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diff --git a/doc/refman/RefMan-tac.tex b/doc/refman/RefMan-tac.tex new file mode 100644 index 00000000..b82cdc2d --- /dev/null +++ b/doc/refman/RefMan-tac.tex @@ -0,0 +1,4303 @@ +% TODO: unify the use of \form and \type to mean a type +% or use \form specifically for a type of type Prop +\chapter{Tactics +\index{Tactics} +\label{Tactics}} + +A deduction rule is a link between some (unique) formula, that we call +the {\em conclusion} and (several) formulas that we call the {\em +premises}. Indeed, a deduction rule can be read in two ways. The first +one has the shape: {\it ``if I know this and this then I can deduce +this''}. For instance, if I have a proof of $A$ and a proof of $B$ +then I have a proof of $A \land B$. This is forward reasoning from +premises to conclusion. The other way says: {\it ``to prove this I +have to prove this and this''}. For instance, to prove $A \land B$, I +have to prove $A$ and I have to prove $B$. This is backward reasoning +which proceeds from conclusion to premises. We say that the conclusion +is {\em the goal}\index{goal} to prove and premises are {\em the +subgoals}\index{subgoal}. The tactics implement {\em backward +reasoning}. When applied to a goal, a tactic replaces this goal with +the subgoals it generates. We say that a tactic reduces a goal to its +subgoal(s). + +Each (sub)goal is denoted with a number. The current goal is numbered +1. By default, a tactic is applied to the current goal, but one can +address a particular goal in the list by writing {\sl n:\tac} which +means {\it ``apply tactic {\tac} to goal number {\sl n}''}. +We can show the list of subgoals by typing {\tt Show} (see +Section~\ref{Show}). + +Since not every rule applies to a given statement, every tactic cannot be +used to reduce any goal. In other words, before applying a tactic to a +given goal, the system checks that some {\em preconditions} are +satisfied. If it is not the case, the tactic raises an error message. + +Tactics are build from atomic tactics and tactic expressions (which +extends the folklore notion of tactical) to combine those atomic +tactics. This chapter is devoted to atomic tactics. The tactic +language will be described in Chapter~\ref{TacticLanguage}. + +There are, at least, three levels of atomic tactics. The simplest one +implements basic rules of the logical framework. The second level is +the one of {\em derived rules} which are built by combination of other +tactics. The third one implements heuristics or decision procedures to +build a complete proof of a goal. + +\section{Invocation of tactics +\label{tactic-syntax} +\index{tactic@{\tac}}} + +A tactic is applied as an ordinary command. If the tactic does not +address the first subgoal, the command may be preceded by the wished +subgoal number as shown below: + +\begin{tabular}{lcl} +{\commandtac} & ::= & {\num} {\tt :} {\tac} {\tt .}\\ + & $|$ & {\tac} {\tt .} +\end{tabular} + +\section{Explicit proof as a term} + +\subsection{\tt exact \term +\tacindex{exact} +\label{exact}} + +This tactic applies to any goal. It gives directly the exact proof +term of the goal. Let {\T} be our goal, let {\tt p} be a term of type +{\tt U} then {\tt exact p} succeeds iff {\tt T} and {\tt U} are +convertible (see Section~\ref{conv-rules}). + +\begin{ErrMsgs} +\item \errindex{Not an exact proof} +\end{ErrMsgs} + +\begin{Variants} + \item \texttt{eexact \term}\tacindex{eexact} + + This tactic behaves like \texttt{exact} but is able to handle terms with meta-variables. + +\end{Variants} + + +\subsection{\tt refine \term +\tacindex{refine} +\label{refine} +\index{?@{\texttt{?}}}} + +This tactic allows to give an exact proof but still with some +holes. The holes are noted ``\texttt{\_}''. + +\begin{ErrMsgs} +\item \errindex{invalid argument}: + the tactic \texttt{refine} doesn't know what to do + with the term you gave. +\item \texttt{Refine passed ill-formed term}: the term you gave is not + a valid proof (not easy to debug in general). + This message may also occur in higher-level tactics, which call + \texttt{refine} internally. +\item \errindex{Cannot infer a term for this placeholder} + there is a hole in the term you gave + which type cannot be inferred. Put a cast around it. +\end{ErrMsgs} + +An example of use is given in Section~\ref{refine-example}. + +\section{Basics +\index{Typing rules}} + +Tactics presented in this section implement the basic typing rules of +{\CIC} given in Chapter~\ref{Cic}. + +\subsection{{\tt assumption} +\tacindex{assumption}} + +This tactic applies to any goal. It implements the +``Var''\index{Typing rules!Var} rule given in +Section~\ref{Typed-terms}. It looks in the local context for an +hypothesis which type is equal to the goal. If it is the case, the +subgoal is proved. Otherwise, it fails. + +\begin{ErrMsgs} +\item \errindex{No such assumption} +\end{ErrMsgs} + +\begin{Variants} +\tacindex{eassumption} + \item \texttt{eassumption} + + This tactic behaves like \texttt{assumption} but is able to handle + goals with meta-variables. + +\end{Variants} + + +\subsection{\tt clear {\ident} +\tacindex{clear} +\label{clear}} + +This tactic erases the hypothesis named {\ident} in the local context +of the current goal. Then {\ident} is no more displayed and no more +usable in the proof development. + +\begin{Variants} + +\item {\tt clear {\ident$_1$} {\ldots} {\ident$_n$}} + + This is equivalent to {\tt clear {\ident$_1$}. {\ldots} clear + {\ident$_n$}.} + +\item {\tt clearbody {\ident}}\tacindex{clearbody} + + This tactic expects {\ident} to be a local definition then clears + its body. Otherwise said, this tactic turns a definition into an + assumption. + +\item \texttt{clear - {\ident$_1$} {\ldots} {\ident$_n$}} + + This tactic clears all hypotheses except the ones depending in + the hypotheses named {\ident$_1$} {\ldots} {\ident$_n$} and in the + goal. + +\item \texttt{clear} + + This tactic clears all hypotheses except the ones depending in + goal. + +\item {\tt clear dependent \ident \tacindex{clear dependent}} + + This clears the hypothesis \ident\ and all hypotheses + which depend on it. + +\end{Variants} + +\begin{ErrMsgs} +\item \errindex{{\ident} not found} +\item \errindexbis{{\ident} is used in the conclusion}{is used in the + conclusion} +\item \errindexbis{{\ident} is used in the hypothesis {\ident'}}{is + used in the hypothesis} +\end{ErrMsgs} + +\subsection{\tt move {\ident$_1$} after {\ident$_2$} +\tacindex{move} +\label{move}} + +This moves the hypothesis named {\ident$_1$} in the local context +after the hypothesis named {\ident$_2$}. + +If {\ident$_1$} comes before {\ident$_2$} in the order of dependences, +then all hypotheses between {\ident$_1$} and {\ident$_2$} which +(possibly indirectly) depend on {\ident$_1$} are moved also. + +If {\ident$_1$} comes after {\ident$_2$} in the order of dependences, +then all hypotheses between {\ident$_1$} and {\ident$_2$} which +(possibly indirectly) occur in {\ident$_1$} are moved also. + +\begin{Variants} + +\item {\tt move {\ident$_1$} before {\ident$_2$}} + +This moves {\ident$_1$} towards and just before the hypothesis named {\ident$_2$}. + +\item {\tt move {\ident} at top} + +This moves {\ident} at the top of the local context (at the beginning of the context). + +\item {\tt move {\ident} at bottom} + +This moves {\ident} at the bottom of the local context (at the end of the context). + +\end{Variants} + +\begin{ErrMsgs} + +\item \errindex{{\ident$_i$} not found} + +\item \errindex{Cannot move {\ident$_1$} after {\ident$_2$}: + it occurs in {\ident$_2$}} + +\item \errindex{Cannot move {\ident$_1$} after {\ident$_2$}: + it depends on {\ident$_2$}} + +\end{ErrMsgs} + +\subsection{\tt rename {\ident$_1$} into {\ident$_2$} +\tacindex{rename}} + +This renames hypothesis {\ident$_1$} into {\ident$_2$} in the current +context\footnote{but it does not rename the hypothesis in the + proof-term...} + +\begin{Variants} + +\item {\tt rename {\ident$_1$} into {\ident$_2$}, \ldots, + {\ident$_{2k-1}$} into {\ident$_{2k}$}} + + Is equivalent to the sequence of the corresponding atomic {\tt rename}. + +\end{Variants} + +\begin{ErrMsgs} + +\item \errindex{{\ident$_1$} not found} + +\item \errindexbis{{\ident$_2$} is already used}{is already used} + +\end{ErrMsgs} + +\subsection{\tt intro +\tacindex{intro} +\label{intro}} + +This tactic applies to a goal which is either a product or starts with +a let binder. If the goal is a product, the tactic implements the +``Lam''\index{Typing rules!Lam} rule given in +Section~\ref{Typed-terms}\footnote{Actually, only the second subgoal will be +generated since the other one can be automatically checked.}. If the +goal starts with a let binder then the tactic implements a mix of the +``Let''\index{Typing rules!Let} and ``Conv''\index{Typing rules!Conv}. + +If the current goal is a dependent product {\tt forall $x$:$T$, $U$} (resp {\tt +let $x$:=$t$ in $U$}) then {\tt intro} puts {\tt $x$:$T$} (resp {\tt $x$:=$t$}) + in the local context. +% Obsolete (quantified names already avoid hypotheses names): +% Otherwise, it puts +% {\tt x}{\it n}{\tt :T} where {\it n} is such that {\tt x}{\it n} is a +%fresh name. +The new subgoal is $U$. +% If the {\tt x} has been renamed {\tt x}{\it n} then it is replaced +% by {\tt x}{\it n} in {\tt U}. + +If the goal is a non dependent product {\tt $T$ -> $U$}, then it puts +in the local context either {\tt H}{\it n}{\tt :$T$} (if $T$ is of +type {\tt Set} or {\tt Prop}) or {\tt X}{\it n}{\tt :$T$} (if the type +of $T$ is {\tt Type}). The optional index {\it n} is such that {\tt +H}{\it n} or {\tt X}{\it n} is a fresh identifier. +In both cases the new subgoal is $U$. + +If the goal is neither a product nor starting with a let definition, +the tactic {\tt intro} applies the tactic {\tt red} until the tactic +{\tt intro} can be applied or the goal is not reducible. + +\begin{ErrMsgs} +\item \errindex{No product even after head-reduction} +\item \errindexbis{{\ident} is already used}{is already used} +\end{ErrMsgs} + +\begin{Variants} + +\item {\tt intros}\tacindex{intros} + + Repeats {\tt intro} until it meets the head-constant. It never reduces + head-constants and it never fails. + +\item {\tt intro {\ident}} + + Applies {\tt intro} but forces {\ident} to be the name of the + introduced hypothesis. + + \ErrMsg \errindex{name {\ident} is already used} + + \Rem If a name used by {\tt intro} hides the base name of a global + constant then the latter can still be referred to by a qualified name + (see \ref{LongNames}). + +\item {\tt intros \ident$_1$ \dots\ \ident$_n$} + + Is equivalent to the composed tactic {\tt intro \ident$_1$; \dots\ ; + intro \ident$_n$}. + + More generally, the \texttt{intros} tactic takes a pattern as + argument in order to introduce names for components of an inductive + definition or to clear introduced hypotheses; This is explained + in~\ref{intros-pattern}. + +\item {\tt intros until {\ident}} \tacindex{intros until} + + Repeats {\tt intro} until it meets a premise of the goal having form + {\tt (} {\ident}~{\tt :}~{\term} {\tt )} and discharges the variable + named {\ident} of the current goal. + + \ErrMsg \errindex{No such hypothesis in current goal} + +\item {\tt intros until {\num}} \tacindex{intros until} + + Repeats {\tt intro} until the {\num}-th non-dependent product. For + instance, on the subgoal % + \verb+forall x y:nat, x=y -> y=x+ the tactic \texttt{intros until 1} + is equivalent to \texttt{intros x y H}, as \verb+x=y -> y=x+ is the + first non-dependent product. And on the subgoal % + \verb+forall x y z:nat, x=y -> y=x+ the tactic \texttt{intros until 1} + is equivalent to \texttt{intros x y z} as the product on \texttt{z} + can be rewritten as a non-dependent product: % + \verb+forall x y:nat, nat -> x=y -> y=x+ + + + \ErrMsg \errindex{No such hypothesis in current goal} + + Happens when {\num} is 0 or is greater than the number of non-dependent + products of the goal. + +\item {\tt intro after \ident} \tacindex{intro after}\\ + {\tt intro before \ident} \tacindex{intro before}\\ + {\tt intro at top} \tacindex{intro at top}\\ + {\tt intro at bottom} \tacindex{intro at bottom} + + Applies {\tt intro} and moves the freshly introduced hypothesis + respectively after the hypothesis \ident{}, before the hypothesis + \ident{}, at the top of the local context, or at the bottom of the + local context. All hypotheses on which the new hypothesis depends + are moved too so as to respect the order of dependencies between + hypotheses. Note that {\tt intro at bottom} is a synonym for {\tt + intro} with no argument. + +\begin{ErrMsgs} +\item \errindex{No product even after head-reduction} +\item \errindex{No such hypothesis} : {\ident} +\end{ErrMsgs} + +\item {\tt intro \ident$_1$ after \ident$_2$}\\ + {\tt intro \ident$_1$ before \ident$_2$}\\ + {\tt intro \ident$_1$ at top}\\ + {\tt intro \ident$_1$ at bottom} + + Behaves as previously but naming the introduced hypothesis + \ident$_1$. It is equivalent to {\tt intro \ident$_1$} followed by + the appropriate call to {\tt move}~(see Section~\ref{move}). + +\begin{ErrMsgs} +\item \errindex{No product even after head-reduction} +\item \errindex{No such hypothesis} : {\ident} +\end{ErrMsgs} + +\end{Variants} + +\subsection{\tt apply \term +\tacindex{apply} +\label{apply}} + +This tactic applies to any goal. The argument {\term} is a term +well-formed in the local context. The tactic {\tt apply} tries to +match the current goal against the conclusion of the type of {\term}. +If it succeeds, then the tactic returns as many subgoals as the number +of non dependent premises of the type of {\term}. If the conclusion of +the type of {\term} does not match the goal {\em and} the conclusion +is an inductive type isomorphic to a tuple type, then each component +of the tuple is recursively matched to the goal in the left-to-right +order. + +The tactic {\tt apply} relies on first-order unification with +dependent types unless the conclusion of the type of {\term} is of the +form {\tt ($P$~ $t_1$~\ldots ~$t_n$)} with $P$ to be instantiated. In +the latter case, the behavior depends on the form of the goal. If the +goal is of the form {\tt (fun $x$ => $Q$)~$u_1$~\ldots~$u_n$} and the +$t_i$ and $u_i$ unifies, then $P$ is taken to be (fun $x$ => $Q$). +Otherwise, {\tt apply} tries to define $P$ by abstracting over +$t_1$~\ldots ~$t_n$ in the goal. See {\tt pattern} in +Section~\ref{pattern} to transform the goal so that it gets the form +{\tt (fun $x$ => $Q$)~$u_1$~\ldots~$u_n$}. + +\begin{ErrMsgs} +\item \errindex{Impossible to unify \dots\ with \dots} + + The {\tt apply} + tactic failed to match the conclusion of {\term} and the current goal. + You can help the {\tt apply} tactic by transforming your + goal with the {\tt change} or {\tt pattern} tactics (see + sections~\ref{pattern},~\ref{change}). + +\item \errindex{Unable to find an instance for the variables +{\ident} \ldots {\ident}} + + This occurs when some instantiations of the premises of {\term} are not + deducible from the unification. This is the case, for instance, when + you want to apply a transitivity property. In this case, you have to + use one of the variants below: + +\end{ErrMsgs} + +\begin{Variants} + +\item{\tt apply {\term} with {\term$_1$} \dots\ {\term$_n$}} + \tacindex{apply \dots\ with} + + Provides {\tt apply} with explicit instantiations for all dependent + premises of the type of {\term} which do not occur in the conclusion + and consequently cannot be found by unification. Notice that + {\term$_1$} \dots\ {\term$_n$} must be given according to the order + of these dependent premises of the type of {\term}. + + \ErrMsg \errindex{Not the right number of missing arguments} + +\item{\tt apply {\term} with ({\vref$_1$} := {\term$_1$}) \dots\ ({\vref$_n$} + := {\term$_n$})} + + This also provides {\tt apply} with values for instantiating + premises. Here, variables are referred by names and non-dependent + products by increasing numbers (see syntax in Section~\ref{Binding-list}). + +\item {\tt apply} {\term$_1$} {\tt ,} \ldots {\tt ,} {\term$_n$} + + This is a shortcut for {\tt apply} {\term$_1$} {\tt ; [ ..~|} + \ldots~{\tt ; [ ..~| {\tt apply} {\term$_n$} ]} \ldots~{\tt ]}, i.e. for the + successive applications of {\term$_{i+1}$} on the last subgoal + generated by {\tt apply} {\term$_i$}, starting from the application + of {\term$_1$}. + +\item {\tt eapply \term}\tacindex{eapply}\label{eapply} + + The tactic {\tt eapply} behaves as {\tt apply} but does not fail + when no instantiation are deducible for some variables in the + premises. Rather, it turns these variables into so-called + existential variables which are variables still to instantiate. An + existential variable is identified by a name of the form {\tt ?$n$} + where $n$ is a number. The instantiation is intended to be found + later in the proof. + + An example of use of {\tt eapply} is given in + Section~\ref{eapply-example}. + +\item {\tt simple apply {\term}} \tacindex{simple apply} + + This behaves like {\tt apply} but it reasons modulo conversion only + on subterms that contain no variables to instantiate. For instance, + if {\tt id := fun x:nat => x} and {\tt H : forall y, id y = y} then + {\tt simple apply H} on goal {\tt O = O} does not succeed because it + would require the conversion of {\tt f ?y} and {\tt O} where {\tt + ?y} is a variable to instantiate. Tactic {\tt simple apply} does not + either traverse tuples as {\tt apply} does. + + Because it reasons modulo a limited amount of conversion, {\tt + simple apply} fails quicker than {\tt apply} and it is then + well-suited for uses in used-defined tactics that backtrack often. + +\item \zeroone{{\tt simple}} {\tt apply} {\term$_1$} \zeroone{{\tt with} + {\bindinglist$_1$}} {\tt ,} \ldots {\tt ,} {\term$_n$} \zeroone{{\tt with} + {\bindinglist$_n$}}\\ + \zeroone{{\tt simple}} {\tt eapply} {\term$_1$} \zeroone{{\tt with} + {\bindinglist$_1$}} {\tt ,} \ldots {\tt ,} {\term$_n$} \zeroone{{\tt with} + {\bindinglist$_n$}} + + This summarizes the different syntaxes for {\tt apply}. + +\item {\tt lapply {\term}} \tacindex{lapply} + + This tactic applies to any goal, say {\tt G}. The argument {\term} + has to be well-formed in the current context, its type being + reducible to a non-dependent product {\tt A -> B} with {\tt B} + possibly containing products. Then it generates two subgoals {\tt + B->G} and {\tt A}. Applying {\tt lapply H} (where {\tt H} has type + {\tt A->B} and {\tt B} does not start with a product) does the same + as giving the sequence {\tt cut B. 2:apply H.} where {\tt cut} is + described below. + + \Warning When {\term} contains more than one non + dependent product the tactic {\tt lapply} only takes into account the + first product. + +\end{Variants} + +\subsection{{\tt set ( {\ident} {\tt :=} {\term} \tt )} +\label{tactic:set} +\tacindex{set} +\tacindex{pose} +\tacindex{remember}} + +This replaces {\term} by {\ident} in the conclusion or in the +hypotheses of the current goal and adds the new definition {\ident +{\tt :=} \term} to the local context. The default is to make this +replacement only in the conclusion. + +\begin{Variants} + +\item {\tt set (} {\ident} {\tt :=} {\term} {\tt ) in {\occgoalset}} + +This notation allows to specify which occurrences of {\term} have to +be substituted in the context. The {\tt in {\occgoalset}} clause is an +occurrence clause whose syntax and behavior is described in +Section~\ref{Occurrences clauses}. + +\item {\tt set (} {\ident} \nelist{\binder}{} {\tt :=} {\term} {\tt )} + + This is equivalent to {\tt set (} {\ident} {\tt :=} {\tt fun} + \nelist{\binder}{} {\tt =>} {\term} {\tt )}. + +\item {\tt set } {\term} + + This behaves as {\tt set (} {\ident} := {\term} {\tt )} but {\ident} + is generated by {\Coq}. This variant also supports an occurrence clause. + +\item {\tt set (} {\ident$_0$} \nelist{\binder}{} {\tt :=} {\term} + {\tt ) in {\occgoalset}}\\ + {\tt set {\term} in {\occgoalset}} + + These are the general forms which combine the previous possibilities. + +\item {\tt remember {\term} {\tt as} {\ident}} + + This behaves as {\tt set (} {\ident} := {\term} {\tt ) in *} and using a + logical (Leibniz's) equality instead of a local definition. + +\item {\tt remember {\term} {\tt as} {\ident} in {\occgoalset}} + + This is a more general form of {\tt remember} that remembers the + occurrences of {\term} specified by an occurrences set. + +\item {\tt pose ( {\ident} {\tt :=} {\term} {\tt )}} + + This adds the local definition {\ident} := {\term} to the current + context without performing any replacement in the goal or in the + hypotheses. It is equivalent to {\tt set ( {\ident} {\tt :=} + {\term} {\tt ) in |-}}. + +\item {\tt pose (} {\ident} \nelist{\binder}{} {\tt :=} {\term} {\tt )} + + This is equivalent to {\tt pose (} {\ident} {\tt :=} {\tt fun} + \nelist{\binder}{} {\tt =>} {\term} {\tt )}. + +\item{\tt pose {\term}} + + This behaves as {\tt pose (} {\ident} := {\term} {\tt )} but + {\ident} is generated by {\Coq}. + +\end{Variants} + +\subsection{{\tt assert ( {\ident} : {\form} \tt )} +\tacindex{assert}} + +This tactic applies to any goal. {\tt assert (H : U)} adds a new +hypothesis of name \texttt{H} asserting \texttt{U} to the current goal +and opens a new subgoal \texttt{U}\footnote{This corresponds to the + cut rule of sequent calculus.}. The subgoal {\texttt U} comes first +in the list of subgoals remaining to prove. + +\begin{ErrMsgs} +\item \errindex{Not a proposition or a type} + + Arises when the argument {\form} is neither of type {\tt Prop}, {\tt + Set} nor {\tt Type}. + +\end{ErrMsgs} + +\begin{Variants} + +\item{\tt assert {\form}} + + This behaves as {\tt assert (} {\ident} : {\form} {\tt )} but + {\ident} is generated by {\Coq}. + +\item{\tt assert (} {\ident} := {\term} {\tt )} + + This behaves as {\tt assert ({\ident} : {\type});[exact + {\term}|idtac]} where {\type} is the type of {\term}. + +\item {\tt cut {\form}}\tacindex{cut} + + This tactic applies to any goal. It implements the non dependent + case of the ``App''\index{Typing rules!App} rule given in + Section~\ref{Typed-terms}. (This is Modus Ponens inference rule.) + {\tt cut U} transforms the current goal \texttt{T} into the two + following subgoals: {\tt U -> T} and \texttt{U}. The subgoal {\tt U + -> T} comes first in the list of remaining subgoal to prove. + +\item \texttt{assert {\form} by {\tac}}\tacindex{assert by} + + This tactic behaves like \texttt{assert} but applies {\tac} + to solve the subgoals generated by \texttt{assert}. + +\item \texttt{assert {\form} as {\intropattern}\tacindex{assert as}} + + If {\intropattern} is a naming introduction pattern (see + Section~\ref{intros-pattern}), the hypothesis is named after this + introduction pattern (in particular, if {\intropattern} is {\ident}, + the tactic behaves like \texttt{assert ({\ident} : {\form})}). + + If {\intropattern} is a disjunctive/conjunctive introduction + pattern, the tactic behaves like \texttt{assert {\form}} then destructing the + resulting hypothesis using the given introduction pattern. + +\item \texttt{assert {\form} as {\intropattern} by {\tac}} + + This combines the two previous variants of {\tt assert}. + +\item \texttt{pose proof {\term} as {\intropattern}\tacindex{pose proof}} + + This tactic behaves like \texttt{assert T as {\intropattern} by + exact {\term}} where \texttt{T} is the type of {\term}. + + In particular, \texttt{pose proof {\term} as {\ident}} behaves as + \texttt{assert ({\ident}:T) by exact {\term}} (where \texttt{T} is + the type of {\term}) and \texttt{pose proof {\term} as + {\disjconjintropattern}\tacindex{pose proof}} behaves + like \texttt{destruct {\term} as {\disjconjintropattern}}. + +\item {\tt specialize ({\ident} \term$_1$ {\ldots} \term$_n$)\tacindex{specialize}} \\ + {\tt specialize {\ident} with \bindinglist} + + The tactic {\tt specialize} works on local hypothesis \ident. + The premises of this hypothesis (either universal + quantifications or non-dependent implications) are instantiated + by concrete terms coming either from arguments \term$_1$ + $\ldots$ \term$_n$ or from a bindings list (see + Section~\ref{Binding-list} for more about bindings lists). In the + second form, all instantiation elements must be given, whereas + in the first form the application to \term$_1$ {\ldots} + \term$_n$ can be partial. The first form is equivalent to + {\tt assert (\ident':=\ident \term$_1$ {\ldots} \term$_n$); + clear \ident; rename \ident' into \ident}. + + The name {\ident} can also refer to a global lemma or + hypothesis. In this case, for compatibility reasons, the + behavior of {\tt specialize} is close to that of {\tt + generalize}: the instantiated statement becomes an additional + premise of the goal. + +%% Moreover, the old syntax allows the use of a number after {\tt specialize} +%% for controlling the number of premises to instantiate. Giving this +%% number should not be mandatory anymore (automatic detection of how +%% many premises can be eaten without leaving meta-variables). Hence +%% no documentation for this integer optional argument of specialize + +\end{Variants} + +\subsection{{\tt apply {\term} in {\ident}} +\tacindex{apply \ldots\ in}} + +This tactic applies to any goal. The argument {\term} is a term +well-formed in the local context and the argument {\ident} is an +hypothesis of the context. The tactic {\tt apply {\term} in {\ident}} +tries to match the conclusion of the type of {\ident} against a non +dependent premise of the type of {\term}, trying them from right to +left. If it succeeds, the statement of hypothesis {\ident} is +replaced by the conclusion of the type of {\term}. The tactic also +returns as many subgoals as the number of other non dependent premises +in the type of {\term} and of the non dependent premises of the type +of {\ident}. If the conclusion of the type of {\term} does not match +the goal {\em and} the conclusion is an inductive type isomorphic to a +tuple type, then the tuple is (recursively) decomposed and the first +component of the tuple of which a non dependent premise matches the +conclusion of the type of {\ident}. Tuples are decomposed in a +width-first left-to-right order (for instance if the type of {\tt H1} +is a \verb=A <-> B= statement, and the type of {\tt H2} is \verb=A= +then {\tt apply H1 in H2} transforms the type of {\tt H2} into {\tt + B}). The tactic {\tt apply} relies on first-order pattern-matching +with dependent types. + +\begin{ErrMsgs} +\item \errindex{Statement without assumptions} + +This happens if the type of {\term} has no non dependent premise. + +\item \errindex{Unable to apply} + +This happens if the conclusion of {\ident} does not match any of the +non dependent premises of the type of {\term}. +\end{ErrMsgs} + +\begin{Variants} +\item {\tt apply \nelist{\term}{,} in {\ident}} + +This applies each of {\term} in sequence in {\ident}. + +\item {\tt apply \nelist{{\term} {\bindinglist}}{,} in {\ident}} + +This does the same but uses the bindings in each {\bindinglist} to +instantiate the parameters of the corresponding type of {\term} +(see syntax of bindings in Section~\ref{Binding-list}). + +\item {\tt eapply \nelist{{\term} {\bindinglist}}{,} in {\ident}} +\tacindex{eapply {\ldots} in} + +This works as {\tt apply \nelist{{\term} {\bindinglist}}{,} in +{\ident}} but turns unresolved bindings into existential variables, if +any, instead of failing. + +\item {\tt apply \nelist{{\term}{,} {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}} + +This works as {\tt apply \nelist{{\term}{,} {\bindinglist}}{,} in +{\ident}} then destructs the hypothesis {\ident} along +{\disjconjintropattern} as {\tt destruct {\ident} as +{\disjconjintropattern}} would. + +\item {\tt eapply \nelist{{\term}{,} {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}} + +This works as {\tt apply \nelist{{\term}{,} {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}} but using {\tt eapply}. + +\item {\tt simple apply {\term} in {\ident}} +\tacindex{simple apply {\ldots} in} +\tacindex{simple eapply {\ldots} in} + +This behaves like {\tt apply {\term} in {\ident}} but it reasons +modulo conversion only on subterms that contain no variables to +instantiate. For instance, if {\tt id := fun x:nat => x} and {\tt H : + forall y, id y = y -> True} and {\tt H0 : O = O} then {\tt simple + apply H in H0} does not succeed because it would require the +conversion of {\tt f ?y} and {\tt O} where {\tt ?y} is a variable to +instantiate. Tactic {\tt simple apply {\term} in {\ident}} does not +either traverse tuples as {\tt apply {\term} in {\ident}} does. + +\item {\tt simple apply \nelist{{\term}{,} {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}}\\ +{\tt simple eapply \nelist{{\term}{,} {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}} + +This are the general forms of {\tt simple apply {\term} in {\ident}} and +{\tt simple eapply {\term} in {\ident}}. +\end{Variants} + +\subsection{\tt generalize \term +\tacindex{generalize} +\label{generalize}} + +This tactic applies to any goal. It generalizes the conclusion w.r.t. +one subterm of it. For example: + +\begin{coq_eval} +Goal forall x y:nat, (0 <= x + y + y). +intros. +\end{coq_eval} +\begin{coq_example} +Show. +generalize (x + y + y). +\end{coq_example} + +\begin{coq_eval} +Abort. +\end{coq_eval} + +If the goal is $G$ and $t$ is a subterm of type $T$ in the goal, then +{\tt generalize} \textit{t} replaces the goal by {\tt forall (x:$T$), $G'$} +where $G'$ is obtained from $G$ by replacing all occurrences of $t$ by +{\tt x}. The name of the variable (here {\tt n}) is chosen based on $T$. + +\begin{Variants} +\item {\tt generalize {\term$_1$ , \dots\ , \term$_n$}} + + Is equivalent to {\tt generalize \term$_n$; \dots\ ; generalize + \term$_1$}. Note that the sequence of \term$_i$'s are processed + from $n$ to $1$. + +\item {\tt generalize {\term} at {\num$_1$ \dots\ \num$_i$}} + + Is equivalent to {\tt generalize \term} but generalizing only over + the specified occurrences of {\term} (counting from left to right on the + expression printed using option {\tt Set Printing All}). + +\item {\tt generalize {\term} as {\ident}} + + Is equivalent to {\tt generalize \term} but use {\ident} to name the + generalized hypothesis. + +\item {\tt generalize {\term$_1$} at {\num$_{11}$ \dots\ \num$_{1i_1}$} + as {\ident$_1$} + , {\ldots} , + {\term$_n$} at {\num$_{n1}$ \dots\ \num$_{ni_n}$} + as {\ident$_2$}} + + This is the most general form of {\tt generalize} that combines the + previous behaviors. + +\item {\tt generalize dependent \term} \tacindex{generalize dependent} + + This generalizes {\term} but also {\em all} hypotheses which depend + on {\term}. It clears the generalized hypotheses. + +\end{Variants} + + +\subsection{\tt revert \ident$_1$ \dots\ \ident$_n$ +\tacindex{revert} +\label{revert}} + +This applies to any goal with variables \ident$_1$ \dots\ \ident$_n$. +It moves the hypotheses (possibly defined) to the goal, if this respects +dependencies. This tactic is the inverse of {\tt intro}. + +\begin{ErrMsgs} +\item \errindexbis{{\ident} is used in the hypothesis {\ident'}}{is + used in the hypothesis} +\end{ErrMsgs} + +\begin{Variants} +\item {\tt revert dependent \ident \tacindex{revert dependent}} + + This moves to the goal the hypothesis \ident\ and all hypotheses + which depend on it. + +\end{Variants} + +\subsection{\tt change \term +\tacindex{change} +\label{change}} + +This tactic applies to any goal. It implements the rule +``Conv''\index{Typing rules!Conv} given in Section~\ref{Conv}. {\tt + change U} replaces the current goal \T\ with \U\ providing that +\U\ is well-formed and that \T\ and \U\ are convertible. + +\begin{ErrMsgs} +\item \errindex{Not convertible} +\end{ErrMsgs} + +\tacindex{change \dots\ in} +\begin{Variants} +\item {\tt change \term$_1$ with \term$_2$} + + This replaces the occurrences of \term$_1$ by \term$_2$ in the + current goal. The terms \term$_1$ and \term$_2$ must be + convertible. + +\item {\tt change \term$_1$ at \num$_1$ \dots\ \num$_i$ with \term$_2$} + + This replaces the occurrences numbered \num$_1$ \dots\ \num$_i$ of + \term$_1$ by \term$_2$ in the current goal. + The terms \term$_1$ and \term$_2$ must be convertible. + + \ErrMsg {\tt Too few occurrences} + +\item {\tt change {\term} in {\ident}} + +\item {\tt change \term$_1$ with \term$_2$ in {\ident}} + +\item {\tt change \term$_1$ at \num$_1$ \dots\ \num$_i$ with \term$_2$ in + {\ident}} + + This applies the {\tt change} tactic not to the goal but to the + hypothesis {\ident}. + +\end{Variants} + +\SeeAlso \ref{Conversion-tactics} + +\subsection{\tt fix {\ident} {\num} +\tacindex{fix} +\label{tactic:fix}} + +This tactic is a primitive tactic to start a proof by induction. In +general, it is easier to rely on higher-level induction tactics such +as the ones described in Section~\ref{Tac-induction}. + +In the syntax of the tactic, the identifier {\ident} is the name given +to the induction hypothesis. The natural number {\num} tells on which +premise of the current goal the induction acts, starting +from 1 and counting both dependent and non dependent +products. Especially, the current lemma must be composed of at least +{\num} products. + +Like in a {\tt fix} expression, the induction +hypotheses have to be used on structurally smaller arguments. +The verification that inductive proof arguments are correct is done +only at the time of registering the lemma in the environment. To know +if the use of induction hypotheses is correct at some +time of the interactive development of a proof, use the command {\tt + Guarded} (see Section~\ref{Guarded}). + +\begin{Variants} + \item {\tt fix} {\ident}$_1$ {\num} {\tt with (} {\ident}$_2$ + \nelist{{\binder}$_{2}$}{} \zeroone{{\tt \{ struct {\ident$'_2$} + \}}} {\tt :} {\type}$_2$ {\tt )} {\ldots} {\tt (} {\ident}$_1$ + \nelist{{\binder}$_n$}{} \zeroone{{\tt \{ struct {\ident$'_n$} \}}} + {\tt :} {\type}$_n$ {\tt )} + +This starts a proof by mutual induction. The statements to be +simultaneously proved are respectively {\tt forall} + \nelist{{\binder}$_2$}{}{\tt ,} {\type}$_2$, {\ldots}, {\tt forall} + \nelist{{\binder}$_n$}{}{\tt ,} {\type}$_n$. The identifiers +{\ident}$_1$ {\ldots} {\ident}$_n$ are the names of the induction +hypotheses. The identifiers {\ident}$'_2$ {\ldots} {\ident}$'_n$ are the +respective names of the premises on which the induction is performed +in the statements to be simultaneously proved (if not given, the +system tries to guess itself what they are). + +\end{Variants} + +\subsection{\tt cofix {\ident} +\tacindex{cofix} +\label{tactic:cofix}} + +This tactic starts a proof by coinduction. The identifier {\ident} is +the name given to the coinduction hypothesis. Like in a {\tt cofix} +expression, the use of induction hypotheses have to guarded by a +constructor. The verification that the use of coinductive hypotheses +is correct is done only at the time of registering the lemma in the +environment. To know if the use of coinduction hypotheses is correct +at some time of the interactive development of a proof, use the +command {\tt Guarded} (see Section~\ref{Guarded}). + + +\begin{Variants} + \item {\tt cofix} {\ident}$_1$ {\tt with (} {\ident}$_2$ + \nelist{{\binder}$_2$}{} {\tt :} {\type}$_2$ {\tt )} {\ldots} {\tt + (} {\ident}$_1$ \nelist{{\binder}$_1$}{} {\tt :} {\type}$_n$ + {\tt )} + +This starts a proof by mutual coinduction. The statements to be +simultaneously proved are respectively {\tt forall} +\nelist{{\binder}$_2$}{}{\tt ,} {\type}$_2$, {\ldots}, {\tt forall} + \nelist{{\binder}$_n$}{}{\tt ,} {\type}$_n$. The identifiers + {\ident}$_1$ {\ldots} {\ident}$_n$ are the names of the + coinduction hypotheses. + +\end{Variants} + +\subsection{\tt evar (\ident:\term) +\tacindex{evar} +\label{evar}} + +The {\tt evar} tactic creates a new local definition named \ident\ with +type \term\ in the context. The body of this binding is a fresh +existential variable. + +\subsection{\tt instantiate (\num:= \term) +\tacindex{instantiate} +\label{instantiate}} + +The {\tt instantiate} tactic allows to solve an existential variable +with the term \term. The \num\ argument is the position of the +existential variable from right to left in the conclusion. This cannot be +the number of the existential variable since this number is different +in every session. + +\begin{Variants} + \item {\tt instantiate (\num:=\term) in \ident} + + \item {\tt instantiate (\num:=\term) in (Value of \ident)} + + \item {\tt instantiate (\num:=\term) in (Type of \ident)} + +These allow to refer respectively to existential variables occurring in +a hypothesis or in the body or the type of a local definition. + + \item {\tt instantiate} + + Without argument, the {\tt instantiate} tactic tries to solve as + many existential variables as possible, using information gathered + from other tactics in the same tactical. This is automatically + done after each complete tactic (i.e. after a dot in proof mode), + but not, for example, between each tactic when they are sequenced + by semicolons. + +\end{Variants} + +\subsection{\tt admit +\tacindex{admit} +\label{admit}} + +The {\tt admit} tactic ``solves'' the current subgoal by an +axiom. This typically allows to temporarily skip a subgoal so as to +progress further in the rest of the proof. To know if some proof still +relies on unproved subgoals, one can use the command {\tt Print +Assumptions} (see Section~\ref{PrintAssumptions}). Admitted subgoals +have names of the form {\ident}\texttt{\_admitted} possibly followed +by a number. + +\subsection{Bindings list +\index{Binding list} +\label{Binding-list}} + +Tactics that take a term as argument may also support a bindings list, so +as to instantiate some parameters of the term by name or position. +The general form of a term equipped with a bindings list is {\tt +{\term} with {\bindinglist}} where {\bindinglist} may be of two +different forms: + +\begin{itemize} +\item In a bindings list of the form {\tt (\vref$_1$ := \term$_1$) + \dots\ (\vref$_n$ := \term$_n$)}, {\vref} is either an {\ident} or a + {\num}. The references are determined according to the type of + {\term}. If \vref$_i$ is an identifier, this identifier has to be + bound in the type of {\term} and the binding provides the tactic + with an instance for the parameter of this name. If \vref$_i$ is + some number $n$, this number denotes the $n$-th non dependent + premise of the {\term}, as determined by the type of {\term}. + + \ErrMsg \errindex{No such binder} + +\item A bindings list can also be a simple list of terms {\tt + \term$_1$ \dots\term$_n$}. In that case the references to + which these terms correspond are determined by the tactic. In case + of {\tt induction}, {\tt destruct}, {\tt elim} and {\tt case} (see + Section~\ref{elim}) the terms have to provide instances for all the + dependent products in the type of \term\ while in the case of {\tt + apply}, or of {\tt constructor} and its variants, only instances for + the dependent products which are not bound in the conclusion of the + type are required. + + \ErrMsg \errindex{Not the right number of missing arguments} + +\end{itemize} + +\subsection{Occurrences sets and occurrences clauses} +\label{Occurrences clauses} +\index{Occurrences clauses} + +An occurrences clause is a modifier to some tactics that obeys the +following syntax: + +$\!\!\!$\begin{tabular}{lcl} +{\occclause} & ::= & {\tt in} {\occgoalset} \\ +{\occgoalset} & ::= & + \zeroone{{\ident$_1$} \zeroone{\atoccurrences} {\tt ,} \\ +& & {\dots} {\tt ,}\\ +& & {\ident$_m$} \zeroone{\atoccurrences}}\\ +& & \zeroone{{\tt |-} \zeroone{{\tt *} \zeroone{\atoccurrences}}}\\ +& | & + {\tt *} {\tt |-} \zeroone{{\tt *} \zeroone{\atoccurrences}}\\ +& | & + {\tt *}\\ +{\atoccurrences} & ::= & {\tt at} {\occlist}\\ +{\occlist} & ::= & \zeroone{{\tt -}} {\num$_1$} \dots\ {\num$_n$} +\end{tabular} + +The role of an occurrence clause is to select a set of occurrences of +a {\term} in a goal. In the first case, the {{\ident$_i$} +\zeroone{{\tt at} {\num$_1^i$} \dots\ {\num$_{n_i}^i$}}} parts +indicate that occurrences have to be selected in the hypotheses named +{\ident$_i$}. If no numbers are given for hypothesis {\ident$_i$}, +then all occurrences of {\term} in the hypothesis are selected. If +numbers are given, they refer to occurrences of {\term} when the term +is printed using option {\tt Set Printing All} (see +Section~\ref{SetPrintingAll}), counting from left to right. In +particular, occurrences of {\term} in implicit arguments (see +Section~\ref{Implicit Arguments}) or coercions (see +Section~\ref{Coercions}) are counted. + +If a minus sign is given between {\tt at} and the list of occurrences, +it negates the condition so that the clause denotes all the occurrences except +the ones explicitly mentioned after the minus sign. + +As an exception to the left-to-right order, the occurrences in the +{\tt return} subexpression of a {\tt match} are considered {\em +before} the occurrences in the matched term. + +In the second case, the {\tt *} on the left of {\tt |-} means that +all occurrences of {\term} are selected in every hypothesis. + +In the first and second case, if {\tt *} is mentioned on the right of +{\tt |-}, the occurrences of the conclusion of the goal have to be +selected. If some numbers are given, then only the occurrences denoted +by these numbers are selected. In no numbers are given, all +occurrences of {\term} in the goal are selected. + +Finally, the last notation is an abbreviation for {\tt * |- *}. Note +also that {\tt |-} is optional in the first case when no {\tt *} is +given. + +Here are some tactics that understand occurrences clauses: +{\tt set}, {\tt remember}, {\tt induction}, {\tt destruct}. + +\SeeAlso~Sections~\ref{tactic:set}, \ref{Tac-induction}, \ref{SetPrintingAll}. + + +\section{Negation and contradiction} + +\subsection{\tt absurd \term +\tacindex{absurd} +\label{absurd}} + +This tactic applies to any goal. The argument {\term} is any +proposition {\tt P} of type {\tt Prop}. This tactic applies {\tt + False} elimination, that is it deduces the current goal from {\tt + False}, and generates as subgoals {\tt $\sim$P} and {\tt P}. It is +very useful in proofs by cases, where some cases are impossible. In +most cases, \texttt{P} or $\sim$\texttt{P} is one of the hypotheses of +the local context. + +\subsection{\tt contradiction +\label{contradiction} +\tacindex{contradiction}} + +This tactic applies to any goal. The {\tt contradiction} tactic +attempts to find in the current context (after all {\tt intros}) one +hypothesis which is equivalent to {\tt False}. It permits to prune +irrelevant cases. This tactic is a macro for the tactics sequence +{\tt intros; elimtype False; assumption}. + +\begin{ErrMsgs} +\item \errindex{No such assumption} +\end{ErrMsgs} + +\begin{Variants} +\item {\tt contradiction \ident} + +The proof of {\tt False} is searched in the hypothesis named \ident. +\end{Variants} + +\subsection {\tt contradict \ident} +\label{contradict} +\tacindex{contradict} + +This tactic allows to manipulate negated hypothesis and goals. The +name \ident\ should correspond to a hypothesis. With +{\tt contradict H}, the current goal and context is transformed in +the following way: +\begin{itemize} +\item {\tt H:$\neg$A $\vd$ B} \ becomes \ {\tt $\vd$ A} +\item {\tt H:$\neg$A $\vd$ $\neg$B} \ becomes \ {\tt H: B $\vd$ A } +\item {\tt H: A $\vd$ B} \ becomes \ {\tt $\vd$ $\neg$A} +\item {\tt H: A $\vd$ $\neg$B} \ becomes \ {\tt H: B $\vd$ $\neg$A} +\end{itemize} + +\subsection{\tt exfalso} +\label{exfalso} +\tacindex{exfalso} + +This tactic implements the ``ex falso quodlibet'' logical principle: +an elimination of {\tt False} is performed on the current goal, and the +user is then required to prove that {\tt False} is indeed provable in +the current context. This tactic is a macro for {\tt elimtype False}. + +\section{Conversion tactics +\index{Conversion tactics} +\label{Conversion-tactics}} + +This set of tactics implements different specialized usages of the +tactic \texttt{change}. + +All conversion tactics (including \texttt{change}) can be +parameterized by the parts of the goal where the conversion can +occur. This is done using \emph{goal clauses} which consists in a list +of hypotheses and, optionally, of a reference to the conclusion of the +goal. For defined hypothesis it is possible to specify if the +conversion should occur on the type part, the body part or both +(default). + +\index{Clauses} +\index{Goal clauses} +Goal clauses are written after a conversion tactic (tactics +\texttt{set}~\ref{tactic:set}, \texttt{rewrite}~\ref{rewrite}, +\texttt{replace}~\ref{tactic:replace} and +\texttt{autorewrite}~\ref{tactic:autorewrite} also use goal clauses) and +are introduced by the keyword \texttt{in}. If no goal clause is provided, +the default is to perform the conversion only in the conclusion. + +The syntax and description of the various goal clauses is the following: +\begin{description} +\item[]\texttt{in {\ident}$_1$ $\ldots$ {\ident}$_n$ |- } only in hypotheses {\ident}$_1$ + \ldots {\ident}$_n$ +\item[]\texttt{in {\ident}$_1$ $\ldots$ {\ident}$_n$ |- *} in hypotheses {\ident}$_1$ \ldots + {\ident}$_n$ and in the conclusion +\item[]\texttt{in * |-} in every hypothesis +\item[]\texttt{in *} (equivalent to \texttt{in * |- *}) everywhere +\item[]\texttt{in (type of {\ident}$_1$) (value of {\ident}$_2$) $\ldots$ |-} in + type part of {\ident}$_1$, in the value part of {\ident}$_2$, etc. +\end{description} + +For backward compatibility, the notation \texttt{in}~{\ident}$_1$\ldots {\ident}$_n$ +performs the conversion in hypotheses {\ident}$_1$\ldots {\ident}$_n$. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%voir reduction__conv_x : histoires d'univers. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection[{\tt cbv \flag$_1$ \dots\ \flag$_n$}, {\tt lazy \flag$_1$ +\dots\ \flag$_n$} and {\tt compute}] +{{\tt cbv \flag$_1$ \dots\ \flag$_n$}, {\tt lazy \flag$_1$ +\dots\ \flag$_n$} and {\tt compute} +\tacindex{cbv} +\tacindex{lazy} +\tacindex{compute} +\tacindex{vm\_compute}\label{vmcompute}} + +These parameterized reduction tactics apply to any goal and perform +the normalization of the goal according to the specified flags. In +correspondence with the kinds of reduction considered in \Coq\, namely +$\beta$ (reduction of functional application), $\delta$ (unfolding of +transparent constants, see \ref{Transparent}), $\iota$ (reduction of +pattern-matching over a constructed term, and unfolding of {\tt fix} +and {\tt cofix} expressions) and $\zeta$ (contraction of local +definitions), the flag are either {\tt beta}, {\tt delta}, {\tt iota} +or {\tt zeta}. The {\tt delta} flag itself can be refined into {\tt +delta [\qualid$_1$\ldots\qualid$_k$]} or {\tt delta +-[\qualid$_1$\ldots\qualid$_k$]}, restricting in the first case the +constants to unfold to the constants listed, and restricting in the +second case the constant to unfold to all but the ones explicitly +mentioned. Notice that the {\tt delta} flag does not apply to +variables bound by a let-in construction inside the term itself (use +here the {\tt zeta} flag). In any cases, opaque constants are not +unfolded (see Section~\ref{Opaque}). + +The goal may be normalized with two strategies: {\em lazy} ({\tt lazy} +tactic), or {\em call-by-value} ({\tt cbv} tactic). The lazy strategy +is a call-by-need strategy, with sharing of reductions: the arguments of a +function call are partially evaluated only when necessary, and if an +argument is used several times then it is computed only once. This +reduction is efficient for reducing expressions with dead code. For +instance, the proofs of a proposition {\tt exists~$x$. $P(x)$} reduce to a +pair of a witness $t$, and a proof that $t$ satisfies the predicate +$P$. Most of the time, $t$ may be computed without computing the proof +of $P(t)$, thanks to the lazy strategy. + +The call-by-value strategy is the one used in ML languages: the +arguments of a function call are evaluated first, using a weak +reduction (no reduction under the $\lambda$-abstractions). Despite the +lazy strategy always performs fewer reductions than the call-by-value +strategy, the latter is generally more efficient for evaluating purely +computational expressions (i.e. with few dead code). + +\begin{Variants} +\item {\tt compute} \tacindex{compute}\\ + {\tt cbv} + + These are synonyms for {\tt cbv beta delta iota zeta}. + +\item {\tt lazy} + + This is a synonym for {\tt lazy beta delta iota zeta}. + +\item {\tt compute [\qualid$_1$\ldots\qualid$_k$]}\\ + {\tt cbv [\qualid$_1$\ldots\qualid$_k$]} + + These are synonyms of {\tt cbv beta delta + [\qualid$_1$\ldots\qualid$_k$] iota zeta}. + +\item {\tt compute -[\qualid$_1$\ldots\qualid$_k$]}\\ + {\tt cbv -[\qualid$_1$\ldots\qualid$_k$]} + + These are synonyms of {\tt cbv beta delta + -[\qualid$_1$\ldots\qualid$_k$] iota zeta}. + +\item {\tt lazy [\qualid$_1$\ldots\qualid$_k$]}\\ + {\tt lazy -[\qualid$_1$\ldots\qualid$_k$]} + + These are respectively synonyms of {\tt cbv beta delta + [\qualid$_1$\ldots\qualid$_k$] iota zeta} and {\tt cbv beta delta + -[\qualid$_1$\ldots\qualid$_k$] iota zeta}. + +\item {\tt vm\_compute} \tacindex{vm\_compute} + + This tactic evaluates the goal using the optimized call-by-value + evaluation bytecode-based virtual machine. This algorithm is + dramatically more efficient than the algorithm used for the {\tt + cbv} tactic, but it cannot be fine-tuned. It is specially + interesting for full evaluation of algebraic objects. This includes + the case of reflexion-based tactics. + +\end{Variants} + +% Obsolete? Anyway not very important message +%\begin{ErrMsgs} +%\item \errindex{Delta must be specified before} +% +% A list of constants appeared before the {\tt delta} flag. +%\end{ErrMsgs} + + +\subsection{{\tt red} +\tacindex{red}} + +This tactic applies to a goal which has the form {\tt + forall (x:T1)\dots(xk:Tk), c t1 \dots\ tn} where {\tt c} is a constant. If +{\tt c} is transparent then it replaces {\tt c} with its definition +(say {\tt t}) and then reduces {\tt (t t1 \dots\ tn)} according to +$\beta\iota\zeta$-reduction rules. + +\begin{ErrMsgs} +\item \errindex{Not reducible} +\end{ErrMsgs} + +\subsection{{\tt hnf} +\tacindex{hnf}} + +This tactic applies to any goal. It replaces the current goal with its +head normal form according to the $\beta\delta\iota\zeta$-reduction +rules, i.e. it reduces the head of the goal until it becomes a +product or an irreducible term. + +\Example +The term \verb+forall n:nat, (plus (S n) (S n))+ is not reduced by {\tt hnf}. + +\Rem The $\delta$ rule only applies to transparent constants +(see Section~\ref{Opaque} on transparency and opacity). + +\subsection{\tt simpl +\tacindex{simpl}} + +This tactic applies to any goal. The tactic {\tt simpl} first applies +$\beta\iota$-reduction rule. Then it expands transparent constants +and tries to reduce {\tt T'} according, once more, to $\beta\iota$ +rules. But when the $\iota$ rule is not applicable then possible +$\delta$-reductions are not applied. For instance trying to use {\tt +simpl} on {\tt (plus n O)=n} changes nothing. Notice that only +transparent constants whose name can be reused as such in the +recursive calls are possibly unfolded. For instance a constant defined +by {\tt plus' := plus} is possibly unfolded and reused in the +recursive calls, but a constant such as {\tt succ := plus (S O)} is +never unfolded. + +\tacindex{simpl \dots\ in} +\begin{Variants} +\item {\tt simpl {\term}} + + This applies {\tt simpl} only to the occurrences of {\term} in the + current goal. + +\item {\tt simpl {\term} at \num$_1$ \dots\ \num$_i$} + + This applies {\tt simpl} only to the \num$_1$, \dots, \num$_i$ + occurrences of {\term} in the current goal. + + \ErrMsg {\tt Too few occurrences} + +\item {\tt simpl {\ident}} + + This applies {\tt simpl} only to the applicative subterms whose head + occurrence is {\ident}. + +\item {\tt simpl {\ident} at \num$_1$ \dots\ \num$_i$} + + This applies {\tt simpl} only to the \num$_1$, \dots, \num$_i$ +applicative subterms whose head occurrence is {\ident}. + +\end{Variants} + +\subsection{\tt unfold \qualid +\tacindex{unfold} +\label{unfold}} + +This tactic applies to any goal. The argument {\qualid} must denote a +defined transparent constant or local definition (see Sections~\ref{Basic-definitions} and~\ref{Transparent}). The tactic {\tt + unfold} applies the $\delta$ rule to each occurrence of the constant +to which {\qualid} refers in the current goal and then replaces it +with its $\beta\iota$-normal form. + +\begin{ErrMsgs} +\item {\qualid} \errindex{does not denote an evaluable constant} + +\end{ErrMsgs} + +\begin{Variants} +\item {\tt unfold {\qualid}$_1$, \dots, \qualid$_n$} + \tacindex{unfold \dots\ in} + + Replaces {\em simultaneously} {\qualid}$_1$, \dots, {\qualid}$_n$ + with their definitions and replaces the current goal with its + $\beta\iota$ normal form. + +\item {\tt unfold {\qualid}$_1$ at \num$_1^1$, \dots, \num$_i^1$, +\dots,\ \qualid$_n$ at \num$_1^n$ \dots\ \num$_j^n$} + + The lists \num$_1^1$, \dots, \num$_i^1$ and \num$_1^n$, \dots, + \num$_j^n$ specify the occurrences of {\qualid}$_1$, \dots, + \qualid$_n$ to be unfolded. Occurrences are located from left to + right. + + \ErrMsg {\tt bad occurrence number of {\qualid}$_i$} + + \ErrMsg {\qualid}$_i$ {\tt does not occur} + +\item {\tt unfold {\qstring}} + + If {\qstring} denotes the discriminating symbol of a notation (e.g. {\tt + "+"}) or an expression defining a notation (e.g. \verb!"_ + _"!), and + this notation refers to an unfoldable constant, then the tactic + unfolds it. + +\item {\tt unfold {\qstring}\%{\delimkey}} + + This is variant of {\tt unfold {\qstring}} where {\qstring} gets its + interpretation from the scope bound to the delimiting key + {\delimkey} instead of its default interpretation (see + Section~\ref{scopechange}). + +\item {\tt unfold \qualidorstring$_1$ at \num$_1^1$, \dots, \num$_i^1$, +\dots,\ \qualidorstring$_n$ at \num$_1^n$ \dots\ \num$_j^n$} + + This is the most general form, where {\qualidorstring} is either a + {\qualid} or a {\qstring} referring to a notation. + +\end{Variants} + +\subsection{{\tt fold} \term +\tacindex{fold}} + +This tactic applies to any goal. The term \term\ is reduced using the {\tt red} +tactic. Every occurrence of the resulting term in the goal is then +replaced by \term. + +\begin{Variants} +\item {\tt fold} \term$_1$ \dots\ \term$_n$ + + Equivalent to {\tt fold} \term$_1${\tt;}\ldots{\tt; fold} \term$_n$. +\end{Variants} + +\subsection{{\tt pattern {\term}} +\tacindex{pattern} +\label{pattern}} + +This command applies to any goal. The argument {\term} must be a free +subterm of the current goal. The command {\tt pattern} performs +$\beta$-expansion (the inverse of $\bt$-reduction) of the current goal +(say \T) by +\begin{enumerate} +\item replacing all occurrences of {\term} in {\T} with a fresh variable +\item abstracting this variable +\item applying the abstracted goal to {\term} +\end{enumerate} + +For instance, if the current goal $T$ is expressible has $\phi(t)$ +where the notation captures all the instances of $t$ in $\phi(t)$, +then {\tt pattern $t$} transforms it into {\tt (fun x:$A$ => $\phi(${\tt +x}$)$) $t$}. This command can be used, for instance, when the tactic +{\tt apply} fails on matching. + +\begin{Variants} +\item {\tt pattern {\term} at {\num$_1$} \dots\ {\num$_n$}} + + Only the occurrences {\num$_1$} \dots\ {\num$_n$} of {\term} are + considered for $\beta$-expansion. Occurrences are located from left + to right. + +\item {\tt pattern {\term} at - {\num$_1$} \dots\ {\num$_n$}} + + All occurrences except the occurrences of indexes {\num$_1$} \dots\ + {\num$_n$} of {\term} are considered for + $\beta$-expansion. Occurrences are located from left to right. + +\item {\tt pattern {\term$_1$}, \dots, {\term$_m$}} + + Starting from a goal $\phi(t_1 \dots\ t_m)$, the tactic + {\tt pattern $t_1$, \dots,\ $t_m$} generates the equivalent goal {\tt + (fun (x$_1$:$A_1$) \dots\ (x$_m$:$A_m$) => $\phi(${\tt x$_1$\dots\ + x$_m$}$)$) $t_1$ \dots\ $t_m$}.\\ If $t_i$ occurs in one of the + generated types $A_j$ these occurrences will also be considered and + possibly abstracted. + +\item {\tt pattern {\term$_1$} at {\num$_1^1$} \dots\ {\num$_{n_1}^1$}, \dots, + {\term$_m$} at {\num$_1^m$} \dots\ {\num$_{n_m}^m$}} + + This behaves as above but processing only the occurrences \num$_1^1$, + \dots, \num$_i^1$ of \term$_1$, \dots, \num$_1^m$, \dots, \num$_j^m$ + of \term$_m$ starting from \term$_m$. + +\item {\tt pattern} {\term$_1$} \zeroone{{\tt at \zeroone{-}} {\num$_1^1$} \dots\ {\num$_{n_1}^1$}} {\tt ,} \dots {\tt ,} + {\term$_m$} \zeroone{{\tt at \zeroone{-}} {\num$_1^m$} \dots\ {\num$_{n_m}^m$}} + + This is the most general syntax that combines the different variants. + +\end{Variants} + +\subsection{Conversion tactics applied to hypotheses} + +{\convtactic} {\tt in} \ident$_1$ \dots\ \ident$_n$ + +Applies the conversion tactic {\convtactic} to the +hypotheses \ident$_1$, \ldots, \ident$_n$. The tactic {\convtactic} is +any of the conversion tactics listed in this section. + +If \ident$_i$ is a local definition, then \ident$_i$ can be replaced +by (Type of \ident$_i$) to address not the body but the type of the +local definition. Example: {\tt unfold not in (Type of H1) (Type of H3).} + +\begin{ErrMsgs} +\item \errindex{No such hypothesis} : {\ident}. +\end{ErrMsgs} + + +\section{Introductions} + +Introduction tactics address goals which are inductive constants. +They are used when one guesses that the goal can be obtained with one +of its constructors' type. + +\subsection{\tt constructor \num +\label{constructor} +\tacindex{constructor}} + +This tactic applies to a goal such that the head of its conclusion is +an inductive constant (say {\tt I}). The argument {\num} must be less +or equal to the numbers of constructor(s) of {\tt I}. Let {\tt ci} be +the {\tt i}-th constructor of {\tt I}, then {\tt constructor i} is +equivalent to {\tt intros; apply ci}. + +\begin{ErrMsgs} +\item \errindex{Not an inductive product} +\item \errindex{Not enough constructors} +\end{ErrMsgs} + +\begin{Variants} +\item \texttt{constructor} + + This tries \texttt{constructor 1} then \texttt{constructor 2}, + \dots\ , then \texttt{constructor} \textit{n} where \textit{n} if + the number of constructors of the head of the goal. + +\item {\tt constructor \num~with} {\bindinglist} + + Let {\tt ci} be the {\tt i}-th constructor of {\tt I}, then {\tt + constructor i with \bindinglist} is equivalent to {\tt intros; + apply ci with \bindinglist}. + + \Warning the terms in the \bindinglist\ are checked + in the context where {\tt constructor} is executed and not in the + context where {\tt apply} is executed (the introductions are not + taken into account). + +% To document? +% \item {\tt constructor {\tactic}} + +\item {\tt split}\tacindex{split} + + Applies if {\tt I} has only one constructor, typically in the case + of conjunction $A\land B$. Then, it is equivalent to {\tt constructor 1}. + +\item {\tt exists {\bindinglist}}\tacindex{exists} + + Applies if {\tt I} has only one constructor, for instance in the + case of existential quantification $\exists x\cdot P(x)$. + Then, it is equivalent to {\tt intros; constructor 1 with \bindinglist}. + +\item {\tt exists \nelist{\bindinglist}{,}} + + This iteratively applies {\tt exists {\bindinglist}}. + +\item {\tt left}\tacindex{left}\\ + {\tt right}\tacindex{right} + + Apply if {\tt I} has two constructors, for instance in the case of + disjunction $A\lor B$. Then, they are respectively equivalent to {\tt + constructor 1} and {\tt constructor 2}. + +\item {\tt left \bindinglist}\\ + {\tt right \bindinglist}\\ + {\tt split \bindinglist} + + As soon as the inductive type has the right number of constructors, + these expressions are equivalent to the corresponding {\tt + constructor $i$ with \bindinglist}. + +\item \texttt{econstructor}\tacindex{econstructor}\\ + \texttt{eexists}\tacindex{eexists}\\ + \texttt{esplit}\tacindex{esplit}\\ + \texttt{eleft}\tacindex{eleft}\\ + \texttt{eright}\tacindex{eright}\\ + + These tactics and their variants behave like \texttt{constructor}, + \texttt{exists}, \texttt{split}, \texttt{left}, \texttt{right} and + their variants but they introduce existential variables instead of + failing when the instantiation of a variable cannot be found (cf + \texttt{eapply} and Section~\ref{eapply-example}). + +\end{Variants} + +\section[Induction and Case Analysis]{Induction and Case Analysis +\label{Tac-induction}} + +The tactics presented in this section implement induction or case +analysis on inductive or coinductive objects (see +Section~\ref{Cic-inductive-definitions}). + +\subsection{\tt induction \term +\tacindex{induction}} + +This tactic applies to any goal. The type of the argument {\term} must +be an inductive constant. Then, the tactic {\tt induction} +generates subgoals, one for each possible form of {\term}, i.e. one +for each constructor of the inductive type. + +The tactic {\tt induction} automatically replaces every occurrences +of {\term} in the conclusion and the hypotheses of the goal. It +automatically adds induction hypotheses (using names of the form {\tt + IHn1}) to the local context. If some hypothesis must not be taken +into account in the induction hypothesis, then it needs to be removed +first (you can also use the tactics {\tt elim} or {\tt simple induction}, +see below). + +There are particular cases: + +\begin{itemize} + +\item If {\term} is an identifier {\ident} denoting a quantified +variable of the conclusion of the goal, then {\tt induction {\ident}} +behaves as {\tt intros until {\ident}; induction {\ident}}. + +\item If {\term} is a {\num}, then {\tt induction {\num}} behaves as +{\tt intros until {\num}} followed by {\tt induction} applied to the +last introduced hypothesis. + +\Rem For simple induction on a numeral, use syntax {\tt induction +({\num})} (not very interesting anyway). + +\end{itemize} + +\Example + +\begin{coq_example} +Lemma induction_test : forall n:nat, n = n -> n <= n. +intros n H. +induction n. +\end{coq_example} + +\begin{ErrMsgs} +\item \errindex{Not an inductive product} +\item \errindex{Unable to find an instance for the variables +{\ident} \ldots {\ident}} + + Use in this case + the variant {\tt elim \dots\ with \dots} below. +\end{ErrMsgs} + +\begin{Variants} +\item{\tt induction {\term} as {\disjconjintropattern}} + + This behaves as {\tt induction {\term}} but uses the names in + {\disjconjintropattern} to name the variables introduced in the context. + The {\disjconjintropattern} must typically be of the form + {\tt [} $p_{11}$ \ldots + $p_{1n_1}$ {\tt |} {\ldots} {\tt |} $p_{m1}$ \ldots $p_{mn_m}$ {\tt + ]} with $m$ being the number of constructors of the type of + {\term}. Each variable introduced by {\tt induction} in the context + of the $i^{th}$ goal gets its name from the list $p_{i1}$ \ldots + $p_{in_i}$ in order. If there are not enough names, {\tt induction} + invents names for the remaining variables to introduce. More + generally, the $p_{ij}$ can be any disjunctive/conjunctive + introduction pattern (see Section~\ref{intros-pattern}). For instance, + for an inductive type with one constructor, the pattern notation + {\tt ($p_{1}$,\ldots,$p_{n}$)} can be used instead of + {\tt [} $p_{1}$ \ldots $p_{n}$ {\tt ]}. + +\item{\tt induction {\term} as {\namingintropattern}} + + This behaves as {\tt induction {\term}} but adds an equation between + {\term} and the value that {\term} takes in each of the induction + case. The name of the equation is built according to + {\namingintropattern} which can be an identifier, a ``?'', etc, as + indicated in Section~\ref{intros-pattern}. + +\item{\tt induction {\term} as {\namingintropattern} {\disjconjintropattern}} + + This combines the two previous forms. + +\item{\tt induction {\term} with \bindinglist} + + This behaves like \texttt{induction {\term}} providing explicit + instances for the premises of the type of {\term} (see the syntax of + bindings in Section~\ref{Binding-list}). + +\item{\tt einduction {\term}\tacindex{einduction}} + + This tactic behaves like \texttt{induction {\term}} excepts that it + does not fail if some dependent premise of the type of {\term} is + not inferable. Instead, the unresolved premises are posed as + existential variables to be inferred later, in the same way as {\tt + eapply} does (see Section~\ref{eapply-example}). + +\item {\tt induction {\term$_1$} using {\term$_2$}} + + This behaves as {\tt induction {\term$_1$}} but using {\term$_2$} as + induction scheme. It does not expect the conclusion of the type of + {\term$_1$} to be inductive. + +\item {\tt induction {\term$_1$} using {\term$_2$} with {\bindinglist}} + + This behaves as {\tt induction {\term$_1$} using {\term$_2$}} but + also providing instances for the premises of the type of {\term$_2$}. + +\item \texttt{induction {\term}$_1$ $\ldots$ {\term}$_n$ using {\qualid}} + + This syntax is used for the case {\qualid} denotes an induction principle + with complex predicates as the induction principles generated by + {\tt Function} or {\tt Functional Scheme} may be. + +\item \texttt{induction {\term} in {\occgoalset}} + + This syntax is used for selecting which occurrences of {\term} the + induction has to be carried on. The {\tt in {\atoccurrences}} clause is an + occurrence clause whose syntax and behavior is described in + Section~\ref{Occurrences clauses}. + + When an occurrence clause is given, an equation between {\term} and + the value it gets in each case of the induction is added to the + context of the subgoals corresponding to the induction cases (even + if no clause {\tt as {\namingintropattern}} is given). + +\item {\tt induction {\term$_1$} with {\bindinglist$_1$} as {\namingintropattern} {\disjconjintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}}\\ + {\tt einduction {\term$_1$} with {\bindinglist$_1$} as {\namingintropattern} {\disjconjintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}} + + These are the most general forms of {\tt induction} and {\tt + einduction}. It combines the effects of the {\tt with}, {\tt as}, + {\tt using}, and {\tt in} clauses. + +\item {\tt elim \term}\label{elim} + + This is a more basic induction tactic. Again, the type of the + argument {\term} must be an inductive type. Then, according to + the type of the goal, the tactic {\tt elim} chooses the appropriate + destructor and applies it as the tactic {\tt apply} + would do. For instance, if the proof context contains {\tt + n:nat} and the current goal is {\tt T} of type {\tt + Prop}, then {\tt elim n} is equivalent to {\tt apply nat\_ind with + (n:=n)}. The tactic {\tt elim} does not modify the context of + the goal, neither introduces the induction loading into the context + of hypotheses. + + More generally, {\tt elim \term} also works when the type of {\term} + is a statement with premises and whose conclusion is inductive. In + that case the tactic performs induction on the conclusion of the + type of {\term} and leaves the non-dependent premises of the type as + subgoals. In the case of dependent products, the tactic tries to + find an instance for which the elimination lemma applies and fails + otherwise. + +\item {\tt elim {\term} with {\bindinglist}} + + Allows to give explicit instances to the premises of the type + of {\term} (see Section~\ref{Binding-list}). + +\item{\tt eelim {\term}\tacindex{eelim}} + + In case the type of {\term} has dependent premises, this turns them into + existential variables to be resolved later on. + +\item{\tt elim {\term$_1$} using {\term$_2$}}\\ + {\tt elim {\term$_1$} using {\term$_2$} with {\bindinglist}\tacindex{elim \dots\ using}} + +Allows the user to give explicitly an elimination predicate +{\term$_2$} which is not the standard one for the underlying inductive +type of {\term$_1$}. The {\bindinglist} clause allows to +instantiate premises of the type of {\term$_2$}. + +\item{\tt elim {\term$_1$} with {\bindinglist$_1$} using {\term$_2$} with {\bindinglist$_2$}}\\ + {\tt eelim {\term$_1$} with {\bindinglist$_1$} using {\term$_2$} with {\bindinglist$_2$}} + + These are the most general forms of {\tt elim} and {\tt eelim}. It + combines the effects of the {\tt using} clause and of the two uses + of the {\tt with} clause. + +\item {\tt elimtype \form}\tacindex{elimtype} + + The argument {\form} must be inductively defined. {\tt elimtype I} + is equivalent to {\tt cut I. intro H{\rm\sl n}; elim H{\rm\sl n}; + clear H{\rm\sl n}}. Therefore the hypothesis {\tt H{\rm\sl n}} will + not appear in the context(s) of the subgoal(s). Conversely, if {\tt + t} is a term of (inductive) type {\tt I} and which does not occur + in the goal then {\tt elim t} is equivalent to {\tt elimtype I; 2: + exact t.} + +\item {\tt simple induction \ident}\tacindex{simple induction} + + This tactic behaves as {\tt intros until + {\ident}; elim {\tt {\ident}}} when {\ident} is a quantified + variable of the goal. + +\item {\tt simple induction {\num}} + + This tactic behaves as {\tt intros until + {\num}; elim {\tt {\ident}}} where {\ident} is the name given by + {\tt intros until {\num}} to the {\num}-th non-dependent premise of + the goal. + +%% \item {\tt simple induction {\term}}\tacindex{simple induction} + +%% If {\term} is an {\ident} corresponding to a quantified variable of +%% the goal then the tactic behaves as {\tt intros until {\ident}; elim +%% {\tt {\ident}}}. If {\term} is a {\num} then the tactic behaves as +%% {\tt intros until {\ident}; elim {\tt {\ident}}}. Otherwise, it is +%% a synonym for {\tt elim {\term}}. + +%% \Rem For simple induction on a numeral, use syntax {\tt simple +%% induction ({\num})}. + +\end{Variants} + +\subsection{\tt destruct \term +\tacindex{destruct}} +\label{destruct} + +The tactic {\tt destruct} is used to perform case analysis without +recursion. Its behavior is similar to {\tt induction} except +that no induction hypothesis is generated. It applies to any goal and +the type of {\term} must be inductively defined. There are particular cases: + +\begin{itemize} + +\item If {\term} is an identifier {\ident} denoting a quantified +variable of the conclusion of the goal, then {\tt destruct {\ident}} +behaves as {\tt intros until {\ident}; destruct {\ident}}. + +\item If {\term} is a {\num}, then {\tt destruct {\num}} behaves as +{\tt intros until {\num}} followed by {\tt destruct} applied to the +last introduced hypothesis. + +\Rem For destruction of a numeral, use syntax {\tt destruct +({\num})} (not very interesting anyway). + +\end{itemize} + +\begin{Variants} +\item{\tt destruct {\term} as {\disjconjintropattern}} + + This behaves as {\tt destruct {\term}} but uses the names in + {\intropattern} to name the variables introduced in the context. + The {\intropattern} must have the form {\tt [} $p_{11}$ \ldots + $p_{1n_1}$ {\tt |} {\ldots} {\tt |} $p_{m1}$ \ldots $p_{mn_m}$ {\tt + ]} with $m$ being the number of constructors of the type of + {\term}. Each variable introduced by {\tt destruct} in the context + of the $i^{th}$ goal gets its name from the list $p_{i1}$ \ldots + $p_{in_i}$ in order. If there are not enough names, {\tt destruct} + invents names for the remaining variables to introduce. More + generally, the $p_{ij}$ can be any disjunctive/conjunctive + introduction pattern (see Section~\ref{intros-pattern}). This + provides a concise notation for nested destruction. + +% It is recommended to use this variant of {\tt destruct} for +% robust proof scripts. + +\item{\tt destruct {\term} as {\disjconjintropattern} \_eqn} + + This behaves as {\tt destruct {\term}} but adds an equation between + {\term} and the value that {\term} takes in each of the possible + cases. The name of the equation is chosen by Coq. If + {\disjconjintropattern} is simply {\tt []}, it is automatically considered + as a disjunctive pattern of the appropriate size. + +\item{\tt destruct {\term} as {\disjconjintropattern} \_eqn: {\namingintropattern}} + + This behaves as {\tt destruct {\term} as + {\disjconjintropattern} \_eqn} but use {\namingintropattern} to + name the equation (see Section~\ref{intros-pattern}). Note that spaces + can generally be removed around {\tt \_eqn}. + +\item{\tt destruct {\term} with \bindinglist} + + This behaves like \texttt{destruct {\term}} providing explicit + instances for the dependent premises of the type of {\term} (see + syntax of bindings in Section~\ref{Binding-list}). + +\item{\tt edestruct {\term}\tacindex{edestruct}} + + This tactic behaves like \texttt{destruct {\term}} excepts that it + does not fail if the instance of a dependent premises of the type of + {\term} is not inferable. Instead, the unresolved instances are left + as existential variables to be inferred later, in the same way as + {\tt eapply} does (see Section~\ref{eapply-example}). + +\item{\tt destruct {\term$_1$} using {\term$_2$}}\\ + {\tt destruct {\term$_1$} using {\term$_2$} with {\bindinglist}} + + These are synonyms of {\tt induction {\term$_1$} using {\term$_2$}} and + {\tt induction {\term$_1$} using {\term$_2$} with {\bindinglist}}. + +\item \texttt{destruct {\term} in {\occgoalset}} + + This syntax is used for selecting which occurrences of {\term} the + case analysis has to be done on. The {\tt in {\occgoalset}} clause is an + occurrence clause whose syntax and behavior is described in + Section~\ref{Occurrences clauses}. + + When an occurrence clause is given, an equation between {\term} and + the value it gets in each case of the analysis is added to the + context of the subgoals corresponding to the cases (even + if no clause {\tt as {\namingintropattern}} is given). + +\item{\tt destruct {\term$_1$} with {\bindinglist$_1$} as {\disjconjintropattern} \_eqn: {\namingintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}}\\ + {\tt edestruct {\term$_1$} with {\bindinglist$_1$} as {\disjconjintropattern} \_eqn: {\namingintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}} + + These are the general forms of {\tt destruct} and {\tt edestruct}. + They combine the effects of the {\tt with}, {\tt as}, {\tt using}, + and {\tt in} clauses. + +\item{\tt case \term}\label{case}\tacindex{case} + + The tactic {\tt case} is a more basic tactic to perform case + analysis without recursion. It behaves as {\tt elim \term} but using + a case-analysis elimination principle and not a recursive one. + +\item{\tt case\_eq \term}\label{case_eq}\tacindex{case\_eq} + + The tactic {\tt case\_eq} is a variant of the {\tt case} tactic that + allow to perform case analysis on a term without completely + forgetting its original form. This is done by generating equalities + between the original form of the term and the outcomes of the case + analysis. The effect of this tactic is similar to the effect of {\tt + destruct {\term} in |- *} with the exception that no new hypotheses + are introduced in the context. + +\item {\tt case {\term} with {\bindinglist}} + + Analogous to {\tt elim {\term} with {\bindinglist}} above. + +\item{\tt ecase {\term}\tacindex{ecase}}\\ + {\tt ecase {\term} with {\bindinglist}} + + In case the type of {\term} has dependent premises, or dependent + premises whose values are not inferable from the {\tt with + {\bindinglist}} clause, {\tt ecase} turns them into existential + variables to be resolved later on. + +\item {\tt simple destruct \ident}\tacindex{simple destruct} + + This tactic behaves as {\tt intros until + {\ident}; case {\tt {\ident}}} when {\ident} is a quantified + variable of the goal. + +\item {\tt simple destruct {\num}} + + This tactic behaves as {\tt intros until + {\num}; case {\tt {\ident}}} where {\ident} is the name given by + {\tt intros until {\num}} to the {\num}-th non-dependent premise of + the goal. + + +\end{Variants} + +\subsection{\tt intros {\intropattern} {\ldots} {\intropattern} +\label{intros-pattern} +\tacindex{intros \intropattern}} +\index{Introduction patterns} +\index{Naming introduction patterns} +\index{Disjunctive/conjunctive introduction patterns} + +This extension of the tactic {\tt intros} combines introduction of +variables or hypotheses and case analysis. An {\em introduction pattern} is +either: +\begin{itemize} +\item A {\em naming introduction pattern}, i.e. either one of: + \begin{itemize} + \item the pattern \texttt{?} + \item the pattern \texttt{?\ident} + \item an identifier + \end{itemize} +\item A {\em disjunctive/conjunctive introduction pattern}, i.e. either one of: + \begin{itemize} + \item a disjunction of lists of patterns: + {\tt [$p_{11}$ {\ldots} $p_{1m_1}$ | {\ldots} | $p_{11}$ {\ldots} $p_{nm_n}$]} + \item a conjunction of patterns: {\tt (} $p_1$ {\tt ,} {\ldots} {\tt ,} $p_n$ {\tt )} + \item a list of patterns {\tt (} $p_1$\ {\tt \&}\ {\ldots}\ {\tt \&}\ $p_n$ {\tt )} + for sequence of right-associative binary constructs + \end{itemize} +\item the wildcard: {\tt \_} +\item the rewriting orientations: {\tt ->} or {\tt <-} +\end{itemize} + +Assuming a goal of type {\tt $Q$ -> $P$} (non dependent product), or +of type {\tt forall $x$:$T$, $P$} (dependent product), the behavior of +{\tt intros $p$} is defined inductively over the structure of the +introduction pattern $p$: +\begin{itemize} +\item introduction on \texttt{?} performs the introduction, and lets {\Coq} + choose a fresh name for the variable; +\item introduction on \texttt{?\ident} performs the introduction, and + lets {\Coq} choose a fresh name for the variable based on {\ident}; +\item introduction on \texttt{\ident} behaves as described in + Section~\ref{intro}; +\item introduction over a disjunction of list of patterns {\tt + [$p_{11}$ {\ldots} $p_{1m_1}$ | {\ldots} | $p_{11}$ {\ldots} + $p_{nm_n}$]} expects the product to be over an inductive type + whose number of constructors is $n$ (or more generally over a type + of conclusion an inductive type built from $n$ constructors, + e.g. {\tt C -> A$\backslash$/B if $n=2$}): it destructs the introduced + hypothesis as {\tt destruct} (see Section~\ref{destruct}) would and + applies on each generated subgoal the corresponding tactic; + \texttt{intros}~$p_{i1}$ {\ldots} $p_{im_i}$; if the disjunctive + pattern is part of a sequence of patterns and is not the last + pattern of the sequence, then {\Coq} completes the pattern so as all + the argument of the constructors of the inductive type are + introduced (for instance, the list of patterns {\tt [$\;$|$\;$] H} + applied on goal {\tt forall x:nat, x=0 -> 0=x} behaves the same as + the list of patterns {\tt [$\,$|$\,$?$\,$] H}); +\item introduction over a conjunction of patterns {\tt ($p_1$, \ldots, + $p_n$)} expects the goal to be a product over an inductive type $I$ with a + single constructor that itself has at least $n$ arguments: it + performs a case analysis over the hypothesis, as {\tt destruct} + would, and applies the patterns $p_1$~\ldots~$p_n$ to the arguments + of the constructor of $I$ (observe that {\tt ($p_1$, {\ldots}, + $p_n$)} is an alternative notation for {\tt [$p_1$ {\ldots} + $p_n$]}); +\item introduction via {\tt ( $p_1$ \& \ldots \& $p_n$ )} + is a shortcut for introduction via + {\tt ($p_1$,(\ldots,(\dots,$p_n$)\ldots))}; it expects the + hypothesis to be a sequence of right-associative binary inductive + constructors such as {\tt conj} or {\tt ex\_intro}; for instance, an + hypothesis with type {\tt A\verb|/\|exists x, B\verb|/\|C\verb|/\|D} can be + introduced via pattern {\tt (a \& x \& b \& c \& d)}; +\item introduction on the wildcard depends on whether the product is + dependent or not: in the non dependent case, it erases the + corresponding hypothesis (i.e. it behaves as an {\tt intro} followed + by a {\tt clear}, cf Section~\ref{clear}) while in the dependent + case, it succeeds and erases the variable only if the wildcard is + part of a more complex list of introduction patterns that also + erases the hypotheses depending on this variable; +\item introduction over {\tt ->} (respectively {\tt <-}) expects the + hypothesis to be an equality and the right-hand-side (respectively + the left-hand-side) is replaced by the left-hand-side (respectively + the right-hand-side) in both the conclusion and the context of the goal; + if moreover the term to substitute is a variable, the hypothesis is + removed. +\end{itemize} + +\Rem {\tt intros $p_1~\ldots~p_n$} is not equivalent to \texttt{intros + $p_1$;\ldots; intros $p_n$} for the following reasons: +\begin{itemize} +\item A wildcard pattern never succeeds when applied isolated on a + dependent product, while it succeeds as part of a list of + introduction patterns if the hypotheses that depends on it are + erased too. +\item A disjunctive or conjunctive pattern followed by an introduction + pattern forces the introduction in the context of all arguments of + the constructors before applying the next pattern while a terminal + disjunctive or conjunctive pattern does not. Here is an example + +\begin{coq_example} +Goal forall n:nat, n = 0 -> n = 0. +intros [ | ] H. +Show 2. +Undo. +intros [ | ]; intros H. +Show 2. +\end{coq_example} + +\end{itemize} + +\begin{coq_example} +Lemma intros_test : forall A B C:Prop, A \/ B /\ C -> (A -> C) -> C. +intros A B C [a| [_ c]] f. +apply (f a). +exact c. +Qed. +\end{coq_example} + +%\subsection[\tt FixPoint \dots]{\tt FixPoint \dots\tacindex{Fixpoint}} +%Not yet documented. + +\subsection{\tt double induction \ident$_1$ \ident$_2$} +%\tacindex{double induction}} +This tactic is deprecated and should be replaced by {\tt induction \ident$_1$; induction \ident$_2$} (or {\tt induction \ident$_1$; destruct \ident$_2$} depending on the exact needs). + +%% This tactic applies to any goal. If the variables {\ident$_1$} and +%% {\ident$_2$} of the goal have an inductive type, then this tactic +%% performs double induction on these variables. For instance, if the +%% current goal is \verb+forall n m:nat, P n m+ then, {\tt double induction n +%% m} yields the four cases with their respective inductive hypotheses. + +%% In particular, for proving \verb+(P (S n) (S m))+, the generated induction +%% hypotheses are \verb+(P (S n) m)+ and \verb+(m:nat)(P n m)+ (of the latter, +%% \verb+(P n m)+ and \verb+(P n (S m))+ are derivable). + +%% \Rem When the induction hypothesis \verb+(P (S n) m)+ is not +%% needed, {\tt induction \ident$_1$; destruct \ident$_2$} produces +%% more concise subgoals. + +\begin{Variant} + +\item {\tt double induction \num$_1$ \num$_2$} + +This tactic is deprecated and should be replaced by {\tt induction + \num$_1$; induction \num$_3$} where \num$_3$ is the result of +\num$_2$-\num$_1$. + +%% This tactic applies to any goal. If the variables {\ident$_1$} and + +%% This applies double induction on the \num$_1^{th}$ and \num$_2^{th}$ {\it +%% non dependent} premises of the goal. More generally, any combination of an +%% {\ident} and a {\num} is valid. + +\end{Variant} + +\subsection{\tt dependent induction \ident + \tacindex{dependent induction} + \label{DepInduction}} + +The \emph{experimental} tactic \texttt{dependent induction} performs +induction-inversion on an instantiated inductive predicate. +One needs to first require the {\tt Coq.Program.Equality} module to use +this tactic. The tactic is based on the BasicElim tactic by Conor +McBride \cite{DBLP:conf/types/McBride00} and the work of Cristina Cornes +around inversion \cite{DBLP:conf/types/CornesT95}. From an instantiated +inductive predicate and a goal it