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%% $Id$
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\chapter{The Classical Reasoner}\label{chap:classical}
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\index{classical reasoner|(}
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\newcommand\ainfer[2]{\begin{array}{r@{\,}l}#2\\ \hline#1\end{array}}
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Although Isabelle is generic, many users will be working in some extension
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of classical first-order logic. Isabelle's set theory~{\tt ZF} is built
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upon theory~{\tt FOL}, while higher-order logic contains first-order logic
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as a fragment. Theorem-proving in predicate logic is undecidable, but many
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researchers have developed strategies to assist in this task.
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Isabelle's classical reasoner is an \ML{} functor that accepts certain
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information about a logic and delivers a suite of automatic tactics. Each
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tactic takes a collection of rules and executes a simple, non-clausal proof
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procedure. They are slow and simplistic compared with resolution theorem
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provers, but they can save considerable time and effort. They can prove
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theorems such as Pelletier's~\cite{pelletier86} problems~40 and~41 in
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seconds:
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\[ (\exists y. \forall x. J(y,x) \bimp \neg J(x,x))
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\imp \neg (\forall x. \exists y. \forall z. J(z,y) \bimp \neg J(z,x)) \]
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\[ (\forall z. \exists y. \forall x. F(x,y) \bimp F(x,z) \conj \neg F(x,x))
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\imp \neg (\exists z. \forall x. F(x,z))
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\]
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%
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The tactics are generic. They are not restricted to first-order logic, and
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have been heavily used in the development of Isabelle's set theory. Few
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interactive proof assistants provide this much automation. The tactics can
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be traced, and their components can be called directly; in this manner,
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any proof can be viewed interactively.
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The simplest way to apply the classical reasoner (to subgoal~$i$) is as
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follows:
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\begin{ttbox}
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by (Fast_tac \(i\));
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\end{ttbox}
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If the subgoal is a simple formula of the predicate calculus or set theory,
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then it should be proved quickly. However, to use the classical reasoner
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effectively, you need to know how it works.
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We shall first discuss the underlying principles, then present the classical
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reasoner. Finally, we shall see how to instantiate it for new logics. The
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logics {\tt FOL}, {\tt HOL} and {\tt ZF} have it already installed.
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\section{The sequent calculus}
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\index{sequent calculus}
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Isabelle supports natural deduction, which is easy to use for interactive
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proof. But natural deduction does not easily lend itself to automation,
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and has a bias towards intuitionism. For certain proofs in classical
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logic, it can not be called natural. The {\bf sequent calculus}, a
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generalization of natural deduction, is easier to automate.
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A {\bf sequent} has the form $\Gamma\turn\Delta$, where $\Gamma$
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and~$\Delta$ are sets of formulae.%
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\footnote{For first-order logic, sequents can equivalently be made from
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lists or multisets of formulae.} The sequent
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\[ P@1,\ldots,P@m\turn Q@1,\ldots,Q@n \]
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is {\bf valid} if $P@1\conj\ldots\conj P@m$ implies $Q@1\disj\ldots\disj
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Q@n$. Thus $P@1,\ldots,P@m$ represent assumptions, each of which is true,
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while $Q@1,\ldots,Q@n$ represent alternative goals. A sequent is {\bf
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basic} if its left and right sides have a common formula, as in $P,Q\turn
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Q,R$; basic sequents are trivially valid.
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Sequent rules are classified as {\bf right} or {\bf left}, indicating which
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side of the $\turn$~symbol they operate on. Rules that operate on the
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right side are analogous to natural deduction's introduction rules, and
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left rules are analogous to elimination rules.
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Recall the natural deduction rules for
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first-order logic,
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\iflabelundefined{fol-fig}{from {\it Introduction to Isabelle}}%
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{Fig.\ts\ref{fol-fig}}.
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The sequent calculus analogue of~$({\imp}I)$ is the rule
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$$ \ainfer{\Gamma &\turn \Delta, P\imp Q}{P,\Gamma &\turn \Delta,Q}
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\eqno({\imp}R) $$
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This breaks down some implication on the right side of a sequent; $\Gamma$
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and $\Delta$ stand for the sets of formulae that are unaffected by the
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inference. The analogue of the pair~$({\disj}I1)$ and~$({\disj}I2)$ is the
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single rule
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$$ \ainfer{\Gamma &\turn \Delta, P\disj Q}{\Gamma &\turn \Delta,P,Q}
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\eqno({\disj}R) $$
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This breaks down some disjunction on the right side, replacing it by both
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disjuncts. Thus, the sequent calculus is a kind of multiple-conclusion logic.
