author  oheimb 
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child 13104  df7aac8543c9 
permissions  rwrr 
104  1 
%% $Id$ 
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\chapter{Tacticals} 

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\index{tacticals(} 

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Tacticals are operations on tactics. Their implementation makes use of 

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functional programming techniques, especially for sequences. Most of the 

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

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

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

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alternation. 

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\begin{ttbox} 

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

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

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

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

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\end{ttbox} 

323  22 
\begin{ttdescription} 
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\item[$tac@1$ \ttindexbold{THEN} $tac@2$] 

104  24 
is the sequential composition of the two tactics. Applied to a proof 
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state, it returns all states reachable in two steps by applying $tac@1$ 

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

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

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concatenates the results. 

29 

323  30 
\item[$tac@1$ \ttindexbold{ORELSE} $tac@2$] 
104  31 
makes a choice between the two tactics. Applied to a state, it 
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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 

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$tac@2$ is excluded. 

35 

323  36 
\item[$tac@1$ \ttindexbold{APPEND} $tac@2$] 
104  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 
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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} 
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TRY : tactic > tactic 
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REPEAT_DETERM : tactic > tactic 
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REPEAT_DETERM_N : int > tactic > tactic 
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REPEAT : tactic > tactic 
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REPEAT1 : tactic > tactic 
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DETERM_UNTIL : (thm > bool) > tactic > tactic 
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trace_REPEAT : bool ref \hfill{\bf initially false} 
104  77 
\end{ttbox} 
323  78 
\begin{ttdescription} 
104  79 
\item[\ttindexbold{TRY} {\it tac}] 
80 
applies {\it tac\/} to the proof state and returns the resulting sequence, 

81 
if nonempty; otherwise it returns the original state. Thus, it applies 

82 
{\it tac\/} at most once. 

83 

84 
\item[\ttindexbold{REPEAT_DETERM} {\it tac}] 

85 
applies {\it tac\/} to the proof state and, recursively, to the head of the 

86 
resulting sequence. It returns the first state to make {\it tac\/} fail. 

87 
It is deterministic, discarding alternative outcomes. 

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\item[\ttindexbold{REPEAT_DETERM_N} {\it n} {\it tac}] 
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is like \hbox{\tt REPEAT_DETERM {\it tac}} but the number of repititions 
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is bound by {\it n} (unless negative). 
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104  93 
\item[\ttindexbold{REPEAT} {\it tac}] 
94 
applies {\it tac\/} to the proof state and, recursively, to each element of 

95 
the resulting sequence. The resulting sequence consists of those states 

96 
that make {\it tac\/} fail. Thus, it applies {\it tac\/} as many times as 

97 
possible (including zero times), and allows backtracking over each 

98 
invocation of {\it tac}. It is more general than {\tt REPEAT_DETERM}, but 

99 
requires more space. 

100 

101 
\item[\ttindexbold{REPEAT1} {\it tac}] 

102 
is like \hbox{\tt REPEAT {\it tac}} but it always applies {\it tac\/} at 

103 
least once, failing if this is impossible. 

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\item[\ttindexbold{DETERM_UNTIL} {\it p} {\it tac}] 
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applies {\it tac\/} to the proof state and, recursively, to the head of the 
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resulting sequence, until the predicate {\it p} (applied on the proof state) 
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yields {\it true}. It fails if {\it tac\/} fails on any of the intermediate 
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states. It is deterministic, discarding alternative outcomes. 
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4317  111 
\item[set \ttindexbold{trace_REPEAT};] 
286  112 
enables an interactive tracing mode for the tacticals {\tt REPEAT_DETERM} 
113 
and {\tt REPEAT}. To view the tracing options, type {\tt h} at the prompt. 

323  114 
\end{ttdescription} 
104  115 

116 

117 
\subsection{Identities for tacticals} 

323  118 
\index{tacticals!identities for} 
104  119 
\begin{ttbox} 
120 
all_tac : tactic 

121 
no_tac : tactic 

122 
\end{ttbox} 

323  123 
\begin{ttdescription} 
104  124 
\item[\ttindexbold{all_tac}] 
125 
maps any proof state to the oneelement sequence containing that state. 

