\chapter{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 fundations, 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\contexthyps{\textrm{\textsl{context\_hyps}}} \def\tacarg{\nterm{tacarg}} \def\cpattern{\nterm{cpattern}} The syntax of the tactic language is given Figures~\ref{ltac} and~\ref{ltac_aux}. See page~\pageref{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 there can be specific variables like {\tt ?id} where {\tt id} is a {\ident} or {\tt \_}, which are metavariables for pattern matching. {\tt ?id} 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, they are used without the question mark. 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 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} \nelist{\letclause}{\tt with} {\tt in} {\atom}\\ & | & {\tt let rec} \nelist{\recclause}{\tt with} {\tt in} {\tacexpr}\\ & | & {\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 abstract} {\atom}\\ & | & {\tt abstract} {\atom} {\tt using} {\ident} \\ & | & {\tt first [} \nelist{\tacexpr}{\tt |} {\tt ]}\\ & | & {\tt solve [} \nelist{\tacexpr}{\tt |} {\tt ]}\\ & | & {\tt idtac} ~|~ {\tt idtac} {\qstring}\\ & | & {\tt fail} ~|~ {\tt fail} {\naturalnumber} {\qstring}\\ & | & {\tt fresh} ~|~ {\tt fresh} {\qstring}\\ & | & {\tt context} {\ident} {\tt [} {\term} {\tt ]}\\ & | & {\tt eval} {\nterm{redexpr}} {\tt in} {\term}\\ & | & {\tt type of} {\term}\\ & | & {\tt constr :} {\term}\\ & | & \atomictac\\ & | & {\qualid} \nelist{\tacarg}{}\\ & | & {\atom}\\ \\ {\atom} & ::= & {\qualid} \\ & | & ()\\ & | & {\tt (} {\tacexpr} {\tt )}\\ \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}\\ \\ \recclause & ::= & {\ident} \nelist{\name}{} {\tt :=} {\tacexpr}\\ \\ \contextrule & ::= & \nelist{\contexthyps}{\tt ,} {\tt |-}{\cpattern} {\tt =>} {\tacexpr}\\ & $|$ & {\tt |-} {\cpattern} {\tt =>} {\tacexpr}\\ & $|$ & {\tt \_ =>} {\tacexpr}\\ \\ \contexthyps & ::= & {\name} {\tt :} {\cpattern}\\ \\ \matchrule & ::= & {\cpattern} {\tt =>} {\tacexpr}\\ & $|$ & {\tt context} {\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} & ::= & {\tt Ltac} \nelist{\nterm{ltac\_def}} {\tt with} \\ \\ \nterm{ltac\_def} & ::= & {\ident} \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 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} \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} {\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 then applied and $v_2$ is applied to every subgoal generated by the application of $v_1$. Sequence is left associating. \subsubsection{General sequence} \tacindex{;[\ldots$\mid$\ldots$\mid$\ldots]} %\tacindex{; [ | ]} %\index{; [ | ]@{\tt ;[\ldots$\mid$\ldots$\mid$\ldots]}} \index{Tacticals!; [ | ]@{\tt {\tac$_0$};[{\tac$_1$}$\mid$\ldots$\mid$\tac$_n$]}} We can generalize the previous sequence operator as \begin{quote} {\tacexpr}$_0$ {\tt ; [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} {\tacexpr}$_n$ {\tt ]} \end{quote} {\tacexpr}$_i$ is evaluated to $v_i$, for $i=0,...,n$. $v_0$ 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. \subsubsection{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} \tacindex{repeat} \index{Tacticals!repeat@{\tt repeat}} We have a repeat loop with: \begin{quote} {\tt repeat} {\tacexpr} \end{quote} {\tacexpr} is evaluated to $v$. $v$ must be a tactic value. $v$ is applied until it fails. Supposing $n>1$, after the first application of $v$, $v$ is applied, at least once, to the generated subgoals and so on. It stops when it fails for all the generated subgoals. It never fails. \subsubsection{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} \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} \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 then $v_2$ is applied. Branching is left associating. \subsubsection{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} \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} \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. \begin{quote} {\tt idtac} and {\tt idtac "message"} \end{quote} The latter variant prints the string on the standard output. \subsubsection{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}. There are three variants: \begin{quote} {\tt fail $n$}, {\tt fail "message"} and {\tt fail $n$ "message"} \end{quote} 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. \ErrMsg \errindex{Tactic Failure "message" (level $n$)}. \subsubsection{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}. Only functions can be defined by recursion, so at least one argument is required. \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} \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} \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 {\tacexpr} is evaluated and should yield a term which is matched (non-linear first order unification) against {\cpattern}$_1$ then {\tacexpr}$_1$ is evaluated into some value by substituting the pattern matching instantiations to the metavariables. If the matching with {\cpattern}$_1$ fails, {\cpattern}$_2$ is used and so on. The pattern {\_} matches any term and shunts all remaining patterns if any. If {\tacexpr}$_1$ evaluates to a tactic, this tactic is not immediately applied to the current goal (in contrast with {\tt match goal}). If all clauses fail (in particular, there is no pattern {\_}) then a no-matching 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} \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. This operator never makes backtracking. If there are several subterms matching the pattern, only the first match is considered. Note that the order of matching is left unspecified. %% TODO: clarify this point! It *should* be specified \subsubsection{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} % TODO: specify order of hypothesis and explain reverse... 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. \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. \subsubsection{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} \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. As before, the following expression returns a term: \begin{quote} {\tt fresh} {\qstring} \end{quote} It evaluates to an identifier unbound in the goal, which is obtained by padding {\qstring} with a number if necessary. If no name is given, the prefix is {\tt H}. \subsubsection{{\tt type of} {\term}} %\tacindex{type of} \index{Ltac!type of} \index{type of!in Ltac} This tactic computes the type of {\term}. \subsubsection{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{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} \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} \section{Tactic toplevel definitions} \comindex{Ltac} Basically, tactics 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 tactic that can be used in any tactic script or new tactic 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} %This definition bloc is a set of definitions (use of %the same previous syntactical sugar) and the other scripts are evaluated as %usual except that the substitutions are lazily carried out (when an identifier %to be evaluated is the name of a recursive definition). \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: