summaryrefslogtreecommitdiff
path: root/doc/refman/RefMan-ltac.tex
diff options
context:
space:
mode:
Diffstat (limited to 'doc/refman/RefMan-ltac.tex')
-rw-r--r--doc/refman/RefMan-ltac.tex1308
1 files changed, 0 insertions, 1308 deletions
diff --git a/doc/refman/RefMan-ltac.tex b/doc/refman/RefMan-ltac.tex
deleted file mode 100644
index d8747254..00000000
--- a/doc/refman/RefMan-ltac.tex
+++ /dev/null
@@ -1,1308 +0,0 @@
-\chapter[The tactic language]{The tactic language\label{TacticLanguage}}
-
-%\geometry{a4paper,body={5in,8in}}
-
-This chapter gives a compact documentation of Ltac, the tactic
-language available in {\Coq}. We start by giving the syntax, and next,
-we present the informal semantics. If you want to know more regarding
-this language and especially about its foundations, you can refer
-to~\cite{Del00}. Chapter~\ref{Tactics-examples} is devoted to giving
-examples of use of this language on small but also with non-trivial
-problems.
-
-
-\section{Syntax}
-
-\def\tacexpr{\textrm{\textsl{expr}}}
-\def\tacexprlow{\textrm{\textsl{tacexpr$_1$}}}
-\def\tacexprinf{\textrm{\textsl{tacexpr$_2$}}}
-\def\tacexprpref{\textrm{\textsl{tacexpr$_3$}}}
-\def\atom{\textrm{\textsl{atom}}}
-%%\def\recclause{\textrm{\textsl{rec\_clause}}}
-\def\letclause{\textrm{\textsl{let\_clause}}}
-\def\matchrule{\textrm{\textsl{match\_rule}}}
-\def\contextrule{\textrm{\textsl{context\_rule}}}
-\def\contexthyp{\textrm{\textsl{context\_hyp}}}
-\def\tacarg{\nterm{tacarg}}
-\def\cpattern{\nterm{cpattern}}
-
-The syntax of the tactic language is given Figures~\ref{ltac}
-and~\ref{ltac_aux}. See Chapter~\ref{BNF-syntax} for a description of
-the BNF metasyntax used in these grammar rules. Various already
-defined entries will be used in this chapter: entries
-{\naturalnumber}, {\integer}, {\ident}, {\qualid}, {\term},
-{\cpattern} and {\atomictac} represent respectively the natural and
-integer numbers, the authorized identificators and qualified names,
-{\Coq}'s terms and patterns and all the atomic tactics described in
-Chapter~\ref{Tactics}. The syntax of {\cpattern} is the same as that
-of terms, but it is extended with pattern matching metavariables. In
-{\cpattern}, a pattern-matching metavariable is represented with the
-syntax {\tt ?id} where {\tt id} is an {\ident}. The notation {\tt \_}
-can also be used to denote metavariable whose instance is
-irrelevant. In the notation {\tt ?id}, the identifier allows us to
-keep instantiations and to make constraints whereas {\tt \_} shows
-that we are not interested in what will be matched. On the right hand
-side of pattern-matching clauses, the named metavariable are used
-without the question mark prefix. There is also a special notation for
-second-order pattern-matching problems: in an applicative pattern of
-the form {\tt @?id id$_1$ \ldots id$_n$}, the variable {\tt id}
-matches any complex expression with (possible) dependencies in the
-variables {\tt id$_1$ \ldots id$_n$} and returns a functional term of
-the form {\tt fun id$_1$ \ldots id$_n$ => {\term}}.
-
-
-The main entry of the grammar is {\tacexpr}. This language is used in
-proof mode but it can also be used in toplevel definitions as shown in
-Figure~\ref{ltactop}.
-
-\begin{Remarks}
-\item The infix tacticals ``\dots\ {\tt ||} \dots'' and ``\dots\ {\tt
- ;} \dots'' are associative.
-
-\item In {\tacarg}, there is an overlap between {\qualid} as a
-direct tactic argument and {\qualid} as a particular case of
-{\term}. The resolution is done by first looking for a reference of
-the tactic language and if it fails, for a reference to a term. To
-force the resolution as a reference of the tactic language, use the
-form {\tt ltac :} {\qualid}. To force the resolution as a reference to
-a term, use the syntax {\tt ({\qualid})}.
-
-\item As shown by the figure, tactical {\tt ||} binds more than the
-prefix tacticals {\tt try}, {\tt repeat}, {\tt do}, {\tt info} and
-{\tt abstract} which themselves bind more than the postfix tactical
-``{\tt \dots\ ;[ \dots\ ]}'' which binds more than ``\dots\ {\tt ;}
-\dots''.
-
-For instance
-\begin{quote}
-{\tt try repeat \tac$_1$ ||
- \tac$_2$;\tac$_3$;[\tac$_{31}$|\dots|\tac$_{3n}$];\tac$_4$.}
-\end{quote}
-is understood as
-\begin{quote}
-{\tt (try (repeat (\tac$_1$ || \tac$_2$)));} \\
-{\tt ((\tac$_3$;[\tac$_{31}$|\dots|\tac$_{3n}$]);\tac$_4$).}
-\end{quote}
-\end{Remarks}
-
-
-\begin{figure}[htbp]
-\begin{centerframe}
-\begin{tabular}{lcl}
-{\tacexpr} & ::= &
- {\tacexpr} {\tt ;} {\tacexpr}\\
-& | & {\tacexpr} {\tt ; [} \nelist{\tacexpr}{|} {\tt ]}\\
-& | & {\tacexprpref}\\
-\\
-{\tacexprpref} & ::= &
- {\tt do} {\it (}{\naturalnumber} {\it |} {\ident}{\it )} {\tacexprpref}\\
-& | & {\tt info} {\tacexprpref}\\
-& | & {\tt progress} {\tacexprpref}\\
-& | & {\tt repeat} {\tacexprpref}\\
-& | & {\tt try} {\tacexprpref}\\
-& | & {\tacexprinf} \\
-\\
-{\tacexprinf} & ::= &
- {\tacexprlow} {\tt ||} {\tacexprpref}\\
-& | & {\tacexprlow}\\
-\\
-{\tacexprlow} & ::= &
-{\tt fun} \nelist{\name}{} {\tt =>} {\atom}\\
-& | &
-{\tt let} \zeroone{\tt rec} \nelist{\letclause}{\tt with} {\tt in}
-{\atom}\\
-& | &
-{\tt match goal with} \nelist{\contextrule}{\tt |} {\tt end}\\
-& | &
-{\tt match reverse goal with} \nelist{\contextrule}{\tt |} {\tt end}\\
