\chapter[Extensions of \Gallina{}]{Extensions of \Gallina{}\label{Gallina-extension}\index{Gallina}} {\gallina} is the kernel language of {\Coq}. We describe here extensions of the Gallina's syntax. \section{Record types \comindex{Record} \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 to define ``signatures''. \begin{figure}[h] \begin{centerframe} \begin{tabular}{lcl} {\sentence} & ++= & {\record}\\ & & \\ {\record} & ::= & {\tt Record} {\ident} \zeroone{\binders} \zeroone{{\tt :} {\sort}} \verb.:=. \\ && ~~~~\zeroone{\ident} \verb!{! \zeroone{\nelist{\field}{;}} \verb!}! \verb:.:\\ & & \\ {\field} & ::= & {\name} : {\type} [ {\tt where} {\it notation} ] \\ & $|$ & {\name} {\typecstr} := {\term} \end{tabular} \end{centerframe} \caption{Syntax for the definition of {\tt Record}} \label{record-syntax} \end{figure} \noindent In the expression \smallskip {\tt Record} {\ident} {\params} \texttt{:} {\sort} := {\ident$_0$} \verb+{+ {\ident$_1$} \texttt{:} {\term$_1$}; \dots {\ident$_n$} \texttt{:} {\term$_n$} \verb+}+. \smallskip \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$_1$}, .., {\ident$_n$} are the names of fields and {\term$_1$}, .., {\term$_n$} their respective types. Remark that the type of {\ident$_i$} may depend on the previous {\ident$_j$} (for $j bottom; Rat_irred_cond : forall x y z:nat, (x * y) = top /\ (x * z) = bottom -> x = 1}. \end{coq_example} Remark here that the field \verb+Rat_cond+ depends on the field \verb+bottom+. %Let us now see the work done by the {\tt Record} macro. %First the macro generates an inductive definition %with just one constructor: % %\medskip %\noindent %{\tt Inductive {\ident} \zeroone{\binders} : {\sort} := \\ %\mbox{}\hspace{0.4cm} {\ident$_0$} : forall ({\ident$_1$}:{\term$_1$}) .. %({\ident$_n$}:{\term$_n$}), {\ident} {\rm\sl params}.} %\medskip Let us now see the work done by the {\tt Record} macro. First the macro generates an inductive definition with just one constructor: \begin{quote} {\tt Inductive {\ident} {\params} :{\sort} :=} \\ \qquad {\tt {\ident$_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*} Require Import Arith. Theorem one_two_irred : forall x y z:nat, x * y = 1 /\ x * z = 2 -> x = 1. \end{coq_example*} \begin{coq_eval} Lemma mult_m_n_eq_m_1 : forall m n:nat, m * n = 1 -> m = 1. destruct m; trivial. intros; apply f_equal with (f := S). destruct m; trivial. destruct n; simpl in H. rewrite <- mult_n_O in H. discriminate. rewrite <- plus_n_Sm in H. discriminate. Qed. intros x y z [H1 H2]. apply mult_m_n_eq_m_1 with (n := y); trivial. \end{coq_eval} \ldots \begin{coq_example*} Qed. \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} The macro generates also, when it is possible, the projection functions for destructuring an object of type {\ident}. These projection functions have the same name that the corresponding fields. If a field is named ``\verb=_='' then no projection is built for it. In our example: \begin{coq_example} Eval compute in half.(top). Eval compute in half.(bottom). Eval compute in half.(Rat_bottom_cond). \end{coq_example} \begin{coq_eval} Reset Initial. \end{coq_eval} \begin{Warnings} \item {\tt Warning: {\ident$_i$} 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{A record cannot be recursive} The record name {\ident} appears in the type of its fields. \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. \Rem {\tt Structure} is a synonym of the keyword {\tt Record}. \Rem Creation of an object of record type can be done by calling {\ident$_0$} and passing arguments in the correct order. \begin{coq_example} Record point := { x : nat; y : nat }. Definition a := Build_point 5 3. \end{coq_example} The following syntax allows to create objects by using named fields. 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 b := {| x := 5; y := 3 |}. Definition c := {| y := 3; x := 5 |}. \end{coq_example} This syntax can also be used for pattern matching. \begin{coq_example} Eval compute in ( match b with | {| y := S n |} => n | _ => 0 end). \end{coq_example} \begin{coq_eval} Reset Initial. \end{coq_eval} \Rem An experimental syntax for projections based on a dot notation is available. The command to activate it is \begin{quote} {\tt Set Printing Projections.} \end{quote} \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 of \texttt{Record} projections} \label{fig:projsyntax} \end{figure} The corresponding grammar rules are given Figure~\ref{fig:projsyntax}. When {\qualid} denotes a projection, the syntax {\tt {\term}.({\qualid})} is equivalent to {\qualid~\term}, the syntax {\tt {\term}.({\qualid}~{\termarg}$_1$~ \ldots~ {\termarg}$_n$)} to {\qualid~{\termarg}$_1$ \ldots {\termarg}$_n$~\term}, and the syntax {\tt {\term}.(@{\qualid}~{\term}$_1$~\ldots~{\term}$_n$)} 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. To deactivate the printing of projections, use {\tt Unset Printing Projections}. \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 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}} Pattern-matching on terms inhabiting inductive type having only one constructor can be alternatively written using {\tt 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} can be alternatively written \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. 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}). If {\term} inhabits an inductive type with one constructor {\tt C}, we have an equivalence between {\tt let ({\ident}$_1$,\ldots,{\ident}$_n$) \zeroone{\ifitem} := {\term} in {\term}'} \noindent and {\tt match {\term} \zeroone{\ifitem} with C {\ident}$_1$ {\ldots} {\ident}$_n$ \verb!