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To illustrate the use of multiple formulae on the right, let us prove
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the classical theorem $(P\imp Q)\disj(Q\imp P)$. Working backwards, we
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reduce this formula to a basic sequent:
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\[ \infer[(\disj)R]{\turn(P\imp Q)\disj(Q\imp P)}
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{\infer[(\imp)R]{\turn(P\imp Q), (Q\imp P)\;}
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{\infer[(\imp)R]{P \turn Q, (Q\imp P)\qquad}
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{P, Q \turn Q, P\qquad\qquad}}}
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\]
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This example is typical of the sequent calculus: start with the desired
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theorem and apply rules backwards in a fairly arbitrary manner. This yields a
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surprisingly effective proof procedure. Quantifiers add few complications,
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since Isabelle handles parameters and schematic variables. See Chapter~10
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of {\em ML for the Working Programmer}~\cite{paulson91} for further
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discussion.
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\section{Simulating sequents by natural deduction}
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Isabelle can represent sequents directly, as in the object-logic~{\tt LK}\@.
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But natural deduction is easier to work with, and most object-logics employ
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it. Fortunately, we can simulate the sequent $P@1,\ldots,P@m\turn
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Q@1,\ldots,Q@n$ by the Isabelle formula
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\[ \List{P@1;\ldots;P@m; \neg Q@2;\ldots; \neg Q@n}\Imp Q@1, \]
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where the order of the assumptions and the choice of~$Q@1$ are arbitrary.
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Elim-resolution plays a key role in simulating sequent proofs.
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We can easily handle reasoning on the left.
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As discussed in
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\iflabelundefined{destruct}{{\it Introduction to Isabelle}}{\S\ref{destruct}},
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elim-resolution with the rules $(\disj E)$, $(\bot E)$ and $(\exists E)$
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achieves a similar effect as the corresponding sequent rules. For the
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other connectives, we use sequent-style elimination rules instead of
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destruction rules such as $({\conj}E1,2)$ and $(\forall E)$. But note that
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the rule $(\neg L)$ has no effect under our representation of sequents!
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$$ \ainfer{\neg P,\Gamma &\turn \Delta}{\Gamma &\turn \Delta,P}
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\eqno({\neg}L) $$
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What about reasoning on the right? Introduction rules can only affect the
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formula in the conclusion, namely~$Q@1$. The other right-side formulae are
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represented as negated assumptions, $\neg Q@2$, \ldots,~$\neg Q@n$.
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\index{assumptions!negated}
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In order to operate on one of these, it must first be exchanged with~$Q@1$.
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Elim-resolution with the {\bf swap} rule has this effect:
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$$ \List{\neg P; \; \neg R\Imp P} \Imp R \eqno(swap)$$
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To ensure that swaps occur only when necessary, each introduction rule is
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converted into a swapped form: it is resolved with the second premise
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of~$(swap)$. The swapped form of~$({\conj}I)$, which might be
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called~$({\neg\conj}E)$, is
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\[ \List{\neg(P\conj Q); \; \neg R\Imp P; \; \neg R\Imp Q} \Imp R. \]
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Similarly, the swapped form of~$({\imp}I)$ is
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\[ \List{\neg(P\imp Q); \; \List{\neg R;P}\Imp Q} \Imp R \]
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Swapped introduction rules are applied using elim-resolution, which deletes
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the negated formula. Our representation of sequents also requires the use
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of ordinary introduction rules. If we had no regard for readability, we
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could treat the right side more uniformly by representing sequents as
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\[ \List{P@1;\ldots;P@m; \neg Q@1;\ldots; \neg Q@n}\Imp \bot. \]
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\section{Extra rules for the sequent calculus}
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As mentioned, destruction rules such as $({\conj}E1,2)$ and $(\forall E)$
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must be replaced by sequent-style elimination rules. In addition, we need
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rules to embody the classical equivalence between $P\imp Q$ and $\neg P\disj
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Q$. The introduction rules~$({\disj}I1,2)$ are replaced by a rule that
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simulates $({\disj}R)$:
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\[ (\neg Q\Imp P) \Imp P\disj Q \]
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The destruction rule $({\imp}E)$ is replaced by
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\[ \List{P\imp Q;\; \neg P\Imp R;\; Q\Imp R} \Imp R. \]
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Quantifier replication also requires special rules. In classical logic,
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$\exists x{.}P$ is equivalent to $\neg\forall x{.}\neg P$; the rules
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$(\exists R)$ and $(\forall L)$ are dual:
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\[ \ainfer{\Gamma &\turn \Delta, \exists x{.}P}
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{\Gamma &\turn \Delta, \exists x{.}P, P[t/x]} \; (\exists R)
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\qquad
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\ainfer{\forall x{.}P, \Gamma &\turn \Delta}
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{P[t/x], \forall x{.}P, \Gamma &\turn \Delta} \; (\forall L)
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\]
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Thus both kinds of quantifier may be replicated. Theorems requiring
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multiple uses of a universal formula are easy to invent; consider
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\[ (\forall x.P(x)\imp P(f(x))) \conj P(a) \imp P(f^n(a)), \]
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for any~$n>1$. Natural examples of the multiple use of an existential
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formula are rare; a standard one is $\exists x.\forall y. P(x)\imp P(y)$.
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Forgoing quantifier replication loses completeness, but gains decidability,
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since the search space becomes finite. Many useful theorems can be proved
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without replication, and the search generally delivers its verdict in a
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reasonable time. To adopt this approach, represent the sequent rules
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$(\exists R)$, $(\exists L)$ and $(\forall R)$ by $(\exists I)$, $(\exists
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E)$ and $(\forall I)$, respectively, and put $(\forall E)$ into elimination
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form:
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$$ \List{\forall x{.}P(x); P(t)\Imp Q} \Imp Q \eqno(\forall E@2) $$
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Elim-resolution with this rule will delete the universal formula after a
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single use. To replicate universal quantifiers, replace the rule by
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$$ \List{\forall x{.}P(x);\; \List{P(t); \forall x{.}P(x)}\Imp Q} \Imp Q.
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\eqno(\forall E@3) $$
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To replicate existential quantifiers, replace $(\exists I)$ by
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\[ \List{\neg(\exists x{.}P(x)) \Imp P(t)} \Imp \exists x{.}P(x). \]
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All introduction rules mentioned above are also useful in swapped form.
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Replication makes the search space infinite; we must apply the rules with
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care. The classical reasoner distinguishes between safe and unsafe
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rules, applying the latter only when there is no alternative. Depth-first
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search may well go down a blind alley; best-first search is better behaved
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in an infinite search space. However, quantifier replication is too
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expensive to prove any but the simplest theorems.
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\section{Classical rule sets}
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\index{classical sets}
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Each automatic tactic takes a {\bf classical set} --- a collection of
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rules, classified as introduction or elimination and as {\bf safe} or {\bf
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unsafe}. In general, safe rules can be attempted blindly, while unsafe
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rules must be used with care. A safe rule must never reduce a provable
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goal to an unprovable set of subgoals.
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The rule~$({\disj}I1)$ is unsafe because it reduces $P\disj Q$ to~$P$. Any
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rule is unsafe whose premises contain new unknowns. The elimination
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rule~$(\forall E@2)$ is unsafe, since it is applied via elim-resolution,
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which discards the assumption $\forall x{.}P(x)$ and replaces it by the
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weaker assumption~$P(\Var{t})$. The rule $({\exists}I)$ is unsafe for
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similar reasons. The rule~$(\forall E@3)$ is unsafe in a different sense:
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since it keeps the assumption $\forall x{.}P(x)$, it is prone to looping.
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In classical first-order logic, all rules are safe except those mentioned
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above.
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The safe/unsafe distinction is vague, and may be regarded merely as a way
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of giving some rules priority over others. One could argue that
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$({\disj}E)$ is unsafe, because repeated application of it could generate
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exponentially many subgoals. Induction rules are unsafe because inductive
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proofs are difficult to set up automatically. Any inference is unsafe that
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instantiates an unknown in the proof state --- thus \ttindex{match_tac}
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must be used, rather than \ttindex{resolve_tac}. Even proof by assumption
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is unsafe if it instantiates unknowns shared with other subgoals --- thus
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\ttindex{eq_assume_tac} must be used, rather than \ttindex{assume_tac}.
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\subsection{Adding rules to classical sets}
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Classical rule sets belong to the abstract type \mltydx{claset}, which
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supports the following operations (provided the classical reasoner is
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installed!):
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\begin{ttbox}
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empty_cs : claset
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print_cs : claset -> unit
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addSIs : claset * thm list -> claset \hfill{\bf infix 4}
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addSEs : claset * thm list -> claset \hfill{\bf infix 4}
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addSDs : claset * thm list -> claset \hfill{\bf infix 4}
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addIs : claset * thm list -> claset \hfill{\bf infix 4}
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addEs : claset * thm list -> claset \hfill{\bf infix 4}
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addDs : claset * thm list -> claset \hfill{\bf infix 4}
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delrules : claset * thm list -> claset \hfill{\bf infix 4}
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\end{ttbox}
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The add operations do not check for repetitions.
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\begin{ttdescription}
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\item[\ttindexbold{empty_cs}] is the empty classical set.
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\item[\ttindexbold{print_cs} $cs$] prints the rules of~$cs$.
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\item[$cs$ addSIs $rules$] \indexbold{*addSIs}
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adds safe introduction~$rules$ to~$cs$.
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\item[$cs$ addSEs $rules$] \indexbold{*addSEs}
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adds safe elimination~$rules$ to~$cs$.
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\item[$cs$ addSDs $rules$] \indexbold{*addSDs}
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adds safe destruction~$rules$ to~$cs$.
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\item[$cs$ addIs $rules$] \indexbold{*addIs}
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adds unsafe introduction~$rules$ to~$cs$.
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\item[$cs$ addEs $rules$] \indexbold{*addEs}
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adds unsafe elimination~$rules$ to~$cs$.
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\item[$cs$ addDs $rules$] \indexbold{*addDs}
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adds unsafe destruction~$rules$ to~$cs$.
