: Set Parsing Explicit. In this case, all arguments of constants, inductive types, constructors, etc, including the arguments declared as implicit, have to be given as if none arguments were implicit. By symmetry, this also affects printing. To restore parsing and normal printing of implicit arguments, use: -\begin{quote} -{\tt Unset Parsing Explicit.} -\end{quote} -\subsection{Canonical structures -\comindex{Canonical Structure}} +.. cmd:: Unset Parsing Explicit. + +Canonical structures +~~~~~~~~~~~~~~~~~~~~ A canonical structure is an instance of a record/structure type that can be used to solve unification problems involving a projection -applied to an unknown structure instance (an implicit argument) and -a value. The complete documentation of canonical structures can be found -in Chapter~\ref{CS-full}, here only a simple example is given. - -Assume that {\qualid} denotes an object $(Build\_struc~ c_1~ \ldots~ c_n)$ in -the -structure {\em struct} of which the fields are $x_1$, ..., -$x_n$. Assume that {\qualid} is declared as a canonical structure -using the command -\begin{quote} -{\tt Canonical Structure {\qualid}.} -\end{quote} -Then, each time an equation of the form $(x_i~ -\_)=_{\beta\delta\iota\zeta}c_i$ has to be solved during the -type-checking process, {\qualid} is used as a solution. Otherwise -said, {\qualid} is canonically used to extend the field $c_i$ into a -complete structure built on $c_i$. - -Canonical structures are particularly useful when mixed with -coercions and strict implicit arguments. Here is an example. -\begin{coq_example*} -Require Import Relations. -Require Import EqNat. -Set Implicit Arguments. -Unset Strict Implicit. -Structure Setoid : Type := - {Carrier :> Set; - Equal : relation Carrier; - Prf_equiv : equivalence Carrier Equal}. -Definition is_law (A B:Setoid) (f:A -> B) := - forall x y:A, Equal x y -> Equal (f x) (f y). -Axiom eq_nat_equiv : equivalence nat eq_nat. -Definition nat_setoid : Setoid := Build_Setoid eq_nat_equiv. -Canonical Structure nat_setoid. -\end{coq_example*} - -Thanks to \texttt{nat\_setoid} declared as canonical, the implicit -arguments {\tt A} and {\tt B} can be synthesized in the next statement. -\begin{coq_example} -Lemma is_law_S : is_law S. -\end{coq_example} - -\Rem If a same field occurs in several canonical structure, then +applied to an unknown structure instance (an implicit argument) and a +value. The complete documentation of canonical structures can be found +in :ref:`canonicalstructures`; here only a simple example is given. + +Assume that `qualid` denotes an object ``(Build_struc`` |c_1| … |c_n| ``)`` in the +structure *struct* of which the fields are |x_1|, …, |x_n|. Assume that +`qualid` is declared as a canonical structure using the command + +.. cmd:: Canonical Structure @qualid. + +Then, each time an equation of the form ``(``\ |x_i| ``_)`` |eq_beta_delta_iota_zeta| |c_i| has to be +solved during the type-checking process, `qualid` is used as a solution. +Otherwise said, `qualid` is canonically used to extend the field |c_i| +into a complete structure built on |c_i|. + +Canonical structures are particularly useful when mixed with coercions +and strict implicit arguments. Here is an example. + +.. coqtop:: all + + Require Import Relations. + + Require Import EqNat. + + Set Implicit Arguments. + + Unset Strict Implicit. + + Structure Setoid : Type := {Carrier :> Set; Equal : relation Carrier; + Prf_equiv : equivalence Carrier Equal}. + + Definition is_law (A B:Setoid) (f:A -> B) := forall x y:A, Equal x y -> Equal (f x) (f y). + + Axiom eq_nat_equiv : equivalence nat eq_nat. + + Definition nat_setoid : Setoid := Build_Setoid eq_nat_equiv. + + Canonical Structure nat_setoid. + +Thanks to ``nat_setoid`` declared as canonical, the implicit arguments ``A`` +and ``B`` can be synthesized in the next statement. + +.. coqtop:: all + + Lemma is_law_S : is_law S. + +Remark: If a same field occurs in several canonical structure, then only the structure declared first as canonical is considered. -\begin{Variants} -\item {\tt Canonical Structure {\ident} := {\term} : {\type}.}\\ - {\tt Canonical Structure {\ident} := {\term}.}\\ - {\tt Canonical Structure {\ident} : {\type} := {\term}.} +.. cmdv:: Canonical Structure @ident := @term : @type. -These are equivalent to a regular definition of {\ident} followed by -the declaration +.. cmdv:: Canonical Structure @ident := @term. -{\tt Canonical Structure {\ident}}. -\end{Variants} +.. cmdv:: Canonical Structure @ident : @type := @term. -\SeeAlso more examples in user contribution \texttt{category} -(\texttt{Rocq/ALGEBRA}). +These are equivalent to a regular definition of `ident` followed by the declaration +``Canonical Structure`` `ident`. -\subsubsection{Print Canonical Projections. -\comindex{Print Canonical Projections}} +See also: more examples in user contribution category (Rocq/ALGEBRA). + + +Print Canonical Projections. +++++++++++++++++++++++++++++ This displays the list of global names that are components of some canonical structure. For each of them, the canonical structure of -which it is a projection is indicated. For instance, the above example +which it is a projection is indicated. For instance, the above example gives the following output: -\begin{coq_example} -Print Canonical Projections. -\end{coq_example} +.. coqtop:: all + + Print Canonical Projections. + -\subsection{Implicit types of variables} -\comindex{Implicit Types} +Implicit types of variables +~~~~~~~~~~~~~~~~~~~~~~~~~~~ It is possible to bind variable names to a given type (e.g. in a -development using arithmetic, it may be convenient to bind the names -{\tt n} or {\tt m} to the type {\tt nat} of natural numbers). The -command for that is -\begin{quote} -\tt Implicit Types \nelist{\ident}{} : {\type} -\end{quote} +development using arithmetic, it may be convenient to bind the names `n` +or `m` to the type ``nat`` of natural numbers). The command for that is + +.. cmd:: Implicit Types {+ @ident } : @type. + The effect of the command is to automatically set the type of bound -variables starting with {\ident} (either {\ident} itself or -{\ident} followed by one or more single quotes, underscore or digits) -to be {\type} (unless the bound variable is already declared with an -explicit type in which case, this latter type is considered). - -\Example -\begin{coq_example} -Require Import List. -Implicit Types m n : nat. -Lemma cons_inj_nat : forall m n l, n :: l = m :: l -> n = m. -intros m n. -Lemma cons_inj_bool : forall (m n:bool) l, n :: l = m :: l -> n = m. -\end{coq_example} - -\begin{Variants} -\item {\tt Implicit Type {\ident} : {\type}}\\ -This is useful for declaring the implicit type of a single variable. -\item - {\tt Implicit Types\,% -(\,{\ident$_{1,1}$}\ldots{\ident$_{1,k_1}$}\,{\tt :}\,{\term$_1$} {\tt )}\,% -\ldots\,{\tt (}\,{\ident$_{n,1}$}\ldots{\ident$_{n,k_n}$}\,{\tt :}\,% -{\term$_n$} {\tt )}.}\\ - Adds $n$ blocks of implicit types with different specifications. -\end{Variants} - - -\subsection{Implicit generalization -\label{implicit-generalization} -\comindex{Generalizable Variables}} -% \textquoteleft since \` doesn't do what we want -\index{0genimpl@{\textquoteleft\{\ldots\}}} -\index{0genexpl@{\textquoteleft(\ldots)}} - -Implicit generalization is an automatic elaboration of a statement with -free variables into a closed statement where these variables are +variables starting with `ident` (either `ident` itself or `ident` followed by +one or more single quotes, underscore or digits) to be `type` (unless +the bound variable is already declared with an explicit type in which +case, this latter type is considered). + +.. example:: + + .. coqtop:: all + + Require Import List. + + Implicit Types m n : nat. + + Lemma cons_inj_nat : forall m n l, n :: l = m :: l -> n = m. + + intros m n. + + Lemma cons_inj_bool : forall (m n:bool) l, n :: l = m :: l -> n = m. + +.. cmdv:: Implicit Type @ident : @type. + + This is useful for declaring the implicit type of a single variable. + +.. cmdv:: Implicit Types {+ ( {+ @ident } : @term ) } + + Adds blocks of implicit types with different specifications. + +Implicit generalization +~~~~~~~~~~~~~~~~~~~~~~~ + +.. index:: `{ } +.. index:: `( ) + +Implicit generalization is an automatic elaboration of a statement +with free variables into a closed statement where these variables are quantified explicitly. Implicit generalization is done inside binders -starting with a \texttt{\`{}} and terms delimited by \texttt{\`{}\{ \}} and -\texttt{\`{}( )}, always introducing maximally inserted implicit arguments for -the generalized variables. Inside implicit generalization -delimiters, free variables in the current context are automatically -quantified using a product or a lambda abstraction to generate a closed -term. In the following statement for example, the variables \texttt{n} -and \texttt{m} are automatically generalized and become explicit -arguments of the lemma as we are using \texttt{\`{}( )}: - -\begin{coq_example} -Generalizable All Variables. -Lemma nat_comm : `(n = n + 0). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} -One can control the set of generalizable identifiers with the -\texttt{Generalizable} vernacular command to avoid unexpected -generalizations when mistyping identifiers. There are three variants of -the command: - -\begin{quote} -{\tt Generalizable (All|No) Variable(s)? ({\ident$_1$ \ident$_n$})?.} -\end{quote} - -\begin{Variants} -\item {\tt Generalizable All Variables.} All variables are candidate for - generalization if they appear free in the context under a - generalization delimiter. This may result in confusing errors in - case of typos. In such cases, the context will probably contain some - unexpected generalized variable. - -\item {\tt Generalizable No Variables.} Disable implicit generalization - entirely. This is the default behavior. - -\item {\tt Generalizable Variable(s)? {\ident$_1$ \ident$_n$}.} - Allow generalization of the given identifiers only. Calling this - command multiple times adds to the allowed identifiers. - -\item {\tt Global Generalizable} Allows to export the choice of - generalizable variables. -\end{Variants} - -One can also use implicit generalization for binders, in which case the -generalized variables are added as binders and set maximally implicit. -\begin{coq_example*} -Definition id `(x : A) : A := x. -\end{coq_example*} -\begin{coq_example} -Print id. -\end{coq_example} - -The generalizing binders \texttt{\`{}\{ \}} and \texttt{\`{}( )} work similarly to -their explicit counterparts, only binding the generalized variables +starting with a \` and terms delimited by \`{ } and \`( ), always +introducing maximally inserted implicit arguments for the generalized +variables. Inside implicit generalization delimiters, free variables +in the current context are automatically quantified using a product or +a lambda abstraction to generate a closed term. In the following +statement for example, the variables n and m are automatically +generalized and become explicit arguments of the lemma as we are using +\`( ): + +.. coqtop:: all + + Generalizable All Variables. + + Lemma nat_comm : `(n = n + 0). + +One can control the set of generalizable identifiers with +the ``Generalizable`` vernacular command to avoid unexpected +generalizations when mistyping identifiers. There are several commands +that specify which variables should be generalizable. + +.. cmd:: Generalizable All Variables. + + All variables are candidate for + generalization if they appear free in the context under a + generalization delimiter. This may result in confusing errors in case + of typos. In such cases, the context will probably contain some + unexpected generalized variable. + +.. cmd:: Generalizable No Variables. + + Disable implicit generalization entirely. This is the default behavior. + +.. cmd:: Generalizable (Variable | Variables) {+ @ident }. + + Allow generalization of the given identifiers only. Calling this command multiple times + adds to the allowed identifiers. + +.. cmd:: Global Generalizable. + + Allows exporting the choice of generalizable variables. + +One can also use implicit generalization for binders, in which case +the generalized variables are added as binders and set maximally +implicit. + +.. coqtop:: all + + Definition id `(x : A) : A := x. + + Print id. + +The generalizing binders \`{ } and \`( ) work similarly to their +explicit counterparts, only binding the generalized variables implicitly, as maximally-inserted arguments. In these binders, the binding name for the bound object is optional, whereas the type is mandatory, dually to regular binders. -\section{Coercions -\label{Coercions} -\index{Coercions}} - -Coercions can be used to implicitly inject terms from one {\em class} in -which they reside into another one. A {\em class} is either a sort -(denoted by the keyword {\tt Sortclass}), a product type (denoted by the -keyword {\tt Funclass}), or a type constructor (denoted by its name), -e.g. an inductive type or any constant with a type of the form -\texttt{forall} $(x_1:A_1) .. (x_n:A_n),~s$ where $s$ is a sort. - -Then the user is able to apply an -object that is not a function, but can be coerced to a function, and -more generally to consider that a term of type A is of type B provided -that there is a declared coercion between A and B. The main command is -\comindex{Coercion} -\begin{quote} -\tt Coercion {\qualid} : {\class$_1$} >-> {\class$_2$}. -\end{quote} -which declares the