-& | &
-{\tt match} {\tacexpr} {\tt with} \nelist{\matchrule}{\tt |} {\tt end}\\
-& | &
-{\tt lazymatch goal with} \nelist{\contextrule}{\tt |} {\tt end}\\
-& | &
-{\tt lazymatch reverse goal with} \nelist{\contextrule}{\tt |} {\tt end}\\
-& | &
-{\tt lazymatch} {\tacexpr} {\tt with} \nelist{\matchrule}{\tt |} {\tt end}\\
-& | & {\tt abstract} {\atom}\\
-& | & {\tt abstract} {\atom} {\tt using} {\ident} \\
-& | & {\tt first [} \nelist{\tacexpr}{\tt |} {\tt ]}\\
-& | & {\tt solve [} \nelist{\tacexpr}{\tt |} {\tt ]}\\
-& | & {\tt idtac} \sequence{\messagetoken}{}\\
-& | & {\tt fail} \zeroone{\naturalnumber} \sequence{\messagetoken}{}\\
-& | & {\tt fresh} ~|~ {\tt fresh} {\qstring}\\
-& | & {\tt context} {\ident} {\tt [} {\term} {\tt ]}\\
-& | & {\tt eval} {\nterm{redexpr}} {\tt in} {\term}\\
-& | & {\tt type of} {\term}\\
-& | & {\tt external} {\qstring} {\qstring} \nelist{\tacarg}{}\\
-& | & {\tt constr :} {\term}\\
-& | & \atomictac\\
-& | & {\qualid} \nelist{\tacarg}{}\\
-& | & {\atom}\\
-\\
-{\atom} & ::= &
- {\qualid} \\
-& | & ()\\
-& | & {\integer}\\
-& | & {\tt (} {\tacexpr} {\tt )}\\
-\\
-{\messagetoken}\!\!\!\!\!\! & ::= & {\qstring} ~|~ {\ident} ~|~ {\integer} \\
-\end{tabular}
-\end{centerframe}
-\caption{Syntax of the tactic language}
-\label{ltac}
-\end{figure}
-
-
-
-\begin{figure}[htbp]
-\begin{centerframe}
-\begin{tabular}{lcl}
-\tacarg & ::= &
- {\qualid}\\
-& $|$ & {\tt ()} \\
-& $|$ & {\tt ltac :} {\atom}\\
-& $|$ & {\term}\\
-\\
-\letclause & ::= & {\ident} \sequence{\name}{} {\tt :=} {\tacexpr}\\
-\\
-\contextrule & ::= &
- \nelist{\contexthyp}{\tt ,} {\tt |-}{\cpattern} {\tt =>} {\tacexpr}\\
-& $|$ & {\tt |-} {\cpattern} {\tt =>} {\tacexpr}\\
-& $|$ & {\tt \_ =>} {\tacexpr}\\
-\\
-\contexthyp & ::= & {\name} {\tt :} {\cpattern}\\
- & $|$ & {\name} {\tt :=} {\cpattern} \zeroone{{\tt :} {\cpattern}}\\
-\\
-\matchrule & ::= &
- {\cpattern} {\tt =>} {\tacexpr}\\
-& $|$ & {\tt context} {\zeroone{\ident}} {\tt [} {\cpattern} {\tt ]}
- {\tt =>} {\tacexpr}\\
-& $|$ & {\tt appcontext} {\zeroone{\ident}} {\tt [} {\cpattern} {\tt ]}
- {\tt =>} {\tacexpr}\\
-& $|$ & {\tt \_ =>} {\tacexpr}\\
-\end{tabular}
-\end{centerframe}
-\caption{Syntax of the tactic language (continued)}
-\label{ltac_aux}
-\end{figure}
-
-\begin{figure}[ht]
-\begin{centerframe}
-\begin{tabular}{lcl}
-\nterm{top} & ::= & \zeroone{\tt Local} {\tt Ltac} \nelist{\nterm{ltac\_def}} {\tt with} \\
-\\
-\nterm{ltac\_def} & ::= & {\ident} \sequence{\ident}{} {\tt :=}
-{\tacexpr}\\
-& $|$ &{\qualid} \sequence{\ident}{} {\tt ::=}{\tacexpr}
-\end{tabular}
-\end{centerframe}
-\caption{Tactic toplevel definitions}
-\label{ltactop}
-\end{figure}
-
-
-%%
-%% Semantics
-%%
-\section{Semantics}
-%\index[tactic]{Tacticals}
-\index{Tacticals}
-%\label{Tacticals}
-
-Tactic expressions can only be applied in the context of a goal. The
-evaluation yields either a term, an integer or a tactic. Intermediary
-results can be terms or integers but the final result must be a tactic
-which is then applied to the current goal.
-
-There is a special case for {\tt match goal} expressions of which
-the clauses evaluate to tactics. Such expressions can only be used as
-end result of a tactic expression (never as argument of a non recursive local
-definition or of an application).
-
-The rest of this section explains the semantics of every construction
-of Ltac.
-
-
-%% \subsection{Values}
-
-%% Values are given by Figure~\ref{ltacval}. All these values are tactic values,
-%% i.e. to be applied to a goal, except {\tt Fun}, {\tt Rec} and $arg$ values.
-
-%% \begin{figure}[ht]
-%% \noindent{}\framebox[6in][l]
-%% {\parbox{6in}
-%% {\begin{center}
-%% \begin{tabular}{lp{0.1in}l}
-%% $vexpr$ & ::= & $vexpr$ {\tt ;} $vexpr$\\
-%% & | & $vexpr$ {\tt ; [} {\it (}$vexpr$ {\tt |}{\it )}$^*$ $vexpr$ {\tt
-%% ]}\\
-%% & | & $vatom$\\
-%% \\
-%% $vatom$ & ::= & {\tt Fun} \nelist{\inputfun}{} {\tt ->} {\tacexpr}\\
-%% %& | & {\tt Rec} \recclause\\
-%% & | &
-%% {\tt Rec} \nelist{\recclause}{\tt And} {\tt In}
-%% {\tacexpr}\\
-%% & | &
-%% {\tt Match Context With} {\it (}$context\_rule$ {\tt |}{\it )}$^*$
-%% $context\_rule$\\
-%% & | & {\tt (} $vexpr$ {\tt )}\\
-%% & | & $vatom$ {\tt Orelse} $vatom$\\
-%% & | & {\tt Do} {\it (}{\naturalnumber} {\it |} {\ident}{\it )} $vatom$\\
-%% & | & {\tt Repeat} $vatom$\\
-%% & | & {\tt Try} $vatom$\\
-%% & | & {\tt First [} {\it (}$vexpr$ {\tt |}{\it )}$^*$ $vexpr$ {\tt ]}\\
-%% & | & {\tt Solve [} {\it (}$vexpr$ {\tt |}{\it )}$^*$ $vexpr$ {\tt ]}\\
-%% & | & {\tt Idtac}\\
-%% & | & {\tt Fail}\\
-%% & | & {\primitivetactic}\\
-%% & | & $arg$
-%% \end{tabular}
-%% \end{center}}}
-%% \caption{Values of ${\cal L}_{tac}$}
-%% \label{ltacval}
-%% \end{figure}
-
-%% \subsection{Evaluation}
-
-\subsubsection[Sequence]{Sequence\tacindex{;}
-\index{Tacticals!;@{\tt {\tac$_1$};\tac$_2$}}}
-
-A sequence is an expression of the following form:
-\begin{quote}
-{\tacexpr}$_1$ {\tt ;} {\tacexpr}$_2$
-\end{quote}
-The expressions {\tacexpr}$_1$ and {\tacexpr}$_2$ are evaluated
-to $v_1$ and $v_2$ which have to be tactic values. The tactic $v_1$ is
-then applied and $v_2$ is applied to every subgoal generated by the
-application of $v_1$. Sequence is left-associative.