=>! {\term}' end} \subsubsection{Second destructuring {\tt let} syntax\index{let '... in}} Another destructuring {\tt 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} 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 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} 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}). \subsection{Controlling pretty-printing of {\tt match} expressions \label{printing-options}} The following commands give some control over the pretty-printing of {\tt match} expressions. \subsubsection{Printing nested patterns \label{SetPrintingMatching} \comindex{Set Printing Matching} \comindex{Unset Printing Matching} \comindex{Test Printing Matching}} 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. \begin{quote} {\tt Set Printing Matching.} \end{quote} 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. \begin{quote} {\tt Test Printing Matching.} \end{quote} This tells if the printing matching mode is on or off. The default is on. \subsubsection{Printing of wildcard pattern \comindex{Set Printing Wildcard} \comindex{Unset Printing Wildcard} \comindex{Test Printing Wildcard}} 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. \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} 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 \comindex{Set Printing Synth} \comindex{Unset Printing Synth} \comindex{Test Printing Synth}} 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. \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} 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 \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 {\tt let} ... {\tt :=} ... {\tt in}~... \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. \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. 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}. \subsubsection{Example} This example emphasizes what the printing options offer. \begin{coq_example} Test Printing Let for prod. Print fst. Remove Printing Let prod. Unset Printing Synth. Unset Printing Wildcard. Print fst. \end{coq_example} % \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 \emph{experimental} command \begin{center} \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} \{decrease\_annot\} : type$_0$ := \term$_0$} \comindex{Function} \label{Function} \end{center} 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 must not necessary 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 enjoys also the {\tt with} extension to define mutually recursive definitions. However, this feature does not work for non structural 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 build 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_example*} Function wrong (C:nat) {\ldots} : nat := List.hd(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} 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. %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 \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}} The sectioning mechanism allows 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 and files \label{Libraries} \index{Libraries} \index{Physical paths} \index{Logical paths}} \paragraph{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} has name {\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 reserved root {\tt Coq} but library file names based on other roots can be obtained by using {\tt coqc} options {\tt -I} 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} or {\tt -notop} is set (see Section~\ref{coqoptions}). As library files are stored on the file system of the underlying operating system, 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. Logical names are mapped to physical paths using the commands {\tt Add LoadPath} or {\tt Add Rec LoadPath} (see Sections~\ref{AddLoadPath} and~\ref{AddRecLoadPath}). \subsection{Qualified names \label{LongNames} \index{Qualified identifiers} \index{Absolute 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 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. A similar table exists for library file names. It is updated by the vernacular commands {\tt Add LoadPath} and {\tt Add Rec LoadPath} (or their equivalent as options of the {\Coq} executables, {\tt -I} and {\tt -R}). 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 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} \SeeAlso Command {\tt Locate} in Section~\ref{Locate} and {\tt Locate Library} in Section~\ref{Locate Library}. %% \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}} 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 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 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 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. \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 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). 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} 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.} 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 are solved based on the structure of their type only. \subsection{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. 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}. \subsection{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 possible, the correct argument will be automatically generated. \begin{ErrMsgs} \item \errindex{Cannot infer a term for this placeholder} {\Coq} was not able to deduce an instantiation of a ``{\tt \_}''. \end{ErrMsgs} \subsection{Declaration of implicit arguments for a constant \comindex{Implicit Arguments}} \label{ImplicitArguments} 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. \subsubsection{Implicit Argument Binders} In the first setting, one wants to explicitly give the implicit arguments of a constant 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 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} One can always specify the parameter if it is not uniform using the usual implicit arguments disambiguation syntax. \subsubsection{The Implicit Arguments Vernacular Command} To set implicit arguments for a constant a-posteriori, one can use the command: \begin{quote} \tt Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ] \end{quote} where the list of {\possiblybracketedident} is the list of parameters to be declared implicit, each of the identifier of the list being optionally surrounded by square brackets, then meaning that this parameter has to be maximally inserted. After the above declaration is issued, implicit arguments can just (and have to) be skipped in any expression involving an application of {\qualid}. \begin{Variants} \item {\tt Global Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ] \comindex{Global Implicit Arguments}} Tells 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 Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ] \comindex{Local Implicit Arguments}} When in a module, tells not to activate the implicit arguments of {\qualid} declared by this commands to contexts that requires the module. \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)). Implicit Arguments cons [A]. Implicit 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. Implicit Arguments map [A B]. Implicit Arguments length [[A]]. (* A has to be maximally inserted *) Check (fun l:list (list nat) => map 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}). \Rem If the list of arguments is empty, the command removes the implicit arguments of {\qualid}. \subsection{Automatic declaration of implicit arguments for a constant} {\Coq} can also automatically detect what are the implicit arguments of a defined object. The command is just \begin{quote} {\tt Implicit Arguments {\qualid} \comindex{Implicit Arguments}} \end{quote} 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 Implicit Arguments {\qualid} \comindex{Global Implicit Arguments}} Tells to recompute the implicit arguments of {\qualid} after ending of the current section if any. \item {\tt Local Implicit Arguments {\qualid} \comindex{Local Implicit Arguments}} When in a module, tells 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} Implicit Arguments cons. Print Implicit cons. Implicit Arguments nil. Print Implicit nil. Set Contextual Implicit. Implicit Arguments nil. 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). Implicit Arguments p. \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} \comindex{Set Implicit Arguments} \comindex{Unset 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 \comindex{Set Strict Implicit} \comindex{Unset Strict Implicit} \label{SetStrictImplicit}} 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: \comindex{Set Strongly Strict Implicit} \comindex{Unset 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 declare the strict implicit arguments as implicit. \subsection{Controlling contextual implicit arguments \comindex{Set Contextual Implicit} \comindex{Unset 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 \comindex{Set Reversible Pattern Implicit} \comindex{Unset 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 \comindex{Set Maximal Implicit Insertion} \comindex{Unset Maximal Implicit Insertion} \label{SetMaximalImplicitInsertion}} 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} 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}. \subsection{Explicit applications \index{Explicitly given implicit arguments} \label{Implicits-explicitation} \index{qualid@{\qualid}}} 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{Displaying what the implicit arguments are \comindex{Print Implicit} \label{PrintImplicit}} 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 \comindex{Set Printing Implicit} \comindex{Unset Printing Implicit} \comindex{Set Printing Implicit Defensive} \comindex{Unset Printing Implicit Defensive}} 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} 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} 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}. \subsection{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*} Check nat = Prop. \end{coq_example*} but succeeds in \begin{coq_example*} Check Prop = nat. \end{coq_example*} \subsection{Canonical structures \comindex{Canonical Structure}} A canonical structure is an instance of a record/structure type that can be used to solve equations involving implicit arguments. 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 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}.} These are equivalent to a regular definition of {\ident} followed by the declaration {\tt Canonical Structure {\ident}}. \end{Variants} \SeeAlso more examples in user contribution \texttt{category} (\texttt{Rocq/ALGEBRA}). \subsubsection{Print Canonical Projections. \comindex{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 gives the following output: \begin{coq_example} Print Canonical Projections. \end{coq_example} \subsection{Implicit types of variables} \comindex{Implicit Types} 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} 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}} 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 \verb|`| and terms delimited by \verb|`{ }| and \verb|`( )|, 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 autamatically generalized and become explicit arguments of the lemma as we are using \verb|`( )|: \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 \verb|`{ }| and \verb|`( )| 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$_2$}. More details and examples, and a description of the commands related to coercions are provided in Chapter~\ref{Coercions-full}. \section[Printing constructions in full]{Printing constructions in full\label{SetPrintingAll} \comindex{Set Printing All} \comindex{Unset Printing All}} 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} 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} \comindex{Set Printing Universes} \comindex{Unset 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} 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 Universes.} \end{quote} %%% Local Variables: %%% mode: latex %%% TeX-master: "Reference-Manual" %%% End: