--- a/doc-src/IsarRef/Thy/document/Generic.tex Sat Jun 04 19:39:45 2011 +0200
+++ b/doc-src/IsarRef/Thy/document/Generic.tex Sat Jun 04 22:09:42 2011 +0200
@@ -924,6 +924,254 @@
}
\isamarkuptrue%
%
+\isamarkupsubsection{Introduction%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Although Isabelle is generic, many users will be working in
+ some extension of classical first-order logic. Isabelle/ZF is built
+ upon theory FOL, while Isabelle/HOL conceptually contains
+ first-order logic as a fragment. Theorem-proving in predicate logic
+ is undecidable, but many automated strategies have been developed to
+ assist in this task.
+
+ Isabelle's classical reasoner is a generic package that accepts
+ certain information about a logic and delivers a suite of automatic
+ proof tools, based on rules that are classified and declared in the
+ context. These proof procedures are slow and simplistic compared
+ with high-end automated theorem provers, but they can save
+ considerable time and effort in practice. They can prove theorems
+ such as Pelletier's \cite{pelletier86} problems 40 and 41 in a few
+ milliseconds (including full proof reconstruction):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ F\ x\ y\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ F\ x\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ F\ z\ y\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ F\ z\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ blast%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}z{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}y{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ y\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ f\ x\ z\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ f\ x\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}z{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ f\ x\ z{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+\ \ %
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ blast%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The proof tools are generic. They are not restricted to
+ first-order logic, and have been heavily used in the development of
+ the Isabelle/HOL library and applications. The tactics can be
+ traced, and their components can be called directly; in this manner,
+ any proof can be viewed interactively.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{The sequent calculus%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle supports natural deduction, which is easy to use for
+ interactive proof. But natural deduction does not easily lend
+ itself to automation, and has a bias towards intuitionism. For
+ certain proofs in classical logic, it can not be called natural.
+ The \emph{sequent calculus}, a generalization of natural deduction,
+ is easier to automate.
+
+ A \textbf{sequent} has the form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{22}{\isachardoublequote}}}, where \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{22}{\isachardoublequote}}}
+ and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{22}{\isachardoublequote}}} are sets of formulae.\footnote{For first-order
+ logic, sequents can equivalently be made from lists or multisets of
+ formulae.} The sequent \isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} is
+ \textbf{valid} if \isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}} implies \isa{{\isaliteral{22}{\isachardoublequote}}Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}. Thus \isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}} represent assumptions, each of which
+ is true, while \isa{{\isaliteral{22}{\isachardoublequote}}Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} represent alternative goals. A
+ sequent is \textbf{basic} if its left and right sides have a common
+ formula, as in \isa{{\isaliteral{22}{\isachardoublequote}}P{\isaliteral{2C}{\isacharcomma}}\ Q\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q{\isaliteral{2C}{\isacharcomma}}\ R{\isaliteral{22}{\isachardoublequote}}}; basic sequents are trivially
+ valid.
+
+ Sequent rules are classified as \textbf{right} or \textbf{left},
+ indicating which side of the \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}{\isaliteral{22}{\isachardoublequote}}} symbol they operate on.
+ Rules that operate on the right side are analogous to natural
+ deduction's introduction rules, and left rules are analogous to
+ elimination rules. The sequent calculus analogue of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}I{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}
+ is the rule
+ \[
+ \infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{2C}{\isacharcomma}}\ P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}P{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{2C}{\isacharcomma}}\ Q{\isaliteral{22}{\isachardoublequote}}}}
+ \]
+ Applying the rule backwards, this breaks down some implication on
+ the right side of a sequent; \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{22}{\isachardoublequote}}} stand for
+ the sets of formulae that are unaffected by the inference. The
+ analogue of the pair \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F723E}{\isasymor}}I{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F723E}{\isasymor}}I{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} is the
+ single rule
+ \[
+ \infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F723E}{\isasymor}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{2C}{\isacharcomma}}\ P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{2C}{\isacharcomma}}\ P{\isaliteral{2C}{\isacharcomma}}\ Q{\isaliteral{22}{\isachardoublequote}}}}
+ \]
+ This breaks down some disjunction on the right side, replacing it by
+ both disjuncts. Thus, the sequent calculus is a kind of
+ multiple-conclusion logic.
+
+ To illustrate the use of multiple formulae on the right, let us
+ prove the classical theorem \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{28}{\isacharparenleft}}Q\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}. Working
+ backwards, we reduce this formula to a basic sequent:
+ \[
+ \infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F723E}{\isasymor}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{28}{\isacharparenleft}}Q\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}
+ {\infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}Q\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}
+ {\infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}P\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}Q\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}
+ {\isa{{\isaliteral{22}{\isachardoublequote}}P{\isaliteral{2C}{\isacharcomma}}\ Q\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q{\isaliteral{2C}{\isacharcomma}}\ P{\isaliteral{22}{\isachardoublequote}}}}}}
+ \]
+
+ This example is typical of the sequent calculus: start with the
+ desired theorem and apply rules backwards in a fairly arbitrary
+ manner. This yields a surprisingly effective proof procedure.
+ Quantifiers add only few complications, since Isabelle handles
+ parameters and schematic variables. See \cite[Chapter
+ 10]{paulson-ml2} for further discussion.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Simulating sequents by natural deduction%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle can represent sequents directly, as in the
+ object-logic LK. But natural deduction is easier to work with, and
+ most object-logics employ it. Fortunately, we can simulate the
+ sequent \isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} by the Isabelle formula
+ \isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} where the order of
+ the assumptions and the choice of \isa{{\isaliteral{22}{\isachardoublequote}}Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} are arbitrary.
+ Elim-resolution plays a key role in simulating sequent proofs.
+
+ We can easily handle reasoning on the left. Elim-resolution with
+ the rules \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F723E}{\isasymor}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} achieves
+ a similar effect as the corresponding sequent rules. For the other
+ connectives, we use sequent-style elimination rules instead of
+ destruction rules such as \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616E643E}{\isasymand}}E{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}.
+ But note that the rule \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}L{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} has no effect under our
+ representation of sequents!
+ \[
+ \infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}L{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{2C}{\isacharcomma}}\ P{\isaliteral{22}{\isachardoublequote}}}}
+ \]
+
+ What about reasoning on the right? Introduction rules can only
+ affect the formula in the conclusion, namely \isa{{\isaliteral{22}{\isachardoublequote}}Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}. The
+ other right-side formulae are represented as negated assumptions,
+ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}. In order to operate on one of these, it
+ must first be exchanged with \isa{{\isaliteral{22}{\isachardoublequote}}Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}. Elim-resolution with the
+ \isa{swap} rule has this effect: \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ R\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{22}{\isachardoublequote}}}
+
+ To ensure that swaps occur only when necessary, each introduction
+ rule is converted into a swapped form: it is resolved with the
+ second premise of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}swap{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}. The swapped form of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616E643E}{\isasymand}}I{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, which might be called \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}{\isaliteral{5C3C616E643E}{\isasymand}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, is
+ \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C616E643E}{\isasymand}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ R\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ R\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}
+
+ Similarly, the swapped form of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}I{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} is
+ \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ R\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}
+
+ Swapped introduction rules are applied using elim-resolution, which
+ deletes the negated formula. Our representation of sequents also
+ requires the use of ordinary introduction rules. If we had no
+ regard for readability of intermediate goal states, we could treat
+ the right side more uniformly by representing sequents as \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C626F74746F6D3E}{\isasymbottom}}{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsubsection{Extra rules for the sequent calculus%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+As mentioned, destruction rules such as \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616E643E}{\isasymand}}E{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and
+ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} must be replaced by sequent-style elimination rules.
+ In addition, we need rules to embody the classical equivalence
+ between \isa{{\isaliteral{22}{\isachardoublequote}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{22}{\isachardoublequote}}}. The introduction
+ rules \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F723E}{\isasymor}}I{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} are replaced by a rule that simulates
+ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6F723E}{\isasymor}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}: \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}
+
+ The destruction rule \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} is replaced by \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}Q\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ R{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}
+
+ Quantifier replication also requires special rules. In classical
+ logic, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{22}{\isachardoublequote}}} is equivalent to \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}};
+ the rules \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}L{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} are dual:
+ \[
+ \infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{2C}{\isacharcomma}}\ P\ t{\isaliteral{22}{\isachardoublequote}}}}
+ \qquad
+ \infer[\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}L{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}P\ t{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C44656C74613E}{\isasymDelta}}{\isaliteral{22}{\isachardoublequote}}}}
+ \]
+ Thus both kinds of quantifier may be replicated. Theorems requiring
+ multiple uses of a universal formula are easy to invent; consider
+ \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P\ x\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}f\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ P\ a\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}f\isaliteral{5C3C5E7375703E}{}\isactrlsup n\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle} for any
+ \isa{{\isaliteral{22}{\isachardoublequote}}n\ {\isaliteral{3E}{\isachargreater}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}. Natural examples of the multiple use of an
+ existential formula are rare; a standard one is \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}y{\isaliteral{2E}{\isachardot}}\ P\ x\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ y{\isaliteral{22}{\isachardoublequote}}}.
+
+ Forgoing quantifier replication loses completeness, but gains
+ decidability, since the search space becomes finite. Many useful
+ theorems can be proved without replication, and the search generally
+ delivers its verdict in a reasonable time. To adopt this approach,
+ represent the sequent rules \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}L{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and
+ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}R{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} by \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}I{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}I{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}},
+ respectively, and put \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}E{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} into elimination form: \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}P\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}
+
+ Elim-resolution with this rule will delete the universal formula
+ after a single use. To replicate universal quantifiers, replace the
+ rule by \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}P\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Q{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}
+
+ To replicate existential quantifiers, replace \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}I{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} by
+ \begin{isabelle}%
+{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{22}{\isachardoublequote}}%
+\end{isabelle}
+
+ All introduction rules mentioned above are also useful in swapped
+ form.
+
+ Replication makes the search space infinite; we must apply the rules
+ with care. The classical reasoner distinguishes between safe and
+ unsafe rules, applying the latter only when there is no alternative.
+ Depth-first search may well go down a blind alley; best-first search
+ is better behaved in an infinite search space. However, quantifier
+ replication is too expensive to prove any but the simplest theorems.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
\isamarkupsubsection{Basic methods%
}
\isamarkuptrue%