lcp@104
|
1 |
%% $Id$
|
lcp@319
|
2 |
\chapter{The Classical Reasoner}\label{chap:classical}
|
lcp@286
|
3 |
\index{classical reasoner|(}
|
lcp@308
|
4 |
\newcommand\ainfer[2]{\begin{array}{r@{\,}l}#2\\ \hline#1\end{array}}
|
lcp@308
|
5 |
|
wenzelm@3108
|
6 |
Although Isabelle is generic, many users will be working in some
|
wenzelm@3108
|
7 |
extension of classical first-order logic. Isabelle's set theory~{\tt
|
wenzelm@3108
|
8 |
ZF} is built upon theory~{\tt FOL}, while higher-order logic
|
wenzelm@3108
|
9 |
conceptually contains first-order logic as a fragment.
|
wenzelm@3108
|
10 |
Theorem-proving in predicate logic is undecidable, but many
|
lcp@308
|
11 |
researchers have developed strategies to assist in this task.
|
lcp@104
|
12 |
|
lcp@286
|
13 |
Isabelle's classical reasoner is an \ML{} functor that accepts certain
|
lcp@104
|
14 |
information about a logic and delivers a suite of automatic tactics. Each
|
lcp@104
|
15 |
tactic takes a collection of rules and executes a simple, non-clausal proof
|
lcp@104
|
16 |
procedure. They are slow and simplistic compared with resolution theorem
|
lcp@104
|
17 |
provers, but they can save considerable time and effort. They can prove
|
lcp@104
|
18 |
theorems such as Pelletier's~\cite{pelletier86} problems~40 and~41 in
|
lcp@104
|
19 |
seconds:
|
lcp@104
|
20 |
\[ (\exists y. \forall x. J(y,x) \bimp \neg J(x,x))
|
lcp@104
|
21 |
\imp \neg (\forall x. \exists y. \forall z. J(z,y) \bimp \neg J(z,x)) \]
|
lcp@104
|
22 |
\[ (\forall z. \exists y. \forall x. F(x,y) \bimp F(x,z) \conj \neg F(x,x))
|
lcp@104
|
23 |
\imp \neg (\exists z. \forall x. F(x,z))
|
lcp@104
|
24 |
\]
|
lcp@308
|
25 |
%
|
lcp@308
|
26 |
The tactics are generic. They are not restricted to first-order logic, and
|
lcp@308
|
27 |
have been heavily used in the development of Isabelle's set theory. Few
|
lcp@308
|
28 |
interactive proof assistants provide this much automation. The tactics can
|
lcp@308
|
29 |
be traced, and their components can be called directly; in this manner,
|
lcp@308
|
30 |
any proof can be viewed interactively.
|
lcp@104
|
31 |
|
paulson@2479
|
32 |
The simplest way to apply the classical reasoner (to subgoal~$i$) is as
|
paulson@2479
|
33 |
follows:
|
paulson@2479
|
34 |
\begin{ttbox}
|
paulson@3089
|
35 |
by (Blast_tac \(i\));
|
paulson@2479
|
36 |
\end{ttbox}
|
paulson@2479
|
37 |
If the subgoal is a simple formula of the predicate calculus or set theory,
|
paulson@2479
|
38 |
then it should be proved quickly. However, to use the classical reasoner
|
paulson@3089
|
39 |
effectively, you need to know how it works and how to choose among the many
|
paulson@3089
|
40 |
tactics available, including {\tt Fast_tac} and {\tt Best_tac}.
|
paulson@2479
|
41 |
|
wenzelm@3108
|
42 |
We shall first discuss the underlying principles, then present the
|
wenzelm@3108
|
43 |
classical reasoner. Finally, we shall see how to instantiate it for
|
wenzelm@3108
|
44 |
new logics. The logics \FOL, \ZF, {\HOL} and {\HOLCF} have it already
|
wenzelm@3108
|
45 |
installed.
|
lcp@104
|
46 |
|
lcp@104
|
47 |
|
lcp@104
|
48 |
\section{The sequent calculus}
|
lcp@104
|
49 |
\index{sequent calculus}
|
lcp@104
|
50 |
Isabelle supports natural deduction, which is easy to use for interactive
|
lcp@104
|
51 |
proof. But natural deduction does not easily lend itself to automation,
|
lcp@104
|
52 |
and has a bias towards intuitionism. For certain proofs in classical
|
lcp@104
|
53 |
logic, it can not be called natural. The {\bf sequent calculus}, a
|
lcp@104
|
54 |
generalization of natural deduction, is easier to automate.
|
lcp@104
|
55 |
|
lcp@104
|
56 |
A {\bf sequent} has the form $\Gamma\turn\Delta$, where $\Gamma$
|
lcp@308
|
57 |
and~$\Delta$ are sets of formulae.%
|
lcp@308
|
58 |
\footnote{For first-order logic, sequents can equivalently be made from
|
lcp@308
|
59 |
lists or multisets of formulae.} The sequent
|
lcp@104
|
60 |
\[ P@1,\ldots,P@m\turn Q@1,\ldots,Q@n \]
|
lcp@104
|
61 |
is {\bf valid} if $P@1\conj\ldots\conj P@m$ implies $Q@1\disj\ldots\disj
|
lcp@104
|
62 |
Q@n$. Thus $P@1,\ldots,P@m$ represent assumptions, each of which is true,
|
lcp@104
|
63 |
while $Q@1,\ldots,Q@n$ represent alternative goals. A sequent is {\bf
|
lcp@104
|
64 |
basic} if its left and right sides have a common formula, as in $P,Q\turn
|
lcp@104
|
65 |
Q,R$; basic sequents are trivially valid.
|
lcp@104
|
66 |
|
lcp@104
|
67 |
Sequent rules are classified as {\bf right} or {\bf left}, indicating which
|
lcp@104
|
68 |
side of the $\turn$~symbol they operate on. Rules that operate on the
|
lcp@104
|
69 |
right side are analogous to natural deduction's introduction rules, and
|
lcp@308
|
70 |
left rules are analogous to elimination rules.
|
lcp@308
|
71 |
Recall the natural deduction rules for
|
lcp@308
|
72 |
first-order logic,
|
lcp@308
|
73 |
\iflabelundefined{fol-fig}{from {\it Introduction to Isabelle}}%
|
lcp@308
|
74 |
{Fig.\ts\ref{fol-fig}}.