-
-\subsubsection[General sequence]{General sequence\tacindex{;[\ldots$\mid$\ldots$\mid$\ldots]}}
-%\tacindex{; [ | ]}
-%\index{; [ | ]@{\tt ;[\ldots$\mid$\ldots$\mid$\ldots]}}
-\index{Tacticals!; [ \mid ]@{\tt {\tac$_0$};[{\tac$_1$}$\mid$\ldots$\mid$\tac$_n$]}}
-
-A general sequence has the following form:
-\begin{quote}
-{\tacexpr}$_0$ {\tt ; [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |}
-{\tacexpr}$_n$ {\tt ]}
-\end{quote}
-The expressions {\tacexpr}$_i$ are evaluated to $v_i$, for $i=0,...,n$
-and all have to be tactics. The tactic $v_0$ is applied and $v_i$ is
-applied to the $i$-th generated subgoal by the application of $v_0$,
-for $=1,...,n$. It fails if the application of $v_0$ does not generate
-exactly $n$ subgoals.
-
-\begin{Variants}
- \item If no tactic is given for the $i$-th generated subgoal, it
-behaves as if the tactic {\tt idtac} were given. For instance, {\tt
-split ; [ | auto ]} is a shortcut for
-{\tt split ; [ idtac | auto ]}.
-
- \item {\tacexpr}$_0$ {\tt ; [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |}
- {\tacexpr}$_i$ {\tt |} {\tt ..} {\tt |} {\tacexpr}$_{i+1+j}$ {\tt |}
- $...$ {\tt |} {\tacexpr}$_n$ {\tt ]}
-
- In this variant, {\tt idtac} is used for the subgoals numbered from
- $i+1$ to $i+j$ (assuming $n$ is the number of subgoals).
-
- Note that {\tt ..} is part of the syntax, while $...$ is the meta-symbol used
- to describe a list of {\tacexpr} of arbitrary length.
-
- \item {\tacexpr}$_0$ {\tt ; [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |}
- {\tacexpr}$_i$ {\tt |} {\tacexpr} {\tt ..} {\tt |}
- {\tacexpr}$_{i+1+j}$ {\tt |} $...$ {\tt |} {\tacexpr}$_n$ {\tt ]}
-
- In this variant, {\tacexpr} is used instead of {\tt idtac} for the
- subgoals numbered from $i+1$ to $i+j$.
-
-\end{Variants}
-
-
-
-\subsubsection[For loop]{For loop\tacindex{do}
-\index{Tacticals!do@{\tt do}}}
-
-There is a for loop that repeats a tactic {\num} times:
-\begin{quote}
-{\tt do} {\num} {\tacexpr}
-\end{quote}
-{\tacexpr} is evaluated to $v$. $v$ must be a tactic value. $v$ is
-applied {\num} times. Supposing ${\num}>1$, after the first
-application of $v$, $v$ is applied, at least once, to the generated
-subgoals and so on. It fails if the application of $v$ fails before
-the {\num} applications have been completed.
-
-\subsubsection[Repeat loop]{Repeat loop\tacindex{repeat}
-\index{Tacticals!repeat@{\tt repeat}}}
-
-We have a repeat loop with:
-\begin{quote}
-{\tt repeat} {\tacexpr}
-\end{quote}
-{\tacexpr} is evaluated to $v$. If $v$ denotes a tactic, this tactic
-is applied to the goal. If the application fails, the tactic is
-applied recursively to all the generated subgoals until it eventually
-fails. The recursion stops in a subgoal when the tactic has failed.
-The tactic {\tt repeat {\tacexpr}} itself never fails.
-
-\subsubsection[Error catching]{Error catching\tacindex{try}
-\index{Tacticals!try@{\tt try}}}
-
-We can catch the tactic errors with:
-\begin{quote}
-{\tt try} {\tacexpr}
-\end{quote}
-{\tacexpr} is evaluated to $v$. $v$ must be a tactic value. $v$ is
-applied. If the application of $v$ fails, it catches the error and
-leaves the goal unchanged. If the level of the exception is positive,
-then the exception is re-raised with its level decremented.
-
-\subsubsection[Detecting progress]{Detecting progress\tacindex{progress}}
-
-We can check if a tactic made progress with:
-\begin{quote}
-{\tt progress} {\tacexpr}
-\end{quote}
-{\tacexpr} is evaluated to $v$. $v$ must be a tactic value. $v$ is
-applied. If the application of $v$ produced one subgoal equal to the
-initial goal (up to syntactical equality), then an error of level 0 is
-raised.
-
-\ErrMsg \errindex{Failed to progress}
-
-\subsubsection[Branching]{Branching\tacindex{$\mid\mid$}
-\index{Tacticals!orelse@{\tt $\mid\mid$}}}
-
-We can easily branch with the following structure:
-\begin{quote}
-{\tacexpr}$_1$ {\tt ||} {\tacexpr}$_2$
-\end{quote}
-{\tacexpr}$_1$ and {\tacexpr}$_2$ are evaluated to $v_1$ and
-$v_2$. $v_1$ and $v_2$ must be tactic values. $v_1$ is applied and if
-it fails to progress then $v_2$ is applied. Branching is left-associative.
-
-\subsubsection[First tactic to work]{First tactic to work\tacindex{first}
-\index{Tacticals!first@{\tt first}}}
-
-We may consider the first tactic to work (i.e. which does not fail) among a
-panel of tactics:
-\begin{quote}
-{\tt first [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} {\tacexpr}$_n$ {\tt ]}
-\end{quote}
-{\tacexpr}$_i$ are evaluated to $v_i$ and $v_i$ must be tactic values, for
-$i=1,...,n$. Supposing $n>1$, it applies $v_1$, if it works, it stops else it
-tries to apply $v_2$ and so on. It fails when there is no applicable tactic.
-
-\ErrMsg \errindex{No applicable tactic}
-
-\subsubsection[Solving]{Solving\tacindex{solve}
-\index{Tacticals!solve@{\tt solve}}}
-
-We may consider the first to solve (i.e. which generates no subgoal) among a
-panel of tactics:
-\begin{quote}
-{\tt solve [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} {\tacexpr}$_n$ {\tt ]}
-\end{quote}
-{\tacexpr}$_i$ are evaluated to $v_i$ and $v_i$ must be tactic values, for
-$i=1,...,n$. Supposing $n>1$, it applies $v_1$, if it solves, it stops else it
-tries to apply $v_2$ and so on. It fails if there is no solving tactic.