generates an equivalent goal where the +hypothesis has been generalized over its indexes which are then +constrained by equalities to be the right instances. This permits to +state lemmas without resorting to manually adding these equalities and +still get enough information in the proofs. +A simple example is the following: + +\begin{coq_eval} +Reset Initial. +\end{coq_eval} +\begin{coq_example} +Lemma le_minus : forall n:nat, n < 1 -> n = 0. +intros n H ; induction H. +\end{coq_example} + +Here we didn't get any information on the indexes to help fulfill this +proof. The problem is that when we use the \texttt{induction} tactic +we lose information on the hypothesis instance, notably that the second +argument is \texttt{1} here. Dependent induction solves this problem by +adding the corresponding equality to the context. + +\begin{coq_eval} +Reset Initial. +\end{coq_eval} +\begin{coq_example} +Require Import Coq.Program.Equality. +Lemma le_minus : forall n:nat, n < 1 -> n = 0. +intros n H ; dependent induction H. +\end{coq_example} + +The subgoal is cleaned up as the tactic tries to automatically +simplify the subgoals with respect to the generated equalities. +In this enriched context it becomes possible to solve this subgoal. +\begin{coq_example} +reflexivity. +\end{coq_example} + +Now we are in a contradictory context and the proof can be solved. +\begin{coq_example} +inversion H. +\end{coq_example} + +This technique works with any inductive predicate. +In fact, the \texttt{dependent induction} tactic is just a wrapper around +the \texttt{induction} tactic. One can make its own variant by just +writing a new tactic based on the definition found in +\texttt{Coq.Program.Equality}. Common useful variants are the following, +defined in the same file: + +\begin{Variants} +\item {\tt dependent induction {\ident} generalizing {\ident$_1$} \dots + {\ident$_n$}}\tacindex{dependent induction \dots\ generalizing} + + Does dependent induction on the hypothesis {\ident} but first + generalizes the goal by the given variables so that they are + universally quantified in the goal. This is generally what one wants + to do with the variables that are inside some constructors in the + induction hypothesis. The other ones need not be further generalized. + +\item {\tt dependent destruction {\ident}}\tacindex{dependent destruction} + + Does the generalization of the instance {\ident} but uses {\tt destruct} + instead of {\tt induction} on the generalized hypothesis. This gives + results equivalent to {\tt inversion} or {\tt dependent inversion} if + the hypothesis is dependent. +\end{Variants} + +A larger example of dependent induction and an explanation of the +underlying technique are developed in section~\ref{dependent-induction-example}. + +\subsection{\tt decompose [ {\qualid$_1$} \dots\ {\qualid$_n$} ] \term +\label{decompose} +\tacindex{decompose}} + +This tactic allows to recursively decompose a +complex proposition in order to obtain atomic ones. +Example: + +\begin{coq_eval} +Reset Initial. +\end{coq_eval} +\begin{coq_example} +Lemma ex1 : forall A B C:Prop, A /\ B /\ C \/ B /\ C \/ C /\ A -> C. +intros A B C H; decompose [and or] H; assumption. +\end{coq_example} +\begin{coq_example*} +Qed. +\end{coq_example*} + +{\tt decompose} does not work on right-hand sides of implications or products. + +\begin{Variants} + +\item {\tt decompose sum \term}\tacindex{decompose sum} + This decomposes sum types (like \texttt{or}). +\item {\tt decompose record \term}\tacindex{decompose record} + This decomposes record types (inductive types with one constructor, + like \texttt{and} and \texttt{exists} and those defined with the + \texttt{Record} macro, see Section~\ref{Record}). +\end{Variants} + + +\subsection{\tt functional induction (\qualid\ \term$_1$ \dots\ \term$_n$). +\tacindex{functional induction} +\label{FunInduction}} + +The \emph{experimental} tactic \texttt{functional induction} performs +case analysis and induction following the definition of a function. It +makes use of a principle generated by \texttt{Function} +(see Section~\ref{Function}) or \texttt{Functional Scheme} +(see Section~\ref{FunScheme}). + +\begin{coq_eval} +Reset Initial. +\end{coq_eval} +\begin{coq_example} +Functional Scheme minus_ind := Induction for minus Sort Prop. + +Lemma le_minus : forall n m:nat, (n - m <= n). +intros n m. +functional induction (minus n m); simpl; auto. +\end{coq_example} +\begin{coq_example*} +Qed. +\end{coq_example*} + +\Rem \texttt{(\qualid\ \term$_1$ \dots\ \term$_n$)} must be a correct +full application of \qualid. In particular, the rules for implicit +arguments are the same as usual. For example use \texttt{@\qualid} if +you want to write implicit arguments explicitly. + +\Rem Parenthesis over \qualid \dots \term$_n$ are mandatory. + +\Rem \texttt{functional induction (f x1 x2 x3)} is actually a wrapper +for \texttt{induction x1 x2 x3 (f x1 x2 x3) using \qualid} followed by +a cleaning phase, where $\qualid$ is the induction principle +registered for $f$ (by the \texttt{Function} (see Section~\ref{Function}) +or \texttt{Functional Scheme} (see Section~\ref{FunScheme}) command) +corresponding to the sort of the goal. Therefore \texttt{functional + induction} may fail if the induction scheme (\texttt{\qualid}) is +not defined. See also Section~\ref{Function} for the function terms +accepted by \texttt{Function}. + +\Rem There is a difference between obtaining an induction scheme for a +function by using \texttt{Function} (see Section~\ref{Function}) and by +using \texttt{Functional Scheme} after a normal definition using +\texttt{Fixpoint} or \texttt{Definition}. See \ref{Function} for +details. + +\SeeAlso{\ref{Function},\ref{FunScheme},\ref{FunScheme-examples}, + \ref{sec:functional-inversion}} + +\begin{ErrMsgs} +\item \errindex{Cannot find induction information on \qualid} + + ~ + +\item \errindex{Not the right number of induction arguments} +\end{ErrMsgs} + +\begin{Variants} +\item {\tt functional induction (\qualid\ \term$_1$ \dots\ \term$_n$) + using \term$_{m+1}$ with {\term$_{n+1}$} \dots {\term$_m$}} + + Similar to \texttt{Induction} and \texttt{elim} + (see Section~\ref{Tac-induction}), allows to give explicitly the + induction principle and the values of dependent premises of the + elimination scheme, including \emph{predicates} for mutual induction + when {\qualid} is part of a mutually recursive definition. + +\item {\tt functional induction (\qualid\ \term$_1$ \dots\ \term$_n$) + using \term$_{m+1}$ with {\vref$_1$} := {\term$_{n+1}$} \dots\ + {\vref$_m$} := {\term$_n$}} + + Similar to \texttt{induction} and \texttt{elim} + (see Section~\ref{Tac-induction}). + +\item All previous variants can be extended by the usual \texttt{as + \intropattern} construction, similar for example to + \texttt{induction} and \texttt{elim} (see Section~\ref{Tac-induction}). + +\end{Variants} + + + +\section{Equality} + +These tactics use the equality {\tt eq:forall A:Type, A->A->Prop} +defined in file {\tt Logic.v} (see Section~\ref{Equality}). The +notation for {\tt eq}~$T~t~u$ is simply {\tt $t$=$u$} dropping the +implicit type of $t$ and $u$. + +\subsection{\tt rewrite \term +\label{rewrite} +\tacindex{rewrite}} + +This tactic applies to any goal. The type of {\term} +must have the form + +\texttt{forall (x$_1$:A$_1$) \dots\ (x$_n$:A$_n$)}\texttt{eq} \term$_1$ \term$_2$. + +\noindent where \texttt{eq} is the Leibniz equality or a registered +setoid equality. + +\noindent Then {\tt rewrite \term} finds the first subterm matching +\term$_1$ in the goal, resulting in instances \term$_1'$ and \term$_2'$ +and then replaces every occurrence of \term$_1'$ by \term$_2'$. +Hence, some of the variables x$_i$ are +solved by unification, and some of the types \texttt{A}$_1$, \dots, +\texttt{A}$_n$ become new subgoals. + +% \Rem In case the type of +% \term$_1$ contains occurrences of variables bound in the +% type of \term, the tactic tries first to find a subterm of the goal +% which matches this term in order to find a closed instance \term$'_1$ +% of \term$_1$, and then all instances of \term$'_1$ will be replaced. + +\begin{ErrMsgs} +\item \errindex{The term provided does not end with an equation} + +\item \errindex{Tactic generated a subgoal identical to the original goal}\\ +This happens if \term$_1$ does not occur in the goal. +\end{ErrMsgs} + +\begin{Variants} +\item {\tt rewrite -> {\term}}\tacindex{rewrite ->}\\ + Is equivalent to {\tt rewrite \term} + +\item {\tt rewrite <- {\term}}\tacindex{rewrite <-}\\ + Uses the equality \term$_1${\tt=}\term$_2$ from right to left + +\item {\tt rewrite {\term} in \textit{clause}} + \tacindex{rewrite \dots\ in}\\ + Analogous to {\tt rewrite {\term}} but rewriting is done following + \textit{clause} (similarly to \ref{Conversion-tactics}). For + instance: + \begin{itemize} + \item \texttt{rewrite H in H1} will rewrite \texttt{H} in the hypothesis + \texttt{H1} instead of the current goal. + \item \texttt{rewrite H in H1 at 1, H2 at - 2 |- *} means \texttt{rewrite H; rewrite H in H1 at 1; + rewrite H in H2 at - 2}. In particular a failure will happen if any of + these three simpler tactics fails. + \item \texttt{rewrite H in * |- } will do \texttt{rewrite H in + H$_i$} for all hypothesis \texttt{H$_i$ <> H}. A success will happen + as soon as at least one of these simpler tactics succeeds. + \item \texttt{rewrite H in *} is a combination of \texttt{rewrite H} + and \texttt{rewrite H in * |-} that succeeds if at + least one of these two tactics succeeds. + \end{itemize} + Orientation {\tt ->} or {\tt <-} can be + inserted before the term to rewrite. + +\item {\tt rewrite {\term} at {\occlist}} + \tacindex{rewrite \dots\ at} + + Rewrite only the given occurrences of \term$_1'$. Occurrences are + specified from left to right as for \texttt{pattern} (\S + \ref{pattern}). The rewrite is always performed using setoid + rewriting, even for Leibniz's equality, so one has to + \texttt{Import Setoid} to use this variant. + +\item {\tt rewrite {\term} by {\tac}} + \tacindex{rewrite \dots\ by} + + Use {\tac} to completely solve the side-conditions arising from the + rewrite. + +\item {\tt rewrite $\term_1$, \ldots, $\term_n$}\\ + Is equivalent to the $n$ successive tactics {\tt rewrite $\term_1$} + up to {\tt rewrite $\term_n$}, each one working on the first subgoal + generated by the previous one. + Orientation {\tt ->} or {\tt <-} can be + inserted before each term to rewrite. One unique \textit{clause} + can be added at the end after the keyword {\tt in}; it will + then affect all rewrite operations. + +\item In all forms of {\tt rewrite} described above, a term to rewrite + can be immediately prefixed by one of the following modifiers: + \begin{itemize} + \item {\tt ?} : the tactic {\tt rewrite ?$\term$} performs the + rewrite of $\term$ as many times as possible (perhaps zero time). + This form never fails. + \item {\tt $n$?} : works similarly, except that it will do at most + $n$ rewrites. + \item {\tt !} : works as {\tt ?}, except that at least one rewrite + should succeed, otherwise the tactic fails. + \item {\tt $n$!} (or simply {\tt $n$}) : precisely $n$ rewrites + of $\term$ will be done, leading to failure if these $n$ rewrites are not possible. + \end{itemize} + +\item {\tt erewrite {\term}\tacindex{erewrite}} + +This tactic works as {\tt rewrite {\term}} but turning unresolved +bindings into existential variables, if any, instead of failing. It has +the same variants as {\tt rewrite} has. + +\end{Variants} + + +\subsection{\tt cutrewrite -> \term$_1$ = \term$_2$ +\label{cutrewrite} +\tacindex{cutrewrite}} + +This tactic acts like {\tt replace {\term$_1$} with {\term$_2$}} +(see below). + +\subsection{\tt replace {\term$_1$} with {\term$_2$} +\label{tactic:replace} +\tacindex{replace \dots\ with}} + +This tactic applies to any goal. It replaces all free occurrences of +{\term$_1$} in the current goal with {\term$_2$} and generates the +equality {\term$_2$}{\tt =}{\term$_1$} as a subgoal. This equality is +automatically solved if it occurs amongst the assumption, or if its +symmetric form occurs. It is equivalent to {\tt cut +\term$_2$=\term$_1$; [intro H{\sl n}; rewrite <- H{\sl n}; clear H{\sl +n}| assumption || symmetry; try assumption]}. + +\begin{ErrMsgs} +\item \errindex{terms do not have convertible types} +\end{ErrMsgs} + +\begin{Variants} +\item {\tt replace {\term$_1$} with {\term$_2$} by \tac}\\ This acts + as {\tt replace {\term$_1$} with {\term$_2$}} but applies {\tt \tac} + to solve the generated subgoal {\tt \term$_2$=\term$_1$}. +\item {\tt replace {\term}}\\ Replace {\term} with {\term'} using the + first assumption whose type has the form {\tt \term=\term'} or {\tt + \term'=\term} +\item {\tt replace -> {\term}}\\ Replace {\term} with {\term'} using the + first assumption whose type has the form {\tt \term=\term'} +\item {\tt replace <- {\term}}\\ Replace {\term} with {\term'} using the + first assumption whose type has the form {\tt \term'=\term} +\item {\tt replace {\term$_1$} with {\term$_2$} \textit{clause} }\\ + {\tt replace {\term$_1$} with {\term$_2$} \textit{clause} by \tac }\\ + {\tt replace {\term} \textit{clause}}\\ + {\tt replace -> {\term} \textit{clause}}\\ + {\tt replace <- {\term} \textit{clause}}\\ + Act as before but the replacements take place in + \textit{clause}~(see Section~\ref{Conversion-tactics}) and not only + in the conclusion of the goal.\\ + The \textit{clause} argument must not contain any \texttt{type of} nor \texttt{value of}. +\end{Variants} + +\subsection{\tt reflexivity +\label{reflexivity} +\tacindex{reflexivity}} + +This tactic applies to a goal which has the form {\tt t=u}. It checks +that {\tt t} and {\tt u} are convertible and then solves the goal. +It is equivalent to {\tt apply refl\_equal}. + +\begin{ErrMsgs} +\item \errindex{The conclusion is not a substitutive equation} +\item \errindex{Impossible to unify \dots\ with \dots.} +\end{ErrMsgs} + +\subsection{\tt symmetry +\tacindex{symmetry} +\tacindex{symmetry in}} +This tactic applies to a goal which has the form {\tt t=u} and changes it +into {\tt u=t}. + +\variant {\tt symmetry in {\ident}}\\ +If the statement of the hypothesis {\ident} has the form {\tt t=u}, +the tactic changes it to {\tt u=t}. + +\subsection{\tt transitivity \term +\tacindex{transitivity}} +This tactic applies to a goal which has the form {\tt t=u} +and transforms it into the two subgoals +{\tt t={\term}} and {\tt {\term}=u}. + +\subsection{\tt subst {\ident} +\tacindex{subst}} + +This tactic applies to a goal which has \ident\ in its context and +(at least) one hypothesis, say {\tt H}, of type {\tt + \ident=t} or {\tt t=\ident}. Then it replaces +\ident\ by {\tt t} everywhere in the goal (in the hypotheses +and in the conclusion) and clears \ident\ and {\tt H} from the context. + +\Rem +When several hypotheses have the form {\tt \ident=t} or {\tt + t=\ident}, the first one is used. + +\begin{Variants} + \item {\tt subst \ident$_1$ \dots \ident$_n$} \\ + Is equivalent to {\tt subst \ident$_1$; \dots; subst \ident$_n$}. + \item {\tt subst} \\ + Applies {\tt subst} repeatedly to all identifiers from the context + for which an equality exists. +\end{Variants} + +\subsection[{\tt stepl {\term}}]{{\tt stepl {\term}}\tacindex{stepl}} + +This tactic is for chaining rewriting steps. It assumes a goal of the +form ``$R$ {\term}$_1$ {\term}$_2$'' where $R$ is a binary relation +and relies on a database of lemmas of the form {\tt forall} $x$ $y$ +$z$, $R$ $x$ $y$ {\tt ->} $eq$ $x$ $z$ {\tt ->} $R$ $z$ $y$ where $eq$ +is typically a setoid equality. The application of {\tt stepl {\term}} +then replaces the goal by ``$R$ {\term} {\term}$_2$'' and adds a new +goal stating ``$eq$ {\term} {\term}$_1$''. + +Lemmas are added to the database using the command +\comindex{Declare Left Step} +\begin{quote} +{\tt Declare Left Step {\term}.} +\end{quote} + +The tactic is especially useful for parametric setoids which are not +accepted as regular setoids for {\tt rewrite} and {\tt + setoid\_replace} (see Chapter~\ref{setoid_replace}). + +\tacindex{stepr} +\comindex{Declare Right Step} +\begin{Variants} +\item{\tt stepl {\term} by {\tac}}\\ +This applies {\tt stepl {\term}} then applies {\tac} to the second goal. + +\item{\tt stepr {\term}}\\ + {\tt stepr {\term} by {\tac}}\\ +This behaves as {\tt stepl} but on the right-hand-side of the binary relation. +Lemmas are expected to be of the form +``{\tt forall} $x$ $y$ +$z$, $R$ $x$ $y$ {\tt ->} $eq$ $y$ $z$ {\tt ->} $R$ $x$ $z$'' +and are registered using the command +\begin{quote} +{\tt Declare Right Step {\term}.} +\end{quote} +\end{Variants} + + +\subsection{\tt f\_equal +\label{f-equal} +\tacindex{f\_equal}} + +This tactic applies to a goal of the form $f\ a_1\ \ldots\ a_n = f'\ +a'_1\ \ldots\ a'_n$. Using {\tt f\_equal} on such a goal leads to +subgoals $f=f'$ and $a_1=a'_1$ and so on up to $a_n=a'_n$. Amongst +these subgoals, the simple ones (e.g. provable by +reflexivity or congruence) are automatically solved by {\tt f\_equal}. + + +\section{Equality and inductive sets} + +We describe in this section some special purpose tactics dealing with +equality and inductive sets or types. These tactics use the equality +{\tt eq:forall (A:Type), A->A->Prop}, simply written with the +infix symbol {\tt =}. + +\subsection{\tt decide equality +\label{decideequality} +\tacindex{decide equality}} + +This tactic solves a goal of the form +{\tt forall $x$ $y$:$R$, \{$x$=$y$\}+\{\verb|~|$x$=$y$\}}, where $R$ +is an inductive type such that its constructors do not take proofs or +functions as arguments, nor objects in dependent types. + +\begin{Variants} +\item {\tt decide equality {\term}$_1$ {\term}$_2$ }.\\ + Solves a goal of the form {\tt \{}\term$_1${\tt =}\term$_2${\tt +\}+\{\verb|~|}\term$_1${\tt =}\term$_2${\tt \}}. +\end{Variants} + +\subsection{\tt compare \term$_1$ \term$_2$ +\tacindex{compare}} + +This tactic compares two given objects \term$_1$ and \term$_2$ +of an inductive datatype. If $G$ is the current goal, it leaves the sub-goals +\term$_1${\tt =}\term$_2$ {\tt ->} $G$ and \verb|~|\term$_1${\tt =}\term$_2$ +{\tt ->} $G$. The type +of \term$_1$ and \term$_2$ must satisfy the same restrictions as in the tactic +\texttt{decide equality}. + +\subsection{\tt discriminate {\term} +\label{discriminate} +\tacindex{discriminate} +\tacindex{ediscriminate}} + +This tactic proves any goal from an assumption stating that two +structurally different terms of an inductive set are equal. For +example, from {\tt (S (S O))=(S O)} we can derive by absurdity any +proposition. + +The argument {\term} is assumed to be a proof of a statement +of conclusion {\tt{\term$_1$} = {\term$_2$}} with {\term$_1$} and +{\term$_2$} being elements of an inductive set. To build the proof, +the tactic traverses the normal forms\footnote{Reminder: opaque + constants will not be expanded by $\delta$ reductions} of +{\term$_1$} and {\term$_2$} looking for a couple of subterms {\tt u} +and {\tt w} ({\tt u} subterm of the normal form of {\term$_1$} and +{\tt w} subterm of the normal form of {\term$_2$}), placed at the same +positions and whose head symbols are two different constructors. If +such a couple of subterms exists, then the proof of the current goal +is completed, otherwise the tactic fails. + +\Rem The syntax {\tt discriminate {\ident}} can be used to refer to a +hypothesis quantified in the goal. In this case, the quantified +hypothesis whose name is {\ident} is first introduced in the local +context using \texttt{intros until \ident}. + +\begin{ErrMsgs} +\item \errindex{No primitive equality found} +\item \errindex{Not a discriminable equality} +\end{ErrMsgs} + +\begin{Variants} +\item \texttt{discriminate} \num + + This does the same thing as \texttt{intros until \num} followed by + \texttt{discriminate \ident} where {\ident} is the identifier for + the last introduced hypothesis. + +\item \texttt{discriminate} {\term} {\tt with} {\bindinglist} + + This does the same thing as \texttt{discriminate {\term}} but using +the given bindings to instantiate parameters or hypotheses of {\term}. + +\item \texttt{ediscriminate} \num\\ + \texttt{ediscriminate} {\term} \zeroone{{\tt with} {\bindinglist}} + + This works the same as {\tt discriminate} but if the type of {\term}, + or the type of the hypothesis referred to by {\num}, has uninstantiated + parameters, these parameters are left as existential variables. + +\item \texttt{discriminate} + + This behaves like {\tt discriminate {\ident}} if {\ident} is the + name of an hypothesis to which {\tt discriminate} is applicable; if + the current goal is of the form {\term$_1$} {\tt <>} {\term$_2$}, + this behaves as {\tt intro {\ident}; injection {\ident}}. + + \begin{ErrMsgs} + \item \errindex{No discriminable equalities} \\ + occurs when the goal does not verify the expected preconditions. + \end{ErrMsgs} +\end{Variants} + +\subsection{\tt injection {\term} +\label{injection} +\tacindex{injection} +\tacindex{einjection}} + +The {\tt injection} tactic is based on the fact that constructors of +inductive sets are injections. That means that if $c$ is a constructor +of an inductive set, and if $(c~\vec{t_1})$ and $(c~\vec{t_2})$ are two +terms that are equal then $~\vec{t_1}$ and $~\vec{t_2}$ are equal +too. + +If {\term} is a proof of a statement of conclusion + {\tt {\term$_1$} = {\term$_2$}}, +then {\tt injection} applies injectivity as deep as possible to +derive the equality of all the subterms of {\term$_1$} and {\term$_2$} +placed in the same positions. For example, from {\tt (S + (S n))=(S (S (S m))} we may derive {\tt n=(S m)}. To use this +tactic {\term$_1$} and {\term$_2$} should be elements of an inductive +set and they should be neither explicitly equal, nor structurally +different. We mean by this that, if {\tt n$_1$} and {\tt n$_2$} are +their respective normal forms, then: +\begin{itemize} +\item {\tt n$_1$} and {\tt n$_2$} should not be syntactically equal, +\item there must not exist any pair of subterms {\tt u} and {\tt w}, + {\tt u} subterm of {\tt n$_1$} and {\tt w} subterm of {\tt n$_2$} , + placed in the same positions and having different constructors as + head symbols. +\end{itemize} +If these conditions are satisfied, then, the tactic derives the +equality of all the subterms of {\term$_1$} and {\term$_2$} placed in +the same positions and puts them as antecedents of the current goal. + +\Example Consider the following goal: + +\begin{coq_example*} +Inductive list : Set := + | nil : list + | cons : nat -> list -> list. +Variable P : list -> Prop. +\end{coq_example*} +\begin{coq_eval} +Lemma ex : + forall (l:list) (n:nat), P nil -> cons n l = cons 0 nil -> P l. +intros l n H H0. +\end{coq_eval} +\begin{coq_example} +Show. +injection H0. +\end{coq_example} +\begin{coq_eval} +Abort. +\end{coq_eval} + +Beware that \texttt{injection} yields always an equality in a sigma type +whenever the injected object has a dependent type. + +\Rem There is a special case for dependent pairs. If we have a decidable +equality over the type of the first argument, then it is safe to do +the projection on the second one, and so {\tt injection} will work fine. +To define such an equality, you have to use the {\tt Scheme} command +(see \ref{Scheme}). + +\Rem If some quantified hypothesis of the goal is named {\ident}, then +{\tt injection {\ident}} first introduces the hypothesis in the local +context using \texttt{intros until \ident}. + +\begin{ErrMsgs} +\item \errindex{Not a projectable equality but a discriminable one} +\item \errindex{Nothing to do, it is an equality between convertible terms} +\item \errindex{Not a primitive equality} +\end{ErrMsgs} + +\begin{Variants} +\item \texttt{injection} \num{} + + This does the same thing as \texttt{intros until \num} followed by +\texttt{injection \ident} where {\ident} is the identifier for the last +introduced hypothesis. + +\item \texttt{injection} \term{} {\tt with} {\bindinglist} + + This does the same as \texttt{injection {\term}} but using + the given bindings to instantiate parameters or hypotheses of {\term}. + +\item \texttt{einjection} \num\\ + \texttt{einjection} \term{} \zeroone{{\tt with} {\bindinglist}} + + This works the same as {\tt injection} but if the type of {\term}, + or the type of the hypothesis referred to by {\num}, has uninstantiated + parameters, these parameters are left as existential variables. + +\item{\tt injection} + + If the current goal is of the form {\term$_1$} {\tt <>} {\term$_2$}, + this behaves as {\tt intro {\ident}; injection {\ident}}. + + \ErrMsg \errindex{goal does not satisfy the expected preconditions} + +\item \texttt{injection} \term{} \zeroone{{\tt with} {\bindinglist}} \texttt{as} \nelist{\intropattern}{}\\ +\texttt{injection} \num{} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ +\texttt{injection} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ +\texttt{einjection} \term{} \zeroone{{\tt with} {\bindinglist}} \texttt{as} \nelist{\intropattern}{}\\ +\texttt{einjection} \num{} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ +\texttt{einjection} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ +\tacindex{injection \ldots{} as} + +These variants apply \texttt{intros} \nelist{\intropattern}{} after +the call to \texttt{injection} or \texttt{einjection}. + +\end{Variants} + +\subsection{\tt simplify\_eq {\term} +\tacindex{simplify\_eq} +\tacindex{esimplify\_eq} +\label{simplify-eq}} + +Let {\term} be the proof of a statement of conclusion {\tt + {\term$_1$}={\term$_2$}}. If {\term$_1$} and +{\term$_2$} are structurally different (in the sense described for the +tactic {\tt discriminate}), then the tactic {\tt simplify\_eq} behaves as {\tt + discriminate {\term}}, otherwise it behaves as {\tt injection + {\term}}. + +\Rem If some quantified hypothesis of the goal is named {\ident}, then +{\tt simplify\_eq {\ident}} first introduces the hypothesis in the local +context using \texttt{intros until \ident}. + +\begin{Variants} +\item \texttt{simplify\_eq} \num + + This does the same thing as \texttt{intros until \num} then +\texttt{simplify\_eq \ident} where {\ident} is the identifier for the last +introduced hypothesis. + +\item \texttt{simplify\_eq} \term{} {\tt with} {\bindinglist} + + This does the same as \texttt{simplify\_eq {\term}} but using + the given bindings to instantiate parameters or hypotheses of {\term}. + +\item \texttt{esimplify\_eq} \num\\ + \texttt{esimplify\_eq} \term{} \zeroone{{\tt with} {\bindinglist}} + + This works the same as {\tt simplify\_eq} but if the type of {\term}, + or the type of the hypothesis referred to by {\num}, has uninstantiated + parameters, these parameters are left as existential variables. + +\item{\tt simplify\_eq} + +If the current goal has form $t_1\verb=<>=t_2$, it behaves as +\texttt{intro {\ident}; simplify\_eq {\ident}}. +\end{Variants} + +\subsection{\tt dependent rewrite -> {\ident} +\tacindex{dependent rewrite ->} +\label{dependent-rewrite}} + +This tactic applies to any goal. If \ident\ has type +\verb+(existT B a b)=(existT B a' b')+ +in the local context (i.e. each term of the +equality has a sigma type $\{ a:A~ \&~(B~a)\}$) this tactic rewrites +\verb+a+ into \verb+a'+ and \verb+b+ into \verb+b'+ in the current +goal. This tactic works even if $B$ is also a sigma type. This kind +of equalities between dependent pairs may be derived by the injection +and inversion tactics. + +\begin{Variants} +\item{\tt dependent rewrite <- {\ident}} +\tacindex{dependent rewrite <-} \\ +Analogous to {\tt dependent rewrite ->} but uses the equality from +right to left. +\end{Variants} + +\section{Inversion +\label{inversion}} + +\subsection{\tt inversion {\ident} +\tacindex{inversion}} + +Let the type of \ident~ in the local context be $(I~\vec{t})$, +where $I$ is a (co)inductive predicate. Then, +\texttt{inversion} applied to \ident~ derives for each possible +constructor $c_i$ of $(I~\vec{t})$, {\bf all} the necessary +conditions that should hold for the instance $(I~\vec{t})$ to be +proved by $c_i$. + +\Rem If {\ident} does not denote a hypothesis in the local context +but refers to a hypothesis quantified in the goal, then the +latter is first introduced in the local context using +\texttt{intros until \ident}. + +\begin{Variants} +\item \texttt{inversion} \num + + This does the same thing as \texttt{intros until \num} then + \texttt{inversion \ident} where {\ident} is the identifier for the + last introduced hypothesis. + +\item \tacindex{inversion\_clear} \texttt{inversion\_clear} \ident + + This behaves as \texttt{inversion} and then erases \ident~ from the + context. + +\item \tacindex{inversion \dots\ as} \texttt{inversion} {\ident} \texttt{as} {\intropattern} + + This behaves as \texttt{inversion} but using names in + {\intropattern} for naming hypotheses. The {\intropattern} must have + the form {\tt [} $p_{11}$ \ldots $p_{1n_1}$ {\tt |} {\ldots} {\tt |} + $p_{m1}$ \ldots $p_{mn_m}$ {\tt ]} with $m$ being the number of + constructors of the type of {\ident}. Be careful that the list must + be of length $m$ even if {\tt inversion} discards some cases (which + is precisely one of its roles): for the discarded cases, just use an + empty list (i.e. $n_i=0$). + + The arguments of the $i^{th}$ constructor and the + equalities that {\tt inversion} introduces in the context of the + goal corresponding to the $i^{th}$ constructor, if it exists, get + their names from the list $p_{i1}$ \ldots $p_{in_i}$ in order. If + there are not enough names, {\tt induction} invents names for the + remaining variables to introduce. In case an equation splits into + several equations (because {\tt inversion} applies {\tt injection} + on the equalities it generates), the corresponding name $p_{ij}$ in + the list must be replaced by a sublist of the form {\tt [$p_{ij1}$ + \ldots $p_{ijq}$]} (or, equivalently, {\tt ($p_{ij1}$, + \ldots, $p_{ijq}$)}) where $q$ is the number of subequalities + obtained from splitting the original equation. Here is an example. + +\begin{coq_eval} +Require Import List. +\end{coq_eval} + +\begin{coq_example} +Inductive contains0 : list nat -> Prop := + | in_hd : forall l, contains0 (0 :: l) + | in_tl : forall l b, contains0 l -> contains0 (b :: l). +Goal forall l:list nat, contains0 (1 :: l) -> contains0 l. +intros l H; inversion H as [ | l' p Hl' [Heqp Heql'] ]. +\end{coq_example} + +\begin{coq_eval} +Abort. +\end{coq_eval} + +\item \texttt{inversion} {\num} {\tt as} {\intropattern} + + This allows to name the hypotheses introduced by + \texttt{inversion} {\num} in the context. + +\item \tacindex{inversion\_cleardots\ as} \texttt{inversion\_clear} + {\ident} {\tt as} {\intropattern} + + This allows to name the hypotheses introduced by + \texttt{inversion\_clear} in the context. + +\item \tacindex{inversion \dots\ in} \texttt{inversion } {\ident} + \texttt{in} \ident$_1$ \dots\ \ident$_n$ + + Let \ident$_1$ \dots\ \ident$_n$, be identifiers in the local context. This + tactic behaves as generalizing \ident$_1$ \dots\ \ident$_n$, and + then performing \texttt{inversion}. + +\item \tacindex{inversion \dots\ as \dots\ in} \texttt{inversion } + {\ident} {\tt as} {\intropattern} \texttt{in} \ident$_1$ \dots\ + \ident$_n$ + + This allows to name the hypotheses introduced in the context by + \texttt{inversion} {\ident} \texttt{in} \ident$_1$ \dots\ + \ident$_n$. + +\item \tacindex{inversion\_clear \dots\ in} \texttt{inversion\_clear} + {\ident} \texttt{in} \ident$_1$ \ldots \ident$_n$ + + Let \ident$_1$ \dots\ \ident$_n$, be identifiers in the local context. This + tactic behaves as generalizing \ident$_1$ \dots\ \ident$_n$, and + then performing {\tt inversion\_clear}. + +\item \tacindex{inversion\_clear \dots\ as \dots\ in} + \texttt{inversion\_clear} {\ident} \texttt{as} {\intropattern} + \texttt{in} \ident$_1$ \ldots \ident$_n$ + + This allows to name the hypotheses introduced in the context by + \texttt{inversion\_clear} {\ident} \texttt{in} \ident$_1$ \ldots + \ident$_n$. + +\item \tacindex{dependent inversion} \texttt{dependent inversion} + {\ident} + + That must be used when \ident\ appears in the current goal. It acts + like \texttt{inversion} and then substitutes \ident\ for the + corresponding term in the goal. + +\item \tacindex{dependent inversion \dots\ as } \texttt{dependent + inversion} {\ident} \texttt{as} {\intropattern} + + This allows to name the hypotheses introduced in the context by + \texttt{dependent inversion} {\ident}. + +\item \tacindex{dependent inversion\_clear} \texttt{dependent + inversion\_clear} {\ident} + + Like \texttt{dependent inversion}, except that {\ident} is cleared + from the local context. + +\item \tacindex{dependent inversion\_clear \dots\ as} + \texttt{dependent inversion\_clear} {\ident}\texttt{as} {\intropattern} + + This allows to name the hypotheses introduced in the context by + \texttt{dependent inversion\_clear} {\ident}. + +\item \tacindex{dependent inversion \dots\ with} \texttt{dependent + inversion } {\ident} \texttt{ with } \term + + This variant allows you to specify the generalization of the goal. It + is useful when the system fails to generalize the goal automatically. If + {\ident} has type $(I~\vec{t})$ and $I$ has type + $forall (\vec{x}:\vec{T}), s$, then \term~ must be of type + $I:forall (\vec{x}:\vec{T}), I~\vec{x}\to s'$ where $s'$ is the + type of the goal. + +\item \tacindex{dependent inversion \dots\ as \dots\ with} + \texttt{dependent inversion } {\ident} \texttt{as} {\intropattern} + \texttt{ with } \term + + This allows to name the hypotheses introduced in the context by + \texttt{dependent inversion } {\ident} \texttt{ with } \term. + +\item \tacindex{dependent inversion\_clear \dots\ with} + \texttt{dependent inversion\_clear } {\ident} \texttt{ with } \term + + Like \texttt{dependent inversion \dots\ with} but clears {\ident} from + the local context. + +\item \tacindex{dependent inversion\_clear \dots\ as \dots\ with} + \texttt{dependent inversion\_clear } {\ident} \texttt{as} + {\intropattern} \texttt{ with } \term + + This allows to name the hypotheses introduced in the context by + \texttt{dependent inversion\_clear } {\ident} \texttt{ with } \term. + +\item \tacindex{simple inversion} \texttt{simple inversion} {\ident} + + It is a very primitive inversion tactic that derives all the necessary + equalities but it does not simplify the constraints as + \texttt{inversion} does. + +\item \tacindex{simple inversion \dots\ as} \texttt{simple inversion} + {\ident} \texttt{as} {\intropattern} + + This allows to name the hypotheses introduced in the context by + \texttt{simple inversion}. + +\item \tacindex{inversion \dots\ using} \texttt{inversion} \ident + \texttt{ using} \ident$'$ + + Let {\ident} have type $(I~\vec{t})$ ($I$ an inductive + predicate) in the local context, and \ident$'$ be a (dependent) inversion + lemma. Then, this tactic refines the current goal with the specified + lemma. + +\item \tacindex{inversion \dots\ using \dots\ in} \texttt{inversion} + {\ident} \texttt{using} \ident$'$ \texttt{in} \ident$_1$\dots\ \ident$_n$ + + This tactic behaves as generalizing \ident$_1$\dots\ \ident$_n$, + then doing \texttt{inversion} {\ident} \texttt{using} \ident$'$. + +\end{Variants} + +\SeeAlso~\ref{inversion-examples} for detailed examples + +\subsection{\tt Derive Inversion {\ident} with + ${\tt forall (}\vec{x}{\tt :}\vec{T}{\tt),} I~\vec{t}$ Sort \sort +\label{Derive-Inversion} +\comindex{Derive Inversion}} + +This command generates an inversion principle for the +\texttt{inversion \dots\ using} tactic. +Let $I$ be an inductive predicate and $\vec{x}$ the variables +occurring in $\vec{t}$. This command generates and stocks the +inversion lemma for the sort \sort~ corresponding to the instance +$forall (\vec{x}:\vec{T}), I~\vec{t}$ with the name {\ident} in the {\bf +global} environment. When applied it is equivalent to have inverted +the instance with the tactic {\tt inversion}. + +\begin{Variants} +\item \texttt{Derive Inversion\_clear} {\ident} \texttt{with} + \comindex{Derive Inversion\_clear} + $forall (\vec{x}:\vec{T}), I~\vec{t}$ \texttt{Sort} \sort~ \\ + \index{Derive Inversion\_clear \dots\ with} + When applied it is equivalent to having + inverted the instance with the tactic \texttt{inversion} + replaced by the tactic \texttt{inversion\_clear}. +\item \texttt{Derive Dependent Inversion} {\ident} \texttt{with} + $forall (\vec{x}:\vec{T}), I~\vec{t}$ \texttt{Sort} \sort~\\ + \comindex{Derive Dependent Inversion} + When applied it is equivalent to having + inverted the instance with the tactic \texttt{dependent inversion}. +\item \texttt{Derive Dependent Inversion\_clear} {\ident} \texttt{with} + $forall (\vec{x}:\vec{T}), I~\vec{t}$ \texttt{Sort} \sort~\\ + \comindex{Derive Dependent Inversion\_clear} + When applied it is equivalent to having + inverted the instance with the tactic \texttt{dependent inversion\_clear}. +\end{Variants} + +\SeeAlso \ref{inversion-examples} for examples + + + +\subsection[\tt functional inversion \ident]{\tt functional inversion \ident\label{sec:functional-inversion}} + +\texttt{functional inversion} is a \emph{highly} experimental tactic +which performs inversion on hypothesis \ident\ of the form +\texttt{\qualid\ \term$_1$\dots\term$_n$\ = \term} or \texttt{\term\ = + \qualid\ \term$_1$\dots\term$_n$} where \qualid\ must have been +defined using \texttt{Function} (see Section~\ref{Function}). + +\begin{ErrMsgs} +\item \errindex{Hypothesis {\ident} must contain at least one Function} +\item \errindex{Cannot find inversion information for hypothesis \ident} + This error may be raised when some inversion lemma failed to be + generated by Function. +\end{ErrMsgs} + +\begin{Variants} +\item {\tt functional inversion \num} + + This does the same thing as \texttt{intros until \num} then + \texttt{functional inversion \ident} where {\ident} is the + identifier for the last introduced hypothesis. +\item {\tt functional inversion \ident\ \qualid}\\ + {\tt functional inversion \num\ \qualid} + + In case the hypothesis {\ident} (or {\num}) has a type of the form + \texttt{\qualid$_1$\ \term$_1$\dots\term$_n$\ =\ \qualid$_2$\ + \term$_{n+1}$\dots\term$_{n+m}$} where \qualid$_1$ and \qualid$_2$ + are valid candidates to functional inversion, this variant allows to + choose which must be inverted. +\end{Variants} + + + +\subsection{\tt quote \ident +\tacindex{quote} +\index{2-level approach}} + +This kind of inversion has nothing to do with the tactic +\texttt{inversion} above. This tactic does \texttt{change (\ident\ + t)}, where \texttt{t} is a term built in order to ensure the +convertibility. In other words, it does inversion of the function +\ident. This function must be a fixpoint on a simple recursive +datatype: see~\ref{quote-examples} for the full details. + +\begin{ErrMsgs} +\item \errindex{quote: not a simple fixpoint}\\ + Happens when \texttt{quote} is not able to perform inversion properly. +\end{ErrMsgs} + +\begin{Variants} +\item \texttt{quote {\ident} [ \ident$_1$ \dots \ident$_n$ ]}\\ + All terms that are built only with \ident$_1$ \dots \ident$_n$ will be + considered by \texttt{quote} as constants rather than variables. +\end{Variants} + +% En attente d'un moyen de valoriser les fichiers de demos +% \SeeAlso file \texttt{theories/DEMOS/DemoQuote.v} in the distribution + +\section[Classical tactics]{Classical tactics\label{ClassicalTactics}} + +In order to ease the proving process, when the {\tt Classical} module is loaded. A few more tactics are available. Make sure to load the module using the \texttt{Require Import} command. + +\subsection{{\tt classical\_left, classical\_right} \tacindex{classical\_left} \tacindex{classical\_right}} + +The tactics \texttt{classical\_left} and \texttt{classical\_right} are the analog of the \texttt{left} and \texttt{right} but using classical logic. They can only be used for disjunctions. +Use \texttt{classical\_left} to prove the left part of the disjunction with the assumption that the negation of right part holds. +Use \texttt{classical\_right} to prove the right part of the disjunction with the assumption that the negation of left part holds. + +\section{Automatizing +\label{Automatizing}} + +\subsection{\tt auto +\label{auto} +\tacindex{auto}} + +This tactic implements a Prolog-like resolution procedure to solve the +current goal. It first tries to solve the goal using the {\tt + assumption} tactic, then it reduces the goal to an atomic one using +{\tt intros} and introducing the newly generated hypotheses as hints. +Then it looks at the list of tactics associated to the head symbol of +the goal and tries to apply one of them (starting from the tactics +with lower cost). This process is recursively applied to the generated +subgoals. + +By default, \texttt{auto} only uses the hypotheses of the current goal and the +hints of the database named {\tt core}. + +\begin{Variants} + +\item {\tt auto \num} + + Forces the search depth to be \num. The maximal search depth is 5 by + default. + +\item {\tt auto with \ident$_1$ \dots\ \ident$_n$} + + Uses the hint databases $\ident_1$ \dots\ $\ident_n$ in addition to + the database {\tt core}. See Section~\ref{Hints-databases} for the + list of pre-defined databases and the way to create or extend a + database. This option can be combined with the previous one. + +\item {\tt auto with *} + + Uses all existing hint databases, minus the special database + {\tt v62}. See Section~\ref{Hints-databases} + +\item \texttt{auto using \nterm{lemma}$_1$ , \ldots , \nterm{lemma}$_n$} + + Uses \nterm{lemma}$_1$, \ldots, \nterm{lemma}$_n$ in addition to + hints (can be combined with the \texttt{with \ident} option). If + $lemma_i$ is an inductive type, it is the collection of its + constructors which is added as hints. + +\item \texttt{auto using \nterm{lemma}$_1$ , \ldots , \nterm{lemma}$_n$ with \ident$_1$ \dots\ \ident$_n$} + + This combines the effects of the {\tt using} and {\tt with} options. + +\item {\tt trivial}\tacindex{trivial} + + This tactic is a restriction of {\tt auto} that is not recursive and + tries only hints which cost 0. Typically it solves trivial + equalities like $X=X$. + +\item \texttt{trivial with \ident$_1$ \dots\ \ident$_n$} + +\item \texttt{trivial with *} + +\end{Variants} + +\Rem {\tt auto} either solves completely the goal or else leaves it +intact. \texttt{auto} and \texttt{trivial} never fail. + +\SeeAlso Section~\ref{Hints-databases} + +\subsection{\tt eauto +\tacindex{eauto} +\label{eauto}} + +This tactic generalizes {\tt auto}. In contrast with +the latter, {\tt eauto} uses unification of the goal +against the hints rather than pattern-matching +(in other words, it uses {\tt eapply} instead of +{\tt apply}). +As a consequence, {\tt eauto} can solve such a goal: + +\begin{coq_example} +Hint Resolve ex_intro. +Goal forall P:nat -> Prop, P 0 -> exists n, P n. +eauto. +\end{coq_example} +\begin{coq_eval} +Abort. +\end{coq_eval} + +Note that {\tt ex\_intro} should be declared as an +hint. + +\SeeAlso Section~\ref{Hints-databases} + +\subsection{\tt autounfold with \ident$_1$ \dots\ \ident$_n$ +\tacindex{autounfold} +\label{autounfold}} + +This tactic unfolds constants that were declared through a {\tt Hint + Unfold} in the given databases. + +\begin{Variants} +\item {\tt autounfold with \ident$_1$ \dots\ \ident$_n$ in \textit{clause}} + + Perform the unfolding in the given clause. + +\item {\tt autounfold with *} + + Uses the unfold hints declared in all the hint databases. +\end{Variants} + + +% EXISTE ENCORE ? +% +% \subsection{\tt Prolog [ \term$_1$ \dots\ \term$_n$ ] \num} +% \tacindex{Prolog}\label{Prolog} +% This tactic, implemented by Chet Murthy, is based upon the concept of +% existential variables of Gilles Dowek, stating that resolution is a +% kind of unification. It tries to solve the current goal using the {\tt +% Assumption} tactic, the {\tt intro} tactic, and applying hypotheses +% of the local context and terms of the given list {\tt [ \term$_1$ +% \dots\ \term$_n$\ ]}. It is more powerful than {\tt auto} since it +% may apply to any theorem, even those of the form {\tt (x:A)(P x) -> Q} +% where {\tt x} does not appear free in {\tt Q}. The maximal search +% depth is {\tt \num}. + +% \begin{ErrMsgs} +% \item \errindex{Prolog failed}\\ +% The Prolog tactic was not able to prove the subgoal. +% \end{ErrMsgs} + +\subsection{\tt tauto +\tacindex{tauto} +\label{tauto}} + +This tactic implements a decision procedure for intuitionistic propositional +calculus based on the contraction-free sequent calculi LJT* of Roy Dyckhoff +\cite{Dyc92}. Note that {\tt tauto} succeeds on any instance of an +intuitionistic tautological proposition. {\tt tauto} unfolds negations +and logical equivalence but does not unfold any other definition. + +The following goal can be proved by {\tt tauto} whereas {\tt auto} +would fail: + +\begin{coq_example} +Goal forall (x:nat) (P:nat -> Prop), x = 0 \/ P x -> x <> 0 -> P x. + intros. + tauto. +\end{coq_example} +\begin{coq_eval} +Abort. +\end{coq_eval} + +Moreover, if it has nothing else to do, {\tt tauto} performs +introductions. Therefore, the use of {\tt intros} in the previous +proof is unnecessary. {\tt tauto} can for instance prove the +following: +\begin{coq_example} +(* auto would fail *) +Goal forall (A:Prop) (P:nat -> Prop), + A \/ (forall x:nat, ~ A -> P x) -> forall x:nat, ~ A -> P x. + + tauto. +\end{coq_example} +\begin{coq_eval} +Abort. +\end{coq_eval} + +\Rem In contrast, {\tt tauto} cannot solve the following goal + +\begin{coq_example*} +Goal forall (A:Prop) (P:nat -> Prop), + A \/ (forall x:nat, ~ A -> P x) -> forall x:nat, ~ ~ (A \/ P x). +\end{coq_example*} +\begin{coq_eval} +Abort. +\end{coq_eval} + +because \verb=(forall x:nat, ~ A -> P x)= cannot be treated as atomic and an +instantiation of \verb=x= is necessary. + +\subsection{\tt intuition {\tac} +\tacindex{intuition} +\label{intuition}} + +The tactic \texttt{intuition} takes advantage of the search-tree built +by the decision procedure involved in the tactic {\tt tauto}. It uses +this information to generate a set of subgoals equivalent to the +original one (but simpler than it) and applies the tactic +{\tac} to them \cite{Mun94}. If this tactic fails on some goals then +{\tt intuition} fails. In fact, {\tt tauto} is simply {\tt intuition + fail}. + +For instance, the tactic {\tt intuition auto} applied to the goal +\begin{verbatim} +(forall (x:nat), P x)/\B -> (forall (y:nat),P y)/\ P O \/B/\ P O +\end{verbatim} +internally replaces it by the equivalent one: +\begin{verbatim} +(forall (x:nat), P x), B |- P O +\end{verbatim} +and then uses {\tt auto} which completes the proof. + +Originally due to C{\'e}sar~Mu{\~n}oz, these tactics ({\tt tauto} and {\tt intuition}) +have been completely re-engineered by David~Delahaye using mainly the tactic +language (see Chapter~\ref{TacticLanguage}). The code is now much shorter and +a significant increase in performance has been noticed. The general behavior +with respect to dependent types, unfolding and introductions has +slightly changed to get clearer semantics. This may lead to some +incompatibilities. + +\begin{Variants} +\item {\tt intuition}\\ + Is equivalent to {\tt intuition auto with *}. +\end{Variants} + +% En attente d'un moyen de valoriser les fichiers de demos +%\SeeAlso file \texttt{contrib/Rocq/DEMOS/Demo\_tauto.v} + + +\subsection{\tt rtauto +\tacindex{rtauto} +\label{rtauto}} + +The {\tt rtauto} tactic solves propositional tautologies similarly to what {\tt tauto} does. The main difference is that the proof term is built using a reflection scheme applied to a sequent calculus proof of the goal. The search procedure is also implemented using a different technique. + +Users should be aware that this difference may result in faster proof-search but slower proof-checking, and {\tt rtauto} might not solve goals that {\tt tauto} would be able to solve (e.g. goals involving universal quantifiers). + +\subsection{{\tt firstorder} +\tacindex{firstorder} +\label{firstorder}} + +The tactic \texttt{firstorder} is an {\it experimental} extension of +\texttt{tauto} to +first-order reasoning, written by Pierre Corbineau. +It is not restricted to usual logical connectives but +instead may reason about any first-order class inductive definition. + +\begin{Variants} + \item {\tt firstorder {\tac}} + \tacindex{firstorder {\tac}} + + Tries to solve the goal with {\tac} when no logical rule may apply. + + \item {\tt firstorder with \ident$_1$ \dots\ \ident$_n$ } + \tacindex{firstorder with} + + Adds lemmas \ident$_1$ \dots\ \ident$_n$ to the proof-search + environment. + + \item {\tt firstorder using {\qualid}$_1$ , \dots\ , {\qualid}$_n$ } + \tacindex{firstorder using} + + Adds lemmas in {\tt auto} hints bases {\qualid}$_1$ \dots\ {\qualid}$_n$ + to the proof-search environment. If {\qualid}$_i$ refers to an inductive + type, it is the collection of its constructors which is added as hints. + +\item \texttt{firstorder using {\qualid}$_1$ , \dots\ , {\qualid}$_n$ with \ident$_1$ \dots\ \ident$_n$} + + This combines the effects of the {\tt using} and {\tt with} options. + +\end{Variants} + +Proof-search is bounded by a depth parameter which can be set by typing the +{\nobreak \tt Set Firstorder Depth $n$} \comindex{Set Firstorder Depth} +vernacular command. + +%% \subsection{{\tt jp} {\em (Jprover)} +%% \tacindex{jp} +%% \label{jprover}} + +%% The tactic \texttt{jp}, due to Huang Guan-Shieng, is an experimental +%% port of the {\em Jprover}\cite{SLKN01} semi-decision procedure for +%% first-order intuitionistic logic implemented in {\em +%% NuPRL}\cite{Kre02}. + +%% The tactic \texttt{jp}, due to Huang Guan-Shieng, is an {\it +%% experimental} port of the {\em Jprover}\cite{SLKN01} semi-decision +%% procedure for first-order intuitionistic logic implemented in {\em +%% NuPRL}\cite{Kre02}. + +%% Search may optionnaly be bounded by a multiplicity parameter +%% indicating how many (at most) copies of a formula may be used in +%% the proof process, its absence may lead to non-termination of the tactic. + +%% %\begin{coq_eval} +%% %Variable S:Set. +%% %Variables P Q:S->Prop. +%% %Variable f:S->S. +%% %\end{coq_eval} + +%% %\begin{coq_example*} +%% %Lemma example: (exists x |P x\/Q x)->(exists x |P x)\/(exists x |Q x). +%% %jp. +%% %Qed. + +%% %Lemma example2: (forall x ,P x->P (f x))->forall x,P x->P (f(f x)). +%% %jp. +%% %Qed. +%% %\end{coq_example*} + +%% \begin{Variants} +%% \item {\tt jp $n$}\\ +%% \tacindex{jp $n$} +%% Tries the {\em Jprover} procedure with multiplicities up to $n$, +%% starting from 1. +%% \item {\tt jp}\\ +%% Tries the {\em Jprover} procedure without multiplicity bound, +%% possibly running forever. +%% \end{Variants} + +%% \begin{ErrMsgs} +%% \item \errindex{multiplicity limit reached}\\ +%% The procedure tried all multiplicities below the limit and +%% failed. Goal might be solved by increasing the multiplicity limit. +%% \item \errindex{formula is not provable}\\ +%% The procedure determined that goal was not provable in +%% intuitionistic first-order logic, no matter how big the +%% multiplicity is. +%% \end{ErrMsgs} + + +% \subsection[\tt Linear]{\tt Linear\tacindex{Linear}\label{Linear}} +% The tactic \texttt{Linear}, due to Jean-Christophe Filli{\^a}atre +% \cite{Fil94}, implements a decision procedure for {\em Direct +% Predicate Calculus}, that is first-order Gentzen's Sequent Calculus +% without contraction rules \cite{KeWe84,BeKe92}. Intuitively, a +% first-order goal is provable in Direct Predicate Calculus if it can be +% proved using each hypothesis at most once. + +% Unlike the previous tactics, the \texttt{Linear} tactic does not belong +% to the initial state of the system, and it must be loaded explicitly +% with the command + +% \begin{coq_example*} +% Require Linear. +% \end{coq_example*} + +% For instance, assuming that \texttt{even} and \texttt{odd} are two +% predicates on natural numbers, and \texttt{a} of type \texttt{nat}, the +% tactic \texttt{Linear} solves the following goal + +% \begin{coq_eval} +% Variables even,odd : nat -> Prop. +% Variable a:nat. +% \end{coq_eval} + +% \begin{coq_example*} +% Lemma example : (even a) +% -> ((x:nat)((even x)->(odd (S x)))) +% -> (EX y | (odd y)). +% \end{coq_example*} + +% You can find examples of the use of \texttt{Linear} in +% \texttt{theories/DEMOS/DemoLinear.v}. +% \begin{coq_eval} +% Abort. +% \end{coq_eval} + +% \begin{Variants} +% \item {\tt Linear with \ident$_1$ \dots\ \ident$_n$}\\ +% \tacindex{Linear with} +% Is equivalent to apply first {\tt generalize \ident$_1$ \dots +% \ident$_n$} (see Section~\ref{generalize}) then the \texttt{Linear} +% tactic. So one can use axioms, lemmas or hypotheses of the local +% context with \texttt{Linear} in this way. +% \end{Variants} + +% \begin{ErrMsgs} +% \item \errindex{Not provable in Direct Predicate Calculus} +% \item \errindex{Found $n$ classical proof(s) but no intuitionistic one}\\ +% The decision procedure looks actually for classical proofs of the +% goals, and then checks that they are intuitionistic. In that case, +% classical proofs have been found, which do not correspond to +% intuitionistic ones. +% \end{ErrMsgs} + +\subsection{\tt congruence +\tacindex{congruence} +\label{congruence}} + +The tactic {\tt congruence}, by Pierre Corbineau, implements the standard Nelson and Oppen +congruence closure algorithm, which is a decision procedure for ground +equalities with uninterpreted symbols. It also include the constructor theory +(see \ref{injection} and \ref{discriminate}). +If the goal is a non-quantified equality, {\tt congruence} tries to +prove it with non-quantified equalities in the context. Otherwise it +tries to infer a discriminable equality from those in the context. Alternatively, congruence tries to prove that a hypothesis is equal to the goal or to the negation of another hypothesis. + +{\tt congruence} is also able to take advantage of hypotheses stating quantified equalities, you have to provide a bound for the number of extra equalities generated that way. Please note that one of the members of the equality must contain all the quantified variables in order for {\tt congruence} to match against it. + +\begin{coq_eval} +Reset Initial. +Variable A:Set. +Variables a b:A. +Variable f:A->A. +Variable g:A->A->A. +\end{coq_eval} + +\begin{coq_example} +Theorem T: + a=(f a) -> (g b (f a))=(f (f a)) -> (g a b)=(f (g b a)) -> (g a b)=a. +intros. +congruence. +\end{coq_example} + +\begin{coq_eval} +Reset Initial. +Variable A:Set. +Variables a c d:A. +Variable f:A->A*A. +\end{coq_eval} + +\begin{coq_example} +Theorem inj : f = pair a -> Some (f c) = Some (f d) -> c=d. +intros. +congruence. +\end{coq_example} + +\begin{Variants} + \item {\tt congruence {\sl n}}\\ + Tries to add at most {\tt \sl n} instances of hypotheses stating quantified equalities to the problem in order to solve it. A bigger value of {\tt \sl n} does not make success slower, only failure. You might consider adding some lemmas as hypotheses using {\tt assert} in order for congruence to use them. + +\end{Variants} + +\begin{Variants} +\item {\tt congruence with \term$_1$ \dots\ \term$_n$}\\ + Adds {\tt \term$_1$ \dots\ \term$_n$} to the pool of terms used by + {\tt congruence}. This helps in case you have partially applied + constructors in your goal. +\end{Variants} + +\begin{ErrMsgs} + \item \errindex{I don't know how to handle dependent equality} \\ + The decision procedure managed to find a proof of the goal or of + a discriminable equality but this proof couldn't be built in {\Coq} + because of dependently-typed functions. + \item \errindex{I couldn't solve goal} \\ + The decision procedure didn't find any way to solve the goal. + \item \errindex{Goal is solvable by congruence but some arguments are missing. Try "congruence with \dots", replacing metavariables by arbitrary terms.} \\ + The decision procedure could solve the goal with the provision + that additional arguments are supplied for some partially applied + constructors. Any term of an appropriate type will allow the + tactic to successfully solve the goal. Those additional arguments + can be given to {\tt congruence} by filling in the holes in the + terms given in the error message, using the {\tt with} variant + described above. +\end{ErrMsgs} + +\subsection{\tt omega +\tacindex{omega} +\label{omega}} + +The tactic \texttt{omega}, due to Pierre Cr{\'e}gut, +is an automatic decision procedure for Presburger +arithmetic. It solves quantifier-free +formulas built with \verb|~|, \verb|\/|, \verb|/\|, +\verb|->| on top of equalities, inequalities and disequalities on +both the type \texttt{nat} of natural numbers and \texttt{Z} of binary +integers. This tactic must be loaded by the command \texttt{Require Import + Omega}. See the additional documentation about \texttt{omega} +(see Chapter~\ref{OmegaChapter}). + +\subsection{{\tt ring} and {\tt ring\_simplify \term$_1$ \dots\ \term$_n$} +\tacindex{ring} +\tacindex{ring\_simplify} +\comindex{Add Ring}} + +The {\tt ring} tactic solves equations upon polynomial expressions of +a ring (or semi-ring) structure. It proceeds by normalizing both hand +sides of the equation (w.r.t. associativity, commutativity and +distributivity, constant propagation) and comparing syntactically the +results. + +{\tt ring\_simplify} applies the normalization procedure described +above to the terms given. The tactic then replaces all occurrences of +the terms given in the conclusion of the goal by their normal +forms. If no term is given, then the conclusion should be an equation +and both hand sides are normalized. + +See Chapter~\ref{ring} for more information on the tactic and how to +declare new ring structures. + +\subsection{{\tt field}, {\tt field\_simplify \term$_1$\dots\ \term$_n$} + and {\tt field\_simplify\_eq} +\tacindex{field} +\tacindex{field\_simplify} +\tacindex{field\_simplify\_eq} +\comindex{Add Field}} + +The {\tt field} tactic is built on the same ideas as {\tt ring}: this +is a reflexive tactic that solves or simplifies equations in a field +structure. The main idea is to reduce a field expression (which is an +extension of ring expressions with the inverse and division +operations) to a fraction made of two polynomial expressions. + +Tactic {\tt field} is used to solve subgoals, whereas {\tt + field\_simplify \term$_1$\dots\term$_n$} replaces the provided terms +by their reduced fraction. {\tt field\_simplify\_eq} applies when the +conclusion is an equation: it simplifies both hand sides and multiplies +so as to cancel denominators. So it produces an equation without +division nor inverse. + +All of these 3 tactics may generate a subgoal in order to prove that +denominators are different from zero. + +See Chapter~\ref{ring} for more information on the tactic and how to +declare new field structures. + +\Example +\begin{coq_example*} +Require Import Reals. +Goal forall x y:R, + (x * y > 0)%R -> + (x * (1 / x + x / (x + y)))%R = + ((- 1 / y) * y * (- x * (x / (x + y)) - 1))%R. +\end{coq_example*} + +\begin{coq_example} +intros; field. +\end{coq_example} + +\begin{coq_eval} +Reset Initial. +\end{coq_eval} + +\SeeAlso file {\tt plugins/setoid\_ring/RealField.v} for an example of instantiation,\\ +\phantom{\SeeAlso}theory {\tt theories/Reals} for many examples of use of {\tt +field}. + +\subsection{\tt fourier +\tacindex{fourier}} + +This tactic written by Lo{\"\i}c Pottier solves linear inequalities on +real numbers using Fourier's method~\cite{Fourier}. This tactic must +be loaded by {\tt Require Import Fourier}. + +\Example +\begin{coq_example*} +Require Import Reals. +Require Import Fourier. +Goal forall x y:R, (x < y)%R -> (y + 1 >= x - 1)%R. +\end{coq_example*} + +\begin{coq_example} +intros; fourier. +\end{coq_example} + +\begin{coq_eval} +Reset Initial. +\end{coq_eval} + +\subsection{\tt autorewrite with \ident$_1$ \dots \ident$_n$. +\label{tactic:autorewrite} +\tacindex{autorewrite}} + +This tactic \footnote{The behavior of this tactic has much changed compared to +the versions available in the previous distributions (V6). This may cause +significant changes in your theories to obtain the same result. As a drawback +of the re-engineering of the code, this tactic has also been completely revised +to get a very compact and readable version.} carries out rewritings according +the rewriting rule bases {\tt \ident$_1$ \dots \ident$_n$}. + +Each rewriting rule of a base \ident$_i$ is applied to the main subgoal until +it fails. Once all the rules have been processed, if the main subgoal has +progressed (e.g., if it is distinct from the initial main goal) then the rules +of this base are processed again. If the main subgoal has not progressed then +the next base is processed. For the bases, the behavior is exactly similar to +the processing of the rewriting rules. + +The rewriting rule bases are built with the {\tt Hint~Rewrite} vernacular +command. + +\Warning{} This tactic may loop if you build non terminating rewriting systems. + +\begin{Variant} +\item {\tt autorewrite with \ident$_1$ \dots \ident$_n$ using \tac}\\ +Performs, in the same way, all the rewritings of the bases {\tt \ident$_1$ $...