construction denoted by {\qualid} as a -coercion between {\class$_1$} and {\class$_2$}. + +Coercions +--------- + +Coercions can be used to implicitly inject terms from one *class* in +which they reside into another one. A *class* is either a sort +(denoted by the keyword ``Sortclass``), a product type (denoted by the +keyword ``Funclass``), or a type constructor (denoted by its name), e.g. +an inductive type or any constant with a type of the form +``forall (`` |x_1| : |A_1| ) … ``(``\ |x_n| : |A_n|\ ``)``, `s` where `s` is a sort. + +Then the user is able to apply an object that is not a function, but +can be coerced to a function, and more generally to consider that a +term of type ``A`` is of type ``B`` provided that there is a declared coercion +between ``A`` and ``B``. The main command is + +.. cmd:: Coercion @qualid : @class >-> @class. + +which declares the construction denoted by qualid as a coercion +between the two given classes. More details and examples, and a description of the commands related -to coercions are provided in Chapter~\ref{Coercions-full}. +to coercions are provided in :ref:`implicitcoercions`. -\section[Printing constructions in full]{Printing constructions in full\label{SetPrintingAll} -\optindex{Printing All}} +.. _printing_constructions_full: + +Printing constructions in full +------------------------------ Coercions, implicit arguments, the type of pattern-matching, but also -notations (see Chapter~\ref{Addoc-syntax}) can obfuscate the behavior -of some tactics (typically the tactics applying to occurrences of -subterms are sensitive to the implicit arguments). The command -\begin{quote} -{\tt Set Printing All.} -\end{quote} +notations (see :ref:`syntaxextensionsandinterpretationscopes`) can obfuscate the behavior of some +tactics (typically the tactics applying to occurrences of subterms are +sensitive to the implicit arguments). The command + +.. cmd:: Set Printing All. + deactivates all high-level printing features such as coercions, implicit arguments, returned type of pattern-matching, notations and various syntactic sugar for pattern-matching or record projections. -Otherwise said, {\tt Set Printing All} includes the effects -of the commands {\tt Set Printing Implicit}, {\tt Set Printing -Coercions}, {\tt Set Printing Synth}, {\tt Unset Printing Projections} -and {\tt Unset Printing Notations}. To reactivate the high-level -printing features, use the command -\begin{quote} -{\tt Unset Printing All.} -\end{quote} - -\section[Printing universes]{Printing universes\label{PrintingUniverses} -\optindex{Printing Universes}} +Otherwise said, ``Set Printing All`` includes the effects of the commands +``Set Printing Implicit``, ``Set Printing Coercions``, ``Set Printing Synth``, +``Unset Printing Projections``, and ``Unset Printing Notations``. To reactivate +the high-level printing features, use the command + +.. cmd:: Unset Printing All. + +Printing universes +------------------ The following command: -\begin{quote} -{\tt Set Printing Universes} -\end{quote} -activates the display of the actual level of each occurrence of -{\Type}. See Section~\ref{Sorts} for details. This wizard option, in -combination with \texttt{Set Printing All} (see -section~\ref{SetPrintingAll}) can help to diagnose failures to unify -terms apparently identical but internally different in the Calculus of -Inductive Constructions. To reactivate the display of the actual level -of the occurrences of {\Type}, use -\begin{quote} -{\tt Unset Printing Universes.} -\end{quote} - -\comindex{Print Universes} -\comindex{Print Sorted Universes} - -The constraints on the internal level of the occurrences of {\Type} -(see Section~\ref{Sorts}) can be printed using the command -\begin{quote} -{\tt Print \zeroone{Sorted} Universes.} -\end{quote} -If the optional {\tt Sorted} option is given, each universe will be -made equivalent to a numbered label reflecting its level (with a -linear ordering) in the universe hierarchy. + +.. cmd:: Set Printing Universes. + +activates the display of the actual level of each occurrence of ``Type``. +See :ref:`TODO-4.1.1-sorts` for details. This wizard option, in combination +with ``Set Printing All`` (see :ref:`printing_constructions_full`) can help to diagnose failures +to unify terms apparently identical but internally different in the +Calculus of Inductive Constructions. To reactivate the display of the +actual level of the occurrences of Type, use + +.. cmd:: Unset Printing Universes. + +The constraints on the