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\item[$cs$ delrules $rules$] \indexbold{*delrules}
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|
256 |
deletes~$rules$ from~$cs$.
|
lcp@308
|
257 |
\end{ttdescription}
|
lcp@308
|
258 |
|
lcp@104
|
259 |
Introduction rules are those that can be applied using ordinary resolution.
|
lcp@104
|
260 |
The classical set automatically generates their swapped forms, which will
|
lcp@104
|
261 |
be applied using elim-resolution. Elimination rules are applied using
|
lcp@286
|
262 |
elim-resolution. In a classical set, rules are sorted by the number of new
|
lcp@286
|
263 |
subgoals they will yield; rules that generate the fewest subgoals will be
|
lcp@286
|
264 |
tried first (see \S\ref{biresolve_tac}).
|
lcp@104
|
265 |
|
lcp@1099
|
266 |
|
lcp@1099
|
267 |
\subsection{Modifying the search step}
|
lcp@104
|
268 |
For a given classical set, the proof strategy is simple. Perform as many
|
lcp@104
|
269 |
safe inferences as possible; or else, apply certain safe rules, allowing
|
lcp@104
|
270 |
instantiation of unknowns; or else, apply an unsafe rule. The tactics may
|
lcp@319
|
271 |
also apply {\tt hyp_subst_tac}, if they have been set up to do so (see
|
lcp@104
|
272 |
below). They may perform a form of Modus Ponens: if there are assumptions
|
lcp@104
|
273 |
$P\imp Q$ and~$P$, then replace $P\imp Q$ by~$Q$.
|
lcp@104
|
274 |
|
lcp@1099
|
275 |
The classical reasoner allows you to modify this basic proof strategy by
|
oheimb@2632
|
276 |
applying two arbitrary {\bf wrapper tacticals} to it. This affects each step of
|
oheimb@2631
|
277 |
the search. Usually they are the identity tacticals, but they could apply
|
oheimb@2631
|
278 |
another tactic before or after the step tactic. The first one, which is
|
oheimb@2631
|
279 |
considered to be safe, affects \ttindex{safe_step_tac} and all the tactics that
|
oheimb@2631
|
280 |
call it. The the second one, which may be unsafe, affects
|
oheimb@2632
|
281 |
\ttindex{step_tac}, \ttindex{slow_step_tac} and the tactics that call them.
|
lcp@1099
|
282 |
|
lcp@1099
|
283 |
\begin{ttbox}
|
oheimb@2632
|
284 |
addss : claset * simpset -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
285 |
addSbefore : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
286 |
addSaltern : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
287 |
setSWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
288 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
289 |
compSWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
290 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
291 |
addbefore : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
292 |
addaltern : claset * (int -> tactic) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
293 |
setWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
294 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
oheimb@2632
|
295 |
compWrapper : claset * ((int -> tactic) ->
|
oheimb@2632
|
296 |
(int -> tactic)) -> claset \hfill{\bf infix 4}
|
lcp@1099
|
297 |
\end{ttbox}
|
lcp@1099
|
298 |
%
|
lcp@1099
|
299 |
\index{simplification!from classical reasoner}
|
oheimb@2631
|
300 |
The wrapper tacticals underly the operator \ttindex{addss}, which combines
|
lcp@1099
|
301 |
each search step by simplification. Strictly speaking, {\tt addss} is not
|
oheimb@2631
|
302 |
part of the classical reasoner. It should be defined (using {\tt addSaltern (CHANGED o (safe_asm_more_full_simp_tac ss)}) when the simplifier is installed.
|
lcp@1099
|
303 |
|
lcp@1099
|
304 |
\begin{ttdescription}
|
lcp@1099
|
305 |
\item[$cs$ addss $ss$] \indexbold{*addss}
|
oheimb@2631
|
306 |
adds the simpset~$ss$ to the classical set. The assumptions and goal will be
|
oheimb@2631
|
307 |
simplified, in a safe way, after the safe steps of the search.
|
oheimb@2631
|
308 |
|
oheimb@2631
|
309 |
\item[$cs$ addSbefore $tac$] \indexbold{*addSbefore}
|
oheimb@2631
|
310 |
changes the safe wrapper tactical to apply the given tactic {\em before}
|
oheimb@2631
|
311 |
each safe step of the search.
|
oheimb@2631
|
312 |
|
oheimb@2631
|
313 |
\item[$cs$ addSaltern $tac$] \indexbold{*addSaltern}
|
oheimb@2631
|
314 |
changes the safe wrapper tactical to apply the given tactic when a safe step
|
oheimb@2631
|
315 |
of the search would fail.
|
oheimb@2631
|
316 |
|
oheimb@2631
|
317 |
\item[$cs$ setSWrapper $tactical$] \indexbold{*setSWrapper}
|
oheimb@2631
|
318 |
specifies a new safe wrapper tactical.
|
oheimb@2631
|
319 |
|
oheimb@2631
|
320 |
\item[$cs$ compSWrapper $tactical$] \indexbold{*compSWrapper}
|
oheimb@2631
|
321 |
composes the $tactical$ with the existing safe wrapper tactical,
|
oheimb@2631
|
322 |
to combine their effects.
|
lcp@1099
|
323 |
|
lcp@1099
|
324 |
\item[$cs$ addbefore $tac$] \indexbold{*addbefore}
|
oheimb@2631
|
325 |
changes the (unsafe) wrapper tactical to apply the given tactic, which should
|
oheimb@2631
|
326 |
be safe, {\em before} each step of the search.
|
lcp@1099
|
327 |
|
oheimb@2631
|
328 |
\item[$cs$ addaltern $tac$] \indexbold{*addaltern}
|
oheimb@2631
|
329 |
changes the (unsafe) wrapper tactical to apply the given tactic
|
oheimb@2631
|
330 |
{\em alternatively} after each step of the search.
|
lcp@1099
|
331 |
|
oheimb@2631
|
332 |
\item[$cs$ setWrapper $tactical$] \indexbold{*setWrapper}
|
oheimb@2631
|
333 |
specifies a new (unsafe) wrapper tactical.
|
lcp@1099
|
334 |
|
oheimb@2631
|
335 |
\item[$cs$ compWrapper $tactical$] \indexbold{*compWrapper}
|
oheimb@2631
|
336 |
composes the $tactical$ with the existing (unsafe) wrapper tactical,
|
oheimb@2631
|
337 |
to combine their effects.
|
lcp@1099
|
338 |
\end{ttdescription}
|
lcp@1099
|
339 |
|
lcp@104
|
340 |
|
lcp@104
|
341 |
\section{The classical tactics}
|
lcp@319
|
342 |
\index{classical reasoner!tactics}
|
lcp@104
|
343 |
If installed, the classical module provides several tactics (and other
|
lcp@104
|
344 |
operations) for simulating the classical sequent calculus.
|
lcp@104
|
345 |
|
lcp@332
|
346 |
\subsection{The automatic tactics}
|
lcp@332
|
347 |
\begin{ttbox}
|
lcp@875
|
348 |
fast_tac : claset -> int -> tactic
|
lcp@875
|
349 |
best_tac : claset -> int -> tactic
|
lcp@875
|
350 |
slow_tac : claset -> int -> tactic
|
lcp@875
|
351 |
slow_best_tac : claset -> int -> tactic
|
lcp@332
|
352 |
\end{ttbox}
|
lcp@875
|
353 |
These tactics work by applying {\tt step_tac} or {\tt slow_step_tac}
|
lcp@875
|
354 |
repeatedly. Their effect is restricted (by {\tt SELECT_GOAL}) to one subgoal;
|
lcp@875
|
355 |
they either solve this subgoal or fail. The {\tt slow_} versions are more
|
lcp@875
|
356 |
powerful but can be much slower.
|
lcp@875
|
357 |
|
lcp@875
|
358 |
The best-first tactics are guided by a heuristic function: typically, the
|
lcp@875
|
359 |
total size of the proof state. This function is supplied in the functor call
|
lcp@875
|
360 |
that sets up the classical reasoner.
|
lcp@332
|
361 |
\begin{ttdescription}
|
lcp@332
|
362 |
\item[\ttindexbold{fast_tac} $cs$ $i$] applies {\tt step_tac} using
|
lcp@332
|
363 |
depth-first search, to solve subgoal~$i$.
|
lcp@332
|
364 |
|
lcp@332
|
365 |
\item[\ttindexbold{best_tac} $cs$ $i$] applies {\tt step_tac} using
|
lcp@875
|
366 |
best-first search, to solve subgoal~$i$.
|
lcp@875
|
367 |
|
lcp@875
|
368 |
\item[\ttindexbold{slow_tac} $cs$ $i$] applies {\tt slow_step_tac} using
|
lcp@875
|
369 |
depth-first search, to solve subgoal~$i$.
|
lcp@875
|
370 |
|
lcp@875
|
371 |
\item[\ttindexbold{slow_best_tac} $cs$ $i$] applies {\tt slow_step_tac} using
|
lcp@875
|
372 |
best-first search, to solve subgoal~$i$.
|
lcp@875
|
373 |
\end{ttdescription}
|
lcp@875
|
374 |
|
lcp@875
|
375 |
|
lcp@875
|
376 |
\subsection{Depth-limited tactics}
|
lcp@875
|
377 |
\begin{ttbox}
|
lcp@875
|
378 |
depth_tac : claset -> int -> int -> tactic
|
lcp@875
|
379 |
deepen_tac : claset -> int -> int -> tactic
|
lcp@875
|
380 |
\end{ttbox}
|
lcp@875
|
381 |
These work by exhaustive search up to a specified depth. Unsafe rules are
|
lcp@875
|
382 |
modified to preserve the formula they act on, so that it be used repeatedly.
|
lcp@1099
|
383 |
They can prove more goals than {\tt fast_tac} can but are much
|
lcp@875
|
384 |
slower, for example if the assumptions have many universal quantifiers.
|
lcp@875
|
385 |
|
lcp@875
|
386 |
The depth limits the number of unsafe steps. If you can estimate the minimum
|
lcp@875
|
387 |
number of unsafe steps needed, supply this value as~$m$ to save time.
|
lcp@875
|
388 |
\begin{ttdescription}
|
lcp@875
|
389 |
\item[\ttindexbold{depth_tac} $cs$ $m$ $i$]
|
lcp@875
|
390 |
tries to solve subgoal~$i$ by exhaustive search up to depth~$m$.
|
lcp@875
|
391 |
|
lcp@875
|
392 |
\item[\ttindexbold{deepen_tac} $cs$ $m$ $i$]
|
lcp@875
|
393 |
tries to solve subgoal~$i$ by iterative deepening. It calls {\tt depth_tac}
|
lcp@875
|
394 |
repeatedly with increasing depths, starting with~$m$.
|
lcp@332
|
395 |
\end{ttdescription}
|
lcp@332
|
396 |
|
lcp@332
|
397 |
|
lcp@104
|
398 |
\subsection{Single-step tactics}
|
lcp@104
|
399 |
\begin{ttbox}
|
lcp@104
|
400 |
safe_step_tac : claset -> int -> tactic
|
lcp@104
|
401 |
safe_tac : claset -> tactic
|
lcp@104
|
402 |
inst_step_tac : claset -> int -> tactic
|
lcp@104
|
403 |
step_tac : claset -> int -> tactic
|
lcp@104
|
404 |
slow_step_tac : claset -> int -> tactic
|
lcp@104
|
405 |
\end{ttbox}
|
lcp@104
|
406 |
The automatic proof procedures call these tactics. By calling them
|
lcp@104
|
407 |
yourself, you can execute these procedures one step at a time.
|
lcp@308
|
408 |
\begin{ttdescription}
|
lcp@104
|
409 |
\item[\ttindexbold{safe_step_tac} $cs$ $i$] performs a safe step on
|
oheimb@2631
|
410 |
subgoal~$i$. The safe wrapper tactical is applied to a tactic that may include
|
oheimb@2631
|
411 |
proof by assumption or Modus Ponens (taking care not to instantiate unknowns),
|
oheimb@2631
|
412 |
or {\tt hyp_subst_tac}.
|
lcp@104
|
413 |
|
lcp@104
|
414 |
\item[\ttindexbold{safe_tac} $cs$] repeatedly performs safe steps on all
|
lcp@104
|
415 |
subgoals. It is deterministic, with at most one outcome. If the automatic
|
lcp@104
|
416 |
tactics fail, try using {\tt safe_tac} to open up your formula; then you
|
lcp@104
|
417 |
can replicate certain quantifiers explicitly by applying appropriate rules.
|
lcp@104
|
418 |
|
lcp@104
|
419 |
\item[\ttindexbold{inst_step_tac} $cs$ $i$] is like {\tt safe_step_tac},
|
lcp@104
|
420 |
but allows unknowns to be instantiated.
|
lcp@104
|
421 |
|
lcp@1099
|
422 |
\item[\ttindexbold{step_tac} $cs$ $i$] is the basic step of the proof
|
oheimb@2631
|
423 |
procedure. The (unsafe) wrapper tactical is applied to a tactic that tries
|
oheimb@2631
|
424 |
{\tt safe_tac}, {\tt inst_step_tac}, or applies an unsafe rule from~$cs$.
|
lcp@104
|
425 |
|
lcp@104
|
426 |
\item[\ttindexbold{slow_step_tac}]
|
lcp@104
|
427 |
resembles {\tt step_tac}, but allows backtracking between using safe
|
lcp@104
|
428 |
rules with instantiation ({\tt inst_step_tac}) and using unsafe rules.
|
lcp@875
|
429 |
The resulting search space is larger.
|
lcp@308
|
430 |
\end{ttdescription}
|
lcp@104
|
431 |
|
berghofe@1869
|
432 |
\subsection{The current claset}
|
paulson@2479
|
433 |
Some logics (\FOL, {\HOL} and \ZF) support the concept of a current
|
paulson@2479
|
434 |
claset\index{claset!current}. This is a default set of classical rules. The
|
paulson@2479
|
435 |
underlying idea is quite similar to that of a current simpset described in
|
paulson@2479
|
436 |
\S\ref{sec:simp-for-dummies}; please read that section, including its
|
paulson@2479
|
437 |
warnings. Just like simpsets, clasets can be associated with theories. The
|
paulson@2479
|
438 |
tactics
|
berghofe@1869
|
439 |
\begin{ttbox}
|
berghofe@1869
|
440 |
Step_tac : int -> tactic
|
berghofe@1869
|
441 |
Fast_tac : int -> tactic
|
berghofe@1869
|
442 |
Best_tac : int -> tactic
|
berghofe@1869
|
443 |
Deepen_tac : int -> int -> tactic
|
berghofe@1869
|
444 |
\end{ttbox}
|
berghofe@1869
|
445 |
\indexbold{*Step_tac} \indexbold{*Best_tac} \indexbold{*Fast_tac}
|
berghofe@1869
|
446 |
\indexbold{*Deepen_tac}
|
berghofe@1869
|
447 |
make use of the current claset. E.g.~{\tt Fast_tac} is defined as follows:
|
berghofe@1869
|
448 |
\begin{ttbox}
|
berghofe@1869
|
449 |
fun Fast_tac i = fast_tac (!claset) i;
|
berghofe@1869
|
450 |
\end{ttbox}
|
berghofe@1869
|
451 |
where \ttindex{!claset} is the current claset.
|
berghofe@1869
|
452 |
The functions
|
berghofe@1869
|
453 |
\begin{ttbox}
|
berghofe@1869
|
454 |
AddSIs, AddSEs, AddSDs, AddIs, AddEs, AddDs: thm list -> unit
|
berghofe@1869
|
455 |
\end{ttbox}
|
berghofe@1869
|
456 |
\indexbold{*AddSIs} \indexbold{*AddSEs} \indexbold{*AddSDs}
|
berghofe@1869
|
457 |
\indexbold{*AddIs} \indexbold{*AddEs} \indexbold{*AddDs}
|
berghofe@1869
|
458 |
are used to add rules to the current claset. They work exactly like their
|
berghofe@1869
|
459 |
lower case counterparts {\tt addSIs} etc.
|
berghofe@1869
|
460 |
\begin{ttbox}
|
berghofe@1869
|
461 |
Delrules : thm list -> unit
|
berghofe@1869
|
462 |
\end{ttbox}
|
berghofe@1869
|
463 |
deletes rules from the current claset. You do not need to worry via which of
|
berghofe@1869
|
464 |
the above {\tt Add} functions the rule was initially added.
|
lcp@104
|
465 |
|
lcp@104
|
466 |
\subsection{Other useful tactics}
|
lcp@319
|
467 |
\index{tactics!for contradiction}
|
lcp@319
|
468 |
\index{tactics!for Modus Ponens}
|
lcp@104
|
469 |
\begin{ttbox}
|
lcp@104
|
470 |
contr_tac : int -> tactic
|
lcp@104
|
471 |
mp_tac : int -> tactic
|
lcp@104
|
472 |
eq_mp_tac : int -> tactic
|
lcp@104
|
473 |
swap_res_tac : thm list -> int -> tactic
|
lcp@104
|
474 |
\end{ttbox}
|
lcp@104
|
475 |
These can be used in the body of a specialized search.
|
lcp@308
|
476 |
\begin{ttdescription}
|
lcp@319
|
477 |
\item[\ttindexbold{contr_tac} {\it i}]\index{assumptions!contradictory}
|
lcp@319
|
478 |
solves subgoal~$i$ by detecting a contradiction among two assumptions of
|
lcp@319
|
479 |
the form $P$ and~$\neg P$, or fail. It may instantiate unknowns. The
|
lcp@319
|
480 |
tactic can produce multiple outcomes, enumerating all possible
|
lcp@319
|
481 |
contradictions.
|
lcp@104
|
482 |
|
lcp@104
|
483 |
\item[\ttindexbold{mp_tac} {\it i}]
|
lcp@104
|
484 |
is like {\tt contr_tac}, but also attempts to perform Modus Ponens in
|
lcp@104
|
485 |
subgoal~$i$. If there are assumptions $P\imp Q$ and~$P$, then it replaces
|
lcp@104
|
486 |
$P\imp Q$ by~$Q$. It may instantiate unknowns. It fails if it can do
|
lcp@104
|
487 |
nothing.
|
lcp@104
|
488 |
|
lcp@104
|
489 |
\item[\ttindexbold{eq_mp_tac} {\it i}]
|
lcp@104
|
490 |
is like {\tt mp_tac} {\it i}, but may not instantiate unknowns --- thus, it
|
lcp@104
|
491 |
is safe.
|
lcp@104
|
492 |
|
lcp@104
|
493 |
\item[\ttindexbold{swap_res_tac} {\it thms} {\it i}] refines subgoal~$i$ of
|
lcp@104
|
494 |
the proof state using {\it thms}, which should be a list of introduction
|
lcp@319
|
495 |
rules. First, it attempts to solve the goal using {\tt assume_tac} or
|
lcp@104
|
496 |
{\tt contr_tac}. It then attempts to apply each rule in turn, attempting
|
lcp@104
|
497 |
resolution and also elim-resolution with the swapped form.
|
lcp@308
|
498 |
\end{ttdescription}
|
lcp@104
|
499 |
|
lcp@104
|
500 |
\subsection{Creating swapped rules}
|
lcp@104
|
501 |
\begin{ttbox}
|
lcp@104
|
502 |
swapify : thm list -> thm list
|
lcp@104
|
503 |
joinrules : thm list * thm list -> (bool * thm) list
|
lcp@104
|
504 |
\end{ttbox}
|
lcp@308
|
505 |
\begin{ttdescription}
|
lcp@104
|
506 |
\item[\ttindexbold{swapify} {\it thms}] returns a list consisting of the
|
lcp@104
|
507 |
swapped versions of~{\it thms}, regarded as introduction rules.
|
lcp@104
|
508 |
|
lcp@308
|
509 |
\item[\ttindexbold{joinrules} ({\it intrs}, {\it elims})]
|
lcp@104
|
510 |
joins introduction rules, their swapped versions, and elimination rules for
|
lcp@104
|
511 |
use with \ttindex{biresolve_tac}. Each rule is paired with~{\tt false}
|
lcp@104
|
512 |
(indicating ordinary resolution) or~{\tt true} (indicating
|
lcp@104
|
513 |
elim-resolution).
|
lcp@308
|
514 |
\end{ttdescription}
|
lcp@104
|
515 |
|
lcp@104
|
516 |
|
lcp@286
|
517 |
\section{Setting up the classical reasoner}
|
lcp@319
|
518 |
\index{classical reasoner!setting up}
|
lcp@104
|
519 |
Isabelle's classical object-logics, including {\tt FOL} and {\tt HOL}, have
|
lcp@286
|
520 |
the classical reasoner already set up. When defining a new classical logic,
|
lcp@286
|
521 |
you should set up the reasoner yourself. It consists of the \ML{} functor
|
lcp@104
|
522 |
\ttindex{ClassicalFun}, which takes the argument
|
lcp@319
|
523 |
signature {\tt CLASSICAL_DATA}:
|
lcp@104
|
524 |
\begin{ttbox}
|
lcp@104
|
525 |
signature CLASSICAL_DATA =
|
lcp@104
|
526 |
sig
|
lcp@104
|
527 |
val mp : thm
|
lcp@104
|
528 |
val not_elim : thm
|
lcp@104
|
529 |
val swap : thm
|
lcp@104
|
530 |
val sizef : thm -> int
|
lcp@104
|
531 |
val hyp_subst_tacs : (int -> tactic) list
|
lcp@104
|
532 |
end;
|
lcp@104
|
533 |
\end{ttbox}
|
lcp@104
|
534 |
Thus, the functor requires the following items:
|
lcp@308
|
535 |
\begin{ttdescription}
|
lcp@319
|
536 |
\item[\tdxbold{mp}] should be the Modus Ponens rule
|
lcp@104
|
537 |
$\List{\Var{P}\imp\Var{Q};\; \Var{P}} \Imp \Var{Q}$.
|
lcp@104
|
538 |
|
lcp@319
|
539 |
\item[\tdxbold{not_elim}] should be the contradiction rule
|
lcp@104
|
540 |
$\List{\neg\Var{P};\; \Var{P}} \Imp \Var{R}$.
|
lcp@104
|
541 |
|
lcp@319
|
542 |
\item[\tdxbold{swap}] should be the swap rule
|
lcp@104
|
543 |
$\List{\neg \Var{P}; \; \neg \Var{R}\Imp \Var{P}} \Imp \Var{R}$.
|
lcp@104
|
544 |
|
lcp@104
|
545 |
\item[\ttindexbold{sizef}] is the heuristic function used for best-first
|
lcp@104
|
546 |
search. It should estimate the size of the remaining subgoals. A good
|
lcp@104
|
547 |
heuristic function is \ttindex{size_of_thm}, which measures the size of the
|
lcp@104
|
548 |
proof state. Another size function might ignore certain subgoals (say,
|
lcp@104
|
549 |
those concerned with type checking). A heuristic function might simply
|
lcp@104
|
550 |
count the subgoals.
|
lcp@104
|
551 |
|
lcp@319
|
552 |
\item[\ttindexbold{hyp_subst_tacs}] is a list of tactics for substitution in
|
lcp@104
|
553 |
the hypotheses, typically created by \ttindex{HypsubstFun} (see
|
lcp@104
|
554 |
Chapter~\ref{substitution}). This list can, of course, be empty. The
|
lcp@104
|
555 |
tactics are assumed to be safe!
|
lcp@308
|
556 |
\end{ttdescription}
|
lcp@104
|
557 |
The functor is not at all sensitive to the formalization of the
|
lcp@104
|
558 |
object-logic. It does not even examine the rules, but merely applies them
|
lcp@104
|
559 |
according to its fixed strategy. The functor resides in {\tt
|
lcp@319
|
560 |
Provers/classical.ML} in the Isabelle distribution directory.
|
lcp@104
|
561 |
|
lcp@319
|
562 |
\index{classical reasoner|)}
|