-
-\ErrMsg \errindex{Cannot solve the goal}
-
-\subsubsection[Identity]{Identity\tacindex{idtac}
-\index{Tacticals!idtac@{\tt idtac}}}
-
-The constant {\tt idtac} is the identity tactic: it leaves any goal
-unchanged but it appears in the proof script.
-
-\variant {\tt idtac \nelist{\messagetoken}{}}
-
-This prints the given tokens. Strings and integers are printed
-literally. If a (term) variable is given, its contents are printed.
-
-
-\subsubsection[Failing]{Failing\tacindex{fail}
-\index{Tacticals!fail@{\tt fail}}}
-
-The tactic {\tt fail} is the always-failing tactic: it does not solve
-any goal. It is useful for defining other tacticals since it can be
-catched by {\tt try} or {\tt match goal}.
-
-\begin{Variants}
-\item {\tt fail $n$}\\
-The number $n$ is the failure level. If no level is specified, it
-defaults to $0$. The level is used by {\tt try} and {\tt match goal}.
-If $0$, it makes {\tt match goal} considering the next clause
-(backtracking). If non zero, the current {\tt match goal} block or
-{\tt try} command is aborted and the level is decremented.
-
-\item {\tt fail \nelist{\messagetoken}{}}\\
-The given tokens are used for printing the failure message.
-
-\item {\tt fail $n$ \nelist{\messagetoken}{}}\\
-This is a combination of the previous variants.
-\end{Variants}
-
-\ErrMsg \errindex{Tactic Failure {\it message} (level $n$)}.
-
-\subsubsection[Local definitions]{Local definitions\index{Ltac!let}
-\index{Ltac!let rec}
-\index{let!in Ltac}
-\index{let rec!in Ltac}}
-
-Local definitions can be done as follows:
-\begin{quote}
-{\tt let} {\ident}$_1$ {\tt :=} {\tacexpr}$_1$\\
-{\tt with} {\ident}$_2$ {\tt :=} {\tacexpr}$_2$\\
-...\\
-{\tt with} {\ident}$_n$ {\tt :=} {\tacexpr}$_n$ {\tt in}\\
-{\tacexpr}
-\end{quote}
-each {\tacexpr}$_i$ is evaluated to $v_i$, then, {\tacexpr} is
-evaluated by substituting $v_i$ to each occurrence of {\ident}$_i$,
-for $i=1,...,n$. There is no dependencies between the {\tacexpr}$_i$
-and the {\ident}$_i$.
-
-Local definitions can be recursive by using {\tt let rec} instead of
-{\tt let}. In this latter case, the definitions are evaluated lazily
-so that the {\tt rec} keyword can be used also in non recursive cases
-so as to avoid the eager evaluation of local definitions.
-
-\subsubsection{Application}
-
-An application is an expression of the following form:
-\begin{quote}
-{\qualid} {\tacarg}$_1$ ... {\tacarg}$_n$
-\end{quote}
-The reference {\qualid} must be bound to some defined tactic
-definition expecting at least $n$ arguments. The expressions
-{\tacexpr}$_i$ are evaluated to $v_i$, for $i=1,...,n$.
-%If {\tacexpr} is a {\tt Fun} or {\tt Rec} value then the body is evaluated by
-%substituting $v_i$ to the formal parameters, for $i=1,...,n$. For recursive
-%clauses, the bodies are lazily substituted (when an identifier to be evaluated
-%is the name of a recursive clause).
-
-%\subsection{Application of tactic values}
-
-\subsubsection[Function construction]{Function construction\index{fun!in Ltac}
-\index{Ltac!fun}}
-
-A parameterized tactic can be built anonymously (without resorting to
-local definitions) with:
-\begin{quote}
-{\tt fun} {\ident${}_1$} ... {\ident${}_n$} {\tt =>} {\tacexpr}
-\end{quote}
-Indeed, local definitions of functions are a syntactic sugar for
-binding a {\tt fun} tactic to an identifier.
-
-\subsubsection[Pattern matching on terms]{Pattern matching on terms\index{Ltac!match}
-\index{match!in Ltac}}
-
-We can carry out pattern matching on terms with:
-\begin{quote}
-{\tt match} {\tacexpr} {\tt with}\\
-~~~{\cpattern}$_1$ {\tt =>} {\tacexpr}$_1$\\
-~{\tt |} {\cpattern}$_2$ {\tt =>} {\tacexpr}$_2$\\
-~...\\
-~{\tt |} {\cpattern}$_n$ {\tt =>} {\tacexpr}$_n$\\
-~{\tt |} {\tt \_} {\tt =>} {\tacexpr}$_{n+1}$\\
-{\tt end}
-\end{quote}
-The expression {\tacexpr} is evaluated and should yield a term which
-is matched against {\cpattern}$_1$. The matching is non-linear: if a
-metavariable occurs more than once, it should match the same
-expression every time. It is first-order except on the
-variables of the form {\tt @?id} that occur in head position of an
-application. For these variables, the matching is second-order and
-returns a functional term.
-
-If the matching with {\cpattern}$_1$ succeeds, then {\tacexpr}$_1$ is
-evaluated into some value by substituting the pattern matching
-instantiations to the metavariables. If {\tacexpr}$_1$ evaluates to a
-tactic and the {\tt match} expression is in position to be applied to
-a goal (e.g. it is not bound to a variable by a {\tt let in}), then
-this tactic is applied. If the tactic succeeds, the list of resulting
-subgoals is the result of the {\tt match} expression. If
-{\tacexpr}$_1$ does not evaluate to a tactic or if the {\tt match}
-expression is not in position to be applied to a goal, then the result
-of the evaluation of {\tacexpr}$_1$ is the result of the {\tt match}
-expression.
-
-If the matching with {\cpattern}$_1$ fails, or if it succeeds but the
-evaluation of {\tacexpr}$_1$ fails, or if the evaluation of
-{\tacexpr}$_1$ succeeds but returns a tactic in execution position
-whose execution fails, then {\cpattern}$_2$ is used and so on. The
-pattern {\_} matches any term and shunts all remaining patterns if
-any. If all clauses fail (in particular, there is no pattern {\_})
-then a no-matching-clause error is raised.
-
-\begin{ErrMsgs}
-
-\item \errindex{No matching clauses for match}
-
- No pattern can be used and, in particular, there is no {\tt \_} pattern.
-
-\item \errindex{Argument of match does not evaluate to a term}
-
- This happens when {\tacexpr} does not denote a term.
-
-\end{ErrMsgs}
-
-\begin{Variants}
-
-\item \index{lazymatch!in Ltac}
-\index{Ltac!lazymatch}
-Using {\tt lazymatch} instead of {\tt match} has an effect if the
-right-hand-side of a clause returns a tactic. With {\tt match}, the
-tactic is applied to the current goal (and the next clause is tried if
-it fails). With {\tt lazymatch}, the tactic is directly returned as
-the result of the whole {\tt lazymatch} block without being first
-tried to be applied to the goal. Typically, if the {\tt lazymatch}
-block is bound to some variable $x$ in a {\tt let in}, then tactic
-expression gets bound to the variable $x$.