internal level of the occurrences of Type +(see :ref:`TODO-4.1.1-sorts`) can be printed using the command + +.. cmd:: Print {? Sorted} Universes. + +If the optional ``Sorted`` option is given, each universe will be made +equivalent to a numbered label reflecting its level (with a linear +ordering) in the universe hierarchy. This command also accepts an optional output filename: -\begin{quote} -\tt Print \zeroone{Sorted} Universes {\str}. -\end{quote} -If {\str} ends in \texttt{.dot} or \texttt{.gv}, the constraints are -printed in the DOT language, and can be processed by Graphviz -tools. The format is unspecified if {\str} doesn't end in -\texttt{.dot} or \texttt{.gv}. -\section[Existential variables]{Existential variables\label{ExistentialVariables}} -\label{evars} +.. cmd:: Print {? Sorted} Universes @string. + +If `string` ends in ``.dot`` or ``.gv``, the constraints are printed in the DOT +language, and can be processed by Graphviz tools. The format is +unspecified if `string` doesn’t end in ``.dot`` or ``.gv``. + -Coq terms can include existential variables which -represents unknown subterms to eventually be replaced by actual -subterms. +Existential variables +--------------------- + +|Coq| terms can include existential variables which represents unknown +subterms to eventually be replaced by actual subterms. Existential variables are generated in place of unsolvable implicit -arguments or ``{\tt \_}'' placeholders when using commands such as -\texttt{Check} (see Section~\ref{Check}) or when using tactics such as -\texttt{refine}~(see Section~\ref{refine}), as well as in place of unsolvable -instances when using tactics such that \texttt{eapply} (see -Section~\ref{eapply}). An existential variable is defined in a -context, which is the context of variables of the placeholder which -generated the existential variable, and a type, which is the expected -type of the placeholder. +arguments or “_” placeholders when using commands such as ``Check`` (see +Section :ref:`TODO-6.3.1-check`) or when using tactics such as ``refine`` (see Section +:ref:`TODO-8.2.3-refine`), as well as in place of unsolvable instances when using +tactics such that ``eapply`` (see Section :ref:`TODO-8.2.4-apply`). An existential +variable is defined in a context, which is the context of variables of +the placeholder which generated the existential variable, and a type, +which is the expected type of the placeholder. As a consequence of typing constraints, existential variables can be duplicated in such a way that they possibly appear in different contexts than their defining context. Thus, any occurrence of a given -existential variable comes with an instance of its original context. In the -simple case, when an existential variable denotes the placeholder -which generated it, or is used in the same context as the one in which -it was generated, the context is not displayed and the existential -variable is represented by ``?'' followed by an identifier. - -\begin{coq_example} -Parameter identity : forall (X:Set), X -> X. -Check identity _ _. -Check identity _ (fun x => _). -\end{coq_example} - -In the general case, when an existential variable ?{\ident} -appears outside of its context of definition, its instance, written under -the form \verb!@{id1:=term1; ...; idn:=termn}!, is appending to its -name, indicating how the variables of its defining context are -instantiated. The variables of the context of the existential -variables which are instantiated by themselves are not written, unless -the flag {\tt Printing Existential Instances} is on (see -Section~\ref{SetPrintingExistentialInstances}), and this is why an -existential variable used in the same context as its context of -definition is written with no instance. - -\begin{coq_example} -Check (fun x y => _) 0 1. -Set Printing Existential Instances. -Check (fun x y => _) 0 1. -\end{coq_example} - -\begin{coq_eval} -Unset Printing Existential Instances. -\end{coq_eval} +existential variable comes with an instance of its original context. +In the simple case, when an existential variable denotes the +placeholder which generated it, or is used in the same context as the +one in which it was generated, the context is not displayed and the +existential variable is represented by “?” followed by an identifier. + +.. coqtop:: all + + Parameter identity : forall (X:Set), X -> X. + + Check identity _ _. + + Check identity _ (fun x => _). + +In the general case, when an existential variable ``?``\ `ident` appears +outside of its context of definition, its instance, written under the +form + +| ``{`` :n:`{*; @ident:=@term}` ``}`` + +is appending to its name, indicating how the variables of its defining context are instantiated. +The variables of the context of the existential variables which are +instantiated by themselves are not written, unless the flag ``Printing Existential Instances`` +is on (see Section :ref:`explicit-display-existentials`), and this is why an +existential variable used in the same context as its context of definition is written with no instance. + +.. coqtop:: all + + Check (fun x y => _) 0 1. + + Set Printing Existential Instances. + + Check (fun x y => _) 0 1. Existential variables can be named by the user upon creation using -the syntax {\tt ?[\ident]}. This is useful when the existential +the syntax ``?``\ `ident`. This is useful when the existential variable needs to be explicitly handled later in the script (e.g. -with a named-goal selector, see~\ref{ltac:selector}). +with a named-goal selector, see :ref:`TODO-9.2-goal-selectors`). -\subsection{Explicit displaying of existential instances for pretty-printing -\label{SetPrintingExistentialInstances} -\optindex{Printing Existential Instances}} +.. _explicit-display-existentials: + +Explicit displaying of existential instances for pretty-printing +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The command: -\begin{quote} -{\tt Set Printing Existential Instances} -\end{quote} -activates the full display of how the context of an existential variable is -instantiated at each of the occurrences of the existential variable. + +.. cmd:: Set Printing Existential Instances. + +activates the full display of how the context of an existential +variable is instantiated at each of the occurrences of the existential +variable. To deactivate the full display of the instances of existential variables, use -\begin{quote} -{\tt Unset Printing Existential Instances.} -\end{quote} - -\subsection{Solving existential variables using tactics} -\ttindex{ltac:( \ldots )} - -\def\expr{\textrm{\textsl{tacexpr}}} - -Instead of letting the unification engine try to solve an existential variable -by itself, one can also provide an explicit hole together with a tactic to solve -it. Using the syntax {\tt ltac:(\expr)}, the user can put a -tactic anywhere a term is expected. The order of resolution is not specified and -is implementation-dependent. The inner tactic may use any variable defined in -its scope, including repeated alternations between variables introduced by term -binding as well as those introduced by tactic binding. The expression {\expr} -can be any tactic expression as described at section~\ref{TacticLanguage}. - -\begin{coq_example*} -Definition foo (x : nat) : nat := ltac:(exact x). -\end{coq_example*} - -This construction is useful when one wants to define complicated terms using -highly automated tactics without resorting to writing the proof-term by means of -the interactive proof engine. - -This mechanism is comparable to the {\tt Declare Implicit Tactic} command -defined at~\ref{DeclareImplicit}, except that the used tactic is local to each -hole instead of being declared globally. - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: + +.. cmd:: Unset Printing Existential Instances. + +Solving existential variables using tactics +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Instead of letting the unification engine try to solve an existential +variable by itself, one can also provide an explicit hole together +with a tactic to solve it. Using the syntax ``ltac:(``\ `tacexpr`\ ``)``, the user +can put a tactic anywhere a term is expected. The order of resolution +is not specified and is implementation-dependent. The inner tactic may +use any variable defined in its scope, including repeated alternations +between variables introduced by term binding as well as those +introduced by tactic binding. The expression `tacexpr` can be any tactic +expression as described in :ref:`thetacticlanguage`. + +.. coqtop:: all + + Definition foo (x : nat) : nat := ltac:(exact x). + +This construction is useful when one wants to define complicated terms +using highly automated tactics without resorting to writing the proof-term +by means of the interactive proof engine. + +This mechanism is comparable to the ``Declare Implicit Tactic`` command +defined at :ref:`TODO-8.9.7-implicit-automation`, except that the used +tactic is local to each hole instead of being declared globally. |