104

1 
%% $Id$


2 
\chapter{Tacticals}


3 
\index{tacticals(}


4 
Tacticals are operations on tactics. Their implementation makes use of


5 
functional programming techniques, especially for sequences. Most of the


6 
time, you may forget about this and regard tacticals as highlevel control


7 
structures.


8 


9 
\section{The basic tacticals}


10 
\subsection{Joining two tactics}

323

11 
\index{tacticals!joining two tactics}

104

12 
The tacticals {\tt THEN} and {\tt ORELSE}, which provide sequencing and


13 
alternation, underlie most of the other control structures in Isabelle.


14 
{\tt APPEND} and {\tt INTLEAVE} provide more sophisticated forms of


15 
alternation.


16 
\begin{ttbox}


17 
THEN : tactic * tactic > tactic \hfill{\bf infix 1}


18 
ORELSE : tactic * tactic > tactic \hfill{\bf infix}


19 
APPEND : tactic * tactic > tactic \hfill{\bf infix}


20 
INTLEAVE : tactic * tactic > tactic \hfill{\bf infix}


21 
\end{ttbox}

323

22 
\begin{ttdescription}


23 
\item[$tac@1$ \ttindexbold{THEN} $tac@2$]

104

24 
is the sequential composition of the two tactics. Applied to a proof


25 
state, it returns all states reachable in two steps by applying $tac@1$


26 
followed by~$tac@2$. First, it applies $tac@1$ to the proof state, getting a


27 
sequence of next states; then, it applies $tac@2$ to each of these and


28 
concatenates the results.


29 

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30 
\item[$tac@1$ \ttindexbold{ORELSE} $tac@2$]

104

31 
makes a choice between the two tactics. Applied to a state, it


32 
tries~$tac@1$ and returns the result if nonempty; if $tac@1$ fails then it


33 
uses~$tac@2$. This is a deterministic choice: if $tac@1$ succeeds then


34 
$tac@2$ is excluded.


35 

323

36 
\item[$tac@1$ \ttindexbold{APPEND} $tac@2$]

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37 
concatenates the results of $tac@1$ and~$tac@2$. By not making a commitment

323

38 
to either tactic, {\tt APPEND} helps avoid incompleteness during


39 
search.\index{search}

104

40 

323

41 
\item[$tac@1$ \ttindexbold{INTLEAVE} $tac@2$]

104

42 
interleaves the results of $tac@1$ and~$tac@2$. Thus, it includes all


43 
possible next states, even if one of the tactics returns an infinite


44 
sequence.

323

45 
\end{ttdescription}

104

46 


47 


48 
\subsection{Joining a list of tactics}

323

49 
\index{tacticals!joining a list of tactics}

104

50 
\begin{ttbox}


51 
EVERY : tactic list > tactic


52 
FIRST : tactic list > tactic


53 
\end{ttbox}


54 
{\tt EVERY} and {\tt FIRST} are block structured versions of {\tt THEN} and


55 
{\tt ORELSE}\@.

323

56 
\begin{ttdescription}

104

57 
\item[\ttindexbold{EVERY} {$[tac@1,\ldots,tac@n]$}]


58 
abbreviates \hbox{\tt$tac@1$ THEN \ldots{} THEN $tac@n$}. It is useful for


59 
writing a series of tactics to be executed in sequence.


60 


61 
\item[\ttindexbold{FIRST} {$[tac@1,\ldots,tac@n]$}]


62 
abbreviates \hbox{\tt$tac@1$ ORELSE \ldots{} ORELSE $tac@n$}. It is useful for


63 
writing a series of tactics to be attempted one after another.

323

64 
\end{ttdescription}

104

65 


66 


67 
\subsection{Repetition tacticals}

323

68 
\index{tacticals!repetition}

104

69 
\begin{ttbox}


70 
TRY : tactic > tactic


71 
REPEAT_DETERM : tactic > tactic


72 
REPEAT : tactic > tactic


73 
REPEAT1 : tactic > tactic


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trace_REPEAT : bool ref \hfill{\bf initially false}


75 
\end{ttbox}

323

76 
\begin{ttdescription}

104

77 
\item[\ttindexbold{TRY} {\it tac}]


78 
applies {\it tac\/} to the proof state and returns the resulting sequence,


79 
if nonempty; otherwise it returns the original state. Thus, it applies


80 
{\it tac\/} at most once.


81 


82 
\item[\ttindexbold{REPEAT_DETERM} {\it tac}]


83 
applies {\it tac\/} to the proof state and, recursively, to the head of the


84 
resulting sequence. It returns the first state to make {\it tac\/} fail.


85 
It is deterministic, discarding alternative outcomes.


86 


87 
\item[\ttindexbold{REPEAT} {\it tac}]


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applies {\it tac\/} to the proof state and, recursively, to each element of


89 
the resulting sequence. The resulting sequence consists of those states


90 
that make {\it tac\/} fail. Thus, it applies {\it tac\/} as many times as


91 
possible (including zero times), and allows backtracking over each


92 
invocation of {\it tac}. It is more general than {\tt REPEAT_DETERM}, but


93 
requires more space.


94 


95 
\item[\ttindexbold{REPEAT1} {\it tac}]


96 
is like \hbox{\tt REPEAT {\it tac}} but it always applies {\it tac\/} at


97 
least once, failing if this is impossible.


98 

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99 
\item[\ttindexbold{trace_REPEAT} := true;]

286

100 
enables an interactive tracing mode for the tacticals {\tt REPEAT_DETERM}


101 
and {\tt REPEAT}. To view the tracing options, type {\tt h} at the prompt.

323

102 
\end{ttdescription}

104

103 


104 


105 
\subsection{Identities for tacticals}

323

106 
\index{tacticals!identities for}

104

107 
\begin{ttbox}


108 
all_tac : tactic


109 
no_tac : tactic


110 
\end{ttbox}

323

111 
\begin{ttdescription}

104

112 
\item[\ttindexbold{all_tac}]


113 
maps any proof state to the oneelement sequence containing that state.


114 
Thus, it succeeds for all states. It is the identity element of the


115 
tactical \ttindex{THEN}\@.


116 


117 
\item[\ttindexbold{no_tac}]


118 
maps any proof state to the empty sequence. Thus it succeeds for no state.


119 
It is the identity element of \ttindex{ORELSE}, \ttindex{APPEND}, and


120 
\ttindex{INTLEAVE}\@. Also, it is a zero element for \ttindex{THEN}, which means that


121 
\hbox{\tt$tac$ THEN no_tac} is equivalent to {\tt no_tac}.

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122 
\end{ttdescription}

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123 
These primitive tactics are useful when writing tacticals. For example,


124 
\ttindexbold{TRY} and \ttindexbold{REPEAT} (ignoring tracing) can be coded


125 
as follows:


126 
\begin{ttbox}


127 
fun TRY tac = tac ORELSE all_tac;


128 

3108

129 
fun REPEAT tac =


130 
(fn state => ((tac THEN REPEAT tac) ORELSE all_tac) state);

104

131 
\end{ttbox}


132 
If $tac$ can return multiple outcomes then so can \hbox{\tt REPEAT $tac$}.


133 
Since {\tt REPEAT} uses \ttindex{ORELSE} and not {\tt APPEND} or {\tt


134 
INTLEAVE}, it applies $tac$ as many times as possible in each


135 
outcome.


136 


137 
\begin{warn}


138 
Note {\tt REPEAT}'s explicit abstraction over the proof state. Recursive


139 
tacticals must be coded in this awkward fashion to avoid infinite


140 
recursion. With the following definition, \hbox{\tt REPEAT $tac$} would

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141 
loop due to \ML's eager evaluation strategy:

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142 
\begin{ttbox}


143 
fun REPEAT tac = (tac THEN REPEAT tac) ORELSE all_tac;


144 
\end{ttbox}


145 
\par\noindent


146 
The builtin {\tt REPEAT} avoids~{\tt THEN}, handling sequences explicitly


147 
and using tail recursion. This sacrifices clarity, but saves much space by


148 
discarding intermediate proof states.