-
-\item \index{context!in pattern}
-There is a special form of patterns to match a subterm against the
-pattern:
-\begin{quote}
-{\tt context} {\ident} {\tt [} {\cpattern} {\tt ]}
-\end{quote}
-It matches any term which one subterm matches {\cpattern}. If there is
-a match, the optional {\ident} is assign the ``matched context'', that
-is the initial term where the matched subterm is replaced by a
-hole. The definition of {\tt context} in expressions below will show
-how to use such term contexts.
-
-If the evaluation of the right-hand-side of a valid match fails, the
-next matching subterm is tried. If no further subterm matches, the
-next clause is tried. Matching subterms are considered top-bottom and
-from left to right (with respect to the raw printing obtained by
-setting option {\tt Printing All}, see Section~\ref{SetPrintingAll}).
-
-\begin{coq_example}
-Ltac f x :=
- match x with
- context f [S ?X] =>
- idtac X; (* To display the evaluation order *)
- assert (p := refl_equal 1 : X=1); (* To filter the case X=1 *)
- let x:= context f[O] in assert (x=O) (* To observe the context *)
- end.
-Goal True.
-f (3+4).
-\end{coq_example}
-
-\item \index{appcontext!in pattern}
-For historical reasons, {\tt context} considers $n$-ary applications
-such as {\tt (f 1 2)} as a whole, and not as a sequence of unary
-applications {\tt ((f 1) 2)}. Hence {\tt context [f ?x]} will fail
-to find a matching subterm in {\tt (f 1 2)}: if the pattern is a partial
-application, the matched subterms will be necessarily be
-applications with exactly the same number of arguments.
-Alternatively, one may now use the following variant of {\tt context}:
-\begin{quote}
-{\tt appcontext} {\ident} {\tt [} {\cpattern} {\tt ]}
-\end{quote}
-The behavior of {\tt appcontext} is the same as the one of {\tt
- context}, except that a matching subterm could be a partial
-part of a longer application. For instance, in {\tt (f 1 2)},
-an {\tt appcontext [f ?x]} will find the matching subterm {\tt (f 1)}.
-
-\end{Variants}
-
-\subsubsection[Pattern matching on goals]{Pattern matching on goals\index{Ltac!match goal}
-\index{Ltac!match reverse goal}
-\index{match goal!in Ltac}
-\index{match reverse goal!in Ltac}}
-
-We can make pattern matching on goals using the following expression:
-\begin{quote}
-\begin{tabbing}
-{\tt match goal with}\\
-~~\={\tt |} $hyp_{1,1}${\tt ,}...{\tt ,}$hyp_{1,m_1}$
- ~~{\tt |-}{\cpattern}$_1${\tt =>} {\tacexpr}$_1$\\
- \>{\tt |} $hyp_{2,1}${\tt ,}...{\tt ,}$hyp_{2,m_2}$
- ~~{\tt |-}{\cpattern}$_2${\tt =>} {\tacexpr}$_2$\\
-~~...\\
- \>{\tt |} $hyp_{n,1}${\tt ,}...{\tt ,}$hyp_{n,m_n}$
- ~~{\tt |-}{\cpattern}$_n${\tt =>} {\tacexpr}$_n$\\
- \>{\tt |\_}~~~~{\tt =>} {\tacexpr}$_{n+1}$\\
-{\tt end}
-\end{tabbing}
-\end{quote}
-
-If each hypothesis pattern $hyp_{1,i}$, with $i=1,...,m_1$
-is matched (non-linear first-order unification) by an hypothesis of
-the goal and if {\cpattern}$_1$ is matched by the conclusion of the
-goal, then {\tacexpr}$_1$ is evaluated to $v_1$ by substituting the
-pattern matching to the metavariables and the real hypothesis names
-bound to the possible hypothesis names occurring in the hypothesis
-patterns. If $v_1$ is a tactic value, then it is applied to the
-goal. If this application fails, then another combination of
-hypotheses is tried with the same proof context pattern. If there is
-no other combination of hypotheses then the second proof context
-pattern is tried and so on. If the next to last proof context pattern
-fails then {\tacexpr}$_{n+1}$ is evaluated to $v_{n+1}$ and $v_{n+1}$
-is applied. Note also that matching against subterms (using the {\tt
-context} {\ident} {\tt [} {\cpattern} {\tt ]}) is available and may
-itself induce extra backtrackings.
-
-\ErrMsg \errindex{No matching clauses for match goal}
-
-No clause succeeds, i.e. all matching patterns, if any,
-fail at the application of the right-hand-side.
-
-\medskip
-
-It is important to know that each hypothesis of the goal can be
-matched by at most one hypothesis pattern. The order of matching is
-the following: hypothesis patterns are examined from the right to the
-left (i.e. $hyp_{i,m_i}$ before $hyp_{i,1}$). For each hypothesis
-pattern, the goal hypothesis are matched in order (fresher hypothesis
-first), but it possible to reverse this order (older first) with
-the {\tt match reverse goal with} variant.
-
-\variant
-\index{lazymatch goal!in Ltac}
-\index{Ltac!lazymatch goal}
-\index{lazymatch reverse goal!in Ltac}
-\index{Ltac!lazymatch reverse goal}
-Using {\tt lazymatch} instead of {\tt match} has an effect if the
-right-hand-side of a clause returns a tactic. With {\tt match}, the
-tactic is applied to the current goal (and the next clause is tried if
-it fails). With {\tt lazymatch}, the tactic is directly returned as
-the result of the whole {\tt lazymatch} block without being first
-tried to be applied to the goal. Typically, if the {\tt lazymatch}
-block is bound to some variable $x$ in a {\tt let in}, then tactic
-expression gets bound to the variable $x$.
-
-\begin{coq_example}
-Ltac test_lazy :=
- lazymatch goal with
- | _ => idtac "here"; fail
- | _ => idtac "wasn't lazy"; trivial
- end.
-Ltac test_eager :=
- match goal with
- | _ => idtac "here"; fail
- | _ => idtac "wasn't lazy"; trivial
- end.
-Goal True.
-test_lazy || idtac "was lazy".
-test_eager || idtac "was lazy".
-\end{coq_example}
-
-\subsubsection[Filling a term context]{Filling a term context\index{context!in expression}}
-
-The following expression is not a tactic in the sense that it does not
-produce subgoals but generates a term to be used in tactic
-expressions:
-\begin{quote}
-{\tt context} {\ident} {\tt [} {\tacexpr} {\tt ]}
-\end{quote}
-{\ident} must denote a context variable bound by a {\tt context}
-pattern of a {\tt match} expression. This expression evaluates
-replaces the hole of the value of {\ident} by the value of
-{\tacexpr}.
-
-\ErrMsg \errindex{not a context variable}
-
-
-\subsubsection[Generating fresh hypothesis names]{Generating fresh hypothesis names\index{Ltac!fresh}
-\index{fresh!in Ltac}}
-
-Tactics sometimes have to generate new names for hypothesis. Letting
-the system decide a name with the {\tt intro} tactic is not so good
-since it is very awkward to retrieve the name the system gave.