$ +\ident$_n$} applying {\tt \tac} to the main subgoal after each rewriting step. + +\item \texttt{autorewrite with {\ident$_1$} \dots \ident$_n$ in {\qualid}} + + Performs all the rewritings in hypothesis {\qualid}. +\item \texttt{autorewrite with {\ident$_1$} \dots \ident$_n$ in {\qualid} using \tac} + + Performs all the rewritings in hypothesis {\qualid} applying {\tt + \tac} to the main subgoal after each rewriting step. + +\item \texttt{autorewrite with {\ident$_1$} \dots \ident$_n$ in \textit{clause}} + Performs all the rewritings in the clause \textit{clause}. \\ + The \textit{clause} argument must not contain any \texttt{type of} nor \texttt{value of}. + +\end{Variant} + +\SeeAlso Section~\ref{HintRewrite} for feeding the database of lemmas used by {\tt autorewrite}. + +\SeeAlso Section~\ref{autorewrite-example} for examples showing the use of +this tactic. + +% En attente d'un moyen de valoriser les fichiers de demos +%\SeeAlso file \texttt{contrib/Rocq/DEMOS/Demo\_AutoRewrite.v} + +\section{Controlling automation} + +\subsection{The hints databases for {\tt auto} and {\tt eauto} +\index{Hints databases} +\label{Hints-databases} +\comindex{Hint}} + +The hints for \texttt{auto} and \texttt{eauto} are stored in +databases. Each database maps head symbols to a list of hints. One can +use the command \texttt{Print Hint \ident} to display the hints +associated to the head symbol \ident{} (see \ref{PrintHint}). Each +hint has a cost that is an nonnegative integer, and an optional pattern. +The hints with lower cost are tried first. A hint is tried by +\texttt{auto} when the conclusion of the current goal +matches its pattern or when it has no pattern. + +\subsubsection*{Creating Hint databases + \label{CreateHintDb}\comindex{CreateHintDb}} + +One can optionally declare a hint database using the command +\texttt{Create HintDb}. If a hint is added to an unknown database, it +will be automatically created. + +\medskip +\texttt{Create HintDb} {\ident} [\texttt{discriminated}] +\medskip + +This command creates a new database named \ident. +The database is implemented by a Discrimination Tree (DT) that serves as +an index of all the lemmas. The DT can use transparency information to decide +if a constant should be indexed or not (c.f. \ref{HintTransparency}), +making the retrieval more efficient. +The legacy implementation (the default one for new databases) uses the +DT only on goals without existentials (i.e., auto goals), for non-Immediate +hints and do not make use of transparency hints, putting more work on the +unification that is run after retrieval (it keeps a list of the lemmas +in case the DT is not used). The new implementation enabled by +the {\tt discriminated} option makes use of DTs in all cases and takes +transparency information into account. However, the order in which hints +are retrieved from the DT may differ from the order in which they were +inserted, making this implementation observationaly different from the +legacy one. + +\begin{Variants} +\item\texttt{Local Hint} \textsl{hint\_definition} \texttt{:} + \ident$_1$ \ldots\ \ident$_n$ + + This is used to declare a hint database that must not be exported to the other + modules that require and import the current module. Inside a + section, the option {\tt Local} is useless since hints do not + survive anyway to the closure of sections. + +\end{Variants} + +The general +command to add a hint to some database \ident$_1$, \dots, \ident$_n$ is: +\begin{tabbing} + \texttt{Hint} \textsl{hint\_definition} \texttt{:} \ident$_1$ \ldots\ \ident$_n$ +\end{tabbing} +where {\sl hint\_definition} is one of the following expressions: + +\begin{itemize} +\item \texttt{Resolve} {\term} + \comindex{Hint Resolve} + + This command adds {\tt apply {\term}} to the hint list + with the head symbol of the type of \term. The cost of that hint is + the number of subgoals generated by {\tt apply {\term}}. + + In case the inferred type of \term\ does not start with a product the + tactic added in the hint list is {\tt exact {\term}}. In case this + type can be reduced to a type starting with a product, the tactic {\tt + apply {\term}} is also stored in the hints list. + + If the inferred type of \term\ contains a dependent + quantification on a predicate, it is added to the hint list of {\tt + eapply} instead of the hint list of {\tt apply}. In this case, a + warning is printed since the hint is only used by the tactic {\tt + eauto} (see \ref{eauto}). A typical example of a hint that is used + only by \texttt{eauto} is a transitivity lemma. + + \begin{ErrMsgs} + \item \errindex{Bound head variable} + + The head symbol of the type of {\term} is a bound variable such + that this tactic cannot be associated to a constant. + + \item \term\ \errindex{cannot be used as a hint} + + The type of \term\ contains products over variables which do not + appear in the conclusion. A typical example is a transitivity axiom. + In that case the {\tt apply} tactic fails, and thus is useless. + + \end{ErrMsgs} + + \begin{Variants} + + \item \texttt{Resolve} {\term$_1$} \dots {\term$_m$} + + Adds each \texttt{Resolve} {\term$_i$}. + + \end{Variants} + +\item \texttt{Immediate {\term}} +\comindex{Hint Immediate} + + This command adds {\tt apply {\term}; trivial} to the hint list + associated with the head symbol of the type of {\ident} in the given + database. This tactic will fail if all the subgoals generated by + {\tt apply {\term}} are not solved immediately by the {\tt trivial} + tactic (which only tries tactics with cost $0$). + + This command is useful for theorems such as the symmetry of equality + or $n+1=m+1 \to n=m$ that we may like to introduce with a + limited use in order to avoid useless proof-search. + + The cost of this tactic (which never generates subgoals) is always 1, + so that it is not used by {\tt trivial} itself. + + \begin{ErrMsgs} + + \item \errindex{Bound head variable} + + \item \term\ \errindex{cannot be used as a hint} + + \end{ErrMsgs} + + \begin{Variants} + + \item \texttt{Immediate} {\term$_1$} \dots {\term$_m$} + + Adds each \texttt{Immediate} {\term$_i$}. + + \end{Variants} + +\item \texttt{Constructors} {\ident} +\comindex{Hint Constructors} + + If {\ident} is an inductive type, this command adds all its + constructors as hints of type \texttt{Resolve}. Then, when the + conclusion of current goal has the form \texttt{({\ident} \dots)}, + \texttt{auto} will try to apply each constructor. + + \begin{ErrMsgs} + + \item {\ident} \errindex{is not an inductive type} + + \item {\ident} \errindex{not declared} + + \end{ErrMsgs} + + \begin{Variants} + + \item \texttt{Constructors} {\ident$_1$} \dots {\ident$_m$} + + Adds each \texttt{Constructors} {\ident$_i$}. + + \end{Variants} + +\item \texttt{Unfold} {\qualid} +\comindex{Hint Unfold} + + This adds the tactic {\tt unfold {\qualid}} to the hint list that + will only be used when the head constant of the goal is \ident. Its + cost is 4. + + \begin{Variants} + + \item \texttt{Unfold} {\ident$_1$} \dots {\ident$_m$} + + Adds each \texttt{Unfold} {\ident$_i$}. + + \end{Variants} + +\item \texttt{Transparent}, \texttt{Opaque} {\qualid} +\label{HintTransparency} +\comindex{Hint Transparent} +\comindex{Hint Opaque} + + This adds a transparency hint to the database, making {\tt {\qualid}} + a transparent or opaque constant during resolution. This information + is used during unification of the goal with any lemma in the database + and inside the discrimination network to relax or constrain it in the + case of \texttt{discriminated} databases. + + \begin{Variants} + + \item \texttt{Transparent}, \texttt{Opaque} {\ident$_1$} \dots {\ident$_m$} + + Declares each {\ident$_i$} as a transparent or opaque constant. + + \end{Variants} + +\item \texttt{Extern \num\ [\pattern]\ => }\textsl{tactic} +\comindex{Hint Extern} + + This hint type is to extend \texttt{auto} with tactics other than + \texttt{apply} and \texttt{unfold}. For that, we must specify a + cost, an optional pattern and a tactic to execute. Here is an example: + +\begin{quotation} +\begin{verbatim} +Hint Extern 4 ~(?=?) => discriminate. +\end{verbatim} +\end{quotation} + + Now, when the head of the goal is a disequality, \texttt{auto} will + try \texttt{discriminate} if it does not manage to solve the goal + with hints with a cost less than 4. + + One can even use some sub-patterns of the pattern in the tactic + script. A sub-pattern is a question mark followed by an ident, like + \texttt{?X1} or \texttt{?X2}. Here is an example: + +% Require EqDecide. +\begin{coq_example*} +Require Import List. +\end{coq_example*} +\begin{coq_example} +Hint Extern 5 ({?X1 = ?X2} + {?X1 <> ?X2}) => + generalize X1, X2; decide equality : eqdec. +Goal +forall a b:list (nat * nat), {a = b} + {a <> b}. +info auto with eqdec. +\end{coq_example} +\begin{coq_eval} +Abort. +\end{coq_eval} + +\end{itemize} + +\Rem One can use an \texttt{Extern} hint with no pattern to do +pattern-matching on hypotheses using \texttt{match goal with} inside +the tactic. + +\begin{Variants} +\item \texttt{Hint} \textsl{hint\_definition} + + No database name is given: the hint is registered in the {\tt core} + database. + +\item\texttt{Hint Local} \textsl{hint\_definition} \texttt{:} + \ident$_1$ \ldots\ \ident$_n$ + + This is used to declare hints that must not be exported to the other + modules that require and import the current module. Inside a + section, the option {\tt Local} is useless since hints do not + survive anyway to the closure of sections. + +\item\texttt{Hint Local} \textsl{hint\_definition} + + Idem for the {\tt core} database. + +\end{Variants} + +% There are shortcuts that allow to define several goal at once: + +% \begin{itemize} +% \item \comindex{Hints Resolve}\texttt{Hints Resolve \ident$_1$ \dots\ \ident$_n$ : \ident.}\\ +% This command is a shortcut for the following ones: +% \begin{quotation} +% \noindent\texttt{Hint \ident$_1$ : \ident\ := Resolve \ident$_1$}\\ +% \dots\\ +% \texttt{Hint \ident$_1$ : \ident := Resolve \ident$_1$} +% \end{quotation} +% Notice that the hint name is the same that the theorem given as +% hint. +% \item \comindex{Hints Immediate}\texttt{Hints Immediate \ident$_1$ \dots\ \ident$_n$ : \ident.}\\ +% \item \comindex{Hints Unfold}\texttt{Hints Unfold \qualid$_1$ \dots\ \qualid$_n$ : \ident.}\\ +% \end{itemize} + +%\begin{Warnings} +% \item \texttt{Overriding hint named \dots\ in database \dots} +%\end{Warnings} + + + +\subsection{Hint databases defined in the \Coq\ standard library} + +Several hint databases are defined in the \Coq\ standard library. The +actual content of a database is the collection of the hints declared +to belong to this database in each of the various modules currently +loaded. Especially, requiring new modules potentially extend a +database. At {\Coq} startup, only the {\tt core} and {\tt v62} +databases are non empty and can be used. + +\begin{description} + +\item[\tt core] This special database is automatically used by + \texttt{auto}. It contains only basic lemmas about negation, + conjunction, and so on from. Most of the hints in this database come + from the \texttt{Init} and \texttt{Logic} directories. + +\item[\tt arith] This database contains all lemmas about Peano's + arithmetic proved in the directories \texttt{Init} and + \texttt{Arith} + +\item[\tt zarith] contains lemmas about binary signed integers from + the directories \texttt{theories/ZArith}. When required, the module + {\tt Omega} also extends the database {\tt zarith} with a high-cost + hint that calls {\tt omega} on equations and inequalities in {\tt + nat} or {\tt Z}. + +\item[\tt bool] contains lemmas about booleans, mostly from directory + \texttt{theories/Bool}. + +\item[\tt datatypes] is for lemmas about lists, streams and so on that + are mainly proved in the \texttt{Lists} subdirectory. + +\item[\tt sets] contains lemmas about sets and relations from the + directories \texttt{Sets} and \texttt{Relations}. + +\item[\tt typeclass\_instances] contains all the type class instances + declared in the environment, including those used for \texttt{setoid\_rewrite}, + from the \texttt{Classes} directory. +\end{description} + +There is also a special database called {\tt v62}. It collects all +hints that were declared in the versions of {\Coq} prior to version +6.2.4 when the databases {\tt core}, {\tt arith}, and so on were +introduced. The purpose of the database {\tt v62} is to ensure +compatibility with further versions of {\Coq} for developments done in +versions prior to 6.2.4 ({\tt auto} being replaced by {\tt auto with v62}). +The database {\tt v62} is intended not to be extended (!). It is not +included in the hint databases list used in the {\tt auto with *} tactic. + +Furthermore, you are advised not to put your own hints in the +{\tt core} database, but use one or several databases specific to your +development. + +\subsection{\tt Print Hint +\label{PrintHint} +\comindex{Print Hint}} + +This command displays all hints that apply to the current goal. It +fails if no proof is being edited, while the two variants can be used at +every moment. + +\begin{Variants} + +\item {\tt Print Hint {\ident} } + + This command displays only tactics associated with \ident\ in the + hints list. This is independent of the goal being edited, so this + command will not fail if no goal is being edited. + +\item {\tt Print Hint *} + + This command displays all declared hints. + +\item {\tt Print HintDb {\ident} } +\label{PrintHintDb} +\comindex{Print HintDb} + + This command displays all hints from database \ident. + +\end{Variants} + +\subsection{\tt Hint Rewrite \term$_1$ \dots \term$_n$ : \ident +\label{HintRewrite} +\comindex{Hint Rewrite}} + +This vernacular command adds the terms {\tt \term$_1$ \dots \term$_n$} +(their types must be equalities) in the rewriting base {\tt \ident} +with the default orientation (left to right). Notice that the +rewriting bases are distinct from the {\tt auto} hint bases and that +{\tt auto} does not take them into account. + +This command is synchronous with the section mechanism (see \ref{Section}): +when closing a section, all aliases created by \texttt{Hint Rewrite} in that +section are lost. Conversely, when loading a module, all \texttt{Hint Rewrite} +declarations at the global level of that module are loaded. + +\begin{Variants} +\item {\tt Hint Rewrite -> \term$_1$ \dots \term$_n$ : \ident}\\ +This is strictly equivalent to the command above (we only make explicit the +orientation which otherwise defaults to {\tt ->}). + +\item {\tt Hint Rewrite <- \term$_1$ \dots \term$_n$ : \ident}\\ +Adds the rewriting rules {\tt \term$_1$ \dots \term$_n$} with a right-to-left +orientation in the base {\tt \ident}. + +\item {\tt Hint Rewrite \term$_1$ \dots \term$_n$ using {\tac} : {\ident}}\\ +When the rewriting rules {\tt \term$_1$ \dots \term$_n$} in {\tt \ident} will +be used, the tactic {\tt \tac} will be applied to the generated subgoals, the +main subgoal excluded. + +%% \item +%% {\tt Hint Rewrite [ \term$_1$ \dots \term$_n$ ] in \ident}\\ +%% {\tt Hint Rewrite [ \term$_1$ \dots \term$_n$ ] in {\ident} using {\tac}}\\ +%% These are deprecated syntactic variants for +%% {\tt Hint Rewrite \term$_1$ \dots \term$_n$ : \ident} and +%% {\tt Hint Rewrite \term$_1$ \dots \term$_n$ using {\tac} : {\ident}}. + +\item \texttt{Print Rewrite HintDb {\ident}} + + This command displays all rewrite hints contained in {\ident}. + +\end{Variants} + +\subsection{Hints and sections +\label{Hint-and-Section}} + +Hints provided by the \texttt{Hint} commands are erased when closing a +section. Conversely, all hints of a module \texttt{A} that are not +defined inside a section (and not defined with option {\tt Local}) become +available when the module {\tt A} is imported (using +e.g. \texttt{Require Import A.}). + +\subsection{Setting implicit automation tactics} + +\subsubsection[\tt Proof with {\tac}.]{\tt Proof with {\tac}.\label{ProofWith} +\comindex{Proof with}} + + This command may be used to start a proof. It defines a default + tactic to be used each time a tactic command {\tac$_1$} is ended by + ``\verb#...#''. In this case the tactic command typed by the user is + equivalent to \tac$_1$;{\tac}. + +\SeeAlso {\tt Proof.} in Section~\ref{BeginProof}. + +\subsubsection[\tt Declare Implicit Tactic {\tac}.]{\tt Declare Implicit Tactic {\tac}.\comindex{Declare Implicit Tactic}} + +This command declares a tactic to be used to solve implicit arguments +that {\Coq} does not know how to solve by unification. It is used +every time the term argument of a tactic has one of its holes not +fully resolved. + +Here is an example: + +\begin{coq_example} +Parameter quo : nat -> forall n:nat, n<>0 -> nat. +Notation "x // y" := (quo x y _) (at level 40). + +Declare Implicit Tactic assumption. +Goal forall n m, m<>0 -> { q:nat & { r | q * m + r = n } }. +intros. +exists (n // m). +\end{coq_example} + +The tactic {\tt exists (n // m)} did not fail. The hole was solved by +{\tt assumption} so that it behaved as {\tt exists (quo n m H)}. + +\section{Generation of induction principles with {\tt Scheme} +\label{Scheme} +\index{Schemes} +\comindex{Scheme}} + +The {\tt Scheme} command is a high-level tool for generating +automatically (possibly mutual) induction principles for given types +and sorts. Its syntax follows the schema: +\begin{quote} +{\tt Scheme {\ident$_1$} := Induction for \ident'$_1$ Sort {\sort$_1$} \\ + with\\ + \mbox{}\hspace{0.1cm} \dots\\ + with {\ident$_m$} := Induction for {\ident'$_m$} Sort + {\sort$_m$}} +\end{quote} +where \ident'$_1$ \dots\ \ident'$_m$ are different inductive type +identifiers belonging to the same package of mutual inductive +definitions. This command generates {\ident$_1$}\dots{} {\ident$_m$} +to be mutually recursive definitions. Each term {\ident$_i$} proves a +general principle of mutual induction for objects in type {\term$_i$}. + +\begin{Variants} +\item {\tt Scheme {\ident$_1$} := Minimality for \ident'$_1$ Sort {\sort$_1$} \\ + with\\ + \mbox{}\hspace{0.1cm} \dots\ \\ + with {\ident$_m$} := Minimality for {\ident'$_m$} Sort + {\sort$_m$}} + + Same as before but defines a non-dependent elimination principle more + natural in case of inductively defined relations. + +\item {\tt Scheme Equality for \ident$_1$\comindex{Scheme Equality}} + + Tries to generate a boolean equality and a proof of the + decidability of the usual equality. + +\item {\tt Scheme Induction for \ident$_1$ Sort {\sort$_1$} \\ + with\\ + \mbox{}\hspace{0.1cm} \dots\\ + with Induction for {\ident$_m$} Sort + {\sort$_m$}} + + If you do not provide the name of the schemes, they will be automatically + computed from the sorts involved (works also with Minimality). + +\end{Variants} + +\SeeAlso Section~\ref{Scheme-examples} + +\subsection{Automatic declaration of schemes} +\comindex{Set Equality Schemes} +\comindex{Set Elimination Schemes} +It is possible to deactivate the automatic declaration of the induction + principles when defining a new inductive type with the + {\tt Unset Elimination Schemes} command. It may be +reactivated at any time with {\tt Set Elimination Schemes}. +\\ + +You can also activate the automatic declaration of those boolean equalities +(see the second variant of {\tt Scheme}) with the {\tt Set Equality Schemes} + command. However you have to be careful with this option since +\Coq~ may now reject well-defined inductive types because it cannot compute +a boolean equality for them. + +\subsection{\tt Combined Scheme\label{CombinedScheme} +\comindex{Combined Scheme}} +The {\tt Combined Scheme} command is a tool for combining +induction principles generated by the {\tt Scheme} command. +Its syntax follows the schema : + +\noindent +{\tt Combined Scheme {\ident$_0$} from {\ident$_1$}, .., {\ident$_n$}}\\ +\ident$_1$ \ldots \ident$_n$ are different inductive principles that must belong to +the same package of mutual inductive principle definitions. This command +generates {\ident$_0$} to be the conjunction of the principles: it is +built from the common premises of the principles and concluded by the +conjunction of their conclusions. + +\SeeAlso Section~\ref{CombinedScheme-examples} + +\section{Generation of induction principles with {\tt Functional Scheme} +\label{FunScheme} +\comindex{Functional Scheme}} + +The {\tt Functional Scheme} command is a high-level experimental +tool for generating automatically induction principles +corresponding to (possibly mutually recursive) functions. Its +syntax follows the schema: +\begin{quote} +{\tt Functional Scheme {\ident$_1$} := Induction for \ident'$_1$ Sort {\sort$_1$} \\ + with\\ + \mbox{}\hspace{0.1cm} \dots\ \\ + with {\ident$_m$} := Induction for {\ident'$_m$} Sort + {\sort$_m$}} +\end{quote} +where \ident'$_1$ \dots\ \ident'$_m$ are different mutually defined function +names (they must be in the same order as when they were defined). +This command generates the induction principles +\ident$_1$\dots\ident$_m$, following the recursive structure and case +analyses of the functions \ident'$_1$ \dots\ \ident'$_m$. + + +\paragraph{\texttt{Functional Scheme}} +There is a difference between obtaining an induction scheme by using +\texttt{Functional Scheme} on a function defined by \texttt{Function} +or not. Indeed \texttt{Function} generally produces smaller +principles, closer to the definition written by the user. + + +\SeeAlso Section~\ref{FunScheme-examples} + + +\section{Simple tactic macros +\index{Tactic macros} +\comindex{Tactic Definition} +\label{TacticDefinition}} + +A simple example has more value than a long explanation: + +\begin{coq_example} +Ltac Solve := simpl; intros; auto. +Ltac ElimBoolRewrite b H1 H2 := + elim b; [ intros; rewrite H1; eauto | intros; rewrite H2; eauto ]. +\end{coq_example} + +The tactics macros are synchronous with the \Coq\ section mechanism: +a tactic definition is deleted from the current environment +when you close the section (see also \ref{Section}) +where it was defined. If you want that a +tactic macro defined in a module is usable in the modules that +require it, you should put it outside of any section. + +Chapter~\ref{TacticLanguage} gives examples of more complex +user-defined tactics. + + +% $Id: RefMan-tac.tex 13344 2010-07-28 15:04:36Z msozeau $ + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "Reference-Manual" +%%% TeX-master: "Reference-Manual" +%%% End: |