126 
Thus, it succeeds for all states. It is the identity element of the 

127 
tactical \ttindex{THEN}\@. 

128 

129 
\item[\ttindexbold{no_tac}] 

130 
maps any proof state to the empty sequence. Thus it succeeds for no state. 

131 
It is the identity element of \ttindex{ORELSE}, \ttindex{APPEND}, and 

132 
\ttindex{INTLEAVE}\@. Also, it is a zero element for \ttindex{THEN}, which means that 

133 
\hbox{\tt$tac$ THEN no_tac} is equivalent to {\tt no_tac}. 

323  134 
\end{ttdescription} 
104  135 
These primitive tactics are useful when writing tacticals. For example, 
136 
\ttindexbold{TRY} and \ttindexbold{REPEAT} (ignoring tracing) can be coded 

137 
as follows: 

138 
\begin{ttbox} 

139 
fun TRY tac = tac ORELSE all_tac; 

140 

3108  141 
fun REPEAT tac = 
142 
(fn state => ((tac THEN REPEAT tac) ORELSE all_tac) state); 

104  143 
\end{ttbox} 
144 
If $tac$ can return multiple outcomes then so can \hbox{\tt REPEAT $tac$}. 

145 
Since {\tt REPEAT} uses \ttindex{ORELSE} and not {\tt APPEND} or {\tt 

146 
INTLEAVE}, it applies $tac$ as many times as possible in each 

147 
outcome. 

148 

149 
\begin{warn} 

150 
Note {\tt REPEAT}'s explicit abstraction over the proof state. Recursive 

151 
tacticals must be coded in this awkward fashion to avoid infinite 

152 
recursion. With the following definition, \hbox{\tt REPEAT $tac$} would 

332  153 
loop due to \ML's eager evaluation strategy: 
104  154 
\begin{ttbox} 
155 
fun REPEAT tac = (tac THEN REPEAT tac) ORELSE all_tac; 

156 
\end{ttbox} 

157 
\par\noindent 

158 
The builtin {\tt REPEAT} avoids~{\tt THEN}, handling sequences explicitly 

159 
and using tail recursion. This sacrifices clarity, but saves much space by 

160 
discarding intermediate proof states. 

161 
\end{warn} 

162 

163 

164 
\section{Control and search tacticals} 

323  165 
\index{search!tacticals(} 
166 

104  167 
A predicate on theorems, namely a function of type \hbox{\tt thm>bool}, 
168 
can test whether a proof state enjoys some desirable property  such as 

169 
having no subgoals. Tactics that search for satisfactory states are easy 

170 
to express. The main search procedures, depthfirst, breadthfirst and 

171 
bestfirst, are provided as tacticals. They generate the search tree by 

172 
repeatedly applying a given tactic. 

173 

174 

175 
\subsection{Filtering a tactic's results} 

323  176 
\index{tacticals!for filtering} 
177 
\index{tactics!filtering results of} 

104  178 
\begin{ttbox} 
179 
FILTER : (thm > bool) > tactic > tactic 

180 
CHANGED : tactic > tactic 

181 
\end{ttbox} 

323  182 
\begin{ttdescription} 
1118  183 
\item[\ttindexbold{FILTER} {\it p} $tac$] 
104  184 
applies $tac$ to the proof state and returns a sequence consisting of those 
185 
result states that satisfy~$p$. 

186 

187 
\item[\ttindexbold{CHANGED} {\it tac}] 

188 
applies {\it tac\/} to the proof state and returns precisely those states 

189 
that differ from the original state. Thus, \hbox{\tt CHANGED {\it tac}} 

190 
always has some effect on the state. 

323  191 
\end{ttdescription} 
104  192 

193 

194 
\subsection{Depthfirst search} 

323  195 
\index{tacticals!searching} 
104  196 
\index{tracing!of searching tacticals} 
197 
\begin{ttbox} 

198 
DEPTH_FIRST : (thm>bool) > tactic > tactic 

332  199 
DEPTH_SOLVE : tactic > tactic 
200 
DEPTH_SOLVE_1 : tactic > tactic 

104  201 
trace_DEPTH_FIRST: bool ref \hfill{\bf initially false} 
202 
\end{ttbox} 

323  203 
\begin{ttdescription} 
104  204 
\item[\ttindexbold{DEPTH_FIRST} {\it satp} {\it tac}] 
205 
returns the proof state if {\it satp} returns true. Otherwise it applies 

206 
{\it tac}, then recursively searches from each element of the resulting 

207 
sequence. The code uses a stack for efficiency, in effect applying 

208 
\hbox{\tt {\it tac} THEN DEPTH_FIRST {\it satp} {\it tac}} to the state. 

209 

210 
\item[\ttindexbold{DEPTH_SOLVE} {\it tac}] 

211 
uses {\tt DEPTH_FIRST} to search for states having no subgoals. 

212 

213 
\item[\ttindexbold{DEPTH_SOLVE_1} {\it tac}] 

214 
uses {\tt DEPTH_FIRST} to search for states having fewer subgoals than the 

215 
given state. Thus, it insists upon solving at least one subgoal. 

216 

4317  217 
\item[set \ttindexbold{trace_DEPTH_FIRST};] 
104  218 
enables interactive tracing for {\tt DEPTH_FIRST}. To view the 
219 
tracing options, type {\tt h} at the prompt. 

323  220 
\end{ttdescription} 
104  221 

222 

223 
\subsection{Other search strategies} 

323  224 
\index{tacticals!searching} 
104  225 
\index{tracing!of searching tacticals} 
226 
\begin{ttbox} 

332  227 
BREADTH_FIRST : (thm>bool) > tactic > tactic 
104  228 
BEST_FIRST : (thm>bool)*(thm>int) > tactic > tactic 
229 
THEN_BEST_FIRST : tactic * ((thm>bool) * (thm>int) * tactic) 

230 
> tactic \hfill{\bf infix 1} 

231 
trace_BEST_FIRST: bool ref \hfill{\bf initially false} 

232 
\end{ttbox} 

233 
These search strategies will find a solution if one exists. However, they 

234 
do not enumerate all solutions; they terminate after the first satisfactory 

235 
result from {\it tac}. 

323  236 
\begin{ttdescription} 
104  237 
\item[\ttindexbold{BREADTH_FIRST} {\it satp} {\it tac}] 
238 
uses breadthfirst search to find states for which {\it satp\/} is true. 

239 
For most applications, it is too slow. 

240 

241 
\item[\ttindexbold{BEST_FIRST} $(satp,distf)$ {\it tac}] 

242 
does a heuristic search, using {\it distf\/} to estimate the distance from 

243 
a satisfactory state. It maintains a list of states ordered by distance. 

244 
It applies $tac$ to the head of this list; if the result contains any 

245 
satisfactory states, then it returns them. Otherwise, {\tt BEST_FIRST} 

246 
adds the new states to the list, and continues. 

247 

248 
The distance function is typically \ttindex{size_of_thm}, which computes 

249 
the size of the state. The smaller the state, the fewer and simpler 

250 
subgoals it has. 

251 

252 
\item[$tac@0$ \ttindexbold{THEN_BEST_FIRST} $(satp,distf,tac)$] 

253 
is like {\tt BEST_FIRST}, except that the priority queue initially 

254 
contains the result of applying $tac@0$ to the proof state. This tactical 

255 
permits separate tactics for starting the search and continuing the search. 

256 

4317  257 
\item[set \ttindexbold{trace_BEST_FIRST};] 
286  258 
enables an interactive tracing mode for the tactical {\tt BEST_FIRST}. To 
259 
view the tracing options, type {\tt h} at the prompt. 

323  260 
\end{ttdescription} 
104  261 

262 

263 
\subsection{Auxiliary tacticals for searching} 

264 
\index{tacticals!conditional} 

265 
\index{tacticals!deterministic} 

266 
\begin{ttbox} 

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COND : (thm>bool) > tactic > tactic > tactic 
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IF_UNSOLVED : tactic > tactic 
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SOLVE : tactic > tactic 
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DETERM : tactic > tactic 
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DETERM_UNTIL_SOLVED : tactic > tactic 
104  272 
\end{ttbox} 
323  273 
\begin{ttdescription} 
1118  274 
\item[\ttindexbold{COND} {\it p} $tac@1$ $tac@2$] 
104  275 
applies $tac@1$ to the proof state if it satisfies~$p$, and applies $tac@2$ 
276 
otherwise. It is a conditional tactical in that only one of $tac@1$ and 

277 
$tac@2$ is applied to a proof state. However, both $tac@1$ and $tac@2$ are 

278 
evaluated because \ML{} uses eager evaluation. 

279 

280 
\item[\ttindexbold{IF_UNSOLVED} {\it tac}] 

281 
applies {\it tac\/} to the proof state if it has any subgoals, and simply 

282 
returns the proof state otherwise. Many common tactics, such as {\tt 

283 
resolve_tac}, fail if applied to a proof state that has no subgoals. 

284 

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\item[\ttindexbold{SOLVE} {\it tac}] 
286 
applies {\it tac\/} to the proof state and then fails iff there are subgoals 

287 
left. 

288 

104  289 
\item[\ttindexbold{DETERM} {\it tac}] 
290 
applies {\it tac\/} to the proof state and returns the head of the 

291 
resulting sequence. {\tt DETERM} limits the search space by making its 

292 
argument deterministic. 

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\item[\ttindexbold{DETERM_UNTIL_SOLVED} {\it tac}] 
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forces repeated deterministic application of {\it tac\/} to the proof state 
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until the goal is solved completely. 
323  297 
\end{ttdescription} 
104  298 

299 

300 
\subsection{Predicates and functions useful for searching} 

301 
\index{theorems!size of} 

302 
\index{theorems!equality of} 

303 
\begin{ttbox} 

304 
has_fewer_prems : int > thm > bool 

305 
eq_thm : thm * thm > bool 

306 
size_of_thm : thm > int 

307 
\end{ttbox} 

323  308 
\begin{ttdescription} 
104  309 
\item[\ttindexbold{has_fewer_prems} $n$ $thm$] 
310 
reports whether $thm$ has fewer than~$n$ premises. By currying, 

311 
\hbox{\tt has_fewer_prems $n$} is a predicate on theorems; it may 

312 
be given to the searching tacticals. 

313 

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\item[\ttindexbold{eq_thm} ($thm@1$, $thm@2$)] reports whether $thm@1$ and 
315 
$thm@2$ are equal. Both theorems must have identical signatures. Both 

316 
theorems must have the same conclusions, and the same hypotheses, in the 

317 
same order. Names of bound variables are ignored. 

104  318 

319 
\item[\ttindexbold{size_of_thm} $thm$] 

320 
computes the size of $thm$, namely the number of variables, constants and 

321 
abstractions in its conclusion. It may serve as a distance function for 

322 
\ttindex{BEST_FIRST}. 

323  323 
\end{ttdescription} 
324 

325 
\index{search!tacticals)} 

104  326 

327 

328 
\section{Tacticals for subgoal numbering} 

329 
When conducting a backward proof, we normally consider one goal at a time. 

330 
A tactic can affect the entire proof state, but many tactics  such as 

331 
{\tt resolve_tac} and {\tt assume_tac}  work on a single subgoal. 

332 
Subgoals are designated by a positive integer, so Isabelle provides 

333 
tacticals for combining values of type {\tt int>tactic}. 

334 

335 

336 
\subsection{Restricting a tactic to one subgoal} 

337 
\index{tactics!restricting to a subgoal} 

338 
\index{tacticals!for restriction to a subgoal} 

339 
\begin{ttbox} 

340 
SELECT_GOAL : tactic > int > tactic 

341 
METAHYPS : (thm list > tactic) > int > tactic 

342 
\end{ttbox} 

323  343 
\begin{ttdescription} 
104  344 
\item[\ttindexbold{SELECT_GOAL} {\it tac} $i$] 
345 
restricts the effect of {\it tac\/} to subgoal~$i$ of the proof state. It 

346 
fails if there is no subgoal~$i$, or if {\it tac\/} changes the main goal 

347 
(do not use {\tt rewrite_tac}). It applies {\it tac\/} to a dummy proof 

348 
state and uses the result to refine the original proof state at 

349 
subgoal~$i$. If {\it tac\/} returns multiple results then so does 

350 
\hbox{\tt SELECT_GOAL {\it tac} $i$}. 

351 

323  352 
{\tt SELECT_GOAL} works by creating a state of the form $\phi\Imp\phi$, 
332  353 
with the one subgoal~$\phi$. If subgoal~$i$ has the form $\psi\Imp\theta$ 
354 
then $(\psi\Imp\theta)\Imp(\psi\Imp\theta)$ is in fact 

355 
$\List{\psi\Imp\theta;\; \psi}\Imp\theta$, a proof state with two subgoals. 

356 
Such a proof state might cause tactics to go astray. Therefore {\tt 

357 
SELECT_GOAL} inserts a quantifier to create the state 

323  358 
\[ (\Forall x.\psi\Imp\theta)\Imp(\Forall x.\psi\Imp\theta). \] 
104  359 

323  360 
\item[\ttindexbold{METAHYPS} {\it tacf} $i$]\index{metaassumptions} 
104  361 
takes subgoal~$i$, of the form 
362 
\[ \Forall x@1 \ldots x@l. \List{\theta@1; \ldots; \theta@k}\Imp\theta, \] 

363 
and creates the list $\theta'@1$, \ldots, $\theta'@k$ of metalevel 

364 
assumptions. In these theorems, the subgoal's parameters ($x@1$, 

365 
\ldots,~$x@l$) become free variables. It supplies the assumptions to 

366 
$tacf$ and applies the resulting tactic to the proof state 

367 
$\theta\Imp\theta$. 

368 

369 
If the resulting proof state is $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, 

370 
possibly containing $\theta'@1,\ldots,\theta'@k$ as assumptions, then it is 

371 
lifted back into the original context, yielding $n$ subgoals. 

372 

286  373 
Metalevel assumptions may not contain unknowns. Unknowns in the 
374 
hypotheses $\theta@1,\ldots,\theta@k$ become free variables in $\theta'@1$, 

375 
\ldots, $\theta'@k$, and are restored afterwards; the {\tt METAHYPS} call 

376 
cannot instantiate them. Unknowns in $\theta$ may be instantiated. New 

323  377 
unknowns in $\phi@1$, \ldots, $\phi@n$ are lifted over the parameters. 
104  378 

379 
Here is a typical application. Calling {\tt hyp_res_tac}~$i$ resolves 

380 
subgoal~$i$ with one of its own assumptions, which may itself have the form 

381 
of an inference rule (these are called {\bf higherlevel assumptions}). 

382 
\begin{ttbox} 

383 
val hyp_res_tac = METAHYPS (fn prems => resolve_tac prems 1); 

384 
\end{ttbox} 

332  385 
The function \ttindex{gethyps} is useful for debugging applications of {\tt 
386 
METAHYPS}. 

323  387 
\end{ttdescription} 
104  388 

389 
\begin{warn} 

390 
{\tt METAHYPS} fails if the context or new subgoals contain type unknowns. 

391 
In principle, the tactical could treat these like ordinary unknowns. 

392 
\end{warn} 

393 

394 

395 
\subsection{Scanning for a subgoal by number} 

323  396 
\index{tacticals!scanning for subgoals} 
104  397 
\begin{ttbox} 
398 
ALLGOALS : (int > tactic) > tactic 

399 
TRYALL : (int > tactic) > tactic 

400 
SOMEGOAL : (int > tactic) > tactic 

401 
FIRSTGOAL : (int > tactic) > tactic 

402 
REPEAT_SOME : (int > tactic) > tactic 

403 
REPEAT_FIRST : (int > tactic) > tactic 

404 
trace_goalno_tac : (int > tactic) > int > tactic 

405 
\end{ttbox} 

406 
These apply a tactic function of type {\tt int > tactic} to all the 

407 
subgoal numbers of a proof state, and join the resulting tactics using 

408 
\ttindex{THEN} or \ttindex{ORELSE}\@. Thus, they apply the tactic to all the 

409 
subgoals, or to one subgoal. 

410 

411 
Suppose that the original proof state has $n$ subgoals. 

412 

323  413 
\begin{ttdescription} 
104  414 
\item[\ttindexbold{ALLGOALS} {\it tacf}] 
415 
is equivalent to 

416 
\hbox{\tt$tacf(n)$ THEN \ldots{} THEN $tacf(1)$}. 

417 

323  418 
It applies {\it tacf} to all the subgoals, counting downwards (to 
104  419 
avoid problems when subgoals are added or deleted). 
420 

421 
\item[\ttindexbold{TRYALL} {\it tacf}] 

422 
is equivalent to 

323  423 
\hbox{\tt TRY$(tacf(n))$ THEN \ldots{} THEN TRY$(tacf(1))$}. 
104  424 

425 
It attempts to apply {\it tacf} to all the subgoals. For instance, 

286  426 
the tactic \hbox{\tt TRYALL assume_tac} attempts to solve all the subgoals by 
104  427 
assumption. 
428 

429 
\item[\ttindexbold{SOMEGOAL} {\it tacf}] 

430 
is equivalent to 

431 
\hbox{\tt$tacf(n)$ ORELSE \ldots{} ORELSE $tacf(1)$}. 

432 

323  433 
It applies {\it tacf} to one subgoal, counting downwards. For instance, 
286  434 
the tactic \hbox{\tt SOMEGOAL assume_tac} solves one subgoal by assumption, 
435 
failing if this is impossible. 

104  436 

437 
\item[\ttindexbold{FIRSTGOAL} {\it tacf}] 

438 
is equivalent to 

439 
\hbox{\tt$tacf(1)$ ORELSE \ldots{} ORELSE $tacf(n)$}. 

440 

323  441 
It applies {\it tacf} to one subgoal, counting upwards. 
104  442 

443 
\item[\ttindexbold{REPEAT_SOME} {\it tacf}] 

323  444 
applies {\it tacf} once or more to a subgoal, counting downwards. 
104  445 

446 
\item[\ttindexbold{REPEAT_FIRST} {\it tacf}] 

323  447 
applies {\it tacf} once or more to a subgoal, counting upwards. 
104  448 

449 
\item[\ttindexbold{trace_goalno_tac} {\it tac} {\it i}] 

450 
applies \hbox{\it tac i\/} to the proof state. If the resulting sequence 

451 
is nonempty, then it is returned, with the sideeffect of printing {\tt 

452 
Subgoal~$i$ selected}. Otherwise, {\tt trace_goalno_tac} returns the empty 

453 
sequence and prints nothing. 

454 

323  455 
It indicates that `the tactic worked for subgoal~$i$' and is mainly used 
104  456 
with {\tt SOMEGOAL} and {\tt FIRSTGOAL}. 
323  457 
\end{ttdescription} 
104  458 

459 

460 
\subsection{Joining tactic functions} 

323  461 
\index{tacticals!joining tactic functions} 
104  462 
\begin{ttbox} 
463 
THEN' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix 1} 

464 
ORELSE' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix} 

465 
APPEND' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix} 

466 
INTLEAVE' : ('a > tactic) * ('a > tactic) > 'a > tactic \hfill{\bf infix} 

467 
EVERY' : ('a > tactic) list > 'a > tactic 

468 
FIRST' : ('a > tactic) list > 'a > tactic 

469 
\end{ttbox} 

470 
These help to express tactics that specify subgoal numbers. The tactic 

471 
\begin{ttbox} 

472 
SOMEGOAL (fn i => resolve_tac rls i ORELSE eresolve_tac erls i) 

473 
\end{ttbox} 

474 
can be simplified to 

475 
\begin{ttbox} 

476 
SOMEGOAL (resolve_tac rls ORELSE' eresolve_tac erls) 

477 
\end{ttbox} 

478 
Note that {\tt TRY'}, {\tt REPEAT'}, {\tt DEPTH_FIRST'}, etc.\ are not 

479 
provided, because function composition accomplishes the same purpose. 

480 
The tactic 

481 
\begin{ttbox} 

482 
ALLGOALS (fn i => REPEAT (etac exE i ORELSE atac i)) 

483 
\end{ttbox} 

484 
can be simplified to 

485 
\begin{ttbox} 

486 
ALLGOALS (REPEAT o (etac exE ORELSE' atac)) 

487 
\end{ttbox} 

488 
These tacticals are polymorphic; $x$ need not be an integer. 

489 
\begin{center} \tt 

490 
\begin{tabular}{r@{\rm\ \ yields\ \ }l} 

323  491 
$(tacf@1$~~THEN'~~$tacf@2)(x)$ \index{*THEN'} & 
104  492 
$tacf@1(x)$~~THEN~~$tacf@2(x)$ \\ 
493 

323  494 
$(tacf@1$ ORELSE' $tacf@2)(x)$ \index{*ORELSE'} & 
104  495 
$tacf@1(x)$ ORELSE $tacf@2(x)$ \\ 
496 

323  497 
$(tacf@1$ APPEND' $tacf@2)(x)$ \index{*APPEND'} & 
104  498 
$tacf@1(x)$ APPEND $tacf@2(x)$ \\ 
499 

323  500 
$(tacf@1$ INTLEAVE' $tacf@2)(x)$ \index{*INTLEAVE'} & 
104  501 
$tacf@1(x)$ INTLEAVE $tacf@2(x)$ \\ 
502 

503 
EVERY' $[tacf@1,\ldots,tacf@n] \; (x)$ \index{*EVERY'} & 

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

505 

506 
FIRST' $[tacf@1,\ldots,tacf@n] \; (x)$ \index{*FIRST'} & 

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

508 
\end{tabular} 

509 
\end{center} 

510 

511 

512 
\subsection{Applying a list of tactics to 1} 

323  513 
\index{tacticals!joining tactic functions} 
104  514 
\begin{ttbox} 
515 
EVERY1: (int > tactic) list > tactic 

516 
FIRST1: (int > tactic) list > tactic 

517 
\end{ttbox} 

518 
A common proof style is to treat the subgoals as a stack, always 

519 
restricting attention to the first subgoal. Such proofs contain long lists 

520 
of tactics, each applied to~1. These can be simplified using {\tt EVERY1} 

521 
and {\tt FIRST1}: 

522 
\begin{center} \tt 

523 
\begin{tabular}{r@{\rm\ \ abbreviates\ \ }l} 

524 
EVERY1 $[tacf@1,\ldots,tacf@n]$ \indexbold{*EVERY1} & 

525 
EVERY $[tacf@1(1),\ldots,tacf@n(1)]$ \\ 

526 

527 
FIRST1 $[tacf@1,\ldots,tacf@n]$ \indexbold{*FIRST1} & 

528 
FIRST $[tacf@1(1),\ldots,tacf@n(1)]$ 

529 
\end{tabular} 

530 
\end{center} 

531 

532 
\index{tacticals)} 

5371  533 

534 

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