149 
\end{warn}


150 


151 


152 
\section{Control and search tacticals}

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153 
\index{search!tacticals(}


154 

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155 
A predicate on theorems, namely a function of type \hbox{\tt thm>bool},


156 
can test whether a proof state enjoys some desirable property  such as


157 
having no subgoals. Tactics that search for satisfactory states are easy


158 
to express. The main search procedures, depthfirst, breadthfirst and


159 
bestfirst, are provided as tacticals. They generate the search tree by


160 
repeatedly applying a given tactic.


161 


162 


163 
\subsection{Filtering a tactic's results}

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164 
\index{tacticals!for filtering}


165 
\index{tactics!filtering results of}

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166 
\begin{ttbox}


167 
FILTER : (thm > bool) > tactic > tactic


168 
CHANGED : tactic > tactic


169 
\end{ttbox}

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\begin{ttdescription}

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171 
\item[\ttindexbold{FILTER} {\it p} $tac$]

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172 
applies $tac$ to the proof state and returns a sequence consisting of those


173 
result states that satisfy~$p$.


174 


175 
\item[\ttindexbold{CHANGED} {\it tac}]


176 
applies {\it tac\/} to the proof state and returns precisely those states


177 
that differ from the original state. Thus, \hbox{\tt CHANGED {\it tac}}


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always has some effect on the state.

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\end{ttdescription}

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180 


181 


182 
\subsection{Depthfirst search}

323

183 
\index{tacticals!searching}

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184 
\index{tracing!of searching tacticals}


185 
\begin{ttbox}


186 
DEPTH_FIRST : (thm>bool) > tactic > tactic

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DEPTH_SOLVE : tactic > tactic


188 
DEPTH_SOLVE_1 : tactic > tactic

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trace_DEPTH_FIRST: bool ref \hfill{\bf initially false}


190 
\end{ttbox}

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191 
\begin{ttdescription}

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192 
\item[\ttindexbold{DEPTH_FIRST} {\it satp} {\it tac}]


193 
returns the proof state if {\it satp} returns true. Otherwise it applies


194 
{\it tac}, then recursively searches from each element of the resulting


195 
sequence. The code uses a stack for efficiency, in effect applying


196 
\hbox{\tt {\it tac} THEN DEPTH_FIRST {\it satp} {\it tac}} to the state.


197 


198 
\item[\ttindexbold{DEPTH_SOLVE} {\it tac}]


199 
uses {\tt DEPTH_FIRST} to search for states having no subgoals.


200 


201 
\item[\ttindexbold{DEPTH_SOLVE_1} {\it tac}]


202 
uses {\tt DEPTH_FIRST} to search for states having fewer subgoals than the


203 
given state. Thus, it insists upon solving at least one subgoal.


204 

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205 
\item[\ttindexbold{trace_DEPTH_FIRST} := true;]

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206 
enables interactive tracing for {\tt DEPTH_FIRST}. To view the


207 
tracing options, type {\tt h} at the prompt.

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208 
\end{ttdescription}

104

209 


210 


211 
\subsection{Other search strategies}

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212 
\index{tacticals!searching}

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213 
\index{tracing!of searching tacticals}


214 
\begin{ttbox}

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BREADTH_FIRST : (thm>bool) > tactic > tactic

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BEST_FIRST : (thm>bool)*(thm>int) > tactic > tactic


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THEN_BEST_FIRST : tactic * ((thm>bool) * (thm>int) * tactic)


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> tactic \hfill{\bf infix 1}


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trace_BEST_FIRST: bool ref \hfill{\bf initially false}


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\end{ttbox}


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These search strategies will find a solution if one exists. However, they


222 
do not enumerate all solutions; they terminate after the first satisfactory


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result from {\it tac}.

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\begin{ttdescription}

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\item[\ttindexbold{BREADTH_FIRST} {\it satp} {\it tac}]


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uses breadthfirst search to find states for which {\it satp\/} is true.


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For most applications, it is too slow.


228 


229 
\item[\ttindexbold{BEST_FIRST} $(satp,distf)$ {\it tac}]


230 
does a heuristic search, using {\it distf\/} to estimate the distance from


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a satisfactory state. It maintains a list of states ordered by distance.


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It applies $tac$ to the head of this list; if the result contains any


233 
satisfactory states, then it returns them. Otherwise, {\tt BEST_FIRST}


234 
adds the new states to the list, and continues.


235 


236 
The distance function is typically \ttindex{size_of_thm}, which computes


237 
the size of the state. The smaller the state, the fewer and simpler


238 
subgoals it has.


239 


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\item[$tac@0$ \ttindexbold{THEN_BEST_FIRST} $(satp,distf,tac)$]


241 
is like {\tt BEST_FIRST}, except that the priority queue initially


242 
contains the result of applying $tac@0$ to the proof state. This tactical


243 
permits separate tactics for starting the search and continuing the search.


244 

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\item[\ttindexbold{trace_BEST_FIRST} := true;]

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246 
enables an interactive tracing mode for the tactical {\tt BEST_FIRST}. To


247 
view the tracing options, type {\tt h} at the prompt.

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248 
\end{ttdescription}

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249 


250 


251 
\subsection{Auxiliary tacticals for searching}


252 
\index{tacticals!conditional}


253 
\index{tacticals!deterministic}


254 
\begin{ttbox}


255 
COND : (thm>bool) > tactic > tactic > tactic


256 
IF_UNSOLVED : tactic > tactic


257 
DETERM : tactic > tactic


258 
\end{ttbox}

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259 
\begin{ttdescription}

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260 
\item[\ttindexbold{COND} {\it p} $tac@1$ $tac@2$]

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261 
applies $tac@1$ to the proof state if it satisfies~$p$, and applies $tac@2$


262 
otherwise. It is a conditional tactical in that only one of $tac@1$ and


263 
$tac@2$ is applied to a proof state. However, both $tac@1$ and $tac@2$ are


264 
evaluated because \ML{} uses eager evaluation.


265 


266 
\item[\ttindexbold{IF_UNSOLVED} {\it tac}]


267 
applies {\it tac\/} to the proof state if it has any subgoals, and simply


268 
returns the proof state otherwise. Many common tactics, such as {\tt


269 
resolve_tac}, fail if applied to a proof state that has no subgoals.


270 


271 
\item[\ttindexbold{DETERM} {\it tac}]


272 
applies {\it tac\/} to the proof state and returns the head of the


273 
resulting sequence. {\tt DETERM} limits the search space by making its


274 
argument deterministic.

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275 
\end{ttdescription}

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276 


277 


278 
\subsection{Predicates and functions useful for searching}


279 
\index{theorems!size of}


280 
\index{theorems!equality of}


281 
\begin{ttbox}


282 
has_fewer_prems : int > thm > bool


283 
eq_thm : thm * thm > bool


284 
size_of_thm : thm > int


285 
\end{ttbox}

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286 
\begin{ttdescription}

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287 
\item[\ttindexbold{has_fewer_prems} $n$ $thm$]


288 
reports whether $thm$ has fewer than~$n$ premises. By currying,


289 
\hbox{\tt has_fewer_prems $n$} is a predicate on theorems; it may


290 
be given to the searching tacticals.


291 


292 
\item[\ttindexbold{eq_thm}($thm1$,$thm2$)]


293 
reports whether $thm1$ and $thm2$ are equal. Both theorems must have


294 
identical signatures. Both theorems must have the same conclusions, and


295 
the same hypotheses, in the same order. Names of bound variables are


296 
ignored.


297 


298 
\item[\ttindexbold{size_of_thm} $thm$]


299 
computes the size of $thm$, namely the number of variables, constants and


300 
abstractions in its conclusion. It may serve as a distance function for


301 
\ttindex{BEST_FIRST}.

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302 
\end{ttdescription}


303 


304 
\index{search!tacticals)}

104

305 


306 


307 
\section{Tacticals for subgoal numbering}


308 
When conducting a backward proof, we normally consider one goal at a time.


309 
A tactic can affect the entire proof state, but many tactics  such as


310 
{\tt resolve_tac} and {\tt assume_tac}  work on a single subgoal.


311 
Subgoals are designated by a positive integer, so Isabelle provides


312 
tacticals for combining values of type {\tt int>tactic}.


313 


314 


315 
\subsection{Restricting a tactic to one subgoal}


316 
\index{tactics!restricting to a subgoal}


317 
\index{tacticals!for restriction to a subgoal}


318 
\begin{ttbox}


319 
SELECT_GOAL : tactic > int > tactic


320 
METAHYPS : (thm list > tactic) > int > tactic


321 
\end{ttbox}

323

322 
\begin{ttdescription}

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323 
\item[\ttindexbold{SELECT_GOAL} {\it tac} $i$]


324 
restricts the effect of {\it tac\/} to subgoal~$i$ of the proof state. It


325 
fails if there is no subgoal~$i$, or if {\it tac\/} changes the main goal


326 
(do not use {\tt rewrite_tac}). It applies {\it tac\/} to a dummy proof


327 
state and uses the result to refine the original proof state at


328 
subgoal~$i$. If {\it tac\/} returns multiple results then so does


329 
\hbox{\tt SELECT_GOAL {\it tac} $i$}.


330 

323

331 
{\tt SELECT_GOAL} works by creating a state of the form $\phi\Imp\phi$,

332

332 
with the one subgoal~$\phi$. If subgoal~$i$ has the form $\psi\Imp\theta$


333 
then $(\psi\Imp\theta)\Imp(\psi\Imp\theta)$ is in fact


334 
$\List{\psi\Imp\theta;\; \psi}\Imp\theta$, a proof state with two subgoals.


335 
Such a proof state might cause tactics to go astray. Therefore {\tt


336 
SELECT_GOAL} inserts a quantifier to create the state

323

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\[ (\Forall x.\psi\Imp\theta)\Imp(\Forall x.\psi\Imp\theta). \]

104

338 

323

339 
\item[\ttindexbold{METAHYPS} {\it tacf} $i$]\index{metaassumptions}

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340 
takes subgoal~$i$, of the form


341 
\[ \Forall x@1 \ldots x@l. \List{\theta@1; \ldots; \theta@k}\Imp\theta, \]


342 
and creates the list $\theta'@1$, \ldots, $\theta'@k$ of metalevel


343 
assumptions. In these theorems, the subgoal's parameters ($x@1$,


344 
\ldots,~$x@l$) become free variables. It supplies the assumptions to


345 
$tacf$ and applies the resulting tactic to the proof state


346 
$\theta\Imp\theta$.


347 


348 
If the resulting proof state is $\List{\phi@1; \ldots; \phi@n} \Imp \phi$,


349 
possibly containing $\theta'@1,\ldots,\theta'@k$ as assumptions, then it is


350 
lifted back into the original context, yielding $n$ subgoals.


351 

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352 
Metalevel assumptions may not contain unknowns. Unknowns in the


353 
hypotheses $\theta@1,\ldots,\theta@k$ become free variables in $\theta'@1$,


354 
\ldots, $\theta'@k$, and are restored afterwards; the {\tt METAHYPS} call


355 
cannot instantiate them. Unknowns in $\theta$ may be instantiated. New

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unknowns in $\phi@1$, \ldots, $\phi@n$ are lifted over the parameters.

104

357 


358 
Here is a typical application. Calling {\tt hyp_res_tac}~$i$ resolves


359 
subgoal~$i$ with one of its own assumptions, which may itself have the form


360 
of an inference rule (these are called {\bf higherlevel assumptions}).


361 
\begin{ttbox}


362 
val hyp_res_tac = METAHYPS (fn prems => resolve_tac prems 1);


363 
\end{ttbox}

332

364 
The function \ttindex{gethyps} is useful for debugging applications of {\tt


365 
METAHYPS}.

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366 
\end{ttdescription}

104

367 


368 
\begin{warn}


369 
{\tt METAHYPS} fails if the context or new subgoals contain type unknowns.


370 
In principle, the tactical could treat these like ordinary unknowns.


371 
\end{warn}


372 


373 


374 
\subsection{Scanning for a subgoal by number}

323

375 
\index{tacticals!scanning for subgoals}

104

376 
\begin{ttbox}


377 
ALLGOALS : (int > tactic) > tactic


378 
TRYALL : (int > tactic) > tactic


379 
SOMEGOAL : (int > tactic) > tactic


380 
FIRSTGOAL : (int > tactic) > tactic


381 
REPEAT_SOME : (int > tactic) > tactic


382 
REPEAT_FIRST : (int > tactic) > tactic


383 
trace_goalno_tac : (int > tactic) > int > tactic


384 
\end{ttbox}


385 
These apply a tactic function of type {\tt int > tactic} to all the


386 
subgoal numbers of a proof state, and join the resulting tactics using


387 
\ttindex{THEN} or \ttindex{ORELSE}\@. Thus, they apply the tactic to all the


388 
subgoals, or to one subgoal.


389 


390 
Suppose that the original proof state has $n$ subgoals.


391 

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392 
\begin{ttdescription}

104

393 
\item[\ttindexbold{ALLGOALS} {\it tacf}]


394 
is equivalent to


395 
\hbox{\tt$tacf(n)$ THEN \ldots{} THEN $tacf(1)$}.


396 

323

397 
It applies {\it tacf} to all the subgoals, counting downwards (to

104

398 
avoid problems when subgoals are added or deleted).


399 


400 
\item[\ttindexbold{TRYALL} {\it tacf}]


401 
is equivalent to

323

402 
\hbox{\tt TRY$(tacf(n))$ THEN \ldots{} THEN TRY$(tacf(1))$}.

104

403 


404 
It attempts to apply {\it tacf} to all the subgoals. For instance,

286

405 
the tactic \hbox{\tt TRYALL assume_tac} attempts to solve all the subgoals by

104

406 
assumption.


407 


408 
\item[\ttindexbold{SOMEGOAL} {\it tacf}]


409 
is equivalent to


410 
\hbox{\tt$tacf(n)$ ORELSE \ldots{} ORELSE $tacf(1)$}.


411 

323

412 
It applies {\it tacf} to one subgoal, counting downwards. For instance,

286

413 
the tactic \hbox{\tt SOMEGOAL assume_tac} solves one subgoal by assumption,


414 
failing if this is impossible.

104

415 


416 
\item[\ttindexbold{FIRSTGOAL} {\it tacf}]


417 
is equivalent to


418 
\hbox{\tt$tacf(1)$ ORELSE \ldots{} ORELSE $tacf(n)$}.


419 

323

420 
It applies {\it tacf} to one subgoal, counting upwards.

104

421 


422 
\item[\ttindexbold{REPEAT_SOME} {\it tacf}]

323

423 
applies {\it tacf} once or more to a subgoal, counting downwards.

104

424 


425 
\item[\ttindexbold{REPEAT_FIRST} {\it tacf}]

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426 
applies {\it tacf} once or more to a subgoal, counting upwards.

104

427 


428 
\item[\ttindexbold{trace_goalno_tac} {\it tac} {\it i}]


429 
applies \hbox{\it tac i\/} to the proof state. If the resulting sequence


430 
is nonempty, then it is returned, with the sideeffect of printing {\tt


431 
Subgoal~$i$ selected}. Otherwise, {\tt trace_goalno_tac} returns the empty


432 
sequence and prints nothing.


433 

323

434 
It indicates that `the tactic worked for subgoal~$i$' and is mainly used

104

435 
with {\tt SOMEGOAL} and {\tt FIRSTGOAL}.