-The following expression returns an identifier:
-\begin{quote}
-{\tt fresh} \nelist{\textrm{\textsl{component}}}{}
-\end{quote}
-It evaluates to an identifier unbound in the goal. This fresh
-identifier is obtained by concatenating the value of the
-\textrm{\textsl{component}}'s (each of them is, either an {\ident} which
-has to refer to a name, or directly a name denoted by a
-{\qstring}). If the resulting name is already used, it is padded
-with a number so that it becomes fresh. If no component is
-given, the name is a fresh derivative of the name {\tt H}.
-
-\subsubsection[Computing in a constr]{Computing in a constr\index{Ltac!eval}
-\index{eval!in Ltac}}
-
-Evaluation of a term can be performed with:
-\begin{quote}
-{\tt eval} {\nterm{redexpr}} {\tt in} {\term}
-\end{quote}
-where \nterm{redexpr} is a reduction tactic among {\tt red}, {\tt
-hnf}, {\tt compute}, {\tt simpl}, {\tt cbv}, {\tt lazy}, {\tt unfold},
-{\tt fold}, {\tt pattern}.
-
-\subsubsection{Type-checking a term}
-%\tacindex{type of}
-\index{Ltac!type of}
-\index{type of!in Ltac}
-
-The following returns the type of {\term}:
-
-\begin{quote}
-{\tt type of} {\term}
-\end{quote}
-
-\subsubsection[Accessing tactic decomposition]{Accessing tactic decomposition\tacindex{info}
-\index{Tacticals!info@{\tt info}}}
-
-Tactical ``{\tt info} {\tacexpr}'' is not really a tactical. For
-elementary tactics, this is equivalent to \tacexpr. For complex tactic
-like \texttt{auto}, it displays the operations performed by the
-tactic.
-
-\subsubsection[Proving a subgoal as a separate lemma]{Proving a subgoal as a separate lemma\tacindex{abstract}
-\index{Tacticals!abstract@{\tt abstract}}}
-
-From the outside ``\texttt{abstract \tacexpr}'' is the same as
-{\tt solve \tacexpr}. Internally it saves an auxiliary lemma called
-{\ident}\texttt{\_subproof}\textit{n} where {\ident} is the name of the
-current goal and \textit{n} is chosen so that this is a fresh name.
-
-This tactical is useful with tactics such as \texttt{omega} or
-\texttt{discriminate} that generate huge proof terms. With that tool
-the user can avoid the explosion at time of the \texttt{Save} command
-without having to cut manually the proof in smaller lemmas.
-
-\begin{Variants}
-\item \texttt{abstract {\tacexpr} using {\ident}}.\\
- Give explicitly the name of the auxiliary lemma.
-\end{Variants}
-
-\ErrMsg \errindex{Proof is not complete}
-
-\subsubsection[Calling an external tactic]{Calling an external tactic\index{Ltac!external}}
-
-The tactic {\tt external} allows to run an executable outside the
-{\Coq} executable. The communication is done via an XML encoding of
-constructions. The syntax of the command is
-
-\begin{quote}
-{\tt external} "\textsl{command}" "\textsl{request}" \nelist{\tacarg}{}
-\end{quote}
-
-The string \textsl{command}, to be interpreted in the default
-execution path of the operating system, is the name of the external
-command. The string \textsl{request} is the name of a request to be
-sent to the external command. Finally the list of tactic arguments
-have to evaluate to terms. An XML tree of the following form is sent
-to the standard input of the external command.
-\medskip
-
-\begin{tabular}{l}
-\texttt{<REQUEST req="}\textsl{request}\texttt{">}\\
-the XML tree of the first argument\\
-{\ldots}\\
-the XML tree of the last argument\\
-\texttt{</REQUEST>}\\
-\end{tabular}
-\medskip
-
-Conversely, the external command must send on its standard output an
-XML tree of the following forms:
-
-\medskip
-\begin{tabular}{l}
-\texttt{<TERM>}\\
-the XML tree of a term\\
-\texttt{</TERM>}\\
-\end{tabular}
-\medskip
-
-\noindent or
-
-\medskip
-\begin{tabular}{l}
-\texttt{<CALL uri="}\textsl{ltac\_qualified\_ident}\texttt{">}\\
-the XML tree of a first argument\\
-{\ldots}\\
-the XML tree of a last argument\\
-\texttt{</CALL>}\\
-\end{tabular}
-
-\medskip
-\noindent where \textsl{ltac\_qualified\_ident} is the name of a
-defined {\ltac} function and each subsequent XML tree is recursively a
-\texttt{CALL} or a \texttt{TERM} node.
-
-The Document Type Definition (DTD) for terms of the Calculus of
-Inductive Constructions is the one developed as part of the MoWGLI
-European project. It can be found in the file {\tt dev/doc/cic.dtd} of
-the {\Coq} source archive.
-
-An example of parser for this DTD, written in the Objective Caml -
-Camlp4 language, can be found in the file {\tt parsing/g\_xml.ml4} of
-the {\Coq} source archive.
-
-\section[Tactic toplevel definitions]{Tactic toplevel definitions\comindex{Ltac}}
-
-\subsection{Defining {\ltac} functions}
-
-Basically, {\ltac} toplevel definitions are made as follows:
-%{\tt Tactic Definition} {\ident} {\tt :=} {\tacexpr}\\
-%
-%{\tacexpr} is evaluated to $v$ and $v$ is associated to {\ident}. Next, every
-%script is evaluated by substituting $v$ to {\ident}.
-%
-%We can define functional definitions by:\\
-\begin{quote}
-{\tt Ltac} {\ident} {\ident}$_1$ ... {\ident}$_n$ {\tt :=}
-{\tacexpr}
-\end{quote}
-This defines a new {\ltac} function that can be used in any tactic
-script or new {\ltac} toplevel definition.
-
-\Rem The preceding definition can equivalently be written:
-\begin{quote}
-{\tt Ltac} {\ident} {\tt := fun} {\ident}$_1$ ... {\ident}$_n$
-{\tt =>} {\tacexpr}
-\end{quote}
-Recursive and mutual recursive function definitions are also
-possible with the syntax:
-\begin{quote}
-{\tt Ltac} {\ident}$_1$ {\ident}$_{1,1}$ ...
-{\ident}$_{1,m_1}$~~{\tt :=} {\tacexpr}$_1$\\
-{\tt with} {\ident}$_2$ {\ident}$_{2,1}$ ... {\ident}$_{2,m_2}$~~{\tt :=}
-{\tacexpr}$_2$\\
-...\\
-{\tt with} {\ident}$_n$ {\ident}$_{n,1}$ ... {\ident}$_{n,m_n}$~~{\tt :=}
-{\tacexpr}$_n$
-\end{quote}
-\medskip
-It is also possible to \emph{redefine} an existing user-defined tactic
-using the syntax:
-\begin{quote}
-{\tt Ltac} {\qualid} {\ident}$_1$ ... {\ident}$_n$ {\tt ::=}
-{\tacexpr}
-\end{quote}
-A previous definition of \qualid must exist in the environment.