|
lcp@308
|
75 |
The sequent calculus analogue of~$({\imp}I)$ is the rule
|
wenzelm@3108
|
76 |
$$
|
wenzelm@3108
|
77 |
\ainfer{\Gamma &\turn \Delta, P\imp Q}{P,\Gamma &\turn \Delta,Q}
|
wenzelm@3108
|
78 |
\eqno({\imp}R)
|
wenzelm@3108
|
79 |
$$
|
lcp@104
|
80 |
This breaks down some implication on the right side of a sequent; $\Gamma$
|
lcp@104
|
81 |
and $\Delta$ stand for the sets of formulae that are unaffected by the
|
lcp@104
|
82 |
inference. The analogue of the pair~$({\disj}I1)$ and~$({\disj}I2)$ is the
|
lcp@104
|
83 |
single rule
|
wenzelm@3108
|
84 |
$$
|
wenzelm@3108
|
85 |
\ainfer{\Gamma &\turn \Delta, P\disj Q}{\Gamma &\turn \Delta,P,Q}
|
wenzelm@3108
|
86 |
\eqno({\disj}R)
|
wenzelm@3108
|
87 |
$$
|
lcp@104
|
88 |
This breaks down some disjunction on the right side, replacing it by both
|
lcp@104
|
89 |
disjuncts. Thus, the sequent calculus is a kind of multiple-conclusion logic.
|
lcp@104
|
90 |
|
lcp@104
|
91 |
To illustrate the use of multiple formulae on the right, let us prove
|
lcp@104
|
92 |
the classical theorem $(P\imp Q)\disj(Q\imp P)$. Working backwards, we
|
lcp@104
|
93 |
reduce this formula to a basic sequent:
|
lcp@104
|
94 |
\[ \infer[(\disj)R]{\turn(P\imp Q)\disj(Q\imp P)}
|
lcp@104
|
95 |
{\infer[(\imp)R]{\turn(P\imp Q), (Q\imp P)\;}
|
lcp@104
|
96 |
{\infer[(\imp)R]{P \turn Q, (Q\imp P)\qquad}
|
lcp@104
|
97 |
{P, Q \turn Q, P\qquad\qquad}}}
|
lcp@104
|
98 |
\]
|
lcp@104
|
99 |
This example is typical of the sequent calculus: start with the desired
|
lcp@104
|
100 |
theorem and apply rules backwards in a fairly arbitrary manner. This yields a
|
lcp@104
|
101 |
surprisingly effective proof procedure. Quantifiers add few complications,
|
lcp@104
|
102 |
since Isabelle handles parameters and schematic variables. See Chapter~10
|
lcp@104
|
103 |
of {\em ML for the Working Programmer}~\cite{paulson91} for further
|
lcp@104
|
104 |
discussion.
|
lcp@104
|
105 |
|
lcp@104
|
106 |
|
lcp@104
|
107 |
\section{Simulating sequents by natural deduction}
|
lcp@308
|
108 |
Isabelle can represent sequents directly, as in the object-logic~{\tt LK}\@.
|
lcp@104
|
109 |
But natural deduction is easier to work with, and most object-logics employ
|
lcp@104
|
110 |
it. Fortunately, we can simulate the sequent $P@1,\ldots,P@m\turn
|
lcp@104
|
111 |
Q@1,\ldots,Q@n$ by the Isabelle formula
|
lcp@104
|
112 |
\[ \List{P@1;\ldots;P@m; \neg Q@2;\ldots; \neg Q@n}\Imp Q@1, \]
|
lcp@104
|
113 |
where the order of the assumptions and the choice of~$Q@1$ are arbitrary.
|
lcp@104
|
114 |
Elim-resolution plays a key role in simulating sequent proofs.
|
lcp@104
|
115 |
|
lcp@104
|
116 |
We can easily handle reasoning on the left.
|
lcp@308
|
117 |
As discussed in
|
lcp@308
|
118 |
\iflabelundefined{destruct}{{\it Introduction to Isabelle}}{\S\ref{destruct}},
|
lcp@104
|
119 |
elim-resolution with the rules $(\disj E)$, $(\bot E)$ and $(\exists E)$
|
lcp@104
|
120 |
achieves a similar effect as the corresponding sequent rules. For the
|
lcp@104
|
121 |
other connectives, we use sequent-style elimination rules instead of
|
lcp@308
|
122 |
destruction rules such as $({\conj}E1,2)$ and $(\forall E)$. But note that
|
lcp@308
|
123 |
the rule $(\neg L)$ has no effect under our representation of sequents!
|
wenzelm@3108
|
124 |
$$
|
wenzelm@3108
|
125 |
\ainfer{\neg P,\Gamma &\turn \Delta}{\Gamma &\turn \Delta,P}\eqno({\neg}L)
|
wenzelm@3108
|
126 |
$$
|
lcp@104
|
127 |
What about reasoning on the right? Introduction rules can only affect the
|
lcp@308
|
128 |
formula in the conclusion, namely~$Q@1$. The other right-side formulae are
|
lcp@319
|
129 |
represented as negated assumptions, $\neg Q@2$, \ldots,~$\neg Q@n$.
|
lcp@319
|
130 |
\index{assumptions!negated}
|
lcp@319
|
131 |
In order to operate on one of these, it must first be exchanged with~$Q@1$.
|
lcp@104
|
132 |
Elim-resolution with the {\bf swap} rule has this effect:
|
wenzelm@3108
|
133 |
$$ \List{\neg P; \; \neg R\Imp P} \Imp R \eqno(swap) $$
|
lcp@104
|
134 |
To ensure that swaps occur only when necessary, each introduction rule is
|
lcp@104
|
135 |
converted into a swapped form: it is resolved with the second premise
|
lcp@104
|
136 |
of~$(swap)$. The swapped form of~$({\conj}I)$, which might be
|
lcp@104
|
137 |
called~$({\neg\conj}E)$, is
|
lcp@104
|
138 |
\[ \List{\neg(P\conj Q); \; \neg R\Imp P; \; \neg R\Imp Q} \Imp R. \]
|
lcp@104
|
139 |
Similarly, the swapped form of~$({\imp}I)$ is
|
lcp@104
|
140 |
\[ \List{\neg(P\imp Q); \; \List{\neg R;P}\Imp Q} \Imp R \]
|
lcp@104
|
141 |
Swapped introduction rules are applied using elim-resolution, which deletes
|
lcp@104
|
142 |
the negated formula. Our representation of sequents also requires the use
|
lcp@104
|
143 |
of ordinary introduction rules. If we had no regard for readability, we
|
lcp@104
|
144 |
could treat the right side more uniformly by representing sequents as
|
lcp@104
|
145 |
\[ \List{P@1;\ldots;P@m; \neg Q@1;\ldots; \neg Q@n}\Imp \bot. \]
|
lcp@104
|
146 |
|
lcp@104
|
147 |
|
lcp@104
|
148 |
\section{Extra rules for the sequent calculus}
|
lcp@104
|
149 |
As mentioned, destruction rules such as $({\conj}E1,2)$ and $(\forall E)$
|
lcp@104
|
150 |
must be replaced by sequent-style elimination rules. In addition, we need
|
lcp@104
|
151 |
rules to embody the classical equivalence between $P\imp Q$ and $\neg P\disj
|
lcp@104
|
152 |
Q$. The introduction rules~$({\disj}I1,2)$ are replaced by a rule that
|
lcp@104
|
153 |
simulates $({\disj}R)$:
|
lcp@104
|
154 |
\[ (\neg Q\Imp P) \Imp P\disj Q \]
|
lcp@104
|
155 |
The destruction rule $({\imp}E)$ is replaced by
|
lcp@332
|
156 |
\[ \List{P\imp Q;\; \neg P\Imp R;\; Q\Imp R} \Imp R. \]
|
lcp@104
|
157 |
Quantifier replication also requires special rules. In classical logic,
|
lcp@308
|
158 |
$\exists x{.}P$ is equivalent to $\neg\forall x{.}\neg P$; the rules
|
lcp@308
|
159 |
$(\exists R)$ and $(\forall L)$ are dual:
|
lcp@104
|
160 |
\[ \ainfer{\Gamma &\turn \Delta, \exists x{.}P}
|
lcp@104
|
161 |
{\Gamma &\turn \Delta, \exists x{.}P, P[t/x]} \; (\exists R)
|
lcp@104
|
162 |
\qquad
|
lcp@104
|
163 |
\ainfer{\forall x{.}P, \Gamma &\turn \Delta}
|
lcp@104
|
164 |
{P[t/x], \forall x{.}P, \Gamma &\turn \Delta} \; (\forall L)
|
lcp@104
|
165 |
\]
|
lcp@104
|
166 |
Thus both kinds of quantifier may be replicated. Theorems requiring
|
lcp@104
|
167 |
multiple uses of a universal formula are easy to invent; consider
|
lcp@308
|
168 |
\[ (\forall x.P(x)\imp P(f(x))) \conj P(a) \imp P(f^n(a)), \]
|
lcp@308
|
169 |
for any~$n>1$. Natural examples of the multiple use of an existential
|
lcp@308
|
170 |
formula are rare; a standard one is $\exists x.\forall y. P(x)\imp P(y)$.
|
lcp@104
|
171 |
|
lcp@104
|
172 |
Forgoing quantifier replication loses completeness, but gains decidability,
|
lcp@104
|
173 |
since the search space becomes finite. Many useful theorems can be proved
|
lcp@104
|
174 |
without replication, and the search generally delivers its verdict in a
|
lcp@104
|
175 |
reasonable time. To adopt this approach, represent the sequent rules
|
lcp@104
|
176 |
$(\exists R)$, $(\exists L)$ and $(\forall R)$ by $(\exists I)$, $(\exists
|
lcp@104
|
177 |
E)$ and $(\forall I)$, respectively, and put $(\forall E)$ into elimination
|
lcp@104
|
178 |
form:
|
lcp@104
|
179 |
$$ \List{\forall x{.}P(x); P(t)\Imp Q} \Imp Q \eqno(\forall E@2) $$
|
lcp@104
|
180 |
Elim-resolution with this rule will delete the universal formula after a
|
lcp@104
|
181 |
single use. To replicate universal quantifiers, replace the rule by
|
wenzelm@3108
|
182 |
$$
|
wenzelm@3108
|
183 |
\List{\forall x{.}P(x);\; \List{P(t); \forall x{.}P(x)}\Imp Q} \Imp Q.
|
wenzelm@3108
|
184 |
\eqno(\forall E@3)
|
wenzelm@3108
|
185 |
$$
|
lcp@104
|
186 |
To replicate existential quantifiers, replace $(\exists I)$ by
|
lcp@332
|
187 |
\[ \List{\neg(\exists x{.}P(x)) \Imp P(t)} \Imp \exists x{.}P(x). \]
|
lcp@104
|
188 |
All introduction rules mentioned above are also useful in swapped form.
|
lcp@104
|
189 |
|
lcp@104
|
190 |
Replication makes the search space infinite; we must apply the rules with
|
lcp@286
|
191 |
care. The classical reasoner distinguishes between safe and unsafe
|
lcp@104
|
192 |
rules, applying the latter only when there is no alternative. Depth-first
|
lcp@104
|
193 |
search may well go down a blind alley; best-first search is better behaved
|
lcp@104
|
194 |
in an infinite search space. However, quantifier replication is too
|
lcp@104
|
195 |
expensive to prove any but the simplest theorems.
|
lcp@104
|
196 |
|
lcp@104
|
197 |
|
lcp@104
|
198 |
\section{Classical rule sets}
|
lcp@319
|
199 |
\index{classical sets}
|
lcp@319
|
200 |
Each automatic tactic takes a {\bf classical set} --- a collection of
|
lcp@104
|
201 |
rules, classified as introduction or elimination and as {\bf safe} or {\bf
|
lcp@104
|
202 |
unsafe}. In general, safe rules can be attempted blindly, while unsafe
|
lcp@104
|
203 |
rules must be used with care. A safe rule must never reduce a provable
|
lcp@308
|
204 |
goal to an unprovable set of subgoals.
|
lcp@104
|
205 |
|
lcp@308
|
206 |
The rule~$({\disj}I1)$ is unsafe because it reduces $P\disj Q$ to~$P$. Any
|
lcp@308
|
207 |
rule is unsafe whose premises contain new unknowns. The elimination
|
lcp@308
|
208 |
rule~$(\forall E@2)$ is unsafe, since it is applied via elim-resolution,
|
lcp@308
|
209 |
which discards the assumption $\forall x{.}P(x)$ and replaces it by the
|
lcp@308
|
210 |
weaker assumption~$P(\Var{t})$. The rule $({\exists}I)$ is unsafe for
|
lcp@308
|
211 |
similar reasons. The rule~$(\forall E@3)$ is unsafe in a different sense:
|
lcp@308
|
212 |
since it keeps the assumption $\forall x{.}P(x)$, it is prone to looping.
|
lcp@308
|
213 |
In classical first-order logic, all rules are safe except those mentioned
|
lcp@308
|
214 |
above.
|
lcp@104
|
215 |
|
lcp@104
|
216 |
The safe/unsafe distinction is vague, and may be regarded merely as a way
|
lcp@104
|
217 |
of giving some rules priority over others. One could argue that
|
lcp@104
|
218 |
$({\disj}E)$ is unsafe, because repeated application of it could generate
|
lcp@104
|
219 |
exponentially many subgoals. Induction rules are unsafe because inductive
|
lcp@104
|
220 |
proofs are difficult to set up automatically. Any inference is unsafe that
|
lcp@104
|
221 |
instantiates an unknown in the proof state --- thus \ttindex{match_tac}
|
lcp@104
|
222 |
must be used, rather than \ttindex{resolve_tac}. Even proof by assumption
|
lcp@104
|
223 |
is unsafe if it instantiates unknowns shared with other subgoals --- thus
|
lcp@104
|
224 |
\ttindex{eq_assume_tac} must be used, rather than \ttindex{assume_tac}.
|
lcp@104
|
225 |
|
lcp@1099
|
226 |
\subsection{Adding rules to classical sets}
|
lcp@319
|
227 |
Classical rule sets belong to the abstract type \mltydx{claset}, which
|
lcp@286
|
228 |
supports the following operations (provided the classical reasoner is
|
lcp@104
|
229 |
installed!):
|
lcp@104
|
230 |
\begin{ttbox}
|
lcp@1099
|
231 |
empty_cs : claset
|
lcp@1099
|
232 |
print_cs : claset -> unit
|
lcp@1099
|
233 |
addSIs : claset * thm list -> claset \hfill{\bf infix 4}
|
lcp@1099
|
234 |
addSEs : claset * thm list -> claset \hfill{\bf infix 4}
|
lcp@1099
|
235 |
addSDs : claset * thm list -> claset \hfill{\bf infix 4}
|
lcp@1099
|
236 |
addIs : claset * thm list -> claset \hfill{\bf infix 4}
|
lcp@1099
|
237 |
addEs : claset * thm list -> claset \hfill{\bf infix 4}
|
lcp@1099
|
238 |
addDs : claset * thm list -> claset \hfill{\bf infix 4}
|
berghofe@1869
|
239 |
delrules : claset * thm list -> claset \hfill{\bf infix 4}
|
lcp@104
|
240 |
\end{ttbox}
|
paulson@3089
|
241 |
The add operations ignore any rule already present in the claset with the same
|
paulson@3089
|
242 |
classification (such as Safe Introduction). They print a warning if the rule
|
paulson@3089
|
243 |
has already been added with some other classification, but add the rule
|
paulson@3089
|
244 |
anyway. Calling {\tt delrules} deletes all occurrences of a rule from the
|
paulson@3089
|
245 |
claset, but see the warning below concerning destruction rules.
|
lcp@308
|
246 |
\begin{ttdescription}
|
lcp@104
|
247 |
\item[\ttindexbold{empty_cs}] is the empty classical set.
|
lcp@104
|
248 |
|
lcp@1099
|
249 |
\item[\ttindexbold{print_cs} $cs$] prints the rules of~$cs$.
|
lcp@1099
|
250 |
|
lcp@308
|
251 |
\item[$cs$ addSIs $rules$] \indexbold{*addSIs}
|
lcp@308
|
252 |
adds safe introduction~$rules$ to~$cs$.
|
lcp@104
|
253 |
|
lcp@308
|
254 |
\item[$cs$ addSEs $rules$] \indexbold{*addSEs}
|
lcp@308
|
255 |
adds safe elimination~$rules$ to~$cs$.
|
lcp@104
|
256 |
|
lcp@308
|
257 |
\item[$cs$ addSDs $rules$] \indexbold{*addSDs}
|
lcp@308
|
258 |
adds safe destruction~$rules$ to~$cs$.
|
lcp@104
|
259 |
|
lcp@308
|
260 |
\item[$cs$ addIs $rules$] \indexbold{*addIs}
|
lcp@308
|
261 |
adds unsafe introduction~$rules$ to~$cs$.
|
lcp@104
|
262 |
|
lcp@308
|
263 |
\item[$cs$ addEs $rules$] \indexbold{*addEs}
|
lcp@308
|
264 |
adds unsafe elimination~$rules$ to~$cs$.
|
lcp@104
|
265 |
|
lcp@308
|
266 |
\item[$cs$ addDs $rules$] \indexbold{*addDs}
|
lcp@308
|
267 |
adds unsafe destruction~$rules$ to~$cs$.
|
berghofe@1869
|
268 |
|
berghofe@1869
|
269 |
\item[$cs$ delrules $rules$] \indexbold{*delrules}
|
paulson@3089
|
270 |
deletes~$rules$ from~$cs$. It prints a warning for those rules that are not
|
paulson@3089
|
271 |
in~$cs$.
|
lcp@308
|
272 |
\end{ttdescription}
|
lcp@308
|
273 |
|
paulson@3089
|
274 |
\begin{warn}
|
paulson@3089
|
275 |
If you added $rule$ using {\tt addSDs} or {\tt addDs}, then you must delete
|
paulson@3089
|
276 |
it as follows:
|
paulson@3089
|
277 |
\begin{ttbox}
|
paulson@3089
|
278 |
\(cs\) delrules [make_elim \(rule\)]
|
paulson@3089
|
279 |
\end{ttbox}
|
paulson@3089
|
280 |
\par\noindent
|
paulson@3089
|
281 |
This is necessary because the operators {\tt addSDs} and {\tt addDs} convert
|
paulson@3089
|
282 |
the destruction rules to elimination rules by applying \ttindex{make_elim},
|
paulson@3089
|
283 |
and then insert them using {\tt addSEs} and {\tt addEs}, respectively.
|
paulson@3089
|
284 |
\end{warn}
|
paulson@3089
|
285 |
|
lcp@104
|
286 |
Introduction rules are those that can be applied using ordinary resolution.
|
lcp@104
|
287 |
The classical set automatically generates their swapped forms, which will
|
lcp@104
|
288 |
be applied using elim-resolution. Elimination rules are applied using
|
lcp@286
|
289 |
elim-resolution. In a classical set, rules are sorted by the number of new
|
lcp@286
|
290 |
subgoals they will yield; rules that generate the fewest subgoals will be
|
lcp@286
|
291 |
tried first (see \S\ref{biresolve_tac}).
|
lcp@104
|
292 |
|
lcp@1099
|
293 |
|
lcp@1099
|
294 |
\subsection{Modifying the search step}
|
lcp@104
|
295 |
For a given classical set, the proof strategy is simple. Perform as many
|
lcp@104
|
296 |
safe inferences as possible; or else, apply certain safe rules, allowing
|
lcp@104
|
297 |
instantiation of unknowns; or else, apply an unsafe rule. The tactics may
|
lcp@319
|
298 |
also apply {\tt hyp_subst_tac}, if they have been set up to do so (see
|
lcp@104
|
299 |
below). They may perform a form of Modus Ponens: if there are assumptions
|
lcp@104
|
300 |
$P\imp Q$ and~$P$, then replace $P\imp Q$ by~$Q$.
|
lcp@104
|
301 |
|
wenzelm@3108
|
302 |
The classical reasoning tactics --- except {\tt blast_tac}! --- allow
|
wenzelm@3108
|
303 |
you to modify this basic proof strategy by applying two arbitrary {\bf
|
wenzelm@3108
|
304 |
wrapper tacticals} to it. This affects each step of the search.
|
wenzelm@3108
|
305 |
Usually they are the identity tacticals, but they could apply another
|
wenzelm@3108
|
306 |
tactic before or after the step tactic. The first one, which is
|
wenzelm@3108
|
307 |
considered to be safe, affects \ttindex{safe_step_tac} and all the
|
wenzelm@3108
|
308 |
tactics that call it. The the second one, which may be unsafe, affects
|
wenzelm@3108
|
309 |
\ttindex{step_tac}, \ttindex{slow_step_tac} and the tactics that call
|
wenzelm@3108
|
310 |
them.
|
lcp@1099
|
311 |
|
lcp@1099
|
312 |
\begin{ttbox}
|
oheimb@2632
|
313 |
addss : claset * simpset -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
314 |
addSbefore : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
315 |
addSaltern : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
316 |
setSWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
317 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
318 |
compSWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
319 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
320 |
addbefore : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
321 |
addaltern : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
322 |
setWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
323 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
324 |
compWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
325 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
lcp@1099
|
326 |
\end{ttbox}
|
lcp@1099
|
327 |
%
|
wenzelm@3108
|
328 |
\index{simplification!from classical reasoner} The wrapper tacticals
|
wenzelm@3108
|
329 |
underly the operator addss, which combines each search step by
|
wenzelm@3108
|
330 |
simplification. Strictly speaking, {\tt addss} is not part of the
|
wenzelm@3108
|
331 |
classical reasoner. It should be defined (using {\tt addSaltern
|
wenzelm@3108
|
332 |
(CHANGED o (safe_asm_more_full_simp_tac ss)}) when the simplifier is
|
wenzelm@3108
|
333 |
installed.
|
lcp@1099
|
334 |
|
lcp@1099
|
335 |
\begin{ttdescription}
|
lcp@1099
|
336 |
\item[$cs$ addss $ss$] \indexbold{*addss}
|
oheimb@2631
|
337 |
adds the simpset~$ss$ to the classical set. The assumptions and goal will be
|
oheimb@2631
|
338 |
simplified, in a safe way, after the safe steps of the search.
|
oheimb@2631
|
339 |
|
oheimb@2631
|
340 |
\item[$cs$ addSbefore $tac$] \indexbold{*addSbefore}
|
oheimb@2631
|
341 |
changes the safe wrapper tactical to apply the given tactic {\em before}
|
oheimb@2631
|
342 |
each safe step of the search.
|
oheimb@2631
|
343 |
|
oheimb@2631
|
344 |
\item[$cs$ addSaltern $tac$] \indexbold{*addSaltern}
|
oheimb@2631
|
345 |
changes the safe wrapper tactical to apply the given tactic when a safe step
|
oheimb@2631
|
346 |
of the search would fail.
|
oheimb@2631
|
347 |
|
oheimb@2631
|
348 |
\item[$cs$ setSWrapper $tactical$] \indexbold{*setSWrapper}
|
oheimb@2631
|
349 |
specifies a new safe wrapper tactical.
|
oheimb@2631
|
350 |
|
oheimb@2631
|
351 |
\item[$cs$ compSWrapper $tactical$] \indexbold{*compSWrapper}
|
oheimb@2631
|
352 |
composes the $tactical$ with the existing safe wrapper tactical,
|
oheimb@2631
|
353 |
to combine their effects.
|
lcp@1099
|
354 |
|
lcp@1099
|
355 |
\item[$cs$ addbefore $tac$] \indexbold{*addbefore}
|
oheimb@2631
|
356 |
changes the (unsafe) wrapper tactical to apply the given tactic, which should
|
oheimb@2631
|
357 |
be safe, {\em before} each step of the search.
|
lcp@1099
|
358 |
|
oheimb@2631
|
359 |
\item[$cs$ addaltern $tac$] \indexbold{*addaltern}
|
oheimb@2631
|
360 |
changes the (unsafe) wrapper tactical to apply the given tactic
|
oheimb@2631
|
361 |
{\em alternatively} after each step of the search.
|
lcp@1099
|
362 |
|
oheimb@2631
|
363 |
\item[$cs$ setWrapper $tactical$] \indexbold{*setWrapper}
|
oheimb@2631
|
364 |
specifies a new (unsafe) wrapper tactical.
|
lcp@1099
|
365 |
|
oheimb@2631
|
366 |
\item[$cs$ compWrapper $tactical$] \indexbold{*compWrapper}
|
oheimb@2631
|
367 |
composes the $tactical$ with the existing (unsafe) wrapper tactical,
|
oheimb@2631
|
368 |
to combine their effects.
|
lcp@1099
|
369 |
\end{ttdescription}
|
lcp@1099
|
370 |
|
lcp@104
|
371 |
|
lcp@104
|
372 |
\section{The classical tactics}
|
lcp@319
|
373 |
\index{classical reasoner!tactics}
|
lcp@104
|
374 |
If installed, the classical module provides several tactics (and other
|
lcp@104
|
375 |
operations) for simulating the classical sequent calculus.
|
lcp@104
|
376 |
|
paulson@3089
|
377 |
\subsection{The automatic tableau prover}
|
paulson@3089
|
378 |
The tactic {\tt blast_tac} finds a proof rapidly using a separate tableau
|
paulson@3089
|
379 |
prover and then reconstructs the proof using Isabelle tactics. It is much
|
paulson@3089
|
380 |
faster and more powerful than the other classical reasoning tactics, but has
|
paulson@3089
|
381 |
major limitations too.
|
paulson@3089
|
382 |
\begin{itemize}
|
paulson@3089
|
383 |
\item It does not use the wrapper tacticals described above, such as
|
paulson@3089
|
384 |
\ttindex{addss}.
|
paulson@3089
|
385 |
\item It ignores types, which can cause problems in \HOL. If it applies a rule
|
paulson@3089
|
386 |
whose types are inappropriate, then proof reconstruction will fail.
|
paulson@3089
|
387 |
\item It does not perform higher-order unification, as needed by the rule {\tt
|
paulson@3089
|
388 |
rangeI} in {\HOL} and {\tt RepFunI} in {\ZF}. There are often
|
paulson@3089
|
389 |
alternatives to such rules, for example {\tt
|
paulson@3089
|
390 |
range_eqI} and {\tt RepFun_eqI}.
|
paulson@3089
|
391 |
\item The message {\small\tt Function Var's argument not a bound variable\ }
|
paulson@3089
|
392 |
relates to the lack of higher-order unification. Function variables
|
paulson@3089
|
393 |
may only be applied to parameters of the subgoal.
|
paulson@3089
|
394 |
\item Its proof strategy is more general than {\tt fast_tac}'s but can be
|
paulson@3089
|
395 |
slower. If {\tt blast_tac} fails or seems to be running forever, try {\tt
|
paulson@3089
|
396 |
fast_tac} and the other tactics described below.
|
paulson@3089
|
397 |
\end{itemize}
|
paulson@3089
|
398 |
%
|
paulson@3089
|
399 |
\begin{ttbox}
|
paulson@3089
|
400 |
blast_tac : claset -> int -> tactic
|
paulson@3089
|
401 |
Blast.depth_tac : claset -> int -> int -> tactic
|
paulson@3089
|
402 |
Blast.trace : bool ref \hfill{\bf initially false}
|
paulson@3089
|
403 |
\end{ttbox}
|
paulson@3089
|
404 |
The two tactics differ on how they bound the number of unsafe steps used in a
|
paulson@3089
|
405 |
proof. While {\tt blast_tac} starts with a bound of zero and increases it
|
paulson@3089
|
406 |
successively to~20, {\tt Blast.depth_tac} applies a user-supplied search bound.
|
paulson@3089
|
407 |
\begin{ttdescription}
|
paulson@3089
|
408 |
\item[\ttindexbold{blast_tac} $cs$ $i$] tries to prove
|
paulson@3089
|
409 |
subgoal~$i$ using iterative deepening to increase the search bound.
|
paulson@3089
|
410 |
|
paulson@3089
|
411 |
\item[\ttindexbold{Blast.depth_tac} $cs$ $lim$ $i$] tries
|
paulson@3089
|
412 |
to prove subgoal~$i$ using a search bound of $lim$. Often a slow
|
paulson@3089
|
413 |
proof using {\tt blast_tac} can be made much faster by supplying the
|
paulson@3089
|
414 |
successful search bound to this tactic instead.
|
paulson@3089
|
415 |
|
paulson@3089
|
416 |
\item[\ttindexbold{Blast.trace} := true;] \index{tracing!of classical prover}
|
paulson@3089
|
417 |
causes the tableau prover to print a trace of its search. At each step it
|
paulson@3089
|
418 |
displays the formula currently being examined and reports whether the branch
|
paulson@3089
|
419 |
has been closed, extended or split.
|
paulson@3089
|
420 |
\end{ttdescription}
|
paulson@3089
|
421 |
|
lcp@332
|
422 |
\subsection{The automatic tactics}
|
lcp@332
|
423 |
\begin{ttbox}
|
lcp@875
|
424 |
fast_tac : claset -> int -> tactic
|
lcp@875
|
425 |
best_tac : claset -> int -> tactic
|
lcp@875
|
426 |
slow_tac : claset -> int -> tactic
|
lcp@875
|
427 |
slow_best_tac : claset -> int -> tactic
|
lcp@332
|
428 |
\end{ttbox}
|
lcp@875
|
429 |
These tactics work by applying {\tt step_tac} or {\tt slow_step_tac}
|
lcp@875
|
430 |
repeatedly. Their effect is restricted (by {\tt SELECT_GOAL}) to one subgoal;
|
paulson@3089
|
431 |
they either prove this subgoal or fail. The {\tt slow_} versions are more
|
lcp@875
|
432 |
powerful but can be much slower.
|
lcp@875
|
433 |
|
lcp@875
|
434 |
The best-first tactics are guided by a heuristic function: typically, the
|
lcp@875
|
435 |
total size of the proof state. This function is supplied in the functor call
|
lcp@875
|
436 |
that sets up the classical reasoner.
|
lcp@332
|
437 |
\begin{ttdescription}
|
lcp@332
|
438 |
\item[\ttindexbold{fast_tac} $cs$ $i$] applies {\tt step_tac} using
|
paulson@3089
|
439 |
depth-first search, to prove subgoal~$i$.
|
lcp@332
|
440 |
|
lcp@332
|
441 |
\item[\ttindexbold{best_tac} $cs$ $i$] applies {\tt step_tac} using
|
paulson@3089
|
442 |
best-first search, to prove subgoal~$i$.
|
lcp@875
|
443 |
|
lcp@875
|
444 |
\item[\ttindexbold{slow_tac} $cs$ $i$] applies {\tt slow_step_tac} using
|
paulson@3089
|
445 |
depth-first search, to prove subgoal~$i$.
|
lcp@875
|
446 |
|
lcp@875
|
447 |
\item[\ttindexbold{slow_best_tac} $cs$ $i$] applies {\tt slow_step_tac} using
|
paulson@3089
|
448 |
best-first search, to prove subgoal~$i$.
|
lcp@875
|
449 |
\end{ttdescription}
|
lcp@875
|
450 |
|
lcp@875
|
451 |
|
lcp@875
|
452 |
\subsection{Depth-limited tactics}
|
lcp@875
|
453 |
\begin{ttbox}
|
lcp@875
|
454 |
depth_tac : claset -> int -> int -> tactic
|
lcp@875
|
455 |
deepen_tac : claset -> int -> int -> tactic
|
lcp@875
|
456 |
\end{ttbox}
|
lcp@875
|
457 |
These work by exhaustive search up to a specified depth. Unsafe rules are
|
lcp@875
|
458 |
modified to preserve the formula they act on, so that it be used repeatedly.
|
lcp@1099
|
459 |
They can prove more goals than {\tt fast_tac} can but are much
|
lcp@875
|
460 |
slower, for example if the assumptions have many universal quantifiers.
|
lcp@875
|
461 |
|
lcp@875
|
462 |
The depth limits the number of unsafe steps. If you can estimate the minimum
|
lcp@875
|
463 |
number of unsafe steps needed, supply this value as~$m$ to save time.
|
lcp@875
|
464 |
\begin{ttdescription}
|
lcp@875
|
465 |
\item[\ttindexbold{depth_tac} $cs$ $m$ $i$]
|
paulson@3089
|
466 |
tries to prove subgoal~$i$ by exhaustive search up to depth~$m$.
|
lcp@875
|
467 |
|
lcp@875
|
468 |
\item[\ttindexbold{deepen_tac} $cs$ $m$ $i$]
|
paulson@3089
|
469 |
tries to prove subgoal~$i$ by iterative deepening. It calls {\tt depth_tac}
|
lcp@875
|
470 |
repeatedly with increasing depths, starting with~$m$.
|
lcp@332
|
471 |
\end{ttdescription}
|
lcp@332
|
472 |
|
lcp@332
|
473 |
|
lcp@104
|
474 |
\subsection{Single-step tactics}
|
lcp@104
|
475 |
\begin{ttbox}
|
lcp@104
|
476 |
safe_step_tac : claset -> int -> tactic
|
lcp@104
|
477 |
safe_tac : claset -> tactic
|
lcp@104
|
478 |
inst_step_tac : claset -> int -> tactic
|
lcp@104
|
479 |
step_tac : claset -> int -> tactic
|
lcp@104
|
480 |
slow_step_tac : claset -> int -> tactic
|
lcp@104
|
481 |
\end{ttbox}
|
lcp@104
|
482 |
The automatic proof procedures call these tactics. By calling them
|
lcp@104
|
483 |
yourself, you can execute these procedures one step at a time.
|
lcp@308
|
484 |
\begin{ttdescription}
|
lcp@104
|
485 |
\item[\ttindexbold{safe_step_tac} $cs$ $i$] performs a safe step on
|
oheimb@2631
|
486 |
subgoal~$i$. The safe wrapper tactical is applied to a tactic that may include
|
oheimb@2631
|
487 |
proof by assumption or Modus Ponens (taking care not to instantiate unknowns),
|
oheimb@2631
|
488 |
or {\tt hyp_subst_tac}.
|
lcp@104
|
489 |
|
lcp@104
|
490 |
\item[\ttindexbold{safe_tac} $cs$] repeatedly performs safe steps on all
|
lcp@104
|
491 |
subgoals. It is deterministic, with at most one outcome. If the automatic
|
lcp@104
|
492 |
tactics fail, try using {\tt safe_tac} to open up your formula; then you
|
lcp@104
|
493 |
can replicate certain quantifiers explicitly by applying appropriate rules.
|
lcp@104
|
494 |
|
lcp@104
|
495 |
\item[\ttindexbold{inst_step_tac} $cs$ $i$] is like {\tt safe_step_tac},
|
lcp@104
|
496 |
but allows unknowns to be instantiated.
|
lcp@104
|
497 |
|
lcp@1099
|
498 |
\item[\ttindexbold{step_tac} $cs$ $i$] is the basic step of the proof
|
oheimb@2631
|
499 |
procedure. The (unsafe) wrapper tactical is applied to a tactic that tries
|
oheimb@2631
|
500 |
{\tt safe_tac}, {\tt inst_step_tac}, or applies an unsafe rule from~$cs$.
|
lcp@104
|
501 |
|
lcp@104
|
502 |
\item[\ttindexbold{slow_step_tac}]
|
lcp@104
|
503 |
resembles {\tt step_tac}, but allows backtracking between using safe
|
lcp@104
|
504 |
rules with instantiation ({\tt inst_step_tac}) and using unsafe rules.
|
lcp@875
|
505 |
The resulting search space is larger.
|
lcp@308
|
506 |
\end{ttdescription}
|
lcp@104
|
507 |
|
berghofe@1869
|
508 |
\subsection{The current claset}
|
paulson@2479
|
509 |
Some logics (\FOL, {\HOL} and \ZF) support the concept of a current
|
paulson@2479
|
510 |
claset\index{claset!current}. This is a default set of classical rules. The
|
paulson@2479
|
511 |
underlying idea is quite similar to that of a current simpset described in
|
paulson@2479
|
512 |
\S\ref{sec:simp-for-dummies}; please read that section, including its
|
paulson@2479
|
513 |
warnings. Just like simpsets, clasets can be associated with theories. The
|
paulson@2479
|
514 |
tactics
|
berghofe@1869
|
515 |
\begin{ttbox}
|
wenzelm@3108
|
516 |
Blast_tac : int -> tactic
|
berghofe@1869
|
517 |
Fast_tac : int -> tactic
|
berghofe@1869
|
518 |
Best_tac : int -> tactic
|
berghofe@1869
|
519 |
Deepen_tac : int -> int -> tactic
|
paulson@3089
|
520 |
Step_tac : int -> tactic
|
berghofe@1869
|
521 |
\end{ttbox}
|
paulson@3089
|
522 |
\indexbold{*Blast_tac}\indexbold{*Best_tac}\indexbold{*Fast_tac}%
|
paulson@3089
|
523 |
\indexbold{*Deepen_tac}\indexbold{*Step_tac}
|
paulson@3089
|
524 |
make use of the current claset. E.g. {\tt Blast_tac} is defined as follows:
|
berghofe@1869
|
525 |
\begin{ttbox}
|
paulson@3089
|
526 |
fun Blast_tac i = fast_tac (!claset) i;
|
berghofe@1869
|
527 |
\end{ttbox}
|
berghofe@1869
|
528 |
where \ttindex{!claset} is the current claset.
|
berghofe@1869
|
529 |
The functions
|
berghofe@1869
|
530 |
\begin{ttbox}
|
berghofe@1869
|
531 |
AddSIs, AddSEs, AddSDs, AddIs, AddEs, AddDs: thm list -> unit
|
berghofe@1869
|
532 |
\end{ttbox}
|
berghofe@1869
|
533 |
\indexbold{*AddSIs} \indexbold{*AddSEs} \indexbold{*AddSDs}
|
berghofe@1869
|
534 |
\indexbold{*AddIs} \indexbold{*AddEs} \indexbold{*AddDs}
|
berghofe@1869
|
535 |
are used to add rules to the current claset. They work exactly like their
|
berghofe@1869
|
536 |
lower case counterparts {\tt addSIs} etc.
|
berghofe@1869
|
537 |
\begin{ttbox}
|
berghofe@1869
|
538 |
Delrules : thm list -> unit
|
berghofe@1869
|
539 |
\end{ttbox}
|
berghofe@1869
|
540 |
deletes rules from the current claset. You do not need to worry via which of
|
berghofe@1869
|
541 |
the above {\tt Add} functions the rule was initially added.
|
lcp@104
|
542 |
|
lcp@104
|
543 |
\subsection{Other useful tactics}
|
lcp@319
|
544 |
\index{tactics!for contradiction}
|
lcp@319
|
545 |
\index{tactics!for Modus Ponens}
|
lcp@104
|
546 |
\begin{ttbox}
|
lcp@104
|
547 |
contr_tac : int -> tactic
|
lcp@104
|
548 |
mp_tac : int -> tactic
|
lcp@104
|
549 |
eq_mp_tac : int -> tactic
|
lcp@104
|
550 |
swap_res_tac : thm list -> int -> tactic
|
lcp@104
|
551 |
\end{ttbox}
|
lcp@104
|
552 |
These can be used in the body of a specialized search.
|
lcp@308
|
553 |
\begin{ttdescription}
|
lcp@319
|
554 |
\item[\ttindexbold{contr_tac} {\it i}]\index{assumptions!contradictory}
|
lcp@319
|
555 |
solves subgoal~$i$ by detecting a contradiction among two assumptions of
|
lcp@319
|
556 |
the form $P$ and~$\neg P$, or fail. It may instantiate unknowns. The
|
lcp@319
|
557 |
tactic can produce multiple outcomes, enumerating all possible
|
lcp@319
|
558 |
contradictions.
|
lcp@104
|
559 |
|
lcp@104
|
560 |
\item[\ttindexbold{mp_tac} {\it i}]
|
lcp@104
|
561 |
is like {\tt contr_tac}, but also attempts to perform Modus Ponens in
|
lcp@104
|
562 |
subgoal~$i$. If there are assumptions $P\imp Q$ and~$P$, then it replaces
|
lcp@104
|
563 |
$P\imp Q$ by~$Q$. It may instantiate unknowns. It fails if it can do
|
lcp@104
|
564 |
nothing.
|
lcp@104
|
565 |
|
lcp@104
|
566 |
\item[\ttindexbold{eq_mp_tac} {\it i}]
|
lcp@104
|
567 |
is like {\tt mp_tac} {\it i}, but may not instantiate unknowns --- thus, it
|
lcp@104
|
568 |
is safe.
|
lcp@104
|
569 |
|
lcp@104
|
570 |
\item[\ttindexbold{swap_res_tac} {\it thms} {\it i}] refines subgoal~$i$ of
|
lcp@104
|
571 |
the proof state using {\it thms}, which should be a list of introduction
|
paulson@3089
|
572 |
rules. First, it attempts to prove the goal using {\tt assume_tac} or
|
lcp@104
|
573 |
{\tt contr_tac}. It then attempts to apply each rule in turn, attempting
|
lcp@104
|
574 |
resolution and also elim-resolution with the swapped form.
|
lcp@308
|
575 |
\end{ttdescription}
|
lcp@104
|
576 |
|
lcp@104
|
577 |
\subsection{Creating swapped rules}
|
lcp@104
|
578 |
\begin{ttbox}
|
lcp@104
|
579 |
swapify : thm list -> thm list
|
lcp@104
|
580 |
joinrules : thm list * thm list -> (bool * thm) list
|
lcp@104
|
581 |
\end{ttbox}
|
lcp@308
|
582 |
\begin{ttdescription}
|
lcp@104
|
583 |
\item[\ttindexbold{swapify} {\it thms}] returns a list consisting of the
|
lcp@104
|
584 |
swapped versions of~{\it thms}, regarded as introduction rules.
|
lcp@104
|
585 |
|
lcp@308
|
586 |
\item[\ttindexbold{joinrules} ({\it intrs}, {\it elims})]
|
lcp@104
|
587 |
joins introduction rules, their swapped versions, and elimination rules for
|
lcp@104
|
588 |
use with \ttindex{biresolve_tac}. Each rule is paired with~{\tt false}
|
lcp@104
|
589 |
(indicating ordinary resolution) or~{\tt true} (indicating
|
lcp@104
|
590 |
elim-resolution).
|
lcp@308
|
591 |
\end{ttdescription}
|
lcp@104
|
592 |
|
lcp@104
|
593 |
|
lcp@286
|
594 |
\section{Setting up the classical reasoner}
|
lcp@319
|
595 |
\index{classical reasoner!setting up}
|
lcp@104
|
596 |
Isabelle's classical object-logics, including {\tt FOL} and {\tt HOL}, have
|
lcp@286
|
597 |
the classical reasoner already set up. When defining a new classical logic,
|
lcp@286
|
598 |
you should set up the reasoner yourself. It consists of the \ML{} functor
|
lcp@104
|
599 |
\ttindex{ClassicalFun}, which takes the argument
|
lcp@319
|
600 |
signature {\tt CLASSICAL_DATA}:
|
lcp@104
|
601 |
\begin{ttbox}
|
lcp@104
|
602 |
signature CLASSICAL_DATA =
|
lcp@104
|
603 |
sig
|
lcp@104
|
604 |
val mp : thm
|
lcp@104
|
605 |
val not_elim : thm
|
lcp@104
|
606 |
val swap : thm
|
lcp@104
|
607 |
val sizef : thm -> int
|
lcp@104
|
608 |
val hyp_subst_tacs : (int -> tactic) list
|
lcp@104
|
609 |
end;
|
lcp@104
|
610 |
\end{ttbox}
|
lcp@104
|
611 |
Thus, the functor requires the following items:
|
lcp@308
|
612 |
\begin{ttdescription}
|
lcp@319
|
613 |
\item[\tdxbold{mp}] should be the Modus Ponens rule
|
lcp@104
|
614 |
$\List{\Var{P}\imp\Var{Q};\; \Var{P}} \Imp \Var{Q}$.
|
lcp@104
|
615 |
|
lcp@319
|
616 |
\item[\tdxbold{not_elim}] should be the contradiction rule
|
lcp@104
|
617 |
$\List{\neg\Var{P};\; \Var{P}} \Imp \Var{R}$.
|
lcp@104
|
618 |
|
lcp@319
|
619 |
\item[\tdxbold{swap}] should be the swap rule
|
lcp@104
|
620 |
$\List{\neg \Var{P}; \; \neg \Var{R}\Imp \Var{P}} \Imp \Var{R}$.
|
lcp@104
|
621 |
|
lcp@104
|
622 |
\item[\ttindexbold{sizef}] is the heuristic function used for best-first
|
lcp@104
|
623 |
search. It should estimate the size of the remaining subgoals. A good
|
lcp@104
|
624 |
heuristic function is \ttindex{size_of_thm}, which measures the size of the
|
lcp@104
|
625 |
proof state. Another size function might ignore certain subgoals (say,
|
lcp@104
|
626 |
those concerned with type checking). A heuristic function might simply
|
lcp@104
|
627 |
count the subgoals.
|
lcp@104
|
628 |
|
lcp@319
|
629 |
\item[\ttindexbold{hyp_subst_tacs}] is a list of tactics for substitution in
|
lcp@104
|
630 |
the hypotheses, typically created by \ttindex{HypsubstFun} (see
|
lcp@104
|
631 |
Chapter~\ref{substitution}). This list can, of course, be empty. The
|
lcp@104
|
632 |
tactics are assumed to be safe!
|
lcp@308
|
633 |
\end{ttdescription}
|
lcp@104
|
634 |
The functor is not at all sensitive to the formalization of the
|
wenzelm@3108
|
635 |
object-logic. It does not even examine the rules, but merely applies
|
wenzelm@3108
|
636 |
them according to its fixed strategy. The functor resides in {\tt
|
wenzelm@3108
|
637 |
Provers/classical.ML} in the Isabelle sources.
|
lcp@104
|
638 |
|
lcp@319
|
639 |
\index{classical reasoner|)}
|