323

436 
\end{ttdescription}

104

437 


438 


439 
\subsection{Joining tactic functions}

323

440 
\index{tacticals!joining tactic functions}

104

441 
\begin{ttbox}


442 
THEN' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix 1}


443 
ORELSE' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix}


444 
APPEND' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix}


445 
INTLEAVE' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix}


446 
EVERY' : ('a > tactic) list > 'a > tactic


447 
FIRST' : ('a > tactic) list > 'a > tactic


448 
\end{ttbox}


449 
These help to express tactics that specify subgoal numbers. The tactic


450 
\begin{ttbox}


451 
SOMEGOAL (fn i => resolve_tac rls i ORELSE eresolve_tac erls i)


452 
\end{ttbox}


453 
can be simplified to


454 
\begin{ttbox}


455 
SOMEGOAL (resolve_tac rls ORELSE' eresolve_tac erls)


456 
\end{ttbox}


457 
Note that {\tt TRY'}, {\tt REPEAT'}, {\tt DEPTH_FIRST'}, etc.\ are not


458 
provided, because function composition accomplishes the same purpose.


459 
The tactic


460 
\begin{ttbox}


461 
ALLGOALS (fn i => REPEAT (etac exE i ORELSE atac i))


462 
\end{ttbox}


463 
can be simplified to


464 
\begin{ttbox}


465 
ALLGOALS (REPEAT o (etac exE ORELSE' atac))


466 
\end{ttbox}


467 
These tacticals are polymorphic; $x$ need not be an integer.


468 
\begin{center} \tt


469 
\begin{tabular}{r@{\rm\ \ yields\ \ }l}

323

470 
$(tacf@1$~~THEN'~~$tacf@2)(x)$ \index{*THEN'} &

104

471 
$tacf@1(x)$~~THEN~~$tacf@2(x)$ \\


472 

323

473 
$(tacf@1$ ORELSE' $tacf@2)(x)$ \index{*ORELSE'} &

104

474 
$tacf@1(x)$ ORELSE $tacf@2(x)$ \\


475 

323

476 
$(tacf@1$ APPEND' $tacf@2)(x)$ \index{*APPEND'} &

104

477 
$tacf@1(x)$ APPEND $tacf@2(x)$ \\


478 

323

479 
$(tacf@1$ INTLEAVE' $tacf@2)(x)$ \index{*INTLEAVE'} &

104

480 
$tacf@1(x)$ INTLEAVE $tacf@2(x)$ \\


481 


482 
EVERY' $[tacf@1,\ldots,tacf@n] \; (x)$ \index{*EVERY'} &


483 
EVERY $[tacf@1(x),\ldots,tacf@n(x)]$ \\


484 


485 
FIRST' $[tacf@1,\ldots,tacf@n] \; (x)$ \index{*FIRST'} &


486 
FIRST $[tacf@1(x),\ldots,tacf@n(x)]$


487 
\end{tabular}


488 
\end{center}


489 


490 


491 
\subsection{Applying a list of tactics to 1}

323

492 
\index{tacticals!joining tactic functions}

104

493 
\begin{ttbox}


494 
EVERY1: (int > tactic) list > tactic


495 
FIRST1: (int > tactic) list > tactic


496 
\end{ttbox}


497 
A common proof style is to treat the subgoals as a stack, always


498 
restricting attention to the first subgoal. Such proofs contain long lists


499 
of tactics, each applied to~1. These can be simplified using {\tt EVERY1}


500 
and {\tt FIRST1}:


501 
\begin{center} \tt


502 
\begin{tabular}{r@{\rm\ \ abbreviates\ \ }l}


503 
EVERY1 $[tacf@1,\ldots,tacf@n]$ \indexbold{*EVERY1} &


504 
EVERY $[tacf@1(1),\ldots,tacf@n(1)]$ \\


505 


506 
FIRST1 $[tacf@1,\ldots,tacf@n]$ \indexbold{*FIRST1} &


507 
FIRST $[tacf@1(1),\ldots,tacf@n(1)]$


508 
\end{tabular}


509 
\end{center}


510 


511 
\index{tacticals)}