-The new definition will always be used instead of the old one and
-it goes accross module boundaries.
-
-If preceded by the keyword {\tt Local} the tactic definition will not
-be exported outside the current module.
-
-\subsection[Printing {\ltac} tactics]{Printing {\ltac} tactics\comindex{Print Ltac}}
-
-Defined {\ltac} functions can be displayed using the command
-
-\begin{quote}
-{\tt Print Ltac {\qualid}.}
-\end{quote}
-
-\section[Debugging {\ltac} tactics]{Debugging {\ltac} tactics\comindex{Set Ltac Debug}
-\comindex{Unset Ltac Debug}
-\comindex{Test Ltac Debug}}
-
-The {\ltac} interpreter comes with a step-by-step debugger. The
-debugger can be activated using the command
-
-\begin{quote}
-{\tt Set Ltac Debug.}
-\end{quote}
-
-\noindent and deactivated using the command
-
-\begin{quote}
-{\tt Unset Ltac Debug.}
-\end{quote}
-
-To know if the debugger is on, use the command \texttt{Test Ltac Debug}.
-When the debugger is activated, it stops at every step of the
-evaluation of the current {\ltac} expression and it prints information
-on what it is doing. The debugger stops, prompting for a command which
-can be one of the following:
-
-\medskip
-\begin{tabular}{ll}
-simple newline: & go to the next step\\
-h: & get help\\
-x: & exit current evaluation\\
-s: & continue current evaluation without stopping\\
-r$n$: & advance $n$ steps further\\
-\end{tabular}
-\endinput
-
-\subsection{Permutation on closed lists}
-
-\begin{figure}[b]
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example*}
-Require Import List.
-Section Sort.
-Variable A : Set.
-Inductive permut : list A -> list A -> Prop :=
- | permut_refl : forall l, permut l l
- | permut_cons :
- forall a l0 l1, permut l0 l1 -> permut (a :: l0) (a :: l1)
- | permut_append : forall a l, permut (a :: l) (l ++ a :: nil)
- | permut_trans :
- forall l0 l1 l2, permut l0 l1 -> permut l1 l2 -> permut l0 l2.
-End Sort.
-\end{coq_example*}
-\end{center}
-\caption{Definition of the permutation predicate}
-\label{permutpred}
-\end{figure}
-
-
-Another more complex example is the problem of permutation on closed
-lists. The aim is to show that a closed list is a permutation of
-another one. First, we define the permutation predicate as shown on
-Figure~\ref{permutpred}.
-
-\begin{figure}[p]
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example}
-Ltac Permut n :=
- match goal with
- | |- (permut _ ?l ?l) => apply permut_refl
- | |- (permut _ (?a :: ?l1) (?a :: ?l2)) =>
- let newn := eval compute in (length l1) in
- (apply permut_cons; Permut newn)
- | |- (permut ?A (?a :: ?l1) ?l2) =>
- match eval compute in n with
- | 1 => fail
- | _ =>
- let l1' := constr:(l1 ++ a :: nil) in
- (apply (permut_trans A (a :: l1) l1' l2);
- [ apply permut_append | compute; Permut (pred n) ])
- end
- end.
-Ltac PermutProve :=
- match goal with
- | |- (permut _ ?l1 ?l2) =>
- match eval compute in (length l1 = length l2) with
- | (?n = ?n) => Permut n
- end
- end.
-\end{coq_example}
-\end{minipage}}
-\end{center}
-\caption{Permutation tactic}
-\label{permutltac}
-\end{figure}
-
-\begin{figure}[p]
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example*}
-Lemma permut_ex1 :
- permut nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil).
-Proof.
-PermutProve.
-Qed.
-
-Lemma permut_ex2 :
- permut nat
- (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil)
- (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil).
-Proof.
-PermutProve.
-Qed.
-\end{coq_example*}
-\end{minipage}}
-\end{center}
-\caption{Examples of {\tt PermutProve} use}
-\label{permutlem}
-\end{figure}
-
-Next, we can write naturally the tactic and the result can be seen on
-Figure~\ref{permutltac}. We can notice that we use two toplevel
-definitions {\tt PermutProve} and {\tt Permut}. The function to be
-called is {\tt PermutProve} which computes the lengths of the two
-lists and calls {\tt Permut} with the length if the two lists have the
-same length. {\tt Permut} works as expected. If the two lists are
-equal, it concludes. Otherwise, if the lists have identical first
-elements, it applies {\tt Permut} on the tail of the lists. Finally,
-if the lists have different first elements, it puts the first element
-of one of the lists (here the second one which appears in the {\tt
- permut} predicate) at the end if that is possible, i.e., if the new
-first element has been at this place previously. To verify that all
-rotations have been done for a list, we use the length of the list as
-an argument for {\tt Permut} and this length is decremented for each
-rotation down to, but not including, 1 because for a list of length
-$n$, we can make exactly $n-1$ rotations to generate at most $n$
-distinct lists. Here, it must be noticed that we use the natural
-numbers of {\Coq} for the rotation counter. On Figure~\ref{ltac}, we
-can see that it is possible to use usual natural numbers but they are
-only used as arguments for primitive tactics and they cannot be
-handled, in particular, we cannot make computations with them. So, a
-natural choice is to use {\Coq} data structures so that {\Coq} makes
-the computations (reductions) by {\tt eval compute in} and we can get
-the terms back by {\tt match}.
-
-With {\tt PermutProve}, we can now prove lemmas such those shown on
-Figure~\ref{permutlem}.
-
-
-\subsection{Deciding intuitionistic propositional logic}
-
-\begin{figure}[tbp]
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example}
-Ltac Axioms :=
- match goal with
- | |- True => trivial
- | _:False |- _ => elimtype False; assumption
- | _:?A |- ?A => auto
- end.
-Ltac DSimplif :=
- repeat
- (intros;
- match goal with
- | id:(~ _) |- _ => red in id
- | id:(_ /\ _) |- _ =>
- elim id; do 2 intro; clear id
- | id:(_ \/ _) |- _ =>
- elim id; intro; clear id
- | id:(?A /\ ?B -> ?C) |- _ =>
- cut (A -> B -> C);
- [ intro | intros; apply id; split; assumption ]
- | id:(?A \/ ?B -> ?C) |- _ =>
- cut (B -> C);
- [ cut (A -> C);
- [ intros; clear id
- | intro; apply id; left; assumption ]
- | intro; apply id; right; assumption ]
- | id0:(?A -> ?B),id1:?A |- _ =>
- cut B; [ intro; clear id0 | apply id0; assumption ]
- | |- (_ /\ _) => split
- | |- (~ _) => red
- end).
-\end{coq_example}
-\end{minipage}}
-\end{center}
-\caption{Deciding intuitionistic propositions (1)}
-\label{tautoltaca}
-\end{figure}
-
-\begin{figure}
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example}
-Ltac TautoProp :=
- DSimplif;
- Axioms ||
- match goal with
- | id:((?A -> ?B) -> ?C) |- _ =>
- cut (B -> C);
- [ intro; cut (A -> B);
- [ intro; cut C;
- [ intro; clear id | apply id; assumption ]
- | clear id ]
- | intro; apply id; intro; assumption ]; TautoProp
- | id:(~ ?A -> ?B) |- _ =>
- cut (False -> B);
- [ intro; cut (A -> False);
- [ intro; cut B;
- [ intro; clear id | apply id; assumption ]
- | clear id ]
- | intro; apply id; red; intro; assumption ]; TautoProp
- | |- (_ \/ _) => (left; TautoProp) || (right; TautoProp)
- end.
-\end{coq_example}
-\end{minipage}}
-\end{center}
-\caption{Deciding intuitionistic propositions (2)}
-\label{tautoltacb}
-\end{figure}
-
-The pattern matching on goals allows a complete and so a powerful
-backtracking when returning tactic values. An interesting application
-is the problem of deciding intuitionistic propositional logic.
-Considering the contraction-free sequent calculi {\tt LJT*} of
-Roy~Dyckhoff (\cite{Dyc92}), it is quite natural to code such a tactic
-using the tactic language. On Figure~\ref{tautoltaca}, the tactic {\tt
- Axioms} tries to conclude using usual axioms. The {\tt DSimplif}
-tactic applies all the reversible rules of Dyckhoff's system.
-Finally, on Figure~\ref{tautoltacb}, the {\tt TautoProp} tactic (the
-main tactic to be called) simplifies with {\tt DSimplif}, tries to
-conclude with {\tt Axioms} and tries several paths using the
-backtracking rules (one of the four Dyckhoff's rules for the left
-implication to get rid of the contraction and the right or).
-
-\begin{figure}[tb]
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example*}
-Lemma tauto_ex1 : forall A B:Prop, A /\ B -> A \/ B.
-Proof.
-TautoProp.
-Qed.
-
-Lemma tauto_ex2 :
- forall A B:Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B.
-Proof.
-TautoProp.
-Qed.
-\end{coq_example*}
-\end{minipage}}
-\end{center}
-\caption{Proofs of tautologies with {\tt TautoProp}}
-\label{tautolem}
-\end{figure}
-
-For example, with {\tt TautoProp}, we can prove tautologies like those of
-Figure~\ref{tautolem}.
-
-
-\subsection{Deciding type isomorphisms}
-
-A more tricky problem is to decide equalities between types and modulo
-isomorphisms. Here, we choose to use the isomorphisms of the simply typed
-$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example,
-\cite{RC95}). The axioms of this $\lb{}$-calculus are given by
-Figure~\ref{isosax}.
-
-\begin{figure}
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-\begin{coq_example*}
-Open Scope type_scope.
-Section Iso_axioms.
-Variables A B C : Set.
-Axiom Com : A * B = B * A.
-Axiom Ass : A * (B * C) = A * B * C.
-Axiom Cur : (A * B -> C) = (A -> B -> C).
-Axiom Dis : (A -> B * C) = (A -> B) * (A -> C).
-Axiom P_unit : A * unit = A.
-Axiom AR_unit : (A -> unit) = unit.
-Axiom AL_unit : (unit -> A) = A.
-Lemma Cons : B = C -> A * B = A * C.
-Proof.
-intro Heq; rewrite Heq; apply refl_equal.
-Qed.
-End Iso_axioms.
-\end{coq_example*}
-\end{minipage}}
-\end{center}
-\caption{Type isomorphism axioms}
-\label{isosax}
-\end{figure}
-
-The tactic to judge equalities modulo this axiomatization can be written as
-shown on Figures~\ref{isosltac1} and~\ref{isosltac2}. The algorithm is quite
-simple. Types are reduced using axioms that can be oriented (this done by {\tt
-MainSimplif}). The normal forms are sequences of Cartesian
-products without Cartesian product in the left component. These normal forms
-are then compared modulo permutation of the components (this is done by {\tt
-CompareStruct}). The main tactic to be called and realizing this algorithm is
-{\tt IsoProve}.
-
-\begin{figure}
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example}
-Ltac DSimplif trm :=
- match trm with
- | (?A * ?B * ?C) =>
- rewrite <- (Ass A B C); try MainSimplif
- | (?A * ?B -> ?C) =>
- rewrite (Cur A B C); try MainSimplif
- | (?A -> ?B * ?C) =>
- rewrite (Dis A B C); try MainSimplif
- | (?A * unit) =>
- rewrite (P_unit A); try MainSimplif
- | (unit * ?B) =>
- rewrite (Com unit B); try MainSimplif
- | (?A -> unit) =>
- rewrite (AR_unit A); try MainSimplif
- | (unit -> ?B) =>
- rewrite (AL_unit B); try MainSimplif
- | (?A * ?B) =>
- (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif)
- | (?A -> ?B) =>
- (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif)
- end
- with MainSimplif :=
- match goal with
- | |- (?A = ?B) => try DSimplif A; try DSimplif B
- end.
-Ltac Length trm :=
- match trm with
- | (_ * ?B) => let succ := Length B in constr:(S succ)
- | _ => constr:1
- end.
-Ltac assoc := repeat rewrite <- Ass.
-\end{coq_example}
-\end{minipage}}
-\end{center}
-\caption{Type isomorphism tactic (1)}
-\label{isosltac1}
-\end{figure}
-
-\begin{figure}
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example}
-Ltac DoCompare n :=
- match goal with
- | [ |- (?A = ?A) ] => apply refl_equal
- | [ |- (?A * ?B = ?A * ?C) ] =>
- apply Cons; let newn := Length B in DoCompare newn
- | [ |- (?A * ?B = ?C) ] =>
- match eval compute in n with
- | 1 => fail
- | _ =>
- pattern (A * B) at 1; rewrite Com; assoc; DoCompare (pred n)
- end
- end.
-Ltac CompareStruct :=
- match goal with
- | [ |- (?A = ?B) ] =>
- let l1 := Length A
- with l2 := Length B in
- match eval compute in (l1 = l2) with
- | (?n = ?n) => DoCompare n
- end
- end.
-Ltac IsoProve := MainSimplif; CompareStruct.
-\end{coq_example}
-\end{minipage}}
-\end{center}
-\caption{Type isomorphism tactic (2)}
-\label{isosltac2}
-\end{figure}
-
-Figure~\ref{isoslem} gives examples of what can be solved by {\tt IsoProve}.
-
-\begin{figure}
-\begin{center}
-\fbox{\begin{minipage}{0.95\textwidth}
-\begin{coq_example*}
-Lemma isos_ex1 :
- forall A B:Set, A * unit * B = B * (unit * A).
-Proof.
-intros; IsoProve.
-Qed.
-
-Lemma isos_ex2 :
- forall A B C:Set,
- (A * unit -> B * (C * unit)) =
- (A * unit -> (C -> unit) * C) * (unit -> A -> B).
-Proof.
-intros; IsoProve.
-Qed.
-\end{coq_example*}
-\end{minipage}}
-\end{center}
-\caption{Type equalities solved by {\tt IsoProve}}
-\label{isoslem}
-\end{figure}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: