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+%% $Id$
+\part{Advanced methods}
+Before continuing, it might be wise to try some of your own examples in
+Isabelle, reinforcing your knowledge of the basic functions.
+This paper is merely an introduction to Isabelle. Two other documents
+exist:
+\begin{itemize}
+ \item {\em The Isabelle Reference Manual\/} contains information about
+most of the facilities of Isabelle, apart from particular object-logics.
+
+ \item {\em Isabelle's Object-Logics\/} describes the various logics
+distributed with Isabelle. It also explains how to define new logics in
+Isabelle.
+\end{itemize}
+Look through {\em Isabelle's Object-Logics\/} and try proving some simple
+theorems. You probably should begin with first-order logic ({\tt FOL}
+or~{\tt LK}). Try working some of the examples provided, and others from
+the literature. Set theory~({\tt ZF}) and Constructive Type Theory~({\tt
+ CTT}) form a richer world for mathematical reasoning and, again, many
+examples are in the literature. Higher-order logic~({\tt HOL}) is
+Isabelle's most sophisticated logic, because its types and functions are
+identified with those of the meta-logic; this may cause difficulties for
+beginners.
+
+Choose a logic that you already understand. Isabelle is a proof
+tool, not a teaching tool; if you do not know how to do a particular proof
+on paper, then you certainly will not be able to do it on the machine.
+Even experienced users plan large proofs on paper.
+
+We have covered only the bare essentials of Isabelle, but enough to perform
+substantial proofs. By occasionally dipping into the {\em Reference
+Manual}, you can learn additional tactics, subgoal commands and tacticals.
+Isabelle's simplifier and classical theorem prover are
+difficult to learn, and can be ignored at first.
+
+
+\section{Deriving rules in Isabelle}
+\index{rules!derived}
+A mathematical development goes through a progression of stages. Each
+stage defines some concepts and derives rules about them. We shall see how
+to derive rules, perhaps involving definitions, using Isabelle. The
+following section will explain how to declare types, constants, axioms and
+definitions.
+
+
+\subsection{Deriving a rule using tactics} \label{deriving-example}
+\index{examples!of deriving rules}
+The subgoal module supports the derivation of rules. The \ttindex{goal}
+command, when supplied a goal of the form $\List{\theta@1; \ldots;
+\theta@k} \Imp \phi$, creates $\phi\Imp\phi$ as the initial proof state and
+returns a list consisting of the theorems
+${\theta@i\;[\theta@i]}$, for $i=1$, \ldots,~$k$. These assumptions are
+also recorded internally, allowing \ttindex{result} to discharge them in the
+original order.
+
+Let us derive $\conj$ elimination~(\S\ref{deriving}) using Isabelle.
+Until now, calling \ttindex{goal} has returned an empty list, which we have
+thrown away. In this example, the list contains the two premises of the
+rule. We bind them to the \ML\ identifiers {\tt major} and {\tt
+minor}:\footnote{Some ML compilers will print a message such as {\em
+binding not exhaustive}. This warns that {\tt goal} must return a
+2-element list. Otherwise, the pattern-match will fail; ML will
+raise exception \ttindex{Match}.}
+\begin{ttbox}
+val [major,minor] = goal FOL.thy
+ "[| P&Q; [| P; Q |] ==> R |] ==> R";
+{\out Level 0}
+{\out R}
+{\out 1. R}
+{\out val major = "P & Q [P & Q]" : thm}
+{\out val minor = "[| P; Q |] ==> R [[| P; Q |] ==> R]" : thm}
+\end{ttbox}
+Look at the minor premise, recalling that meta-level assumptions are
+shown in brackets. Using {\tt minor}, we reduce $R$ to the subgoals
+$P$ and~$Q$:
+\begin{ttbox}
+by (resolve_tac [minor] 1);
+{\out Level 1}
+{\out R}
+{\out 1. P}
+{\out 2. Q}
+\end{ttbox}
+Deviating from~\S\ref{deriving}, we apply $({\conj}E1)$ forwards from the
+assumption $P\conj Q$ to obtain the theorem~$P\;[P\conj Q]$.
+\begin{ttbox}
+major RS conjunct1;
+{\out val it = "P [P & Q]" : thm}
+\ttbreak
+by (resolve_tac [major RS conjunct1] 1);
+{\out Level 2}
+{\out R}
+{\out 1. Q}
+\end{ttbox}
+Similarly, we solve the subgoal involving~$Q$.
+\begin{ttbox}
+major RS conjunct2;
+{\out val it = "Q [P & Q]" : thm}
+by (resolve_tac [major RS conjunct2] 1);
+{\out Level 3}
+{\out R}
+{\out No subgoals!}
+\end{ttbox}
+Calling \ttindex{topthm} returns the current proof state as a theorem.
+Note that it contains assumptions. Calling \ttindex{result} discharges the
+assumptions --- both occurrences of $P\conj Q$ are discharged as one ---
+and makes the variables schematic.
+\begin{ttbox}
+topthm();
+{\out val it = "R [P & Q, P & Q, [| P; Q |] ==> R]" : thm}
+val conjE = result();
+{\out val conjE = "[| ?P & ?Q; [| ?P; ?Q |] ==> ?R |] ==> ?R" : thm}
+\end{ttbox}
+
+
+\subsection{Definitions and derived rules} \label{definitions}
+\index{rules!derived}
+\index{Isabelle!definitions in}
+\index{definitions!reasoning about|bold}
+Definitions are expressed as meta-level equalities. Let us define negation
+and the if-and-only-if connective:
+\begin{eqnarray*}
+ \neg \Var{P} & \equiv & \Var{P}\imp\bot \\
+ \Var{P}\bimp \Var{Q} & \equiv &
+ (\Var{P}\imp \Var{Q}) \conj (\Var{Q}\imp \Var{P})
+\end{eqnarray*}
+\index{rewriting!meta-level|bold}
+\index{unfolding|bold}\index{folding|bold}
+Isabelle permits {\bf meta-level rewriting} using definitions such as
+these. {\bf Unfolding} replaces every instance
+of $\neg \Var{P}$ by the corresponding instance of $\Var{P}\imp\bot$. For
+example, $\forall x.\neg (P(x)\conj \neg R(x,0))$ unfolds to
+\[ \forall x.(P(x)\conj R(x,0)\imp\bot)\imp\bot. \]
+{\bf Folding} a definition replaces occurrences of the right-hand side by
+the left. The occurrences need not be free in the entire formula.
+
+\begin{warn}
+Isabelle does not distinguish sensible definitions, like $1\equiv Suc(0)$, from
+equations like $1\equiv Suc(1)$. However, meta-rewriting fails for
+equations like ${f(\Var{x})\equiv g(\Var{x},\Var{y})}$: all variables on
+the right-hand side must also be present on the left.
+\index{rewriting!meta-level}
+\end{warn}
+
+When you define new concepts, you should derive rules asserting their
+abstract properties, and then forget their definitions. This supports
+modularity: if you later change the definitions, without affecting their
+abstract properties, then most of your proofs will carry through without
+change. Indiscriminate unfolding makes a subgoal grow exponentially,
+becoming unreadable.
+
+Taking this point of view, Isabelle does not unfold definitions
+automatically during proofs. Rewriting must be explicit and selective.
+Isabelle provides tactics and meta-rules for rewriting, and a version of
+the {\tt goal} command that unfolds the conclusion and premises of the rule
+being derived.
+
+For example, the intuitionistic definition of negation given above may seem
+peculiar. Using Isabelle, we shall derive pleasanter negation rules:
+\[ \infer[({\neg}I)]{\neg P}{\infer*{\bot}{[P]}} \qquad
+ \infer[({\neg}E)]{Q}{\neg P & P} \]
+This requires proving the following formulae:
+$$ (P\Imp\bot) \Imp \neg P \eqno(\neg I)$$
+$$ \List{\neg P; P} \Imp Q. \eqno(\neg E)$$
+
+
+\subsubsection{Deriving the introduction rule}
+To derive $(\neg I)$, we may call \ttindex{goal} with the appropriate
+formula. Again, {\tt goal} returns a list consisting of the rule's
+premises. We bind this list, which contains the one element $P\Imp\bot$,
+to the \ML\ identifier {\tt prems}.
+\begin{ttbox}
+val prems = goal FOL.thy "(P ==> False) ==> ~P";
+{\out Level 0}
+{\out ~P}
+{\out 1. ~P}
+{\out val prems = ["P ==> False [P ==> False]"] : thm list}
+\end{ttbox}
+Calling \ttindex{rewrite_goals_tac} with \ttindex{not_def}, which is the
+definition of negation, unfolds that definition in the subgoals. It leaves
+the main goal alone.
+\begin{ttbox}
+not_def;
+{\out val it = "~?P == ?P --> False" : thm}
+by (rewrite_goals_tac [not_def]);
+{\out Level 1}
+{\out ~P}
+{\out 1. P --> False}
+\end{ttbox}
+Using \ttindex{impI} and the premise, we reduce subgoal~1 to a triviality:
+\begin{ttbox}
+by (resolve_tac [impI] 1);
+{\out Level 2}
+{\out ~P}
+{\out 1. P ==> False}
+\ttbreak
+by (resolve_tac prems 1);
+{\out Level 3}
+{\out ~P}
+{\out 1. P ==> P}
+\end{ttbox}
+The rest of the proof is routine.
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 4}
+{\out ~P}
+{\out No subgoals!}
+val notI = result();
+{\out val notI = "(?P ==> False) ==> ~?P" : thm}
+\end{ttbox}
+\indexbold{*notI}
+
+\medskip
+There is a simpler way of conducting this proof. The \ttindex{goalw}
+command starts a backward proof, as does \ttindex{goal}, but it also
+unfolds definitions:
+\begin{ttbox}
+val prems = goalw FOL.thy [not_def]
+ "(P ==> False) ==> ~P";
+{\out Level 0}
+{\out ~P}
+{\out 1. P --> False}
+{\out val prems = ["P ==> False [P ==> False]"] : thm list}
+\end{ttbox}
+The proof continues as above, but without calling \ttindex{rewrite_goals_tac}.
+
+
+\subsubsection{Deriving the elimination rule}
+Let us derive $(\neg E)$. The proof follows that of~{\tt conjE}
+(\S\ref{deriving-example}), with an additional step to unfold negation in
+the major premise. Although the {\tt goalw} command is best for this, let
+us try~\ttindex{goal}. As usual, we bind the premises to \ML\ identifiers.
+We then apply \ttindex{FalseE}, which stands for~$(\bot E)$:
+\begin{ttbox}
+val [major,minor] = goal FOL.thy "[| ~P; P |] ==> R";
+{\out Level 0}
+{\out R}
+{\out 1. R}
+{\out val major = "~ P [~ P]" : thm}
+{\out val minor = "P [P]" : thm}
+\ttbreak
+by (resolve_tac [FalseE] 1);
+{\out Level 1}
+{\out R}
+{\out 1. False}
+\ttbreak
+by (resolve_tac [mp] 1);
+{\out Level 2}
+{\out R}
+{\out 1. ?P1 --> False}
+{\out 2. ?P1}
+\end{ttbox}
+For subgoal~1, we transform the major premise from~$\neg P$
+to~${P\imp\bot}$. The function \ttindex{rewrite_rule}, given a list of
+definitions, unfolds them in a theorem. Rewriting does {\bf not}
+affect the theorem's hypothesis, which remains~$\neg P$:
+\begin{ttbox}
+rewrite_rule [not_def] major;
+{\out val it = "P --> False [~P]" : thm}
+by (resolve_tac [it] 1);
+{\out Level 3}
+{\out R}
+{\out 1. P}
+\end{ttbox}
+Now {\tt?P1} has changed to~{\tt P}; we need only use the minor premise:
+\begin{ttbox}
+by (resolve_tac [minor] 1);
+{\out Level 4}
+{\out R}
+{\out No subgoals!}
+val notE = result();
+{\out val notE = "[| ~?P; ?P |] ==> ?R" : thm}
+\end{ttbox}
+\indexbold{*notE}
+
+\medskip
+Again, there is a simpler way of conducting this proof. The
+\ttindex{goalw} command starts unfolds definitions in the premises, as well
+as the conclusion:
+\begin{ttbox}
+val [major,minor] = goalw FOL.thy [not_def]
+ "[| ~P; P |] ==> R";
+{\out val major = "P --> False [~ P]" : thm}
+{\out val minor = "P [P]" : thm}
+\end{ttbox}
+Observe the difference in {\tt major}; the premises are now {\bf unfolded}
+and we need not call~\ttindex{rewrite_rule}. Incidentally, the four calls
+to \ttindex{resolve_tac} above can be collapsed to one, with the help
+of~\ttindex{RS}\@:
+\begin{ttbox}
+minor RS (major RS mp RS FalseE);
+{\out val it = "?P [P, ~P]" : thm}
+by (resolve_tac [it] 1);
+{\out Level 1}
+{\out R}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\medskip Finally, here is a trick that is sometimes useful. If the goal
+has an outermost meta-quantifier, then \ttindex{goal} and \ttindex{goalw}
+do not return the rule's premises in the list of theorems. Instead, the
+premises become assumptions in subgoal~1:
+\begin{ttbox}
+goalw FOL.thy [not_def] "!!P R. [| ~P; P |] ==> R";
+{\out Level 0}
+{\out !!P R. [| ~ P; P |] ==> R}
+{\out 1. !!P R. [| P --> False; P |] ==> R}
+val it = [] : thm list
+\end{ttbox}
+The proof continues as before. But instead of referring to \ML\
+identifiers, we refer to assumptions using \ttindex{eresolve_tac} or
+\ttindex{assume_tac}:
+\begin{ttbox}
+by (resolve_tac [FalseE] 1);
+{\out Level 1}
+{\out !!P R. [| ~ P; P |] ==> R}
+{\out 1. !!P R. [| P --> False; P |] ==> False}
+\ttbreak
+by (eresolve_tac [mp] 1);
+{\out Level 2}
+{\out !!P R. [| ~ P; P |] ==> R}
+{\out 1. !!P R. P ==> P}
+\ttbreak
+by (assume_tac 1);
+{\out Level 3}
+{\out !!P R. [| ~ P; P |] ==> R}
+{\out No subgoals!}
+\end{ttbox}
+Calling \ttindex{result} strips the meta-quantifiers, so the resulting
+theorem is the same as before.
+\begin{ttbox}
+val notE = result();
+{\out val notE = "[| ~?P; ?P |] ==> ?R" : thm}
+\end{ttbox}
+Do not use the {\tt!!}\ trick if the premises contain meta-level
+connectives, because \ttindex{eresolve_tac} and \ttindex{assume_tac} would
+not be able to handle the resulting assumptions. The trick is not suitable
+for deriving the introduction rule~$(\neg I)$.
+
+
+\section{Defining theories}
+\index{theories!defining|(}
+Isabelle makes no distinction between simple extensions of a logic --- like
+defining a type~$bool$ with constants~$true$ and~$false$ --- and defining
+an entire logic. A theory definition has the form
+\begin{ttbox}
+\(T\) = \(S@1\) + \(\cdots\) + \(S@n\) +
+classes {\it class declarations}
+default {\it sort}
+types {\it type declarations}
+arities {\it arity declarations}
+consts {\it constant declarations}
+rules {\it rule declarations}
+translations {\it translation declarations}
+end
+ML {\it ML code}
+\end{ttbox}
+This declares the theory $T$ to extend the existing theories
+$S@1$,~\ldots,~$S@n$. It may declare new classes, types, arities
+(overloadings of existing types), constants and rules; it can specify the
+default sort for type variables. A constant declaration can specify an
+associated concrete syntax. The translations section specifies rewrite
+rules on abstract syntax trees, for defining notations and abbreviations.
+The {\ML} section contains code to perform arbitrary syntactic
+transformations. The main declaration forms are discussed below; see {\em
+ Isabelle's Object-Logics} for full details and examples.
+
+All the declaration parts can be omitted. In the simplest case, $T$ is
+just the union of $S@1$,~\ldots,~$S@n$. New theories always extend one
+or more other theories, inheriting their types, constants, syntax, etc.
+The theory \ttindexbold{Pure} contains nothing but Isabelle's meta-logic.
+
+Each theory definition must reside in a separate file, whose name is
+determined as follows: the theory name, say {\tt ListFn}, is converted to
+lower case and {\tt.thy} is appended, yielding the filename {\tt
+ listfn.thy}. Isabelle uses this convention to locate the file containing
+a given theory; \ttindexbold{use_thy} automatically loads a theory's
+parents before loading the theory itself.
+
+Calling \ttindexbold{use_thy}~{\tt"}{\it tf\/}{\tt"} reads a theory from
+the file {\it tf}{\tt.thy}, writes the corresponding {\ML} code to the file
+{\tt.}{\it tf}{\tt.thy.ML}, and reads the latter file. This declares the
+{\ML} structure~$T$, which contains a component {\tt thy} denoting the new
+theory, a component for each rule, and everything declared in {\it ML
+ code}.
+
+Errors may arise during the translation to {\ML} (say, a misspelled keyword)
+or during creation of the new theory (say, a type error in a rule). But if
+all goes well, {\tt use_thy} will finally read the file {\it tf}{\tt.ML}, if
+it exists. This file typically begins with the {\ML} declaration {\tt
+open}~$T$ and contains proofs that refer to the components of~$T$.
+Theories can be defined directly by issuing {\ML} declarations to Isabelle,
+but the calling sequences are extremely cumbersome.
+
+If theory~$T$ is later redeclared in order to delete an incorrect rule,
+bindings to the old rule may persist. Isabelle ensures that the old and
+new versions of~$T$ are not involved in the same proof. Attempting to
+combine different versions of~$T$ yields the fatal error
+\begin{ttbox}
+Attempt to merge different versions of theory: \(T\)
+\end{ttbox}
+
+\subsection{Declaring constants and rules}
+\indexbold{constants!declaring}\indexbold{rules!declaring}
+Most theories simply declare constants and some rules. The {\bf constant
+declaration part} has the form
+\begin{ttbox}
+consts \(c@1\) :: "\(\tau@1\)"
+ \vdots
+ \(c@n\) :: "\(\tau@n\)"
+\end{ttbox}
+where $c@1$, \ldots, $c@n$ are constants and $\tau@1$, \ldots, $\tau@n$ are
+types. Each type {\em must\/} be enclosed in quotation marks. Each
+constant must be enclosed in quotation marks unless it is a valid
+identifier. To declare $c@1$, \ldots, $c@n$ as constants of type $\tau$,
+the $n$ declarations may be abbreviated to a single line:
+\begin{ttbox}
+ \(c@1\), \ldots, \(c@n\) :: "\(\tau\)"
+\end{ttbox}
+The {\bf rule declaration part} has the form
+\begin{ttbox}
+rules \(id@1\) "\(rule@1\)"
+ \vdots
+ \(id@n\) "\(rule@n\)"
+\end{ttbox}
+where $id@1$, \ldots, $id@n$ are \ML{} identifiers and $rule@1$, \ldots,
+$rule@n$ are expressions of type~$prop$. {\bf Definitions} are rules of
+the form $t\equiv u$. Each rule {\em must\/} be enclosed in quotation marks.
+
+\index{examples!of theories}
+This theory extends first-order logic with two constants {\em nand} and
+{\em xor}, and two rules defining them:
+\begin{ttbox}
+Gate = FOL +
+consts nand,xor :: "[o,o] => o"
+rules nand_def "nand(P,Q) == ~(P & Q)"
+ xor_def "xor(P,Q) == P & ~Q | ~P & Q"
+end
+\end{ttbox}
+
+
+\subsection{Declaring type constructors}
+\indexbold{type constructors!declaring}\indexbold{arities!declaring}
+Types are composed of type variables and {\bf type constructors}. Each
+type constructor has a fixed number of argument places. For example,
+$list$ is a 1-place type constructor and $nat$ is a 0-place type
+constructor.
+
+The {\bf type declaration part} has the form
+\begin{ttbox}
+types \(id@1\) \(k@1\)
+ \vdots
+ \(id@n\) \(k@n\)
+\end{ttbox}
+where $id@1$, \ldots, $id@n$ are identifiers and $k@1$, \ldots, $k@n$ are
+natural numbers. It declares each $id@i$ as a type constructor with $k@i$
+argument places.
+
+The {\bf arity declaration part} has the form
+\begin{ttbox}
+arities \(tycon@1\) :: \(arity@1\)
+ \vdots
+ \(tycon@n\) :: \(arity@n\)
+\end{ttbox}
+where $tycon@1$, \ldots, $tycon@n$ are identifiers and $arity@1$, \ldots,
+$arity@n$ are arities. Arity declarations add arities to existing
+types; they complement type declarations.
+
+In the simplest case, for an 0-place type constructor, an arity is simply
+the type's class. Let us declare a type~$bool$ of class $term$, with
+constants $tt$ and~$ff$:\footnote{In first-order logic, booleans are
+distinct from formulae, which have type $o::logic$.}
+\index{examples!of theories}
+\begin{ttbox}
+Bool = FOL +
+types bool 0
+arities bool :: term
+consts tt,ff :: "bool"
+end
+\end{ttbox}
+In the general case, type constructors take arguments. Each type
+constructor has an {\bf arity} with respect to
+classes~(\S\ref{polymorphic}). A $k$-place type constructor may have
+arities of the form $(s@1,\ldots,s@k)c$, where $s@1,\ldots,s@n$ are sorts
+and $c$ is a class. Each sort specifies a type argument; it has the form
+$\{c@1,\ldots,c@m\}$, where $c@1$, \dots,~$c@m$ are classes. Mostly we
+deal with singleton sorts, and may abbreviate them by dropping the braces.
+The arity declaration $list{::}(term)term$ is short for
+$list{::}(\{term\})term$.
+
+A type constructor may be overloaded (subject to certain conditions) by
+appearing in several arity declarations. For instance, the built-in type
+constructor~$\To$ has the arity $(logic,logic)logic$; in higher-order
+logic, it is declared also to have arity $(term,term)term$.
+
+Theory {\tt List} declares the 1-place type constructor $list$, gives
+it arity $list{::}(term)term$, and declares constants $Nil$ and $Cons$ with
+polymorphic types:
+\index{examples!of theories}
+\begin{ttbox}
+List = FOL +
+types list 1
+arities list :: (term)term
+consts Nil :: "'a list"
+ Cons :: "['a, 'a list] => 'a list"
+end
+\end{ttbox}
+Multiple type and arity declarations may be abbreviated to a single line:
+\begin{ttbox}
+types \(id@1\), \ldots, \(id@n\) \(k\)
+arities \(tycon@1\), \ldots, \(tycon@n\) :: \(arity\)
+\end{ttbox}
+
+\begin{warn}
+Arity declarations resemble constant declarations, but there are {\it no\/}
+quotation marks! Types and rules must be quoted because the theory
+translator passes them verbatim to the {\ML} output file.
+\end{warn}
+
+\subsection{Infixes and Mixfixes}
+\indexbold{infix operators}\index{examples!of theories}
+The constant declaration part of the theory
+\begin{ttbox}
+Gate2 = FOL +
+consts "~&" :: "[o,o] => o" (infixl 35)
+ "#" :: "[o,o] => o" (infixl 30)
+rules nand_def "P ~& Q == ~(P & Q)"
+ xor_def "P # Q == P & ~Q | ~P & Q"
+end
+\end{ttbox}
+declares two left-associating infix operators: $\nand$ of precedence~35 and
+$\xor$ of precedence~30. Hence $P \xor Q \xor R$ is parsed as $(P\xor
+Q) \xor R$ and $P \xor Q \nand R$ as $P \xor (Q \nand R)$. Note the
+quotation marks in \verb|"~&"| and \verb|"#"|.
+
+The constants \hbox{\verb|op ~&|} and \hbox{\verb|op #|} are declared
+automatically, just as in \ML. Hence you may write propositions like
+\verb|op #(True) == op ~&(True)|, which asserts that the functions $\lambda
+Q.True \xor Q$ and $\lambda Q.True \nand Q$ are identical.
+
+\indexbold{mixfix operators}
+{\bf Mixfix} operators may have arbitrary context-free syntaxes. For example
+\begin{ttbox}
+ If :: "[o,o,o] => o" ("if _ then _ else _")
+\end{ttbox}
+declares a constant $If$ of type $[o,o,o] \To o$ with concrete syntax
+$if~P~then~Q~else~R$ instead of $If(P,Q,R)$. Underscores denote argument
+positions. Pretty-printing information can be specified in order to
+improve the layout of formulae with mixfix operations. For details, see
+{\em Isabelle's Object-Logics}.
+
+Mixfix declarations can be annotated with precedences, just like
+infixes. The example above is just a shorthand for
+\begin{ttbox}
+ If :: "[o,o,o] => o" ("if _ then _ else _" [0,0,0] 1000)
+\end{ttbox}
+The numeric components determine precedences. The list of integers
+defines, for each argument position, the minimal precedence an expression
+at that position must have. The final integer is the precedence of the
+construct itself. In the example above, any argument expression is
+acceptable because precedences are non-negative, and conditionals may
+appear everywhere because 1000 is the highest precedence. On the other
+hand,
+\begin{ttbox}
+ If :: "[o,o,o] => o" ("if _ then _ else _" [100,0,0] 99)
+\end{ttbox}
+defines a conditional whose first argument cannot be a conditional because it
+must have a precedence of at least 100. Precedences only apply to
+mixfix syntax: we may write $If(If(P,Q,R),S,T)$.
+
+Binary type constructors, like products and sums, may also be declared as
+infixes. The type declaration below introduces a type constructor~$*$ with
+infix notation $\alpha*\beta$, together with the mixfix notation
+${<}\_,\_{>}$ for pairs.
+\index{examples!of theories}
+\begin{ttbox}
+Prod = FOL +
+types "*" 2 (infixl 20)
+arities "*" :: (term,term)term
+consts fst :: "'a * 'b => 'a"
+ snd :: "'a * 'b => 'b"
+ Pair :: "['a,'b] => 'a * 'b" ("(1<_,/_>)")
+rules fst "fst(<a,b>) = a"
+ snd "snd(<a,b>) = b"
+end
+\end{ttbox}
+
+\begin{warn}
+The name of the type constructor is~{\tt *} and not {\tt op~*}, as it would
+be in the case of an infix constant. Only infix type constructors can have
+symbolic names like~{\tt *}. There is no general mixfix syntax for types.
+\end{warn}
+
+
+\subsection{Overloading}
+\index{overloading}\index{examples!of theories}
+The {\bf class declaration part} has the form
+\begin{ttbox}
+classes \(id@1\) < \(c@1\)
+ \vdots
+ \(id@n\) < \(c@n\)
+\end{ttbox}
+where $id@1$, \ldots, $id@n$ are identifiers and $c@1$, \ldots, $c@n$ are
+existing classes. It declares each $id@i$ as a new class, a subclass
+of~$c@i$. In the general case, an identifier may be declared to be a
+subclass of $k$ existing classes:
+\begin{ttbox}
+ \(id\) < \(c@1\), \ldots, \(c@k\)
+\end{ttbox}
+Type classes allow constants to be overloaded~(\S\ref{polymorphic}). As an
+example, we define the class $arith$ of ``arithmetic'' types with the
+constants ${+} :: [\alpha,\alpha]\To \alpha$ and $0,1 :: \alpha$, for
+$\alpha{::}arith$. We introduce $arith$ as a subclass of $term$ and add
+the three polymorphic constants of this class.
+\index{examples!of theories}
+\begin{ttbox}
+Arith = FOL +
+classes arith < term
+consts "0" :: "'a::arith" ("0")
+ "1" :: "'a::arith" ("1")
+ "+" :: "['a::arith,'a] => 'a" (infixl 60)
+end
+\end{ttbox}
+No rules are declared for these constants: we merely introduce their
+names without specifying properties. On the other hand, classes
+with rules make it possible to prove {\bf generic} theorems. Such
+theorems hold for all instances, all types in that class.
+
+We can now obtain distinct versions of the constants of $arith$ by
+declaring certain types to be of class $arith$. For example, let us
+declare the 0-place type constructors $bool$ and $nat$:
+\index{examples!of theories}
+\begin{ttbox}
+BoolNat = Arith +
+types bool,nat 0
+arities bool,nat :: arith
+consts Suc :: "nat=>nat"
+rules add0 "0 + n = n::nat"
+ addS "Suc(m)+n = Suc(m+n)"
+ nat1 "1 = Suc(0)"
+ or0l "0 + x = x::bool"
+ or0r "x + 0 = x::bool"
+ or1l "1 + x = 1::bool"
+ or1r "x + 1 = 1::bool"
+end
+\end{ttbox}
+Because $nat$ and $bool$ have class $arith$, we can use $0$, $1$ and $+$ at
+either type. The type constraints in the axioms are vital. Without
+constraints, the $x$ in $1+x = x$ would have type $\alpha{::}arith$
+and the axiom would hold for any type of class $arith$. This would
+collapse $nat$:
+\[ Suc(1) = Suc(0+1) = Suc(0)+1 = 1+1 = 1! \]
+The class $arith$ as defined above is more specific than necessary. Many
+types come with a binary operation and identity~(0). On lists,
+$+$ could be concatenation and 0 the empty list --- but what is 1? Hence it
+may be better to define $+$ and 0 on $arith$ and introduce a separate
+class, say $k$, containing~1. Should $k$ be a subclass of $term$ or of
+$arith$? This depends on the structure of your theories; the design of an
+appropriate class hierarchy may require some experimentation.
+
+We will now work through a small example of formalized mathematics
+demonstrating many of the theory extension features.
+
+
+\subsection{Extending first-order logic with the natural numbers}
+\index{examples!of theories}
+
+The early part of this paper defines a first-order logic, including a
+type~$nat$ and the constants $0::nat$ and $Suc::nat\To nat$. Let us
+introduce the Peano axioms for mathematical induction and the freeness of
+$0$ and~$Suc$:
+\[ \vcenter{\infer[(induct)*]{P[n/x]}{P[0/x] & \infer*{P[Suc(x)/x]}{[P]}}}
+ \qquad \parbox{4.5cm}{provided $x$ is not free in any assumption except~$P$}
+\]
+\[ \infer[(Suc\_inject)]{m=n}{Suc(m)=Suc(n)} \qquad
+ \infer[(Suc\_neq\_0)]{R}{Suc(m)=0}
+\]
+Mathematical induction asserts that $P(n)$ is true, for any $n::nat$,
+provided $P(0)$ holds and that $P(x)$ implies $P(Suc(x))$ for all~$x$.
+Some authors express the induction step as $\forall x. P(x)\imp P(Suc(x))$.
+To avoid making induction require the presence of other connectives, we
+formalize mathematical induction as
+$$ \List{P(0); \Forall x. P(x)\Imp P(Suc(x))} \Imp P(n). \eqno(induct) $$
+
+\noindent
+Similarly, to avoid expressing the other rules using~$\forall$, $\imp$
+and~$\neg$, we take advantage of the meta-logic;\footnote
+{On the other hand, the axioms $Suc(m)=Suc(n) \bimp m=n$
+and $\neg(Suc(m)=0)$ are logically equivalent to those given, and work
+better with Isabelle's simplifier.}
+$(Suc\_neq\_0)$ is
+an elimination rule for $Suc(m)=0$:
+$$ Suc(m)=Suc(n) \Imp m=n \eqno(Suc\_inject) $$
+$$ Suc(m)=0 \Imp R \eqno(Suc\_neq\_0) $$
+
+\noindent
+We shall also define a primitive recursion operator, $rec$. Traditionally,
+primitive recursion takes a natural number~$a$ and a 2-place function~$f$,
+and obeys the equations
+\begin{eqnarray*}
+ rec(0,a,f) & = & a \\
+ rec(Suc(m),a,f) & = & f(m, rec(m,a,f))
+\end{eqnarray*}
+Addition, defined by $m+n \equiv rec(m,n,\lambda x\,y.Suc(y))$,
+should satisfy
+\begin{eqnarray*}
+ 0+n & = & n \\
+ Suc(m)+n & = & Suc(m+n)
+\end{eqnarray*}
+This appears to pose difficulties: first-order logic has no functions.
+Following the previous examples, we take advantage of the meta-logic, which
+does have functions. We also generalise primitive recursion to be
+polymorphic over any type of class~$term$, and declare the addition
+function:
+\begin{eqnarray*}
+ rec & :: & [nat, \alpha{::}term, [nat,\alpha]\To\alpha] \To\alpha \\
+ + & :: & [nat,nat]\To nat
+\end{eqnarray*}
+
+
+\subsection{Declaring the theory to Isabelle}
+\index{examples!of theories}
+Let us create the theory \ttindexbold{Nat} starting from theory~\verb$FOL$,
+which contains only classical logic with no natural numbers. We declare
+the 0-place type constructor $nat$ and the constants $rec$ and~$Suc$:
+\begin{ttbox}
+Nat = FOL +
+types nat 0
+arities nat :: term
+consts "0" :: "nat" ("0")
+ Suc :: "nat=>nat"
+ rec :: "[nat, 'a, [nat,'a]=>'a] => 'a"
+ "+" :: "[nat, nat] => nat" (infixl 60)
+rules induct "[| P(0); !!x. P(x) ==> P(Suc(x)) |] ==> P(n)"
+ Suc_inject "Suc(m)=Suc(n) ==> m=n"
+ Suc_neq_0 "Suc(m)=0 ==> R"
+ rec_0 "rec(0,a,f) = a"
+ rec_Suc "rec(Suc(m), a, f) = f(m, rec(m,a,f))"
+ add_def "m+n == rec(m, n, %x y. Suc(y))"
+end
+\end{ttbox}
+In axiom {\tt add_def}, recall that \verb|%| stands for~$\lambda$.
+Opening the \ML\ structure {\tt Nat} permits reference to the axioms by \ML\
+identifiers; we may write {\tt induct} instead of {\tt Nat.induct}.
+\begin{ttbox}
+open Nat;
+\end{ttbox}
+File {\tt FOL/ex/nat.ML} contains proofs involving this theory of the
+natural numbers. As a trivial example, let us derive recursion equations
+for \verb$+$. Here is the zero case:
+\begin{ttbox}
+goalw Nat.thy [add_def] "0+n = n";
+{\out Level 0}
+{\out 0 + n = n}
+{\out 1. rec(0,n,%x y. Suc(y)) = n}
+\ttbreak
+by (resolve_tac [rec_0] 1);
+{\out Level 1}
+{\out 0 + n = n}
+{\out No subgoals!}
+val add_0 = result();
+\end{ttbox}
+And here is the successor case:
+\begin{ttbox}
+goalw Nat.thy [add_def] "Suc(m)+n = Suc(m+n)";
+{\out Level 0}
+{\out Suc(m) + n = Suc(m + n)}
+{\out 1. rec(Suc(m),n,%x y. Suc(y)) = Suc(rec(m,n,%x y. Suc(y)))}
+\ttbreak
+by (resolve_tac [rec_Suc] 1);
+{\out Level 1}
+{\out Suc(m) + n = Suc(m + n)}
+{\out No subgoals!}
+val add_Suc = result();
+\end{ttbox}
+The induction rule raises some complications, which are discussed next.
+\index{theories!defining|)}
+
+
+\section{Refinement with explicit instantiation}
+\index{refinement!with instantiation|bold}
+\index{instantiation!explicit|bold}
+In order to employ mathematical induction, we need to refine a subgoal by
+the rule~$(induct)$. The conclusion of this rule is $\Var{P}(\Var{n})$,
+which is highly ambiguous in higher-order unification. It matches every
+way that a formula can be regarded as depending on a subterm of type~$nat$.
+To get round this problem, we could make the induction rule conclude
+$\forall n.\Var{P}(n)$ --- but putting a subgoal into this form requires
+refinement by~$(\forall E)$, which is equally hard!
+
+The tactic {\tt res_inst_tac}, like {\tt resolve_tac}, refines a subgoal by
+a rule. But it also accepts explicit instantiations for the rule's
+schematic variables.
+\begin{description}
+\item[\ttindexbold{res_inst_tac} {\it insts} {\it thm} {\it i}]
+instantiates the rule {\it thm} with the instantiations {\it insts}, and
+then performs resolution on subgoal~$i$.
+
+\item[\ttindexbold{eres_inst_tac}]
+and \ttindexbold{dres_inst_tac} are similar, but perform elim-resolution
+and destruct-resolution, respectively.
+\end{description}
+The list {\it insts} consists of pairs $[(v@1,e@1), \ldots, (v@n,e@n)]$,
+where $v@1$, \ldots, $v@n$ are names of schematic variables in the rule ---
+with {\bf no} leading question marks!! --- and $e@1$, \ldots, $e@n$ are
+expressions giving their instantiations. The expressions are type-checked
+in the context of a particular subgoal: free variables receive the same
+types as they have in the subgoal, and parameters may appear. Type
+variable instantiations may appear in~{\it insts}, but they are seldom
+required: {\tt res_inst_tac} instantiates type variables automatically
+whenever the type of~$e@i$ is an instance of the type of~$\Var{v@i}$.
+
+\subsection{A simple proof by induction}
+\index{proof!by induction}\index{examples!of induction}
+Let us prove that no natural number~$k$ equals its own successor. To
+use~$(induct)$, we instantiate~$\Var{n}$ to~$k$; Isabelle finds a good
+instantiation for~$\Var{P}$.
+\begin{ttbox}
+goal Nat.thy "~ (Suc(k) = k)";
+{\out Level 0}
+{\out ~Suc(k) = k}
+{\out 1. ~Suc(k) = k}
+\ttbreak
+by (res_inst_tac [("n","k")] induct 1);
+{\out Level 1}
+{\out ~Suc(k) = k}
+{\out 1. ~Suc(0) = 0}
+{\out 2. !!x. ~Suc(x) = x ==> ~Suc(Suc(x)) = Suc(x)}
+\end{ttbox}
+We should check that Isabelle has correctly applied induction. Subgoal~1
+is the base case, with $k$ replaced by~0. Subgoal~2 is the inductive step,
+with $k$ replaced by~$Suc(x)$ and with an induction hypothesis for~$x$.
+The rest of the proof demonstrates~\ttindex{notI}, \ttindex{notE} and the
+other rules of~\ttindex{Nat.thy}. The base case holds by~\ttindex{Suc_neq_0}:
+\begin{ttbox}
+by (resolve_tac [notI] 1);
+{\out Level 2}
+{\out ~Suc(k) = k}
+{\out 1. Suc(0) = 0 ==> False}
+{\out 2. !!x. ~Suc(x) = x ==> ~Suc(Suc(x)) = Suc(x)}
+\ttbreak
+by (eresolve_tac [Suc_neq_0] 1);
+{\out Level 3}
+{\out ~Suc(k) = k}
+{\out 1. !!x. ~Suc(x) = x ==> ~Suc(Suc(x)) = Suc(x)}
+\end{ttbox}
+The inductive step holds by the contrapositive of~\ttindex{Suc_inject}.
+Using the negation rule, we assume $Suc(Suc(x)) = Suc(x)$ and prove $Suc(x)=x$:
+\begin{ttbox}
+by (resolve_tac [notI] 1);
+{\out Level 4}
+{\out ~Suc(k) = k}
+{\out 1. !!x. [| ~Suc(x) = x; Suc(Suc(x)) = Suc(x) |] ==> False}
+\ttbreak
+by (eresolve_tac [notE] 1);
+{\out Level 5}
+{\out ~Suc(k) = k}
+{\out 1. !!x. Suc(Suc(x)) = Suc(x) ==> Suc(x) = x}
+\ttbreak
+by (eresolve_tac [Suc_inject] 1);
+{\out Level 6}
+{\out ~Suc(k) = k}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\subsection{An example of ambiguity in {\tt resolve_tac}}
+\index{examples!of induction}\index{unification!higher-order}
+If you try the example above, you may observe that {\tt res_inst_tac} is
+not actually needed. Almost by chance, \ttindex{resolve_tac} finds the right
+instantiation for~$(induct)$ to yield the desired next state. With more
+complex formulae, our luck fails.
+\begin{ttbox}
+goal Nat.thy "(k+m)+n = k+(m+n)";
+{\out Level 0}
+{\out k + m + n = k + (m + n)}
+{\out 1. k + m + n = k + (m + n)}
+\ttbreak
+by (resolve_tac [induct] 1);
+{\out Level 1}
+{\out k + m + n = k + (m + n)}
+{\out 1. k + m + n = 0}
+{\out 2. !!x. k + m + n = x ==> k + m + n = Suc(x)}
+\end{ttbox}
+This proof requires induction on~$k$. But the 0 in subgoal~1 indicates
+that induction has been applied to the term~$k+(m+n)$. The
+\ttindex{back} command causes backtracking to an alternative
+outcome of the tactic.
+\begin{ttbox}
+back();
+{\out Level 1}
+{\out k + m + n = k + (m + n)}
+{\out 1. k + m + n = k + 0}
+{\out 2. !!x. k + m + n = k + x ==> k + m + n = k + Suc(x)}
+\end{ttbox}
+Now induction has been applied to~$m+n$. Let us call \ttindex{back}
+again.
+\begin{ttbox}
+back();
+{\out Level 1}
+{\out k + m + n = k + (m + n)}
+{\out 1. k + m + 0 = k + (m + 0)}
+{\out 2. !!x. k + m + x = k + (m + x) ==> k + m + Suc(x) = k + (m + Suc(x))}
+\end{ttbox}
+Now induction has been applied to~$n$. What is the next alternative?
+\begin{ttbox}
+back();
+{\out Level 1}
+{\out k + m + n = k + (m + n)}
+{\out 1. k + m + n = k + (m + 0)}
+{\out 2. !!x. k + m + n = k + (m + x) ==> k + m + n = k + (m + Suc(x))}
+\end{ttbox}
+Inspecting subgoal~1 reveals that induction has been applied to just the
+second occurrence of~$n$. This perfectly legitimate induction is useless
+here. The main goal admits fourteen different applications of induction.
+The number is exponential in the size of the formula.
+
+\subsection{Proving that addition is associative}
+\index{associativity of addition}
+\index{examples!of rewriting}
+Let us do the proof properly, using~\ttindex{res_inst_tac}. At the same
+time, we shall have a glimpse at Isabelle's rewriting tactics, which are
+described in the {\em Reference Manual}.
+
+\index{rewriting!object-level}
+Isabelle's rewriting tactics repeatedly applies equations to a subgoal,
+simplifying or proving it. For efficiency, the rewriting rules must be
+packaged into a \bfindex{simplification set}. Let us include the equations
+for~{\tt add} proved in the previous section, namely $0+n=n$ and ${\tt
+ Suc}(m)+n={\tt Suc}(m+n)$:
+\begin{ttbox}
+val add_ss = FOL_ss addrews [add_0, add_Suc];
+\end{ttbox}
+We state the goal for associativity of addition, and
+use \ttindex{res_inst_tac} to invoke induction on~$k$:
+\begin{ttbox}
+goal Nat.thy "(k+m)+n = k+(m+n)";
+{\out Level 0}
+{\out k + m + n = k + (m + n)}
+{\out 1. k + m + n = k + (m + n)}
+\ttbreak
+by (res_inst_tac [("n","k")] induct 1);
+{\out Level 1}
+{\out k + m + n = k + (m + n)}
+{\out 1. 0 + m + n = 0 + (m + n)}
+{\out 2. !!x. x + m + n = x + (m + n) ==> Suc(x) + m + n = Suc(x) + (m + n)}
+\end{ttbox}
+The base case holds easily; both sides reduce to $m+n$. The
+tactic~\ttindex{simp_tac} rewrites with respect to the given simplification
+set, applying the rewrite rules for~{\tt +}:
+\begin{ttbox}
+by (simp_tac add_ss 1);
+{\out Level 2}
+{\out k + m + n = k + (m + n)}
+{\out 1. !!x. x + m + n = x + (m + n) ==> Suc(x) + m + n = Suc(x) + (m + n)}
+\end{ttbox}
+The inductive step requires rewriting by the equations for~{\tt add}
+together the induction hypothesis, which is also an equation. The
+tactic~\ttindex{asm_simp_tac} rewrites using a simplification set and any
+useful assumptions:
+\begin{ttbox}
+by (asm_simp_tac add_ss 1);
+{\out Level 3}
+{\out k + m + n = k + (m + n)}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\section{A {\sc Prolog} interpreter}
+\index{Prolog interpreter|bold}
+To demonstrate the power of tacticals, let us construct a {\sc Prolog}
+interpreter and execute programs involving lists.\footnote{To run these
+examples, see the file {\tt FOL/ex/prolog.ML}.} The {\sc Prolog} program
+consists of a theory. We declare a type constructor for lists, with an
+arity declaration to say that $(\tau)list$ is of class~$term$
+provided~$\tau$ is:
+\begin{eqnarray*}
+ list & :: & (term)term
+\end{eqnarray*}
+We declare four constants: the empty list~$Nil$; the infix list
+constructor~{:}; the list concatenation predicate~$app$; the list reverse
+predicate~$rev$. (In {\sc Prolog}, functions on lists are expressed as
+predicates.)
+\begin{eqnarray*}
+ Nil & :: & \alpha list \\
+ {:} & :: & [\alpha,\alpha list] \To \alpha list \\
+ app & :: & [\alpha list,\alpha list,\alpha list] \To o \\
+ rev & :: & [\alpha list,\alpha list] \To o
+\end{eqnarray*}
+The predicate $app$ should satisfy the {\sc Prolog}-style rules
+\[ {app(Nil,ys,ys)} \qquad
+ {app(xs,ys,zs) \over app(x:xs, ys, x:zs)} \]
+We define the naive version of $rev$, which calls~$app$:
+\[ {rev(Nil,Nil)} \qquad
+ {rev(xs,ys)\quad app(ys, x:Nil, zs) \over
+ rev(x:xs, zs)}
+\]
+
+\index{examples!of theories}
+Theory \ttindex{Prolog} extends first-order logic in order to make use
+of the class~$term$ and the type~$o$. The interpreter does not use the
+rules of~\ttindex{FOL}.
+\begin{ttbox}
+Prolog = FOL +
+types list 1
+arities list :: (term)term
+consts Nil :: "'a list"
+ ":" :: "['a, 'a list]=> 'a list" (infixr 60)
+ app :: "['a list, 'a list, 'a list] => o"
+ rev :: "['a list, 'a list] => o"
+rules appNil "app(Nil,ys,ys)"
+ appCons "app(xs,ys,zs) ==> app(x:xs, ys, x:zs)"
+ revNil "rev(Nil,Nil)"
+ revCons "[| rev(xs,ys); app(ys,x:Nil,zs) |] ==> rev(x:xs,zs)"
+end
+\end{ttbox}
+\subsection{Simple executions}
+Repeated application of the rules solves {\sc Prolog} goals. Let us
+append the lists $[a,b,c]$ and~$[d,e]$. As the rules are applied, the
+answer builds up in~{\tt ?x}.
+\begin{ttbox}
+goal Prolog.thy "app(a:b:c:Nil, d:e:Nil, ?x)";
+{\out Level 0}
+{\out app(a : b : c : Nil, d : e : Nil, ?x)}
+{\out 1. app(a : b : c : Nil, d : e : Nil, ?x)}
+\ttbreak
+by (resolve_tac [appNil,appCons] 1);
+{\out Level 1}
+{\out app(a : b : c : Nil, d : e : Nil, a : ?zs1)}
+{\out 1. app(b : c : Nil, d : e : Nil, ?zs1)}
+\ttbreak
+by (resolve_tac [appNil,appCons] 1);
+{\out Level 2}
+{\out app(a : b : c : Nil, d : e : Nil, a : b : ?zs2)}
+{\out 1. app(c : Nil, d : e : Nil, ?zs2)}
+\end{ttbox}
+At this point, the first two elements of the result are~$a$ and~$b$.
+\begin{ttbox}
+by (resolve_tac [appNil,appCons] 1);
+{\out Level 3}
+{\out app(a : b : c : Nil, d : e : Nil, a : b : c : ?zs3)}
+{\out 1. app(Nil, d : e : Nil, ?zs3)}
+\ttbreak
+by (resolve_tac [appNil,appCons] 1);
+{\out Level 4}
+{\out app(a : b : c : Nil, d : e : Nil, a : b : c : d : e : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+
+{\sc Prolog} can run functions backwards. Which list can be appended
+with $[c,d]$ to produce $[a,b,c,d]$?
+Using \ttindex{REPEAT}, we find the answer at once, $[a,b]$:
+\begin{ttbox}
+goal Prolog.thy "app(?x, c:d:Nil, a:b:c:d:Nil)";
+{\out Level 0}
+{\out app(?x, c : d : Nil, a : b : c : d : Nil)}
+{\out 1. app(?x, c : d : Nil, a : b : c : d : Nil)}
+\ttbreak
+by (REPEAT (resolve_tac [appNil,appCons] 1));
+{\out Level 1}
+{\out app(a : b : Nil, c : d : Nil, a : b : c : d : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\subsection{Backtracking}
+\index{backtracking}
+Which lists $x$ and $y$ can be appended to form the list $[a,b,c,d]$?
+Using \ttindex{REPEAT} to apply the rules, we quickly find the solution
+$x=[]$ and $y=[a,b,c,d]$:
+\begin{ttbox}
+goal Prolog.thy "app(?x, ?y, a:b:c:d:Nil)";
+{\out Level 0}
+{\out app(?x, ?y, a : b : c : d : Nil)}
+{\out 1. app(?x, ?y, a : b : c : d : Nil)}
+\ttbreak
+by (REPEAT (resolve_tac [appNil,appCons] 1));
+{\out Level 1}
+{\out app(Nil, a : b : c : d : Nil, a : b : c : d : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+The \ttindex{back} command returns the tactic's next outcome,
+$x=[a]$ and $y=[b,c,d]$:
+\begin{ttbox}
+back();
+{\out Level 1}
+{\out app(a : Nil, b : c : d : Nil, a : b : c : d : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+The other solutions are generated similarly.
+\begin{ttbox}
+back();
+{\out Level 1}
+{\out app(a : b : Nil, c : d : Nil, a : b : c : d : Nil)}
+{\out No subgoals!}
+\ttbreak
+back();
+{\out Level 1}
+{\out app(a : b : c : Nil, d : Nil, a : b : c : d : Nil)}
+{\out No subgoals!}
+\ttbreak
+back();
+{\out Level 1}
+{\out app(a : b : c : d : Nil, Nil, a : b : c : d : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\subsection{Depth-first search}
+\index{search!depth-first}
+Now let us try $rev$, reversing a list.
+Bundle the rules together as the \ML{} identifier {\tt rules}. Naive
+reverse requires 120 inferences for this 14-element list, but the tactic
+terminates in a few seconds.
+\begin{ttbox}
+goal Prolog.thy "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:Nil, ?w)";
+{\out Level 0}
+{\out rev(a : b : c : d : e : f : g : h : i : j : k : l : m : n : Nil, ?w)}
+{\out 1. rev(a : b : c : d : e : f : g : h : i : j : k : l : m : n : Nil, ?w)}
+val rules = [appNil,appCons,revNil,revCons];
+\ttbreak
+by (REPEAT (resolve_tac rules 1));
+{\out Level 1}
+{\out rev(a : b : c : d : e : f : g : h : i : j : k : l : m : n : Nil,}
+{\out n : m : l : k : j : i : h : g : f : e : d : c : b : a : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+We may execute $rev$ backwards. This, too, should reverse a list. What
+is the reverse of $[a,b,c]$?
+\begin{ttbox}
+goal Prolog.thy "rev(?x, a:b:c:Nil)";
+{\out Level 0}
+{\out rev(?x, a : b : c : Nil)}
+{\out 1. rev(?x, a : b : c : Nil)}
+\ttbreak
+by (REPEAT (resolve_tac rules 1));
+{\out Level 1}
+{\out rev(?x1 : Nil, a : b : c : Nil)}
+{\out 1. app(Nil, ?x1 : Nil, a : b : c : Nil)}
+\end{ttbox}
+The tactic has failed to find a solution! It reached a dead end at
+subgoal~1: there is no~$\Var{x1}$ such that [] appended with~$[\Var{x1}]$
+equals~$[a,b,c]$. Backtracking explores other outcomes.
+\begin{ttbox}
+back();
+{\out Level 1}
+{\out rev(?x1 : a : Nil, a : b : c : Nil)}
+{\out 1. app(Nil, ?x1 : Nil, b : c : Nil)}
+\end{ttbox}
+This too is a dead end, but the next outcome is successful.
+\begin{ttbox}
+back();
+{\out Level 1}
+{\out rev(c : b : a : Nil, a : b : c : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+\ttindex{REPEAT} stops when it cannot continue, regardless of which state
+is reached. The tactical \ttindex{DEPTH_FIRST} searches for a satisfactory
+state, as specified by an \ML{} predicate. Below,
+\ttindex{has_fewer_prems} specifies that the proof state should have no
+subgoals.
+\begin{ttbox}
+val prolog_tac = DEPTH_FIRST (has_fewer_prems 1)
+ (resolve_tac rules 1);
+\end{ttbox}
+Since {\sc Prolog} uses depth-first search, this tactic is a (slow!) {\sc
+Prolog} interpreter. We return to the start of the proof (using
+\ttindex{choplev}), and apply {\tt prolog_tac}:
+\begin{ttbox}
+choplev 0;
+{\out Level 0}
+{\out rev(?x, a : b : c : Nil)}
+{\out 1. rev(?x, a : b : c : Nil)}
+\ttbreak
+by (DEPTH_FIRST (has_fewer_prems 1) (resolve_tac rules 1));
+{\out Level 1}
+{\out rev(c : b : a : Nil, a : b : c : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+Let us try {\tt prolog_tac} on one more example, containing four unknowns:
+\begin{ttbox}
+goal Prolog.thy "rev(a:?x:c:?y:Nil, d:?z:b:?u)";
+{\out Level 0}
+{\out rev(a : ?x : c : ?y : Nil, d : ?z : b : ?u)}
+{\out 1. rev(a : ?x : c : ?y : Nil, d : ?z : b : ?u)}
+\ttbreak
+by prolog_tac;
+{\out Level 1}
+{\out rev(a : b : c : d : Nil, d : c : b : a : Nil)}
+{\out No subgoals!}
+\end{ttbox}
+Although Isabelle is much slower than a {\sc Prolog} system, many
+Isabelle tactics exploit logic programming techniques. For instance, the
+simplification tactics prove subgoals of the form $t=\Var{x}$, rewriting~$t$
+and placing the normalised result in~$\Var{x}$.
+% Local Variables:
+% mode: latex
+% TeX-master: t
+% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/arith.thy Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,6 @@
+Arith = FOL +
+classes arith < term
+consts "0" :: "'a::arith" ("0")
+ "1" :: "'a::arith" ("1")
+ "+" :: "['a::arith,'a] => 'a" (infixl 60)
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/bool.thy Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,5 @@
+Bool = FOL +
+types bool 0
+arities bool :: term
+consts tt,ff :: "bool"
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/bool_nat.thy Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,12 @@
+BoolNat = Arith +
+types bool,nat 0
+arities bool,nat :: arith
+consts Suc :: "nat=>nat"
+rules add0 "0 + n = n::nat"
+ addS "Suc(m)+n = Suc(m+n)"
+ nat1 "1 = Suc(0)"
+ or0l "0 + x = x::bool"
+ or0r "x + 0 = x::bool"
+ or1l "1 + x = 1::bool"
+ or1r "x + 1 = 1::bool"
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/deriv.txt Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,210 @@
+(* Deriving an inference rule *)
+
+Pretty.setmargin 72; (*existing macros just allow this margin*)
+print_depth 0;
+
+
+val [major,minor] = goal Int_Rule.thy
+ "[| P&Q; [| P; Q |] ==> R |] ==> R";
+prth minor;
+by (resolve_tac [minor] 1);
+prth major;
+prth (major RS conjunct1);
+by (resolve_tac [major RS conjunct1] 1);
+prth (major RS conjunct2);
+by (resolve_tac [major RS conjunct2] 1);
+prth (topthm());
+val conjE = prth(result());
+
+
+- val [major,minor] = goal Int_Rule.thy
+= "[| P&Q; [| P; Q |] ==> R |] ==> R";
+Level 0
+R
+ 1. R
+val major = Thm {hyps=#,maxidx=#,prop=#,sign=#} : thm
+val minor = Thm {hyps=#,maxidx=#,prop=#,sign=#} : thm
+- prth minor;
+[| P; Q |] ==> R [[| P; Q |] ==> R]
+- by (resolve_tac [minor] 1);
+Level 1
+R
+ 1. P
+ 2. Q
+- prth major;
+P & Q [P & Q]
+- prth (major RS conjunct1);
+P [P & Q]
+- by (resolve_tac [major RS conjunct1] 1);
+Level 2
+R
+ 1. Q
+- prth (major RS conjunct2);
+Q [P & Q]
+- by (resolve_tac [major RS conjunct2] 1);
+Level 3
+R
+No subgoals!
+- prth (topthm());
+R [P & Q, P & Q, [| P; Q |] ==> R]
+- val conjE = prth(result());
+[| ?P & ?Q; [| ?P; ?Q |] ==> ?R |] ==> ?R
+val conjE = Thm {hyps=#,maxidx=#,prop=#,sign=#} : thm
+
+
+(*** Derived rules involving definitions ***)
+
+(** notI **)
+
+val prems = goal Int_Rule.thy "(P ==> False) ==> ~P";
+prth not_def;
+by (rewrite_goals_tac [not_def]);
+by (resolve_tac [impI] 1);
+by (resolve_tac prems 1);
+by (assume_tac 1);
+val notI = prth(result());
+
+val prems = goalw Int_Rule.thy [not_def]
+ "(P ==> False) ==> ~P";
+
+
+- prth not_def;
+~?P == ?P --> False
+- val prems = goal Int_Rule.thy "(P ==> False) ==> ~P";
+Level 0
+~P
+ 1. ~P
+- by (rewrite_goals_tac [not_def]);
+Level 1
+~P
+ 1. P --> False
+- by (resolve_tac [impI] 1);
+Level 2
+~P
+ 1. P ==> False
+- by (resolve_tac prems 1);
+Level 3
+~P
+ 1. P ==> P
+- by (assume_tac 1);
+Level 4
+~P
+No subgoals!
+- val notI = prth(result());
+(?P ==> False) ==> ~?P
+val notI = # : thm
+
+- val prems = goalw Int_Rule.thy [not_def]
+= "(P ==> False) ==> ~P";
+Level 0
+~P
+ 1. P --> False
+
+
+(** notE **)
+
+val [major,minor] = goal Int_Rule.thy "[| ~P; P |] ==> R";
+by (resolve_tac [FalseE] 1);
+by (resolve_tac [mp] 1);
+prth (rewrite_rule [not_def] major);
+by (resolve_tac [it] 1);
+by (resolve_tac [minor] 1);
+val notE = prth(result());
+
+val [major,minor] = goalw Int_Rule.thy [not_def]
+ "[| ~P; P |] ==> R";
+prth (minor RS (major RS mp RS FalseE));
+by (resolve_tac [it] 1);
+
+
+val prems = goalw Int_Rule.thy [not_def]
+ "[| ~P; P |] ==> R";
+prths prems;
+by (resolve_tac [FalseE] 1);
+by (resolve_tac [mp] 1);
+by (resolve_tac prems 1);
+by (resolve_tac prems 1);
+val notE = prth(result());
+
+
+- val [major,minor] = goal Int_Rule.thy "[| ~P; P |] ==> R";
+Level 0
+R
+ 1. R
+val major = # : thm
+val minor = # : thm
+- by (resolve_tac [FalseE] 1);
+Level 1
+R
+ 1. False
+- by (resolve_tac [mp] 1);
+Level 2
+R
+ 1. ?P1 --> False
+ 2. ?P1
+- prth (rewrite_rule [not_def] major);
+P --> False [~P]
+- by (resolve_tac [it] 1);
+Level 3
+R
+ 1. P
+- by (resolve_tac [minor] 1);
+Level 4
+R
+No subgoals!
+- val notE = prth(result());
+[| ~?P; ?P |] ==> ?R
+val notE = # : thm
+- val [major,minor] = goalw Int_Rule.thy [not_def]
+= "[| ~P; P |] ==> R";
+Level 0
+R
+ 1. R
+val major = # : thm
+val minor = # : thm
+- prth (minor RS (major RS mp RS FalseE));
+?P [P, ~P]
+- by (resolve_tac [it] 1);
+Level 1
+R
+No subgoals!
+
+
+
+
+- goal Int_Rule.thy "[| ~P; P |] ==> R";
+Level 0
+R
+ 1. R
+- prths (map (rewrite_rule [not_def]) it);
+P --> False [~P]
+
+P [P]
+
+- val prems = goalw Int_Rule.thy [not_def]
+= "[| ~P; P |] ==> R";
+Level 0
+R
+ 1. R
+val prems = # : thm list
+- prths prems;
+P --> False [~P]
+
+P [P]
+
+- by (resolve_tac [mp RS FalseE] 1);
+Level 1
+R
+ 1. ?P1 --> False
+ 2. ?P1
+- by (resolve_tac prems 1);
+Level 2
+R
+ 1. P
+- by (resolve_tac prems 1);
+Level 3
+R
+No subgoals!
+- val notE = prth(result());
+[| ~?P; ?P |] ==> ?R
+val notE = # : thm
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/foundations.tex Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,1123 @@
+%% $Id$
+\part{Foundations}
+This Part presents Isabelle's logical foundations in detail:
+representing logical syntax in the typed $\lambda$-calculus; expressing
+inference rules in Isabelle's meta-logic; combining rules by resolution.
+Readers wishing to use Isabelle immediately may prefer to skip straight to
+Part~II, using this Part (via the index) for reference only.
+
+\begin{figure}
+\begin{eqnarray*}
+ \neg P & \hbox{abbreviates} & P\imp\bot \\
+ P\bimp Q & \hbox{abbreviates} & (P\imp Q) \conj (Q\imp P)
+\end{eqnarray*}
+\vskip 4ex
+
+\(\begin{array}{c@{\qquad\qquad}c}
+ \infer[({\conj}I)]{P\conj Q}{P & Q} &
+ \infer[({\conj}E1)]{P}{P\conj Q} \qquad
+ \infer[({\conj}E2)]{Q}{P\conj Q} \\[4ex]
+
+ \infer[({\disj}I1)]{P\disj Q}{P} \qquad
+ \infer[({\disj}I2)]{P\disj Q}{Q} &
+ \infer[({\disj}E)]{R}{P\disj Q & \infer*{R}{[P]} & \infer*{R}{[Q]}}\\[4ex]
+
+ \infer[({\imp}I)]{P\imp Q}{\infer*{Q}{[P]}} &
+ \infer[({\imp}E)]{Q}{P\imp Q & P} \\[4ex]
+
+ &
+ \infer[({\bot}E)]{P}{\bot}\\[4ex]
+
+ \infer[({\forall}I)*]{\forall x.P}{P} &
+ \infer[({\forall}E)]{P[t/x]}{\forall x.P} \\[3ex]
+
+ \infer[({\exists}I)]{\exists x.P}{P[t/x]} &
+ \infer[({\exists}E)*]{Q}{{\exists x.P} & \infer*{Q}{[P]} } \\[3ex]
+
+ {t=t} \,(refl) & \vcenter{\infer[(subst)]{P[u/x]}{t=u & P[t/x]}}
+\end{array} \)
+
+\bigskip\bigskip
+*{\em Eigenvariable conditions\/}:
+
+$\forall I$: provided $x$ is not free in the assumptions
+
+$\exists E$: provided $x$ is not free in $Q$ or any assumption except $P$
+\caption{Intuitionistic first-order logic} \label{fol-fig}
+\end{figure}
+
+\section{Formalizing logical syntax in Isabelle}
+\index{Isabelle!formalizing syntax|bold}
+Figure~\ref{fol-fig} presents intuitionistic first-order logic,
+including equality and the natural numbers. Let us see how to formalize
+this logic in Isabelle, illustrating the main features of Isabelle's
+polymorphic meta-logic.
+
+Isabelle represents syntax using the typed $\lambda$-calculus. We declare
+a type for each syntactic category of the logic. We declare a constant for
+each symbol of the logic, giving each $n$-ary operation an $n$-argument
+curried function type. Most importantly, $\lambda$-abstraction represents
+variable binding in quantifiers.
+
+\index{$\To$|bold}\index{types}
+Isabelle has \ML-style type constructors such as~$(\alpha)list$, where
+$(bool)list$ is the type of lists of booleans. Function types have the
+form $\sigma\To\tau$, where $\sigma$ and $\tau$ are types. Functions
+taking $n$~arguments require curried function types; we may abbreviate
+\[ \sigma@1\To (\cdots \sigma@n\To \tau\cdots) \quad \hbox{as} \quad
+ [\sigma@1, \ldots, \sigma@n] \To \tau. $$
+
+The syntax for terms is summarised below. Note that function application is
+written $t(u)$ rather than the usual $t\,u$.
+\[
+\begin{array}{ll}
+ t :: \sigma & \hbox{type constraint, on a term or variable} \\
+ \lambda x.t & \hbox{abstraction} \\
+ \lambda x@1\ldots x@n.t
+ & \hbox{curried abstraction, $\lambda x@1. \ldots \lambda x@n.t$} \\
+ t(u) & \hbox{application} \\
+ t (u@1, \ldots, u@n) & \hbox{curried application, $t(u@1)\ldots(u@n)$}
+\end{array}
+\]
+
+
+\subsection{Simple types and constants}
+\index{types!simple|bold}
+The syntactic categories of our logic (Figure~\ref{fol-fig}) are
+{\bf formulae} and {\bf terms}. Formulae denote truth
+values, so (following logical tradition) we call their type~$o$. Terms can
+be constructed using~0 and~$Suc$, requiring a type~$nat$ of natural
+numbers. Later, we shall see how to admit terms of other types.
+
+After declaring the types~$o$ and~$nat$, we may declare constants for the
+symbols of our logic. Since $\bot$ denotes a truth value (falsity) and 0
+denotes a number, we put \begin{eqnarray*}
+ \bot & :: & o \\
+ 0 & :: & nat.
+\end{eqnarray*}
+If a symbol requires operands, the corresponding constant must have a
+function type. In our logic, the successor function
+($Suc$) is from natural numbers to natural numbers, negation ($\neg$) is a
+function from truth values to truth values, and the binary connectives are
+curried functions taking two truth values as arguments:
+\begin{eqnarray*}
+ Suc & :: & nat\To nat \\
+ {\neg} & :: & o\To o \\
+ \conj,\disj,\imp,\bimp & :: & [o,o]\To o
+\end{eqnarray*}
+
+Isabelle allows us to declare the binary connectives as infixes, with
+appropriate precedences, so that we write $P\conj Q\disj R$ instead of
+$\disj(\conj(P,Q), R)$.
+
+
+\subsection{Polymorphic types and constants} \label{polymorphic}
+\index{types!polymorphic|bold}
+Which type should we assign to the equality symbol? If we tried
+$[nat,nat]\To o$, then equality would be restricted to the natural
+numbers; we would have to declare different equality symbols for each
+type. Isabelle's type system is polymorphic, so we could declare
+\begin{eqnarray*}
+ {=} & :: & [\alpha,\alpha]\To o.
+\end{eqnarray*}
+But this is also wrong. The declaration is too polymorphic; $\alpha$
+ranges over all types, including~$o$ and $nat\To nat$. Thus, it admits
+$\bot=\neg(\bot)$ and $Suc=Suc$ as formulae, which is acceptable in
+higher-order logic but not in first-order logic.
+
+Isabelle's \bfindex{classes} control polymorphism. Each type variable
+belongs to a class, which denotes a set of types. Classes are partially
+ordered by the subclass relation, which is essentially the subset relation
+on the sets of types. They closely resemble the classes of the functional
+language Haskell~\cite{haskell-tutorial,haskell-report}. Nipkow and
+Prehofer discuss type inference for type classes~\cite{nipkow-prehofer}.
+
+Isabelle provides the built-in class $logic$, which consists of the logical
+types: the ones we want to reason about. Let us declare a class $term$, to
+consist of all legal types of terms in our logic. The subclass structure
+is now $term\le logic$.
+
+We declare $nat$ to be in class $term$. Type variables of class~$term$
+should resemble Standard~\ML's equality type variables, which range over
+those types that possess an equality test. Thus, we declare the equality
+constant by
+\begin{eqnarray*}
+ {=} & :: & [\alpha{::}term,\alpha]\To o
+\end{eqnarray*}
+We give function types and~$o$ the class $logic$ rather than~$term$, since
+they are not legal types for terms. We may introduce new types of class
+$term$ --- for instance, type $string$ or $real$ --- at any time. We can
+even declare type constructors such as $(\alpha)list$, and state that type
+$(\sigma)list$ belongs to class~$term$ provided $\sigma$ does; equality
+applies to lists of natural numbers but not to lists of formulae. We may
+summarize this paragraph by a set of {\bf arity declarations} for type
+constructors:
+\index{$\To$|bold}\index{arities!declaring}
+\begin{eqnarray*}
+ o & :: & logic \\
+ {\To} & :: & (logic,logic)logic \\
+ nat, string, real & :: & term \\
+ list & :: & (term)term
+\end{eqnarray*}
+Higher-order logic, where equality does apply to truth values and
+functions, would require different arity declarations, namely ${o::term}$
+and ${{\To}::(term,term)term}$. The class system can also handle
+overloading.\index{overloading|bold} We could declare $arith$ to be the
+subclass of $term$ consisting of the `arithmetic' types, such as~$nat$.
+Then we could declare the operators
+\begin{eqnarray*}
+ {+},{-},{\times},{/} & :: & [\alpha{::}arith,\alpha]\To \alpha
+\end{eqnarray*}
+If we declare new types $real$ and $complex$ of class $arith$, then we
+effectively have three sets of operators:
+\begin{eqnarray*}
+ {+},{-},{\times},{/} & :: & [nat,nat]\To nat \\
+ {+},{-},{\times},{/} & :: & [real,real]\To real \\
+ {+},{-},{\times},{/} & :: & [complex,complex]\To complex
+\end{eqnarray*}
+Isabelle will regard these as distinct constants, each of which can be defined
+separately. We could even introduce the type $(\alpha)vector$, make
+$(\sigma)vector$ belong to $arith$ provided $\sigma$ is in $arith$, and define
+\begin{eqnarray*}
+ {+} & :: & [(\sigma)vector,(\sigma)vector]\To (\sigma)vector
+\end{eqnarray*}
+in terms of ${+} :: [\sigma,\sigma]\To \sigma$.
+
+Although we have pretended so far that a type variable belongs to only one
+class --- Isabelle's concrete syntax helps to uphold this illusion --- it
+may in fact belong to any finite number of classes. For example suppose
+that we had declared yet another class $ord \le term$, the class of all
+`ordered' types, and a constant
+\begin{eqnarray*}
+ {\le} & :: & [\alpha{::}ord,\alpha]\To o.
+\end{eqnarray*}
+In this context the variable $x$ in $x \le (x+x)$ will be assigned type
+$\alpha{::}\{arith,ord\}$, i.e.\ $\alpha$ belongs to both $arith$ and $ord$.
+Semantically the set $\{arith,ord\}$ should be understood
+as the intersection of the sets of types represented by $arith$ and $ord$.
+Such intersections of classes are called \bfindex{sorts}. The empty
+intersection of classes, $\{\}$, contains all types and is thus the
+{\bf universal sort}.
+
+The type checker handles overloading, assigning each term a unique type. For
+this to be possible, the class and type declarations must satisfy certain
+technical constraints~\cite{nipkow-prehofer}.
+
+\subsection{Higher types and quantifiers}
+\index{types!higher|bold}
+Quantifiers are regarded as operations upon functions. Ignoring polymorphism
+for the moment, consider the formula $\forall x. P(x)$, where $x$ ranges
+over type~$nat$. This is true if $P(x)$ is true for all~$x$. Abstracting
+$P(x)$ into a function, this is the same as saying that $\lambda x.P(x)$
+returns true for all arguments. Thus, the universal quantifier can be
+represented by a constant
+\begin{eqnarray*}
+ \forall & :: & (nat\To o) \To o,
+\end{eqnarray*}
+which is essentially an infinitary truth table. The representation of $\forall
+x. P(x)$ is $\forall(\lambda x. P(x))$.
+
+The existential quantifier is treated
+in the same way. Other binding operators are also easily handled; for
+instance, the summation operator $\Sigma@{k=i}^j f(k)$ can be represented as
+$\Sigma(i,j,\lambda k.f(k))$, where
+\begin{eqnarray*}
+ \Sigma & :: & [nat,nat, nat\To nat] \To nat.
+\end{eqnarray*}
+Quantifiers may be polymorphic. We may define $\forall$ and~$\exists$ over
+all legal types of terms, not just the natural numbers, and
+allow summations over all arithmetic types:
+\begin{eqnarray*}
+ \forall,\exists & :: & (\alpha{::}term\To o) \To o \\
+ \Sigma & :: & [nat,nat, nat\To \alpha{::}arith] \To \alpha
+\end{eqnarray*}
+Observe that the index variables still have type $nat$, while the values
+being summed may belong to any arithmetic type.
+
+
+\section{Formalizing logical rules in Isabelle}
+\index{meta-logic|bold}
+\index{Isabelle!formalizing rules|bold}
+\index{$\Imp$|bold}\index{$\Forall$|bold}\index{$\equiv$|bold}
+\index{implication!meta-level|bold}
+\index{quantifiers!meta-level|bold}
+\index{equality!meta-level|bold}
+Object-logics are formalized by extending Isabelle's meta-logic, which is
+intuitionistic higher-order logic. The meta-level connectives are {\bf
+implication}, the {\bf universal quantifier}, and {\bf equality}.
+\begin{itemize}
+ \item The implication \(\phi\Imp \psi\) means `\(\phi\) implies
+\(\psi\)', and expresses logical {\bf entailment}.
+
+ \item The quantification \(\Forall x.\phi\) means `\(\phi\) is true for
+all $x$', and expresses {\bf generality} in rules and axiom schemes.
+
+\item The equality \(a\equiv b\) means `$a$ equals $b$', for expressing
+ \bfindex{definitions} (see~\S\ref{definitions}). Equalities left over
+ from the unification process, so called \bfindex{flex-flex equations},
+ are written $a\qeq b$. The two equality symbols have the same logical
+ meaning.
+
+\end{itemize}
+The syntax of the meta-logic is formalized in precisely in the same manner
+as object-logics, using the typed $\lambda$-calculus. Analogous to
+type~$o$ above, there is a built-in type $prop$ of meta-level truth values.
+Meta-level formulae will have this type. Type $prop$ belongs to
+class~$logic$; also, $\sigma\To\tau$ belongs to $logic$ provided $\sigma$
+and $\tau$ do. Here are the types of the built-in connectives:
+\begin{eqnarray*}
+ \Imp & :: & [prop,prop]\To prop \\
+ \Forall & :: & (\alpha{::}logic\To prop) \To prop \\
+ {\equiv} & :: & [\alpha{::}\{\},\alpha]\To prop \\
+ \qeq & :: & [\alpha{::}\{\},\alpha]\To prop c
+\end{eqnarray*}
+The restricted polymorphism in $\Forall$ excludes certain types, those used
+just for parsing.
+
+In our formalization of first-order logic, we declared a type~$o$ of
+object-level truth values, rather than using~$prop$ for this purpose. If
+we declared the object-level connectives to have types such as
+${\neg}::prop\To prop$, then these connectives would be applicable to
+meta-level formulae. Keeping $prop$ and $o$ as separate types maintains
+the distinction between the meta-level and the object-level. To formalize
+the inference rules, we shall need to relate the two levels; accordingly,
+we declare the constant
+\index{Trueprop@{$Trueprop$}}
+\begin{eqnarray*}
+ Trueprop & :: & o\To prop.
+\end{eqnarray*}
+We may regard $Trueprop$ as a meta-level predicate, reading $Trueprop(P)$ as
+`$P$ is true at the object-level.' Put another way, $Trueprop$ is a coercion
+from $o$ to $prop$.
+
+
+\subsection{Expressing propositional rules}
+\index{rules!propositional|bold}
+We shall illustrate the use of the meta-logic by formalizing the rules of
+Figure~\ref{fol-fig}. Each object-level rule is expressed as a meta-level
+axiom.
+
+One of the simplest rules is $(\conj E1)$. Making
+everything explicit, its formalization in the meta-logic is
+$$ \Forall P\;Q. Trueprop(P\conj Q) \Imp Trueprop(P). \eqno(\conj E1) $$
+This may look formidable, but it has an obvious reading: for all object-level
+truth values $P$ and~$Q$, if $P\conj Q$ is true then so is~$P$. The
+reading is correct because the meta-logic has simple models, where
+types denote sets and $\Forall$ really means `for all.'
+
+\index{Trueprop@{$Trueprop$}}
+Isabelle adopts notational conventions to ease the writing of rules. We may
+hide the occurrences of $Trueprop$ by making it an implicit coercion.
+Outer universal quantifiers may be dropped. Finally, the nested implication
+\[ \phi@1\Imp(\cdots \phi@n\Imp\psi\cdots) \]
+may be abbreviated as $\List{\phi@1; \ldots; \phi@n} \Imp \psi$, which
+formalizes a rule of $n$~premises.
+
+Using these conventions, the conjunction rules become the following axioms.
+These fully specify the properties of~$\conj$:
+$$ \List{P; Q} \Imp P\conj Q \eqno(\conj I) $$
+$$ P\conj Q \Imp P \qquad P\conj Q \Imp Q \eqno(\conj E1,2) $$
+
+\noindent
+Next, consider the disjunction rules. The discharge of assumption in
+$(\disj E)$ is expressed using $\Imp$:
+$$ P \Imp P\disj Q \qquad Q \Imp P\disj Q \eqno(\disj I1,2) $$
+$$ \List{P\disj Q; P\Imp R; Q\Imp R} \Imp R \eqno(\disj E) $$
+
+\noindent
+To understand this treatment of assumptions\index{assumptions} in natural
+deduction, look at implication. The rule $({\imp}I)$ is the classic
+example of natural deduction: to prove that $P\imp Q$ is true, assume $P$
+is true and show that $Q$ must then be true. More concisely, if $P$
+implies $Q$ (at the meta-level), then $P\imp Q$ is true (at the
+object-level). Showing the coercion explicitly, this is formalized as
+\[ (Trueprop(P)\Imp Trueprop(Q)) \Imp Trueprop(P\imp Q). \]
+The rule $({\imp}E)$ is straightforward; hiding $Trueprop$, the axioms to
+specify $\imp$ are
+$$ (P \Imp Q) \Imp P\imp Q \eqno({\imp}I) $$
+$$ \List{P\imp Q; P} \Imp Q. \eqno({\imp}E) $$
+
+\noindent
+Finally, the intuitionistic contradiction rule is formalized as the axiom
+$$ \bot \Imp P. \eqno(\bot E) $$
+
+\begin{warn}
+Earlier versions of Isabelle, and certain
+papers~\cite{paulson89,paulson700}, use $\List{P}$ to mean $Trueprop(P)$.
+\index{Trueprop@{$Trueprop$}}
+\end{warn}
+
+\subsection{Quantifier rules and substitution}
+\index{rules!quantifier|bold}\index{substitution|bold}
+\index{variables!bound}
+Isabelle expresses variable binding using $\lambda$-abstraction; for instance,
+$\forall x.P$ is formalized as $\forall(\lambda x.P)$. Recall that $F(t)$
+is Isabelle's syntax for application of the function~$F$ to the argument~$t$;
+it is not a meta-notation for substitution. On the other hand, a substitution
+will take place if $F$ has the form $\lambda x.P$; Isabelle transforms
+$(\lambda x.P)(t)$ to~$P[t/x]$ by $\beta$-conversion. Thus, we can express
+inference rules that involve substitution for bound variables.
+
+\index{parameters|bold}\index{eigenvariables|see{parameters}}
+A logic may attach provisos to certain of its rules, especially quantifier
+rules. We cannot hope to formalize arbitrary provisos. Fortunately, those
+typical of quantifier rules always have the same form, namely `$x$ not free in
+\ldots {\it (some set of formulae)},' where $x$ is a variable (called a {\bf
+parameter} or {\bf eigenvariable}) in some premise. Isabelle treats
+provisos using~$\Forall$, its inbuilt notion of `for all'.
+
+\index{$\Forall$}
+The purpose of the proviso `$x$ not free in \ldots' is
+to ensure that the premise may not make assumptions about the value of~$x$,
+and therefore holds for all~$x$. We formalize $(\forall I)$ by
+\[ \left(\Forall x. Trueprop(P(x))\right) \Imp Trueprop(\forall x.P(x)). \]
+This means, `if $P(x)$ is true for all~$x$, then $\forall x.P(x)$ is true.'
+The $\forall E$ rule exploits $\beta$-conversion. Hiding $Trueprop$, the
+$\forall$ axioms are
+$$ \left(\Forall x. P(x)\right) \Imp \forall x.P(x) \eqno(\forall I) $$
+$$ \forall x.P(x) \Imp P(t). \eqno(\forall E)$$
+
+\noindent
+We have defined the object-level universal quantifier~($\forall$)
+using~$\Forall$. But we do not require meta-level counterparts of all the
+connectives of the object-logic! Consider the existential quantifier:
+$$ P(t) \Imp \exists x.P(x) \eqno(\exists I)$$
+$$ \List{\exists x.P(x);\; \Forall x. P(x)\Imp Q} \Imp Q \eqno(\exists E) $$
+Let us verify $(\exists E)$ semantically. Suppose that the premises
+hold; since $\exists x.P(x)$ is true, we may choose $a$ such that $P(a)$ is
+true. Instantiating $\Forall x. P(x)\Imp Q$ with $a$ yields $P(a)\Imp Q$, and
+we obtain the desired conclusion, $Q$.
+
+The treatment of substitution deserves mention. The rule
+\[ \infer{P[u/t]}{t=u & P} \]
+would be hard to formalize in Isabelle. It calls for replacing~$t$ by $u$
+throughout~$P$, which cannot be expressed using $\beta$-conversion. Our
+rule~$(subst)$ uses the occurrences of~$x$ in~$P$ as a template for
+substitution, inferring $P[u/x]$ from~$P[t/x]$. When we formalize this as
+an axiom, the template becomes a function variable:
+$$ \List{t=u; P(t)} \Imp P(u). \eqno(subst)$$
+
+
+\subsection{Signatures and theories}
+\index{signatures|bold}\index{theories|bold}
+A {\bf signature} contains the information necessary for type checking,
+parsing and pretty printing. It specifies classes and their
+relationships; types, with their arities, and constants, with
+their types. It also contains syntax rules, specified using mixfix
+declarations.
+
+Two signatures can be merged provided their specifications are compatible ---
+they must not, for example, assign different types to the same constant.
+Under similar conditions, a signature can be extended. Signatures are
+managed internally by Isabelle; users seldom encounter them.
+
+A {\bf theory} consists of a signature plus a collection of axioms. The
+{\bf pure} theory contains only the meta-logic. Theories can be combined
+provided their signatures are compatible. A theory definition extends an
+existing theory with further signature specifications --- classes, types,
+constants and mixfix declarations --- plus a list of axioms, expressed as
+strings to be parsed. A theory can formalize a small piece of mathematics,
+such as lists and their operations, or an entire logic. A mathematical
+development typically involves many theories in a hierarchy. For example,
+the pure theory could be extended to form a theory for
+Figure~\ref{fol-fig}; this could be extended in two separate ways to form a
+theory for natural numbers and a theory for lists; the union of these two
+could be extended into a theory defining the length of a list:
+\[ \bf
+\begin{array}{c@{}c@{}c@{}c@{}c}
+ {} & {} & \hbox{Length} & {} & {} \\
+ {} & {} & \uparrow & {} & {} \\
+ {} & {} &\hbox{Nat}+\hbox{List}& {} & {} \\
+ {} & \nearrow & {} & \nwarrow & {} \\
+ \hbox{Nat} & {} & {} & {} & \hbox{List} \\
+ {} & \nwarrow & {} & \nearrow & {} \\
+ {} & {} &\hbox{FOL} & {} & {} \\
+ {} & {} & \uparrow & {} & {} \\
+ {} & {} &\hbox{Pure}& {} & {}
+\end{array}
+\]
+Each Isabelle proof typically works within a single theory, which is
+associated with the proof state. However, many different theories may
+coexist at the same time, and you may work in each of these during a single
+session.
+
+
+\section{Proof construction in Isabelle}
+\index{Isabelle!proof construction in|bold}
+There is a one-to-one correspondence between meta-level proofs and
+object-level proofs~\cite{paulson89}. To each use of a meta-level axiom,
+such as $(\forall I)$, there is a use of the corresponding object-level
+rule. Object-level assumptions and parameters have meta-level
+counterparts. The meta-level formalization is {\bf faithful}, admitting no
+incorrect object-level inferences, and {\bf adequate}, admitting all
+correct object-level inferences. These properties must be demonstrated
+separately for each object-logic.
+
+The meta-logic is defined by a collection of inference rules, including
+equational rules for the $\lambda$-calculus, and logical rules. The rules
+for~$\Imp$ and~$\Forall$ resemble those for~$\imp$ and~$\forall$ in
+Figure~\ref{fol-fig}. Proofs performed using the primitive meta-rules
+would be lengthy; Isabelle proofs normally use certain derived rules.
+{\bf Resolution}, in particular, is convenient for backward proof.
+
+Unification is central to theorem proving. It supports quantifier
+reasoning by allowing certain `unknown' terms to be instantiated later,
+possibly in stages. When proving that the time required to sort $n$
+integers is proportional to~$n^2$, we need not state the constant of
+proportionality; when proving that a hardware adder will deliver the sum of
+its inputs, we need not state how many clock ticks will be required. Such
+quantities often emerge from the proof.
+
+\index{variables!schematic|see{unknowns}}
+Isabelle provides {\bf schematic variables}, or \bfindex{unknowns}, for
+unification. Logically, unknowns are free variables. Pragmatically, they
+may be instantiated during a proof, while ordinary variables remain fixed.
+Unknowns are written with a ?\ prefix and are frequently subscripted:
+$\Var{a}$, $\Var{a@1}$, $\Var{a@2}$, \ldots, $\Var{P}$, $\Var{P@1}$, \ldots.
+
+Recall that an inference rule of the form
+\[ \infer{\phi}{\phi@1 & \ldots & \phi@n} \]
+is formalized in Isabelle's meta-logic as the axiom
+$\List{\phi@1; \ldots; \phi@n} \Imp \phi$.
+Such axioms resemble {\sc Prolog}'s Horn clauses, and can be combined by
+resolution --- Isabelle's principal proof method. Resolution yields both
+forward and backward proof. Backward proof works by unifying a goal with
+the conclusion of a rule, whose premises become new subgoals. Forward proof
+works by unifying theorems with the premises of a rule, deriving a new theorem.
+
+Isabelle axioms will require an extended notion of resolution, because
+they differ from Horn clauses in two major respects:
+\begin{itemize}
+ \item They are written in the typed $\lambda$-calculus, and therefore must be
+resolved using higher-order unification.
+
+ \item Horn clauses are composed of atomic formulae. Any formula of the form
+$Trueprop(\cdots)$ is atomic, but axioms such as ${\imp}I$ and $\forall I$
+contain non-atomic formulae.
+\index{Trueprop@{$Trueprop$}}
+\end{itemize}
+Isabelle should not be confused with classical resolution theorem provers
+such as Otter~\cite{wos-bledsoe}. At the meta-level, Isabelle proves
+theorems in their positive form, not by refutation. However, an
+object-logic that includes a contradiction rule may employ a refutation
+proof procedure.
+
+
+\subsection{Higher-order unification}
+\index{unification!higher-order|bold}
+Unification is equation solving. The solution of $f(\Var{x},c) \qeq
+f(d,\Var{y})$ is $\Var{x}\equiv d$ and $\Var{y}\equiv c$. {\bf
+Higher-order unification} is equation solving for typed $\lambda$-terms.
+To handle $\beta$-conversion, it must reduce $(\lambda x.t)u$ to $t[u/x]$.
+That is easy --- in the typed $\lambda$-calculus, all reduction sequences
+terminate at a normal form. But it must guess the unknown
+function~$\Var{f}$ in order to solve the equation
+\begin{equation} \label{hou-eqn}
+ \Var{f}(t) \qeq g(u@1,\ldots,u@k).
+\end{equation}
+Huet's~\cite{huet75} search procedure solves equations by imitation and
+projection. {\bf Imitation}\index{imitation|bold} makes~$\Var{f}$ apply
+leading symbol (if a constant) of the right-hand side. To solve
+equation~(\ref{hou-eqn}), it guesses
+\[ \Var{f} \equiv \lambda x. g(\Var{h@1}(x),\ldots,\Var{h@k}(x)), \]
+where $\Var{h@1}$, \ldots, $\Var{h@k}$ are new unknowns. Assuming there are no
+other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the
+set of equations
+\[ \Var{h@1}(t)\qeq u@1 \quad\ldots\quad \Var{h@k}(t)\qeq u@k. \]
+If the procedure solves these equations, instantiating $\Var{h@1}$, \ldots,
+$\Var{h@k}$, then it yields an instantiation for~$\Var{f}$.
+
+\index{projection|bold}
+{\bf Projection} makes $\Var{f}$ apply one of its arguments. To solve
+equation~(\ref{hou-eqn}), if $t$ expects~$m$ arguments and delivers a
+result of suitable type, it guesses
+\[ \Var{f} \equiv \lambda x. x(\Var{h@1}(x),\ldots,\Var{h@m}(x)), \]
+where $\Var{h@1}$, \ldots, $\Var{h@m}$ are new unknowns. Assuming there are no
+other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the
+equation
+\[ t(\Var{h@1}(t),\ldots,\Var{h@m}(t)) \qeq g(u@1,\ldots,u@k). $$
+
+\begin{warn}
+Huet's unification procedure is complete. Isabelle's polymorphic version,
+which solves for type unknowns as well as for term unknowns, is incomplete.
+The problem is that projection requires type information. In
+equation~(\ref{hou-eqn}), if the type of~$t$ is unknown, then projections
+are possible for all~$m\geq0$, and the types of the $\Var{h@i}$ will be
+similarly unconstrained. Therefore, Isabelle never attempts such
+projections, and may fail to find unifiers where a type unknown turns out
+to be a function type.
+\end{warn}
+
+\index{unknowns!of function type|bold}
+Given $\Var{f}(t@1,\ldots,t@n)\qeq u$, Huet's procedure could make up to
+$n+1$ guesses. The search tree and set of unifiers may be infinite. But
+higher-order unification can work effectively, provided you are careful
+with {\bf function unknowns}:
+\begin{itemize}
+ \item Equations with no function unknowns are solved using first-order
+unification, extended to treat bound variables. For example, $\lambda x.x
+\qeq \lambda x.\Var{y}$ has no solution because $\Var{y}\equiv x$ would
+capture the free variable~$x$.
+
+ \item An occurrence of the term $\Var{f}(x,y,z)$, where the arguments are
+distinct bound variables, causes no difficulties. Its projections can only
+match the corresponding variables.
+
+ \item Even an equation such as $\Var{f}(a)\qeq a+a$ is all right. It has
+four solutions, but Isabelle evaluates them lazily, trying projection before
+imitation. The first solution is usually the one desired:
+\[ \Var{f}\equiv \lambda x. x+x \quad
+ \Var{f}\equiv \lambda x. a+x \quad
+ \Var{f}\equiv \lambda x. x+a \quad
+ \Var{f}\equiv \lambda x. a+a \]
+ \item Equations such as $\Var{f}(\Var{x},\Var{y})\qeq t$ and
+$\Var{f}(\Var{g}(x))\qeq t$ admit vast numbers of unifiers, and must be
+avoided.
+\end{itemize}
+In problematic cases, you may have to instantiate some unknowns before
+invoking unification.
+
+
+\subsection{Joining rules by resolution} \label{joining}
+\index{resolution|bold}
+Let $\List{\psi@1; \ldots; \psi@m} \Imp \psi$ and $\List{\phi@1; \ldots;
+\phi@n} \Imp \phi$ be two Isabelle theorems, representing object-level rules.
+Choosing some~$i$ from~1 to~$n$, suppose that $\psi$ and $\phi@i$ have a
+higher-order unifier. Writing $Xs$ for the application of substitution~$s$ to
+expression~$X$, this means there is some~$s$ such that $\psi s\equiv \phi@i s$.
+By resolution, we may conclude
+\[ (\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;
+ \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s.
+\]
+The substitution~$s$ may instantiate unknowns in both rules. In short,
+resolution is the following rule:
+\[ \infer[(\psi s\equiv \phi@i s)]
+ {(\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;
+ \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s}
+ {\List{\psi@1; \ldots; \psi@m} \Imp \psi & &
+ \List{\phi@1; \ldots; \phi@n} \Imp \phi}
+\]
+It operates at the meta-level, on Isabelle theorems, and is justified by
+the properties of $\Imp$ and~$\Forall$. It takes the number~$i$ (for
+$1\leq i\leq n$) as a parameter and may yield infinitely many conclusions,
+one for each unifier of $\psi$ with $\phi@i$. Isabelle returns these
+conclusions as a sequence (lazy list).
+
+Resolution expects the rules to have no outer quantifiers~($\Forall$). It
+may rename or instantiate any schematic variables, but leaves free
+variables unchanged. When constructing a theory, Isabelle puts the rules
+into a standard form containing no free variables; for instance, $({\imp}E)$
+becomes
+\[ \List{\Var{P}\imp \Var{Q}; \Var{P}} \Imp \Var{Q}.
+\]
+When resolving two rules, the unknowns in the first rule are renamed, by
+subscripting, to make them distinct from the unknowns in the second rule. To
+resolve $({\imp}E)$ with itself, the first copy of the rule would become
+\[ \List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}} \Imp \Var{Q@1}. \]
+Resolving this with $({\imp}E)$ in the first premise, unifying $\Var{Q@1}$ with
+$\Var{P}\imp \Var{Q}$, is the meta-level inference
+\[ \infer{\List{\Var{P@1}\imp (\Var{P}\imp \Var{Q}); \Var{P@1}; \Var{P}}
+ \Imp\Var{Q}.}
+ {\List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}} \Imp \Var{Q@1} & &
+ \List{\Var{P}\imp \Var{Q}; \Var{P}} \Imp \Var{Q}}
+\]
+Renaming the unknowns in the resolvent, we have derived the
+object-level rule
+\[ \infer{Q.}{R\imp (P\imp Q) & R & P} \]
+\index{rules!derived}
+Joining rules in this fashion is a simple way of proving theorems. The
+derived rules are conservative extensions of the object-logic, and may permit
+simpler proofs. Let us consider another example. Suppose we have the axiom
+$$ \forall x\,y. Suc(x)=Suc(y)\imp x=y. \eqno (inject) $$
+
+\noindent
+The standard form of $(\forall E)$ is
+$\forall x.\Var{P}(x) \Imp \Var{P}(\Var{t})$.
+Resolving $(inject)$ with $(\forall E)$ replaces $\Var{P}$ by
+$\lambda x. \forall y. Suc(x)=Suc(y)\imp x=y$ and leaves $\Var{t}$
+unchanged, yielding
+\[ \forall y. Suc(\Var{t})=Suc(y)\imp \Var{t}=y. \]
+Resolving this with $(\forall E)$ puts a subscript on~$\Var{t}$
+and yields
+\[ Suc(\Var{t@1})=Suc(\Var{t})\imp \Var{t@1}=\Var{t}. \]
+Resolving this with $({\imp}E)$ increases the subscripts and yields
+\[ Suc(\Var{t@2})=Suc(\Var{t@1})\Imp \Var{t@2}=\Var{t@1}.
+\]
+We have derived the rule
+\[ \infer{m=n,}{Suc(m)=Suc(n)} \]
+which goes directly from $Suc(m)=Suc(n)$ to $m=n$. It is handy for simplifying
+an equation like $Suc(Suc(Suc(m)))=Suc(Suc(Suc(0)))$.
+
+
+\subsection{Lifting a rule into a context}
+The rules $({\imp}I)$ and $(\forall I)$ may seem unsuitable for
+resolution. They have non-atomic premises, namely $P\Imp Q$ and $\Forall
+x.P(x)$, while the conclusions of all the rules are atomic (they have the form
+$Trueprop(\cdots)$). Isabelle gets round the problem through a meta-inference
+called \bfindex{lifting}. Let us consider how to construct proofs such as
+\[ \infer[({\imp}I)]{P\imp(Q\imp R)}
+ {\infer[({\imp}I)]{Q\imp R}
+ {\infer*{R}{[P] & [Q]}}}
+ \qquad
+ \infer[(\forall I)]{\forall x\,y.P(x,y)}
+ {\infer[(\forall I)]{\forall y.P(x,y)}{P(x,y)}}
+\]
+
+\subsubsection{Lifting over assumptions}
+\index{lifting!over assumptions|bold}
+Lifting over $\theta\Imp{}$ is the following meta-inference rule:
+\[ \infer{\List{\theta\Imp\phi@1; \ldots; \theta\Imp\phi@n} \Imp
+ (\theta \Imp \phi)}
+ {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
+This is clearly sound: if $\List{\phi@1; \ldots; \phi@n} \Imp \phi$ is true and
+$\theta\Imp\phi@1$, \ldots, $\theta\Imp\phi@n$, $\theta$ are all true then
+$\phi$ must be true. Iterated lifting over a series of meta-formulae
+$\theta@k$, \ldots, $\theta@1$ yields an object-rule whose conclusion is
+$\List{\theta@1; \ldots; \theta@k} \Imp \phi$. Typically the $\theta@i$ are
+the assumptions in a natural deduction proof; lifting copies them into a rule's
+premises and conclusion.
+
+When resolving two rules, Isabelle lifts the first one if necessary. The
+standard form of $({\imp}I)$ is
+\[ (\Var{P} \Imp \Var{Q}) \Imp \Var{P}\imp \Var{Q}. \]
+To resolve this rule with itself, Isabelle modifies one copy as follows: it
+renames the unknowns to $\Var{P@1}$ and $\Var{Q@1}$, then lifts the rule over
+$\Var{P}\Imp{}$:
+\[ \List{\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}; \Var{P}}
+ \Imp \Var{P@1}\imp \Var{Q@1}. \]
+Unifying $\Var{P}\Imp \Var{P@1}\imp\Var{Q@1}$ with $\Var{P} \Imp
+\Var{Q}$ instantiates $\Var{Q}$ to ${\Var{P@1}\imp\Var{Q@1}}$.
+Resolution yields
+\[ (\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}) \Imp
+\Var{P}\imp(\Var{P@1}\imp\Var{Q@1}). \]
+This represents the derived rule
+\[ \infer{P\imp(Q\imp R).}{\infer*{R}{[P,Q]}} \]
+
+\subsubsection{Lifting over parameters}
+\index{lifting!over parameters|bold}
+An analogous form of lifting handles premises of the form $\Forall x\ldots\,$.
+Here, lifting prefixes an object-rule's premises and conclusion with $\Forall
+x$. At the same time, lifting introduces a dependence upon~$x$. It replaces
+each unknown $\Var{a}$ in the rule by $\Var{a'}(x)$, where $\Var{a'}$ is a new
+unknown (by subscripting) of suitable type --- necessarily a function type. In
+short, lifting is the meta-inference
+\[ \infer{\List{\Forall x.\phi@1^x; \ldots; \Forall x.\phi@n^x}
+ \Imp \Forall x.\phi^x,}
+ {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
+where $\phi^x$ stands for the result of lifting unknowns over~$x$ in $\phi$.
+It is not hard to verify that this meta-inference is sound.
+For example, $(\disj I)$ might be lifted to
+\[ (\Forall x.\Var{P@1}(x)) \Imp (\Forall x. \Var{P@1}(x)\disj \Var{Q@1}(x))\]
+and $(\forall I)$ to
+\[ (\Forall x\,y.\Var{P@1}(x,y)) \Imp (\Forall x. \forall y.\Var{P@1}(x,y)). \]
+Isabelle has renamed a bound variable in $(\forall I)$ from $x$ to~$y$,
+avoiding a clash. Resolving the above with $(\forall I)$ is the meta-inference
+\[ \infer{\Forall x\,y.\Var{P@1}(x,y)) \Imp \forall x\,y.\Var{P@1}(x,y)) }
+ {(\Forall x\,y.\Var{P@1}(x,y)) \Imp
+ (\Forall x. \forall y.\Var{P@1}(x,y)) &
+ (\Forall x.\Var{P}(x)) \Imp (\forall x.\Var{P}(x))} \]
+Here, $\Var{P}$ is replaced by $\lambda x.\forall y.\Var{P@1}(x,y)$; the
+resolvent expresses the derived rule
+\[ \vcenter{ \infer{\forall x\,y.Q(x,y)}{Q(x,y)} }
+ \quad\hbox{provided $x$, $y$ not free in the assumptions}
+\]
+I discuss lifting and parameters at length elsewhere~\cite{paulson89}.
+Miller goes into even greater detail~\cite{miller-jsc}.
+
+
+\section{Backward proof by resolution}
+\index{resolution!in backward proof}\index{proof!backward|bold}
+Resolution is convenient for deriving simple rules and for reasoning
+forward from facts. It can also support backward proof, where we start
+with a goal and refine it to progressively simpler subgoals until all have
+been solved. {\sc lcf} (and its descendants {\sc hol} and Nuprl) provide
+tactics and tacticals, which constitute a high-level language for
+expressing proof searches. \bfindex{Tactics} perform primitive refinements
+while \bfindex{tacticals} combine tactics.
+
+\index{LCF}
+Isabelle's tactics and tacticals work differently from {\sc lcf}'s. An
+Isabelle rule is {\bf bidirectional}: there is no distinction between
+inputs and outputs. {\sc lcf} has a separate tactic for each rule;
+Isabelle performs refinement by any rule in a uniform fashion, using
+resolution.
+
+\index{subgoals|bold}\index{main goal|bold}
+Isabelle works with meta-level theorems of the form
+\( \List{\phi@1; \ldots; \phi@n} \Imp \phi \).
+We have viewed this as the {\bf rule} with premises
+$\phi@1$,~\ldots,~$\phi@n$ and conclusion~$\phi$. It can also be viewed as
+the \bfindex{proof state} with subgoals $\phi@1$,~\ldots,~$\phi@n$ and main
+goal~$\phi$.
+
+To prove the formula~$\phi$, take $\phi\Imp \phi$ as the initial proof
+state. This assertion is, trivially, a theorem. At a later stage in the
+backward proof, a typical proof state is $\List{\phi@1; \ldots; \phi@n}
+\Imp \phi$. This proof state is a theorem, guaranteeing that the subgoals
+$\phi@1$,~\ldots,~$\phi@n$ imply~$\phi$. If $m=0$ then we have
+proved~$\phi$ outright. If $\phi$ contains unknowns, they may become
+instantiated during the proof; a proof state may be $\List{\phi@1; \ldots;
+\phi@n} \Imp \phi'$, where $\phi'$ is an instance of~$\phi$.
+
+\subsection{Refinement by resolution}
+\index{refinement|bold}
+To refine subgoal~$i$ of a proof state by a rule, perform the following
+resolution:
+\[ \infer{\hbox{new proof state}}{\hbox{rule} & & \hbox{proof state}} \]
+If the rule is $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$ after lifting
+over subgoal~$i$, and the proof state is $\List{\phi@1; \ldots; \phi@n}
+\Imp \phi$, then the new proof state is (for~$1\leq i\leq n$)
+\[ (\List{\phi@1; \ldots; \phi@{i-1}; \psi'@1;
+ \ldots; \psi'@m; \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s. \]
+Substitution~$s$ unifies $\psi'$ with~$\phi@i$. In the proof state,
+subgoal~$i$ is replaced by $m$ new subgoals, the rule's instantiated premises.
+If some of the rule's unknowns are left un-instantiated, they become new
+unknowns in the proof state. Refinement by~$(\exists I)$, namely
+\[ \Var{P}(\Var{t}) \Imp \exists x. \Var{P}(x), \]
+inserts a new unknown derived from~$\Var{t}$ by subscripting and lifting.
+We do not have to specify an `existential witness' when
+applying~$(\exists I)$. Further resolutions may instantiate unknowns in
+the proof state.
+
+\subsection{Proof by assumption}
+\index{proof!by assumption|bold}
+In the course of a natural deduction proof, parameters $x@1$, \ldots,~$x@l$ and
+assumptions $\theta@1$, \ldots, $\theta@k$ accumulate, forming a context for
+each subgoal. Repeated lifting steps can lift a rule into any context. To
+aid readability, Isabelle puts contexts into a normal form, gathering the
+parameters at the front:
+\begin{equation} \label{context-eqn}
+\Forall x@1 \ldots x@l. \List{\theta@1; \ldots; \theta@k}\Imp\theta.
+\end{equation}
+Under the usual reading of the connectives, this expresses that $\theta$
+follows from $\theta@1$,~\ldots~$\theta@k$ for arbitrary
+$x@1$,~\ldots,~$x@l$. It is trivially true if $\theta$ equals any of
+$\theta@1$,~\ldots~$\theta@k$, or is unifiable with any of them. This
+models proof by assumption in natural deduction.
+
+Isabelle automates the meta-inference for proof by assumption. Its arguments
+are the meta-theorem $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, and some~$i$
+from~1 to~$n$, where $\phi@i$ has the form~(\ref{context-eqn}). Its results
+are meta-theorems of the form
+\[ (\List{\phi@1; \ldots; \phi@{i-1}; \phi@{i+1}; \phi@n} \Imp \phi)s \]
+for each $s$ and~$j$ such that $s$ unifies $\lambda x@1 \ldots x@l. \theta@j$
+with $\lambda x@1 \ldots x@l. \theta$. Isabelle supplies the parameters
+$x@1$,~\ldots,~$x@l$ to higher-order unification as bound variables, which
+regards them as unique constants with a limited scope --- this enforces
+parameter provisos~\cite{paulson89}.
+
+The premise represents a proof state with~$n$ subgoals, of which the~$i$th is
+to be solved by assumption. Isabelle searches the subgoal's context for an
+assumption, say $\theta@j$, that can solve it by unification. For each
+unifier, the meta-inference returns an instantiated proof state from which the
+$i$th subgoal has been removed. Isabelle searches for a unifying assumption;
+for readability and robustness, proofs do not refer to assumptions by number.
+
+Consider the proof state $(\List{P(a); P(b)} \Imp P(\Var{x})) \Imp Q(\Var{x})$.
+Proof by assumption (with $i=1$, the only possibility) yields two results:
+\begin{itemize}
+ \item $Q(a)$, instantiating $\Var{x}\equiv a$
+ \item $Q(b)$, instantiating $\Var{x}\equiv b$
+\end{itemize}
+Here, proof by assumption affects the main goal. It could also affect
+other subgoals. Consider the effect of having the
+additional subgoal ${\List{P(b); P(c)} \Imp P(\Var{x})}$.
+
+\subsection{A propositional proof} \label{prop-proof}
+\index{examples!propositional}
+Our first example avoids quantifiers. Given the main goal $P\disj P\imp
+P$, Isabelle creates the initial state
+\[ (P\disj P\imp P)\Imp (P\disj P\imp P). \]
+Bear in mind that every proof state we derive will be a meta-theorem,
+expressing that the subgoals imply the main goal. The first step is to refine
+subgoal~1 by (${\imp}I)$, creating a new state where $P\disj P$ is an
+assumption:
+\[ (P\disj P\Imp P)\Imp (P\disj P\imp P) \]
+The next step is $(\disj E)$, which replaces subgoal~1 by three new subgoals.
+Because of lifting, each subgoal contains a copy of the context --- the
+assumption $P\disj P$. (In fact, this assumption is now redundant; we shall
+shortly see how to get rid of it!) The new proof state is the following
+meta-theorem, laid out for clarity:
+\[ \begin{array}{l@{}l@{\qquad\qquad}l}
+ \lbrakk\;& P\disj P\Imp \Var{P@1}\disj\Var{Q@1}; & \hbox{(subgoal 1)} \\
+ & \List{P\disj P; \Var{P@1}} \Imp P; & \hbox{(subgoal 2)} \\
+ & \List{P\disj P; \Var{Q@1}} \Imp P & \hbox{(subgoal 3)} \\
+ \rbrakk\;& \Imp (P\disj P\imp P) & \hbox{(main goal)}
+ \end{array}
+\]
+Notice the unknowns in the proof state. Because we have applied $(\disj E)$,
+we must prove some disjunction, $\Var{P@1}\disj\Var{Q@1}$. Of course,
+subgoal~1 is provable by assumption. This instantiates both $\Var{P@1}$ and
+$\Var{Q@1}$ to~$P$ throughout the proof state:
+\[ \begin{array}{l@{}l@{\qquad\qquad}l}
+ \lbrakk\;& \List{P\disj P; P} \Imp P; & \hbox{(subgoal 1)} \\
+ & \List{P\disj P; P} \Imp P & \hbox{(subgoal 2)} \\
+ \rbrakk\;& \Imp (P\disj P\imp P) & \hbox{(main goal)}
+ \end{array} \]
+Both of the remaining subgoals can be proved by assumption. After two such
+steps, the proof state is simply the meta-theorem $P\disj P\imp P$. This is
+our desired result.
+
+\subsection{A quantifier proof}
+\index{examples!with quantifiers}
+To illustrate quantifiers and $\Forall$-lifting, let us prove
+$(\exists x.P(f(x)))\imp(\exists x.P(x))$. The initial proof
+state is the trivial meta-theorem
+\[ (\exists x.P(f(x)))\imp(\exists x.P(x)) \Imp
+ (\exists x.P(f(x)))\imp(\exists x.P(x)). \]
+As above, the first step is refinement by (${\imp}I)$:
+\[ (\exists x.P(f(x))\Imp \exists x.P(x)) \Imp
+ (\exists x.P(f(x)))\imp(\exists x.P(x))
+\]
+The next step is $(\exists E)$, which replaces subgoal~1 by two new subgoals.
+Both have the assumption $\exists x.P(f(x))$. The new proof
+state is the meta-theorem
+\[ \begin{array}{l@{}l@{\qquad\qquad}l}
+ \lbrakk\;& \exists x.P(f(x)) \Imp \exists x.\Var{P@1}(x); & \hbox{(subgoal 1)} \\
+ & \Forall x.\List{\exists x.P(f(x)); \Var{P@1}(x)} \Imp
+ \exists x.P(x) & \hbox{(subgoal 2)} \\
+ \rbrakk\;& \Imp (\exists x.P(f(x)))\imp(\exists x.P(x)) & \hbox{(main goal)}
+ \end{array}
+\]
+The unknown $\Var{P@1}$ appears in both subgoals. Because we have applied
+$(\exists E)$, we must prove $\exists x.\Var{P@1}(x)$, where $\Var{P@1}(x)$ may
+become any formula possibly containing~$x$. Proving subgoal~1 by assumption
+instantiates $\Var{P@1}$ to~$\lambda x.P(f(x))$:
+\[ \left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp
+ \exists x.P(x)\right)
+ \Imp (\exists x.P(f(x)))\imp(\exists x.P(x))
+\]
+The next step is refinement by $(\exists I)$. The rule is lifted into the
+context of the parameter~$x$ and the assumption $P(f(x))$. This ensures that
+the context is copied to the subgoal and allows the existential witness to
+depend upon~$x$:
+\[ \left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp
+ P(\Var{x@2}(x))\right)
+ \Imp (\exists x.P(f(x)))\imp(\exists x.P(x))
+\]
+The existential witness, $\Var{x@2}(x)$, consists of an unknown
+applied to a parameter. Proof by assumption unifies $\lambda x.P(f(x))$
+with $\lambda x.P(\Var{x@2}(x))$, instantiating $\Var{x@2}$ to $f$. The final
+proof state contains no subgoals: $(\exists x.P(f(x)))\imp(\exists x.P(x))$.
+
+
+\subsection{Tactics and tacticals}
+\index{tactics|bold}\index{tacticals|bold}
+{\bf Tactics} perform backward proof. Isabelle tactics differ from those
+of {\sc lcf}, {\sc hol} and Nuprl by operating on entire proof states,
+rather than on individual subgoals. An Isabelle tactic is a function that
+takes a proof state and returns a sequence (lazy list) of possible
+successor states. Sequences are coded in ML as functions, a standard
+technique~\cite{paulson91}. Isabelle represents proof states by theorems.
+
+Basic tactics execute the meta-rules described above, operating on a
+given subgoal. The {\bf resolution tactics} take a list of rules and
+return next states for each combination of rule and unifier. The {\bf
+assumption tactic} examines the subgoal's assumptions and returns next
+states for each combination of assumption and unifier. Lazy lists are
+essential because higher-order resolution may return infinitely many
+unifiers. If there are no matching rules or assumptions then no next
+states are generated; a tactic application that returns an empty list is
+said to {\bf fail}.
+
+Sequences realize their full potential with {\bf tacticals} --- operators
+for combining tactics. Depth-first search, breadth-first search and
+best-first search (where a heuristic function selects the best state to
+explore) return their outcomes as a sequence. Isabelle provides such
+procedures in the form of tacticals. Simpler procedures can be expressed
+directly using the basic tacticals {\it THEN}, {\it ORELSE} and {\it REPEAT}:
+\begin{description}
+\item[$tac1\;THEN\;tac2$] is a tactic for sequential composition. Applied
+to a proof state, it returns all states reachable in two steps by applying
+$tac1$ followed by~$tac2$.
+
+\item[$tac1\;ORELSE\;tac2$] is a choice tactic. Applied to a state, it
+tries~$tac1$ and returns the result if non-empty; otherwise, it uses~$tac2$.
+
+\item[$REPEAT\;tac$] is a repetition tactic. Applied to a state, it
+returns all states reachable by applying~$tac$ as long as possible (until
+it would fail). $REPEAT$ can be expressed in a few lines of \ML{} using
+{\it THEN} and~{\it ORELSE}.
+\end{description}
+For instance, this tactic repeatedly applies $tac1$ and~$tac2$, giving
+$tac1$ priority:
+\[ REPEAT(tac1\;ORELSE\;tac2) \]
+
+
+\section{Variations on resolution}
+In principle, resolution and proof by assumption suffice to prove all
+theorems. However, specialized forms of resolution are helpful for working
+with elimination rules. Elim-resolution applies an elimination rule to an
+assumption; destruct-resolution is similar, but applies a rule in a forward
+style.
+
+The last part of the section shows how the techniques for proving theorems
+can also serve to derive rules.
+
+\subsection{Elim-resolution}
+\index{elim-resolution|bold}
+Consider proving the theorem $((R\disj R)\disj R)\disj R\imp R$. By
+$({\imp}I)$, we prove $R$ from the assumption $((R\disj R)\disj R)\disj R$.
+Applying $(\disj E)$ to this assumption yields two subgoals, one that
+assumes~$R$ (and is therefore trivial) and one that assumes $(R\disj
+R)\disj R$. This subgoal admits another application of $(\disj E)$. Since
+natural deduction never discards assumptions, we eventually generate a
+subgoal containing much that is redundant:
+\[ \List{((R\disj R)\disj R)\disj R; (R\disj R)\disj R; R\disj R; R} \Imp R. \]
+In general, using $(\disj E)$ on the assumption $P\disj Q$ creates two new
+subgoals with the additional assumption $P$ or~$Q$. In these subgoals,
+$P\disj Q$ is redundant. It wastes space; worse, it could make a theorem
+prover repeatedly apply~$(\disj E)$, looping. Other elimination rules pose
+similar problems. In first-order logic, only universally quantified
+assumptions are sometimes needed more than once --- say, to prove
+$P(f(f(a)))$ from the assumptions $\forall x.P(x)\imp P(f(x))$ and~$P(a)$.
+
+Many logics can be formulated as sequent calculi that delete redundant
+assumptions after use. The rule $(\disj E)$ might become
+\[ \infer[\disj\hbox{-left}]
+ {\Gamma,P\disj Q,\Delta \turn \Theta}
+ {\Gamma,P,\Delta \turn \Theta && \Gamma,Q,\Delta \turn \Theta} \]
+In backward proof, a goal containing $P\disj Q$ on the left of the~$\turn$
+(that is, as an assumption) splits into two subgoals, replacing $P\disj Q$
+by $P$ or~$Q$. But the sequent calculus, with its explicit handling of
+assumptions, can be tiresome to use.
+
+Elim-resolution is Isabelle's way of getting sequent calculus behaviour
+from natural deduction rules. It lets an elimination rule consume an
+assumption. Elim-resolution takes a rule $\List{\psi@1; \ldots; \psi@m}
+\Imp \psi$ a proof state $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, and a
+subgoal number~$i$. The rule must have at least one premise, thus $m>0$.
+Write the rule's lifted form as $\List{\psi'@1; \ldots; \psi'@m} \Imp
+\psi'$. Ordinary resolution would attempt to reduce~$\phi@i$,
+replacing subgoal~$i$ by $m$ new ones. Elim-resolution tries {\bf
+simultaneously} to reduce~$\phi@i$ and to solve~$\psi'@1$ by assumption; it
+returns a sequence of next states. Each of these replaces subgoal~$i$ by
+instances of $\psi'@2$, \ldots, $\psi'@m$ from which the selected
+assumption has been deleted. Suppose $\phi@i$ has the parameter~$x$ and
+assumptions $\theta@1$,~\ldots,~$\theta@k$. Then $\psi'@1$, the rule's first
+premise after lifting, will be
+\( \Forall x. \List{\theta@1; \ldots; \theta@k}\Imp \psi^{x}@1 \).
+Elim-resolution tries to unify simultaneously $\psi'\qeq\phi@i$ and
+$\lambda x. \theta@j \qeq \lambda x. \psi^{x}@1$, for $j=1$,~\ldots,~$k$.
+
+Let us redo the example from~\S\ref{prop-proof}. The elimination rule
+is~$(\disj E)$,
+\[ \List{\Var{P}\disj \Var{Q};\; \Var{P}\Imp \Var{R};\; \Var{Q}\Imp \Var{R}}
+ \Imp \Var{R} \]
+and the proof state is $(P\disj P\Imp P)\Imp (P\disj P\imp P)$. The
+lifted rule would be
+\[ \begin{array}{l@{}l}
+ \lbrakk\;& P\disj P \Imp \Var{P@1}\disj\Var{Q@1}; \\
+ & \List{P\disj P ;\; \Var{P@1}} \Imp \Var{R@1}; \\
+ & \List{P\disj P ;\; \Var{Q@1}} \Imp \Var{R@1} \\
+ \rbrakk\;& \Imp \Var{R@1}
+ \end{array}
+\]
+Unification would take the simultaneous equations
+$P\disj P \qeq \Var{P@1}\disj\Var{Q@1}$ and $\Var{R@1} \qeq P$, yielding
+$\Var{P@1}\equiv\Var{Q@1}\equiv\Var{R@1} \equiv P$. The new proof state
+would be simply
+\[ \List{P \Imp P;\; P \Imp P} \Imp (P\disj P\imp P).
+\]
+Elim-resolution's simultaneous unification gives better control
+than ordinary resolution. Recall the substitution rule:
+$$ \List{\Var{t}=\Var{u}; \Var{P}(\Var{t})} \Imp \Var{P}(\Var{u})
+ \eqno(subst) $$
+Unsuitable for ordinary resolution because $\Var{P}(\Var{u})$ admits many
+unifiers, $(subst)$ works well with elim-resolution. It deletes some
+assumption of the form $x=y$ and replaces every~$y$ by~$x$ in the subgoal
+formula. The simultaneous unification instantiates $\Var{u}$ to~$y$; if
+$y$ is not an unknown, then $\Var{P}(y)$ can easily be unified with another
+formula.
+
+In logical parlance, the premise containing the connective to be eliminated
+is called the \bfindex{major premise}. Elim-resolution expects the major
+premise to come first. The order of the premises is significant in
+Isabelle.
+
+\subsection{Destruction rules} \label{destruct}
+\index{destruction rules|bold}\index{proof!forward}
+Looking back to Figure~\ref{fol-fig}, notice that there are two kinds of
+elimination rule. The rules $({\conj}E1)$, $({\conj}E2)$, $({\imp}E)$ and
+$({\forall}E)$ extract the conclusion from the major premise. In Isabelle
+parlance, such rules are called \bfindex{destruction rules}; they are readable
+and easy to use in forward proof. The rules $({\disj}E)$, $({\bot}E)$ and
+$({\exists}E)$ work by discharging assumptions; they support backward proof
+in a style reminiscent of the sequent calculus.
+
+The latter style is the most general form of elimination rule. In natural
+deduction, there is no way to recast $({\disj}E)$, $({\bot}E)$ or
+$({\exists}E)$ as destruction rules. But we can write general elimination
+rules for $\conj$, $\imp$ and~$\forall$:
+\[
+\infer{R}{P\conj Q & \infer*{R}{[P,Q]}} \qquad
+\infer{R}{P\imp Q & P & \infer*{R}{[Q]}} \qquad
+\infer{Q}{\forall x.P & \infer*{Q}{[P[t/x]]}}
+\]
+Because they are concise, destruction rules are simpler to derive than the
+corresponding elimination rules. To facilitate their use in backward
+proof, Isabelle provides a means of transforming a destruction rule such as
+\[ \infer[\quad\hbox{to the elimination rule}\quad]{Q}{P@1 & \ldots & P@m}
+ \infer{R.}{P@1 & \ldots & P@m & \infer*{R}{[Q]}}
+\]
+\index{destruct-resolution|bold}
+{\bf Destruct-resolution} combines this transformation with
+elim-resolution. It applies a destruction rule to some assumption of a
+subgoal. Given the rule above, it replaces the assumption~$P@1$ by~$Q$,
+with new subgoals of showing instances of $P@2$, \ldots,~$P@m$.
+Destruct-resolution works forward from a subgoal's assumptions. Ordinary
+resolution performs forward reasoning from theorems, as illustrated
+in~\S\ref{joining}.
+
+
+\subsection{Deriving rules by resolution} \label{deriving}
+\index{rules!derived|bold}
+The meta-logic, itself a form of the predicate calculus, is defined by a
+system of natural deduction rules. Each theorem may depend upon
+meta-assumptions. The theorem that~$\phi$ follows from the assumptions
+$\phi@1$, \ldots, $\phi@n$ is written
+\[ \phi \quad [\phi@1,\ldots,\phi@n]. \]
+A more conventional notation might be $\phi@1,\ldots,\phi@n \turn \phi$,
+but Isabelle's notation is more readable with large formulae.
+
+Meta-level natural deduction provides a convenient mechanism for deriving
+new object-level rules. To derive the rule
+\[ \infer{\phi,}{\theta@1 & \ldots & \theta@k} \]
+assume the premises $\theta@1$,~\ldots,~$\theta@k$ at the
+meta-level. Then prove $\phi$, possibly using these assumptions.
+Starting with a proof state $\phi\Imp\phi$, assumptions may accumulate,
+reaching a final state such as
+\[ \phi \quad [\theta@1,\ldots,\theta@k]. \]
+The meta-rule for $\Imp$ introduction discharges an assumption.
+Discharging them in the order $\theta@k,\ldots,\theta@1$ yields the
+meta-theorem $\List{\theta@1; \ldots; \theta@k} \Imp \phi$, with no
+assumptions. This represents the desired rule.
+Let us derive the general $\conj$ elimination rule:
+$$ \infer{R}{P\conj Q & \infer*{R}{[P,Q]}} \eqno(\conj E)$$
+We assume $P\conj Q$ and $\List{P;Q}\Imp R$, and commence backward proof in
+the state $R\Imp R$. Resolving this with the second assumption yields the
+state
+\[ \phantom{\List{P\conj Q;\; P\conj Q}}
+ \llap{$\List{P;Q}$}\Imp R \quad [\,\List{P;Q}\Imp R\,]. \]
+Resolving subgoals~1 and~2 with $({\conj}E1)$ and $({\conj}E2)$,
+respectively, yields the state
+\[ \List{P\conj Q;\; P\conj Q}\Imp R \quad [\,\List{P;Q}\Imp R\,]. \]
+Resolving both subgoals with the assumption $P\conj Q$ yields
+\[ R \quad [\, \List{P;Q}\Imp R, P\conj Q \,]. \]
+The proof may use the meta-assumptions in any order, and as often as
+necessary; when finished, we discharge them in the correct order to
+obtain the desired form:
+\[ \List{P\conj Q;\; \List{P;Q}\Imp R} \Imp R \]
+We have derived the rule using free variables, which prevents their
+premature instantiation during the proof; we may now replace them by
+schematic variables.
+
+\begin{warn}
+Schematic variables are not allowed in (meta) assumptions because they would
+complicate lifting. Assumptions remain fixed throughout a proof.
+\end{warn}
+
+% Local Variables:
+% mode: latex
+% TeX-master: t
+% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/gate.thy Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,5 @@
+Gate = FOL +
+consts nand,xor :: "[o,o] => o"
+rules nand_def "nand(P,Q) == ~(P & Q)"
+ xor_def "xor(P,Q) == P & ~Q | ~P & Q"
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/gate2.thy Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,8 @@
+Gate2 = FOL +
+consts "~&" :: "[o,o] => o" (infixl 35)
+ "#" :: "[o,o] => o" (infixl 30)
+ If :: "[o,o,o] => o" ("if _ then _ else _")
+rules nand_def "P ~& Q == ~(P & Q)"
+ xor_def "P # Q == P & ~Q | ~P & Q"
+ If_def "if P then Q else R == P&Q | ~P&R"
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/getting.tex Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,896 @@
+%% $Id$
+\part{Getting started with Isabelle}
+This Part describes how to perform simple proofs using Isabelle. Although
+it frequently refers to concepts from the previous Part, a user can get
+started without understanding them in detail.
+
+As of this writing, Isabelle's user interface is \ML. Proofs are conducted
+by applying certain \ML{} functions, which update a stored proof state.
+Logics are combined and extended by calling \ML{} functions. All syntax
+must be expressed using {\sc ascii} characters. Menu-driven graphical
+interfaces are under construction, but Isabelle users will always need to
+know some \ML, at least to use tacticals.
+
+Object-logics are built upon Pure Isabelle, which implements the meta-logic
+and provides certain fundamental data structures: types, terms, signatures,
+theorems and theories, tactics and tacticals. These data structures have
+the corresponding \ML{} types {\tt typ}, {\tt term}, {\tt Sign.sg}, {\tt thm},
+{\tt theory} and {\tt tactic}; tacticals have function types such as {\tt
+tactic->tactic}. Isabelle users can operate on these data structures by
+writing \ML{} programs.
+
+\section{Forward proof}
+\index{Isabelle!getting started}\index{ML}
+This section describes the concrete syntax for types, terms and theorems,
+and demonstrates forward proof.
+
+\subsection{Lexical matters}
+\index{identifiers|bold}\index{reserved words|bold}
+An {\bf identifier} is a string of letters, digits, underscores~(\verb|_|)
+and single quotes~({\tt'}), beginning with a letter. Single quotes are
+regarded as primes; for instance {\tt x'} is read as~$x'$. Identifiers are
+separated by white space and special characters. {\bf Reserved words} are
+identifiers that appear in Isabelle syntax definitions.
+
+An Isabelle theory can declare symbols composed of special characters, such
+as {\tt=}, {\tt==}, {\tt=>} and {\tt==>}. (The latter three are part of
+the syntax of the meta-logic.) Such symbols may be run together; thus if
+\verb|}| and \verb|{| are used for set brackets then \verb|{{a},{a,b}}| is
+valid notation for a set of sets --- but only if \verb|}}| and \verb|{{|
+have not been declared as symbols! The parser resolves any ambiguity by
+taking the longest possible symbol that has been declared. Thus the string
+{\tt==>} is read as a single symbol. But \hbox{\tt= =>} is read as two
+symbols, as is \verb|}}|, as discussed above.
+
+Identifiers that are not reserved words may serve as free variables or
+constants. A type identifier consists of an identifier prefixed by a
+prime, for example {\tt'a} and \hbox{\tt'hello}. An unknown (or type
+unknown) consists of a question mark, an identifier (or type identifier),
+and a subscript. The subscript, a non-negative integer, allows the
+renaming of unknowns prior to unification.
+
+The subscript may appear after the identifier, separated by a dot; this
+prevents ambiguity when the identifier ends with a digit. Thus {\tt?z6.0}
+has identifier \verb|"z6"| and subscript~0, while {\tt?a0.5} has identifier
+\verb|"a0"| and subscript~5. If the identifier does not end with a digit,
+then no dot appears and a subscript of~0 is omitted; for example,
+{\tt?hello} has identifier \verb|"hello"| and subscript zero, while
+{\tt?z6} has identifier \verb|"z"| and subscript~6. The same conventions
+apply to type unknowns. Note that the question mark is {\bf not} part of the
+identifier!
+
+
+\subsection{Syntax of types and terms}
+\index{Isabelle!syntax of}
+\index{classes!built-in|bold}
+Classes are denoted by identifiers; the built-in class \ttindex{logic}
+contains the `logical' types. Sorts are lists of classes enclosed in
+braces~\{ and \}; singleton sorts may be abbreviated by dropping the braces.
+
+\index{types!syntax|bold}
+Types are written with a syntax like \ML's. The built-in type \ttindex{prop}
+is the type of propositions. Type variables can be constrained to particular
+classes or sorts, for example {\tt 'a::term} and {\tt ?'b::\{ord,arith\}}.
+\[\dquotes
+\begin{array}{lll}
+ \multicolumn{3}{c}{\hbox{ASCII Notation for Types}} \\ \hline
+ t "::" C & t :: C & \hbox{class constraint} \\
+ t "::" "\{" C@1 "," \ldots "," C@n "\}" &
+ t :: \{C@1,\dots,C@n\} & \hbox{sort constraint} \\
+ \sigma"=>"\tau & \sigma\To\tau & \hbox{function type} \\
+ "[" \sigma@1 "," \ldots "," \sigma@n "] => " \tau &
+ [\sigma@1,\ldots,\sigma@n] \To\tau & \hbox{curried function type} \\
+ "(" \tau@1"," \ldots "," \tau@n ")" tycon &
+ (\tau@1, \ldots, \tau@n)tycon & \hbox{type construction}
+\end{array}
+\]
+Terms are those of the typed $\lambda$-calculus.
+\index{terms!syntax|bold}
+\[\dquotes
+\begin{array}{lll}
+ \multicolumn{3}{c}{\hbox{ASCII Notation for Terms}} \\ \hline
+ t "::" \sigma & t :: \sigma & \hbox{type constraint} \\
+ "\%" x "." t & \lambda x.t & \hbox{abstraction} \\
+ "\%" x@1\ldots x@n "." t & \lambda x@1\ldots x@n.t &
+ \hbox{curried abstraction} \\
+ t "(" u@1"," \ldots "," u@n ")" &
+ t (u@1, \ldots, u@n) & \hbox{curried application}
+\end{array}
+\]
+The theorems and rules of an object-logic are represented by theorems in
+the meta-logic, which are expressed using meta-formulae. Since the
+meta-logic is higher-order, meta-formulae~$\phi$, $\psi$, $\theta$,~\ldots{}
+are just terms of type~\ttindex{prop}.
+\index{meta-formulae!syntax|bold}
+\[\dquotes
+ \begin{array}{l@{\quad}l@{\quad}l}
+ \multicolumn{3}{c}{\hbox{ASCII Notation for Meta-Formulae}} \\ \hline
+ a " == " b & a\equiv b & \hbox{meta-equality} \\
+ a " =?= " b & a\qeq b & \hbox{flex-flex constraint} \\
+ \phi " ==> " \psi & \phi\Imp \psi & \hbox{meta-implication} \\
+ "[|" \phi@1 ";" \ldots ";" \phi@n "|] ==> " \psi &
+ \List{\phi@1;\ldots;\phi@n} \Imp \psi & \hbox{nested implication} \\
+ "!!" x "." \phi & \Forall x.\phi & \hbox{meta-quantification} \\
+ "!!" x@1\ldots x@n "." \phi &
+ \Forall x@1. \ldots \Forall x@n.\phi & \hbox{nested quantification}
+ \end{array}
+\]
+Flex-flex constraints are meta-equalities arising from unification; they
+require special treatment. See~\S\ref{flexflex}.
+\index{flex-flex equations}
+
+Most logics define the implicit coercion $Trueprop$ from object-formulae to
+propositions.
+\index{Trueprop@{$Trueprop$}}
+This could cause an ambiguity: in $P\Imp Q$, do the variables $P$ and $Q$
+stand for meta-formulae or object-formulae? If the latter, $P\Imp Q$
+really abbreviates $Trueprop(P)\Imp Trueprop(Q)$. To prevent such
+ambiguities, Isabelle's syntax does not allow a meta-formula to consist of
+a variable. Variables of type~\ttindex{prop} are seldom useful, but you
+can make a variable stand for a meta-formula by prefixing it with the
+keyword \ttindex{PROP}:
+\begin{ttbox}
+PROP ?psi ==> PROP ?theta
+\end{ttbox}
+
+Symbols of object-logics also must be rendered into {\sc ascii}, typically
+as follows:
+\[ \begin{tabular}{l@{\quad}l@{\quad}l}
+ \tt True & $\top$ & true \\
+ \tt False & $\bot$ & false \\
+ \tt $P$ \& $Q$ & $P\conj Q$ & conjunction \\
+ \tt $P$ | $Q$ & $P\disj Q$ & disjunction \\
+ \verb'~' $P$ & $\neg P$ & negation \\
+ \tt $P$ --> $Q$ & $P\imp Q$ & implication \\
+ \tt $P$ <-> $Q$ & $P\bimp Q$ & bi-implication \\
+ \tt ALL $x\,y\,z$ .\ $P$ & $\forall x\,y\,z.P$ & for all \\
+ \tt EX $x\,y\,z$ .\ $P$ & $\exists x\,y\,z.P$ & there exists
+ \end{tabular}
+\]
+To illustrate the notation, consider two axioms for first-order logic:
+$$ \List{P; Q} \Imp P\conj Q \eqno(\conj I) $$
+$$ \List{\exists x.P(x); \Forall x. P(x)\imp Q} \Imp Q \eqno(\exists E) $$
+Using the {\tt [|\ldots|]} shorthand, $({\conj}I)$ translates literally into
+{\sc ascii} characters as
+\begin{ttbox}
+[| ?P; ?Q |] ==> ?P & ?Q
+\end{ttbox}
+The schematic variables let unification instantiate the rule. To
+avoid cluttering rules with question marks, Isabelle converts any free
+variables in a rule to schematic variables; we normally write $({\conj}I)$ as
+\begin{ttbox}
+[| P; Q |] ==> P & Q
+\end{ttbox}
+This variables convention agrees with the treatment of variables in goals.
+Free variables in a goal remain fixed throughout the proof. After the
+proof is finished, Isabelle converts them to scheme variables in the
+resulting theorem. Scheme variables in a goal may be replaced by terms
+during the proof, supporting answer extraction, program synthesis, and so
+forth.
+
+For a final example, the rule $(\exists E)$ is rendered in {\sc ascii} as
+\begin{ttbox}
+[| EX x.P(x); !!x. P(x) ==> Q |] ==> Q
+\end{ttbox}
+
+
+\subsection{Basic operations on theorems}
+\index{theorems!basic operations on|bold}
+\index{LCF}
+Meta-level theorems have type \ttindex{thm} and represent the theorems and
+inference rules of object-logics. Isabelle's meta-logic is implemented
+using the {\sc lcf} approach: each meta-level inference rule is represented by
+a function from theorems to theorems. Object-level rules are taken as
+axioms.
+
+The main theorem printing commands are {\tt prth}, {\tt prths} and~{\tt
+ prthq}. Of the other operations on theorems, most useful are {\tt RS}
+and {\tt RSN}, which perform resolution.
+
+\index{printing commands|bold}
+\begin{description}
+\item[\ttindexbold{prth} {\it thm}] pretty-prints {\it thm\/} at the terminal.
+
+\item[\ttindexbold{prths} {\it thms}] pretty-prints {\it thms}, a list of
+theorems.
+
+\item[\ttindexbold{prthq} {\it thmq}] pretty-prints {\it thmq}, a sequence of
+theorems; this is useful for inspecting the output of a tactic.
+
+\item[\tt$thm1$ RS $thm2$] \indexbold{*RS} resolves the conclusion of $thm1$
+with the first premise of~$thm2$.
+
+\item[\tt$thm1$ RSN $(i,thm2)$] \indexbold{*RSN} resolves the conclusion of $thm1$
+with the $i$th premise of~$thm2$.
+
+\item[\ttindexbold{standard} $thm$] puts $thm$ into a standard
+format. It also renames schematic variables to have subscript zero,
+improving readability and reducing subscript growth.
+\end{description}
+\index{ML}
+The rules of a theory are normally bound to \ML\ identifiers. Suppose we
+are running an Isabelle session containing natural deduction first-order
+logic. Let us try an example given in~\S\ref{joining}. We first print
+\ttindex{mp}, which is the rule~$({\imp}E)$, then resolve it with itself.
+\begin{ttbox}
+prth mp;
+{\out [| ?P --> ?Q; ?P |] ==> ?Q}
+{\out val it = "[| ?P --> ?Q; ?P |] ==> ?Q" : thm}
+prth (mp RS mp);
+{\out [| ?P1 --> ?P --> ?Q; ?P1; ?P |] ==> ?Q}
+{\out val it = "[| ?P1 --> ?P --> ?Q; ?P1; ?P |] ==> ?Q" : thm}
+\end{ttbox}
+In the Isabelle documentation, user's input appears in {\tt typewriter
+ characters}, and output appears in {\sltt slanted typewriter characters}.
+\ML's response {\out val }~\ldots{} is compiler-dependent and will
+sometimes be suppressed. This session illustrates two formats for the
+display of theorems. Isabelle's top-level displays theorems as ML values,
+enclosed in quotes.\footnote{This works under both Poly/ML and Standard ML
+ of New Jersey.} Printing functions like {\tt prth} omit the quotes and
+the surrounding {\tt val \ldots :\ thm}.
+
+To contrast {\tt RS} with {\tt RSN}, we resolve
+\ttindex{conjunct1}, which stands for~$(\conj E1)$, with~\ttindex{mp}.
+\begin{ttbox}
+conjunct1 RS mp;
+{\out val it = "[| (?P --> ?Q) & ?Q1; ?P |] ==> ?Q" : thm}
+conjunct1 RSN (2,mp);
+{\out val it = "[| ?P --> ?Q; ?P & ?Q1 |] ==> ?Q" : thm}
+\end{ttbox}
+These correspond to the following proofs:
+\[ \infer[({\imp}E)]{Q}{\infer[({\conj}E1)]{P\imp Q}{(P\imp Q)\conj Q@1} & P}
+ \qquad
+ \infer[({\imp}E)]{Q}{P\imp Q & \infer[({\conj}E1)]{P}{P\conj Q@1}}
+\]
+The printing commands return their argument; the \ML{} identifier~{\tt it}
+denotes the value just printed. You may derive a rule by pasting other
+rules together. Below, \ttindex{spec} stands for~$(\forall E)$:
+\begin{ttbox}
+spec;
+{\out val it = "ALL x. ?P(x) ==> ?P(?x)" : thm}
+it RS mp;
+{\out val it = "[| ALL x. ?P3(x) --> ?Q2(x); ?P3(?x1) |] ==> ?Q2(?x1)" : thm}
+it RS conjunct1;
+{\out val it = "[| ALL x. ?P4(x) --> ?P6(x) & ?Q5(x); ?P4(?x2) |] ==> ?P6(?x2)"}
+standard it;
+{\out val it = "[| ALL x. ?P(x) --> ?Pa(x) & ?Q(x); ?P(?x) |] ==> ?Pa(?x)"}
+\end{ttbox}
+By resolving $(\forall E)$ with (${\imp}E)$ and (${\conj}E1)$, we have
+derived a destruction rule for formulae of the form $\forall x.
+P(x)\imp(Q(x)\conj R(x))$. Used with destruct-resolution, such specialized
+rules provide a way of referring to particular assumptions.
+
+\subsection{Flex-flex equations} \label{flexflex}
+\index{flex-flex equations|bold}\index{unknowns!of function type}
+In higher-order unification, {\bf flex-flex} equations are those where both
+sides begin with a function unknown, such as $\Var{f}(0)\qeq\Var{g}(0)$.
+They admit a trivial unifier, here $\Var{f}\equiv \lambda x.\Var{a}$ and
+$\Var{g}\equiv \lambda y.\Var{a}$, where $\Var{a}$ is a new unknown. They
+admit many other unifiers, such as $\Var{f} \equiv \lambda x.\Var{g}(0)$
+and $\{\Var{f} \equiv \lambda x.x,\, \Var{g} \equiv \lambda x.0\}$. Huet's
+procedure does not enumerate the unifiers; instead, it retains flex-flex
+equations as constraints on future unifications. Flex-flex constraints
+occasionally become attached to a proof state; more frequently, they appear
+during use of {\tt RS} and~{\tt RSN}:
+\begin{ttbox}
+refl;
+{\out val it = "?a = ?a" : thm}
+exI;
+{\out val it = "?P(?x) ==> EX x. ?P(x)" : thm}
+refl RS exI;
+{\out val it = "?a3(?x) =?= ?a2(?x) ==> EX x. ?a3(x) = ?a2(x)" : thm}
+\end{ttbox}
+
+\noindent
+Renaming variables, this is $\exists x.\Var{f}(x)=\Var{g}(x)$ with
+the constraint ${\Var{f}(\Var{u})\qeq\Var{g}(\Var{u})}$. Instances
+satisfying the constraint include $\exists x.\Var{f}(x)=\Var{f}(x)$ and
+$\exists x.x=\Var{u}$. Calling \ttindex{flexflex_rule} removes all
+constraints by applying the trivial unifier:\index{*prthq}
+\begin{ttbox}
+prthq (flexflex_rule it);
+{\out EX x. ?a4 = ?a4}
+\end{ttbox}
+Isabelle simplifies flex-flex equations to eliminate redundant bound
+variables. In $\lambda x\,y.\Var{f}(k(y),x) \qeq \lambda x\,y.\Var{g}(y)$,
+there is no bound occurrence of~$x$ on the right side; thus, there will be
+none on the left, in a common instance of these terms. Choosing a new
+variable~$\Var{h}$, Isabelle assigns $\Var{f}\equiv \lambda u\,v.?h(u)$,
+simplifying the left side to $\lambda x\,y.\Var{h}(k(y))$. Dropping $x$
+from the equation leaves $\lambda y.\Var{h}(k(y)) \qeq \lambda
+y.\Var{g}(y)$. By $\eta$-conversion, this simplifies to the assignment
+$\Var{g}\equiv\lambda y.?h(k(y))$.
+
+\begin{warn}
+\ttindex{RS} and \ttindex{RSN} fail (by raising exception {\tt THM}) unless
+the resolution delivers {\bf exactly one} resolvent. For multiple results,
+use \ttindex{RL} and \ttindex{RLN}, which operate on theorem lists. The
+following example uses \ttindex{read_instantiate} to create an instance
+of \ttindex{refl} containing no schematic variables.
+\begin{ttbox}
+val reflk = read_instantiate [("a","k")] refl;
+{\out val reflk = "k = k" : thm}
+\end{ttbox}
+
+\noindent
+A flex-flex constraint is no longer possible; resolution does not find a
+unique unifier:
+\begin{ttbox}
+reflk RS exI;
+{\out uncaught exception THM}
+\end{ttbox}
+Using \ttindex{RL} this time, we discover that there are four unifiers, and
+four resolvents:
+\begin{ttbox}
+[reflk] RL [exI];
+{\out val it = ["EX x. x = x", "EX x. k = x",}
+{\out "EX x. x = k", "EX x. k = k"] : thm list}
+\end{ttbox}
+\end{warn}
+
+
+\section{Backward proof}
+Although {\tt RS} and {\tt RSN} are fine for simple forward reasoning,
+large proofs require tactics. Isabelle provides a suite of commands for
+conducting a backward proof using tactics.
+
+\subsection{The basic tactics}
+\index{tactics!basic|bold}
+The tactics {\tt assume_tac}, {\tt
+resolve_tac}, {\tt eresolve_tac}, and {\tt dresolve_tac} suffice for most
+single-step proofs. Although {\tt eresolve_tac} and {\tt dresolve_tac} are
+not strictly necessary, they simplify proofs involving elimination and
+destruction rules. All the tactics act on a subgoal designated by a
+positive integer~$i$, failing if~$i$ is out of range. The resolution
+tactics try their list of theorems in left-to-right order.
+
+\begin{description}
+\item[\ttindexbold{assume_tac} {\it i}] is the tactic that attempts to solve
+subgoal~$i$ by assumption. Proof by assumption is not a trivial step; it
+can falsify other subgoals by instantiating shared variables. There may be
+several ways of solving the subgoal by assumption.
+
+\item[\ttindexbold{resolve_tac} {\it thms} {\it i}]
+is the basic resolution tactic, used for most proof steps. The $thms$
+represent object-rules, which are resolved against subgoal~$i$ of the proof
+state. For each rule, resolution forms next states by unifying the
+conclusion with the subgoal and inserting instantiated premises in its
+place. A rule can admit many higher-order unifiers. The tactic fails if
+none of the rules generates next states.
+
+\item[\ttindexbold{eresolve_tac} {\it thms} {\it i}]
+performs elim-resolution. Like
+\hbox{\tt resolve_tac {\it thms} {\it i}} followed by \hbox{\tt assume_tac
+{\it i}}, it applies a rule then solves its first premise by assumption.
+But {\tt eresolve_tac} additionally deletes that assumption from any
+subgoals arising from the resolution.
+
+
+\item[\ttindexbold{dresolve_tac} {\it thms} {\it i}]
+performs destruct-resolution with the~$thms$, as described
+in~\S\ref{destruct}. It is useful for forward reasoning from the
+assumptions.
+\end{description}
+
+\subsection{Commands for backward proof}
+\index{proof!commands for|bold}
+Tactics are normally applied using the subgoal module, which maintains a
+proof state and manages the proof construction. It allows interactive
+backtracking through the proof space, going away to prove lemmas, etc.; of
+its many commands, most important are the following:
+\begin{description}
+\item[\ttindexbold{goal} {\it theory} {\it formula}; ]
+begins a new proof, where $theory$ is usually an \ML\ identifier
+and the {\it formula\/} is written as an \ML\ string.
+
+\item[\ttindexbold{by} {\it tactic}; ]
+applies the {\it tactic\/} to the current proof
+state, raising an exception if the tactic fails.
+
+\item[\ttindexbold{undo}(); ]
+reverts to the previous proof state. Undo can be repeated but cannot be
+undone. Do not omit the parentheses; typing {\tt undo;} merely causes \ML\
+to echo the value of that function.
+
+\item[\ttindexbold{result}()]
+returns the theorem just proved, in a standard format. It fails if
+unproved subgoals are left or if the main goal does not match the one you
+started with.
+\end{description}
+The commands and tactics given above are cumbersome for interactive use.
+Although our examples will use the full commands, you may prefer Isabelle's
+shortcuts:
+\begin{center} \tt
+\indexbold{*br} \indexbold{*be} \indexbold{*bd} \indexbold{*ba}
+\begin{tabular}{l@{\qquad\rm abbreviates\qquad}l}
+ ba {\it i}; & by (assume_tac {\it i}); \\
+
+ br {\it thm} {\it i}; & by (resolve_tac [{\it thm}] {\it i}); \\
+
+ be {\it thm} {\it i}; & by (eresolve_tac [{\it thm}] {\it i}); \\
+
+ bd {\it thm} {\it i}; & by (dresolve_tac [{\it thm}] {\it i});
+\end{tabular}
+\end{center}
+
+\subsection{A trivial example in propositional logic}
+\index{examples!propositional}
+Directory {\tt FOL} of the Isabelle distribution defines the \ML\
+identifier~\ttindex{FOL.thy}, which denotes the theory of first-order
+logic. Let us try the example from~\S\ref{prop-proof}, entering the goal
+$P\disj P\imp P$ in that theory.\footnote{To run these examples, see the
+file {\tt FOL/ex/intro.ML}.}
+\begin{ttbox}
+goal FOL.thy "P|P --> P";
+{\out Level 0}
+{\out P | P --> P}
+{\out 1. P | P --> P}
+\end{ttbox}
+Isabelle responds by printing the initial proof state, which has $P\disj
+P\imp P$ as the main goal and the only subgoal. The \bfindex{level} of the
+state is the number of {\tt by} commands that have been applied to reach
+it. We now use \ttindex{resolve_tac} to apply the rule \ttindex{impI},
+or~$({\imp}I)$, to subgoal~1:
+\begin{ttbox}
+by (resolve_tac [impI] 1);
+{\out Level 1}
+{\out P | P --> P}
+{\out 1. P | P ==> P}
+\end{ttbox}
+In the new proof state, subgoal~1 is $P$ under the assumption $P\disj P$.
+(The meta-implication {\tt==>} indicates assumptions.) We apply
+\ttindex{disjE}, or~(${\disj}E)$, to that subgoal:
+\begin{ttbox}
+by (resolve_tac [disjE] 1);
+{\out Level 2}
+{\out P | P --> P}
+{\out 1. P | P ==> ?P1 | ?Q1}
+{\out 2. [| P | P; ?P1 |] ==> P}
+{\out 3. [| P | P; ?Q1 |] ==> P}
+\end{ttbox}
+At Level~2 there are three subgoals, each provable by
+assumption. We deviate from~\S\ref{prop-proof} by tackling subgoal~3
+first, using \ttindex{assume_tac}. This updates {\tt?Q1} to~{\tt P}.
+\begin{ttbox}
+by (assume_tac 3);
+{\out Level 3}
+{\out P | P --> P}
+{\out 1. P | P ==> ?P1 | P}
+{\out 2. [| P | P; ?P1 |] ==> P}
+\end{ttbox}
+Next we tackle subgoal~2, instantiating {\tt?P1} to~{\tt P}.
+\begin{ttbox}
+by (assume_tac 2);
+{\out Level 4}
+{\out P | P --> P}
+{\out 1. P | P ==> P | P}
+\end{ttbox}
+Lastly we prove the remaining subgoal by assumption:
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 5}
+{\out P | P --> P}
+{\out No subgoals!}
+\end{ttbox}
+Isabelle tells us that there are no longer any subgoals: the proof is
+complete. Calling \ttindex{result} returns the theorem.
+\begin{ttbox}
+val mythm = result();
+{\out val mythm = "?P | ?P --> ?P" : thm}
+\end{ttbox}
+Isabelle has replaced the free variable~{\tt P} by the scheme
+variable~{\tt?P}\@. Free variables in the proof state remain fixed
+throughout the proof. Isabelle finally converts them to scheme variables
+so that the resulting theorem can be instantiated with any formula.
+
+
+\subsection{Proving a distributive law}
+\index{examples!propositional}
+To demonstrate the tactics \ttindex{eresolve_tac}, \ttindex{dresolve_tac}
+and the tactical \ttindex{REPEAT}, we shall prove part of the distributive
+law $(P\conj Q)\disj R \iff (P\disj R)\conj (Q\disj R)$.
+
+We begin by stating the goal to Isabelle and applying~$({\imp}I)$ to it:
+\begin{ttbox}
+goal FOL.thy "(P & Q) | R --> (P | R)";
+{\out Level 0}
+{\out P & Q | R --> P | R}
+{\out 1. P & Q | R --> P | R}
+by (resolve_tac [impI] 1);
+{\out Level 1}
+{\out P & Q | R --> P | R}
+{\out 1. P & Q | R ==> P | R}
+\end{ttbox}
+Previously we applied~(${\disj}E)$ using {\tt resolve_tac}, but
+\ttindex{eresolve_tac} deletes the assumption after use. The resulting proof
+state is simpler.
+\begin{ttbox}
+by (eresolve_tac [disjE] 1);
+{\out Level 2}
+{\out P & Q | R --> P | R}
+{\out 1. P & Q ==> P | R}
+{\out 2. R ==> P | R}
+\end{ttbox}
+Using \ttindex{dresolve_tac}, we can apply~(${\conj}E1)$ to subgoal~1,
+replacing the assumption $P\conj Q$ by~$P$. Normally we should apply the
+rule~(${\conj}E)$, given in~\S\ref{destruct}. That is an elimination rule
+and requires {\tt eresolve_tac}; it would replace $P\conj Q$ by the two
+assumptions~$P$ and~$Q$. The present example does not need~$Q$.
+\begin{ttbox}
+by (dresolve_tac [conjunct1] 1);
+{\out Level 3}
+{\out P & Q | R --> P | R}
+{\out 1. P ==> P | R}
+{\out 2. R ==> P | R}
+\end{ttbox}
+The next two steps apply~(${\disj}I1$) and~(${\disj}I2$) in an obvious manner.
+\begin{ttbox}
+by (resolve_tac [disjI1] 1);
+{\out Level 4}
+{\out P & Q | R --> P | R}
+{\out 1. P ==> P}
+{\out 2. R ==> P | R}
+\ttbreak
+by (resolve_tac [disjI2] 2);
+{\out Level 5}
+{\out P & Q | R --> P | R}
+{\out 1. P ==> P}
+{\out 2. R ==> R}
+\end{ttbox}
+Two calls of~\ttindex{assume_tac} can finish the proof. The
+tactical~\ttindex{REPEAT} expresses a tactic that calls {\tt assume_tac~1}
+as many times as possible. We can restrict attention to subgoal~1 because
+the other subgoals move up after subgoal~1 disappears.
+\begin{ttbox}
+by (REPEAT (assume_tac 1));
+{\out Level 6}
+{\out P & Q | R --> P | R}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\section{Quantifier reasoning}
+\index{quantifiers!reasoning about}\index{parameters}\index{unknowns}
+This section illustrates how Isabelle enforces quantifier provisos.
+Quantifier rules create terms such as~$\Var{f}(x,z)$, where~$\Var{f}$ is a
+function unknown and $x$ and~$z$ are parameters. This may be replaced by
+any term, possibly containing free occurrences of $x$ and~$z$.
+
+\subsection{Two quantifier proofs, successful and not}
+\index{examples!with quantifiers}
+Let us contrast a proof of the theorem $\forall x.\exists y.x=y$ with an
+attempted proof of the non-theorem $\exists y.\forall x.x=y$. The former
+proof succeeds, and the latter fails, because of the scope of quantified
+variables~\cite{paulson89}. Unification helps even in these trivial
+proofs. In $\forall x.\exists y.x=y$ the $y$ that `exists' is simply $x$,
+but we need never say so. This choice is forced by the reflexive law for
+equality, and happens automatically.
+
+\subsubsection{The successful proof}
+The proof of $\forall x.\exists y.x=y$ demonstrates the introduction rules
+$(\forall I)$ and~$(\exists I)$. We state the goal and apply $(\forall I)$:
+\begin{ttbox}
+goal FOL.thy "ALL x. EX y. x=y";
+{\out Level 0}
+{\out ALL x. EX y. x = y}
+{\out 1. ALL x. EX y. x = y}
+\ttbreak
+by (resolve_tac [allI] 1);
+{\out Level 1}
+{\out ALL x. EX y. x = y}
+{\out 1. !!x. EX y. x = y}
+\end{ttbox}
+The variable~{\tt x} is no longer universally quantified, but is a
+parameter in the subgoal; thus, it is universally quantified at the
+meta-level. The subgoal must be proved for all possible values of~{\tt x}.
+We apply the rule $(\exists I)$:
+\begin{ttbox}
+by (resolve_tac [exI] 1);
+{\out Level 2}
+{\out ALL x. EX y. x = y}
+{\out 1. !!x. x = ?y1(x)}
+\end{ttbox}
+The bound variable {\tt y} has become {\tt?y1(x)}. This term consists of
+the function unknown~{\tt?y1} applied to the parameter~{\tt x}.
+Instances of {\tt?y1(x)} may or may not contain~{\tt x}. We resolve the
+subgoal with the reflexivity axiom.
+\begin{ttbox}
+by (resolve_tac [refl] 1);
+{\out Level 3}
+{\out ALL x. EX y. x = y}
+{\out No subgoals!}
+\end{ttbox}
+Let us consider what has happened in detail. The reflexivity axiom is
+lifted over~$x$ to become $\Forall x.\Var{f}(x)=\Var{f}(x)$, which is
+unified with $\Forall x.x=\Var{y@1}(x)$. The function unknowns $\Var{f}$
+and~$\Var{y@1}$ are both instantiated to the identity function, and
+$x=\Var{y@1}(x)$ collapses to~$x=x$ by $\beta$-reduction.
+
+\subsubsection{The unsuccessful proof}
+We state the goal $\exists y.\forall x.x=y$, which is {\bf not} a theorem, and
+try~$(\exists I)$:
+\begin{ttbox}
+goal FOL.thy "EX y. ALL x. x=y";
+{\out Level 0}
+{\out EX y. ALL x. x = y}
+{\out 1. EX y. ALL x. x = y}
+\ttbreak
+by (resolve_tac [exI] 1);
+{\out Level 1}
+{\out EX y. ALL x. x = y}
+{\out 1. ALL x. x = ?y}
+\end{ttbox}
+The unknown {\tt ?y} may be replaced by any term, but this can never
+introduce another bound occurrence of~{\tt x}. We now apply~$(\forall I)$:
+\begin{ttbox}
+by (resolve_tac [allI] 1);
+{\out Level 2}
+{\out EX y. ALL x. x = y}
+{\out 1. !!x. x = ?y}
+\end{ttbox}
+Compare our position with the previous Level~2. Instead of {\tt?y1(x)} we
+have~{\tt?y}, whose instances may not contain the bound variable~{\tt x}.
+The reflexivity axiom does not unify with subgoal~1.
+\begin{ttbox}
+by (resolve_tac [refl] 1);
+{\out by: tactic returned no results}
+\end{ttbox}
+No other choice of rules seems likely to complete the proof. Of course,
+this is no guarantee that Isabelle cannot prove $\exists y.\forall x.x=y$
+or other invalid assertions. We must appeal to the soundness of
+first-order logic and the faithfulness of its encoding in
+Isabelle~\cite{paulson89}, and must trust the implementation.
+
+
+\subsection{Nested quantifiers}
+\index{examples!with quantifiers}
+Multiple quantifiers create complex terms. Proving $(\forall x\,y.P(x,y))
+\imp (\forall z\,w.P(w,z))$, will demonstrate how parameters and
+unknowns develop. If they appear in the wrong order, the proof will fail.
+This section concludes with a demonstration of {\tt REPEAT}
+and~{\tt ORELSE}.
+
+The start of the proof is routine.
+\begin{ttbox}
+goal FOL.thy "(ALL x y.P(x,y)) --> (ALL z w.P(w,z))";
+{\out Level 0}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+\ttbreak
+by (resolve_tac [impI] 1);
+{\out Level 1}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. ALL x y. P(x,y) ==> ALL z w. P(w,z)}
+\end{ttbox}
+
+\subsubsection{The wrong approach}
+Using \ttindex{dresolve_tac}, we apply the rule $(\forall E)$, bound to the
+\ML\ identifier \ttindex{spec}. Then we apply $(\forall I)$.
+\begin{ttbox}
+by (dresolve_tac [spec] 1);
+{\out Level 2}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. ALL y. P(?x1,y) ==> ALL z w. P(w,z)}
+\ttbreak
+by (resolve_tac [allI] 1);
+{\out Level 3}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. !!z. ALL y. P(?x1,y) ==> ALL w. P(w,z)}
+\end{ttbox}
+The unknown {\tt ?u} and the parameter {\tt z} have appeared. We again
+apply $(\forall I)$ and~$(\forall E)$.
+\begin{ttbox}
+by (dresolve_tac [spec] 1);
+{\out Level 4}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. !!z. P(?x1,?y3(z)) ==> ALL w. P(w,z)}
+\ttbreak
+by (resolve_tac [allI] 1);
+{\out Level 5}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. !!z w. P(?x1,?y3(z)) ==> P(w,z)}
+\end{ttbox}
+The unknown {\tt ?y3} and the parameter {\tt w} have appeared. Each
+unknown is applied to the parameters existing at the time of its creation;
+instances of {\tt ?x1} cannot contain~{\tt z} or~{\tt w}, while instances
+of {\tt?y3(z)} can only contain~{\tt z}. Because of these restrictions,
+proof by assumption will fail.
+\begin{ttbox}
+by (assume_tac 1);
+{\out by: tactic returned no results}
+{\out uncaught exception ERROR}
+\end{ttbox}
+
+\subsubsection{The right approach}
+To do this proof, the rules must be applied in the correct order.
+Eigenvariables should be created before unknowns. The
+\ttindex{choplev} command returns to an earlier stage of the proof;
+let us return to the result of applying~$({\imp}I)$:
+\begin{ttbox}
+choplev 1;
+{\out Level 1}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. ALL x y. P(x,y) ==> ALL z w. P(w,z)}
+\end{ttbox}
+Previously, we made the mistake of applying $(\forall E)$; this time, we
+apply $(\forall I)$ twice.
+\begin{ttbox}
+by (resolve_tac [allI] 1);
+{\out Level 2}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. !!z. ALL x y. P(x,y) ==> ALL w. P(w,z)}
+\ttbreak
+by (resolve_tac [allI] 1);
+{\out Level 3}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. !!z w. ALL x y. P(x,y) ==> P(w,z)}
+\end{ttbox}
+The parameters {\tt z} and~{\tt w} have appeared. We now create the
+unknowns:
+\begin{ttbox}
+by (dresolve_tac [spec] 1);
+{\out Level 4}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. !!z w. ALL y. P(?x3(z,w),y) ==> P(w,z)}
+\ttbreak
+by (dresolve_tac [spec] 1);
+{\out Level 5}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. !!z w. P(?x3(z,w),?y4(z,w)) ==> P(w,z)}
+\end{ttbox}
+Both {\tt?x3(z,w)} and~{\tt?y4(z,w)} could become any terms containing {\tt
+z} and~{\tt w}:
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 6}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out No subgoals!}
+\end{ttbox}
+
+\subsubsection{A one-step proof using tacticals}
+\index{tacticals}
+\index{examples!of tacticals}
+Repeated application of rules can be an effective theorem-proving
+procedure, but the rules should be attempted in an order that delays the
+creation of unknowns. As we have just seen, \ttindex{allI} should be
+attempted before~\ttindex{spec}, while \ttindex{assume_tac} generally can
+be attempted first. Such priorities can easily be expressed
+using~\ttindex{ORELSE}, and repeated using~\ttindex{REPEAT}. Let us return
+to the original goal using \ttindex{choplev}:
+\begin{ttbox}
+choplev 0;
+{\out Level 0}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out 1. (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+\end{ttbox}
+A repetitive procedure proves it:
+\begin{ttbox}
+by (REPEAT (assume_tac 1
+ ORELSE resolve_tac [impI,allI] 1
+ ORELSE dresolve_tac [spec] 1));
+{\out Level 1}
+{\out (ALL x y. P(x,y)) --> (ALL z w. P(w,z))}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\subsection{A realistic quantifier proof}
+\index{examples!with quantifiers}
+A proof of $(\forall x. P(x) \imp Q) \imp (\exists x. P(x)) \imp Q$
+demonstrates the practical use of parameters and unknowns.
+Since $\imp$ is nested to the right, $({\imp}I)$ can be applied twice; we
+use \ttindex{REPEAT}:
+\begin{ttbox}
+goal FOL.thy "(ALL x.P(x) --> Q) --> (EX x.P(x)) --> Q";
+{\out Level 0}
+{\out (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q}
+{\out 1. (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q}
+\ttbreak
+by (REPEAT (resolve_tac [impI] 1));
+{\out Level 1}
+{\out (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q}
+{\out 1. [| ALL x. P(x) --> Q; EX x. P(x) |] ==> Q}
+\end{ttbox}
+We can eliminate the universal or the existential quantifier. The
+existential quantifier should be eliminated first, since this creates a
+parameter. The rule~$(\exists E)$ is bound to the
+identifier~\ttindex{exE}.
+\begin{ttbox}
+by (eresolve_tac [exE] 1);
+{\out Level 2}
+{\out (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q}
+{\out 1. !!x. [| ALL x. P(x) --> Q; P(x) |] ==> Q}
+\end{ttbox}
+The only possibility now is $(\forall E)$, a destruction rule. We use
+\ttindex{dresolve_tac}, which discards the quantified assumption; it is
+only needed once.
+\begin{ttbox}
+by (dresolve_tac [spec] 1);
+{\out Level 3}
+{\out (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q}
+{\out 1. !!x. [| P(x); P(?x3(x)) --> Q |] ==> Q}
+\end{ttbox}
+Because the parameter~{\tt x} appeared first, the unknown
+term~{\tt?x3(x)} may depend upon it. Had we eliminated the universal
+quantifier before the existential, this would not be so.
+
+Although $({\imp}E)$ is a destruction rule, it works with
+\ttindex{eresolve_tac} to perform backward chaining. This technique is
+frequently useful.
+\begin{ttbox}
+by (eresolve_tac [mp] 1);
+{\out Level 4}
+{\out (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q}
+{\out 1. !!x. P(x) ==> P(?x3(x))}
+\end{ttbox}
+The tactic has reduced~{\tt Q} to~{\tt P(?x3(x))}, deleting the
+implication. The final step is trivial, thanks to the occurrence of~{\tt x}.
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 5}
+{\out (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\subsection{The classical reasoning package}
+\index{classical reasoning package}
+Although Isabelle cannot compete with fully automatic theorem provers, it
+provides enough automation to tackle substantial examples. The classical
+reasoning package can be set up for any classical natural deduction logic
+--- see the {\em Reference Manual}.
+
+Rules are packaged into bundles called \bfindex{classical sets}. The package
+provides several tactics, which apply rules using naive algorithms, using
+unification to handle quantifiers. The most useful tactic
+is~\ttindex{fast_tac}.
+
+Let us solve problems~40 and~60 of Pelletier~\cite{pelletier86}. (The
+backslashes~\hbox{\verb|\|\ldots\verb|\|} are an \ML{} string escape
+sequence, to break the long string over two lines.)
+\begin{ttbox}
+goal FOL.thy "(EX y. ALL x. J(y,x) <-> ~J(x,x)) \ttback
+\ttback --> ~ (ALL x. EX y. ALL z. J(z,y) <-> ~ J(z,x))";
+{\out Level 0}
+{\out (EX y. ALL x. J(y,x) <-> ~J(x,x)) -->}
+{\out ~(ALL x. EX y. ALL z. J(z,y) <-> ~J(z,x))}
+{\out 1. (EX y. ALL x. J(y,x) <-> ~J(x,x)) -->}
+{\out ~(ALL x. EX y. ALL z. J(z,y) <-> ~J(z,x))}
+\end{ttbox}
+The rules of classical logic are bundled as {\tt FOL_cs}. We may solve
+subgoal~1 at a stroke, using~\ttindex{fast_tac}.
+\begin{ttbox}
+by (fast_tac FOL_cs 1);
+{\out Level 1}
+{\out (EX y. ALL x. J(y,x) <-> ~J(x,x)) -->}
+{\out ~(ALL x. EX y. ALL z. J(z,y) <-> ~J(z,x))}
+{\out No subgoals!}
+\end{ttbox}
+Sceptics may examine the proof by calling the package's single-step
+tactics, such as~{\tt step_tac}. This would take up much space, however,
+so let us proceed to the next example:
+\begin{ttbox}
+goal FOL.thy "ALL x. P(x,f(x)) <-> \ttback
+\ttback (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
+{\out Level 0}
+{\out ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))}
+{\out 1. ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))}
+\end{ttbox}
+Again, subgoal~1 succumbs immediately.
+\begin{ttbox}
+by (fast_tac FOL_cs 1);
+{\out Level 1}
+{\out ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))}
+{\out No subgoals!}
+\end{ttbox}
+The classical reasoning package is not restricted to the usual logical
+connectives. The natural deduction rules for unions and intersections in
+set theory resemble those for disjunction and conjunction, and in the
+infinitary case, for quantifiers. The package is valuable for reasoning in
+set theory.
+
+
+% Local Variables:
+% mode: latex
+% TeX-master: t
+% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/intro.bbl Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,108 @@
+\begin{thebibliography}{10}
+
+\bibitem{galton90}
+Antony Galton.
+\newblock {\em Logic for Information Technology}.
+\newblock Wiley, 1990.
+
+\bibitem{gordon88a}
+Michael J.~C. Gordon.
+\newblock {HOL}: A proof generating system for higher-order logic.
+\newblock In Graham Birtwistle and P.~A. Subrahmanyam, editors, {\em {VLSI}
+ Specification, Verification and Synthesis}, pages 73--128. Kluwer Academic
+ Publishers, 1988.
+
+\bibitem{gordon79}
+Michael J.~C. Gordon, Robin Milner, and Christopher~P. Wadsworth.
+\newblock {\em Edinburgh {LCF}: A Mechanised Logic of Computation}.
+\newblock Springer LNCS 78, 1979.
+
+\bibitem{haskell-tutorial}
+Paul Hudak and Joseph~H. Fasel.
+\newblock A gentle introduction to {Haskell}.
+\newblock {\em {ACM} {SIGPLAN} Notices}, 27(5), May 1992.
+
+\bibitem{haskell-report}
+Paul Hudak, Simon~Peyton Jones, and Philip Wadler.
+\newblock Report on the programming language {Haskell}: A non-strict, purely
+ functional language.
+\newblock {\em {ACM} {SIGPLAN} Notices}, 27(5), May 1992.
+\newblock Version 1.2.
+
+\bibitem{huet75}
+G.~P. Huet.
+\newblock A unification algorithm for typed $\lambda$-calculus.
+\newblock {\em Theoretical Computer Science}, 1:27--57, 1975.
+
+\bibitem{miller-jsc}
+Dale Miller.
+\newblock Unification under a mixed prefix.
+\newblock {\em Journal of Symbolic Computation}, 14(4):321--358, 1992.
+
+\bibitem{nipkow-prehofer}
+Tobias Nipkow and Christian Prehofer.
+\newblock Type checking type classes.
+\newblock In {\em 20th ACM Symp.\ Principles of Programming Languages}, 1993.
+\newblock To appear.
+
+\bibitem{nordstrom90}
+Bengt {Nordstr\"om}, Kent Petersson, and Jan Smith.
+\newblock {\em Programming in {Martin-L\"of}'s Type Theory. An Introduction}.
+\newblock Oxford, 1990.
+
+\bibitem{paulson86}
+Lawrence~C. Paulson.
+\newblock Natural deduction as higher-order resolution.
+\newblock {\em Journal of Logic Programming}, 3:237--258, 1986.
+
+\bibitem{paulson87}
+Lawrence~C. Paulson.
+\newblock {\em Logic and Computation: Interactive proof with Cambridge LCF}.
+\newblock Cambridge University Press, 1987.
+
+\bibitem{paulson89}
+Lawrence~C. Paulson.
+\newblock The foundation of a generic theorem prover.
+\newblock {\em Journal of Automated Reasoning}, 5:363--397, 1989.
+
+\bibitem{paulson700}
+Lawrence~C. Paulson.
+\newblock {Isabelle}: The next 700 theorem provers.
+\newblock In P.~Odifreddi, editor, {\em Logic and Computer Science}, pages
+ 361--386. Academic Press, 1990.
+
+\bibitem{paulson91}
+Lawrence~C. Paulson.
+\newblock {\em {ML} for the Working Programmer}.
+\newblock Cambridge University Press, 1991.
+
+\bibitem{paulson-handbook}
+Lawrence~C. Paulson.
+\newblock Designing a theorem prover.
+\newblock In S.~Abramsky, D.~M. Gabbay, and T.~S.~E. Maibaum, editors, {\em
+ Handbook of Logic in Computer Science}, volume~2, pages 415--475. Oxford
+ University Press, 1992.
+
+\bibitem{pelletier86}
+F.~J. Pelletier.
+\newblock Seventy-five problems for testing automatic theorem provers.
+\newblock {\em Journal of Automated Reasoning}, 2:191--216, 1986.
+\newblock Errata, JAR 4 (1988), 235--236.
+
+\bibitem{reeves90}
+Steve Reeves and Michael Clarke.
+\newblock {\em Logic for Computer Science}.
+\newblock Addison-Wesley, 1990.
+
+\bibitem{suppes72}
+Patrick Suppes.
+\newblock {\em Axiomatic Set Theory}.
+\newblock Dover, 1972.
+
+\bibitem{wos-bledsoe}
+Larry Wos.
+\newblock Automated reasoning and {Bledsoe's} dream for the field.
+\newblock In Robert~S. Boyer, editor, {\em Automated Reasoning: Essays in Honor
+ of {Woody Bledsoe}}, pages 297--342. Kluwer Academic Publishers, 1991.
+
+\end{thebibliography}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/intro.ind Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,251 @@
+\begin{theindex}
+
+ \item $\Forall$, \bold{5}, 7
+ \item $\Imp$, \bold{5}
+ \item $\To$, \bold{1}, \bold{3}
+ \item $\equiv$, \bold{5}
+
+ \indexspace
+
+ \item {\tt allI}, 35
+ \item arities
+ \subitem declaring, 3, \bold{46}
+ \item {\tt asm_simp_tac}, 55
+ \item associativity of addition, 54
+ \item {\tt assume_tac}, \bold{27}, 29, 30, 35, 43, 44
+ \item assumptions, 6
+
+ \indexspace
+
+ \item {\tt ba}, \bold{28}
+ \item {\tt back}, 54, 57
+ \item backtracking, 57
+ \item {\tt bd}, \bold{28}
+ \item {\tt be}, \bold{28}
+ \item {\tt br}, \bold{28}
+ \item {\tt by}, \bold{28}
+
+ \indexspace
+
+ \item {\tt choplev}, 34, 35, 59
+ \item classes, \bold{3}
+ \subitem built-in, \bold{23}
+ \item classical reasoning package, 36
+ \item classical sets, \bold{36}
+ \item {\tt conjunct1}, 25
+ \item constants
+ \subitem declaring, \bold{45}
+
+ \indexspace
+
+ \item definitions, \bold{5}
+ \subitem reasoning about, \bold{40}
+ \item {\tt DEPTH_FIRST}, 59
+ \item destruct-resolution, \bold{20}
+ \item destruction rules, \bold{20}
+ \item {\tt disjE}, 29
+ \item {\tt dres_inst_tac}, \bold{52}
+ \item {\tt dresolve_tac}, \bold{28}, 30, 33, 36
+
+ \indexspace
+
+ \item eigenvariables, \see{parameters}{7}
+ \item elim-resolution, \bold{18}
+ \item equality
+ \subitem meta-level, \bold{5}
+ \item {\tt eres_inst_tac}, \bold{52}
+ \item {\tt eresolve_tac}, \bold{27}, 30, 36, 43, 44
+ \item examples
+ \subitem of deriving rules, 38
+ \subitem of induction, 52, 53
+ \subitem of rewriting, 54
+ \subitem of tacticals, 35
+ \subitem of theories, 45--51, 56
+ \subitem propositional, 16, 28, 30
+ \subitem with quantifiers, 17, 31, 33, 35
+ \item {\tt exE}, 35
+
+ \indexspace
+
+ \item {\tt FalseE}, 42
+ \item {\tt fast_tac}, 36, 37
+ \item flex-flex equations, \bold{5}, 23, \bold{26}
+ \item {\tt flexflex_rule}, 26
+ \item {\tt FOL}, 56
+ \item {\tt FOL.thy}, 28
+ \item folding, \bold{40}
+
+ \indexspace
+
+ \item {\tt goal}, \bold{28}, 38, 39, 41--43
+ \item {\tt goalw}, 42, 43
+
+ \indexspace
+
+ \item {\tt has_fewer_prems}, 59
+
+ \indexspace
+
+ \item identifiers, \bold{22}
+ \item imitation, \bold{10}
+ \item {\tt impI}, 29, 41
+ \item implication
+ \subitem meta-level, \bold{5}
+ \item infix operators, \bold{47}
+ \item instantiation
+ \subitem explicit, \bold{52}
+ \item Isabelle
+ \subitem definitions in, 40
+ \subitem formalizing rules, \bold{5}
+ \subitem formalizing syntax, \bold{1}
+ \subitem getting started, 22
+ \subitem object-logics supported, i
+ \subitem overview, i
+ \subitem proof construction in, \bold{9}
+ \subitem release history, i
+ \subitem syntax of, 23
+
+ \indexspace
+
+ \item LCF, i, 14, 24
+ \item level, \bold{29}
+ \item lifting, \bold{13}
+ \subitem over assumptions, \bold{13}
+ \subitem over parameters, \bold{13}
+ \item {\tt logic}, 23
+
+ \indexspace
+
+ \item main goal, \bold{14}
+ \item major premise, \bold{19}
+ \item {\tt Match}, 39
+ \item meta-formulae
+ \subitem syntax, \bold{23}
+ \item meta-logic, \bold{5}
+ \item mixfix operators, \bold{47}
+ \item ML, i, 22, 25
+ \item {\tt mp}, 25
+
+ \indexspace
+
+ \item {\tt Nat}, \bold{51}
+ \item {\tt Nat.thy}, 53
+ \item {\tt not_def}, 41
+ \item {\tt notE}, \bold{43}, 53
+ \item {\tt notI}, \bold{41}, 53
+
+ \indexspace
+
+ \item {\tt ORELSE}, 35
+ \item overloading, \bold{4}, 48
+
+ \indexspace
+
+ \item parameters, \bold{7}, 31
+ \item printing commands, \bold{25}
+ \item projection, \bold{10}
+ \item {\tt Prolog}, 56
+ \item Prolog interpreter, \bold{55}
+ \item proof
+ \subitem backward, \bold{14}
+ \subitem by assumption, \bold{15}
+ \subitem by induction, 52
+ \subitem commands for, \bold{28}
+ \subitem forward, 20
+ \item proof state, \bold{14}
+ \item {\tt PROP}, 24
+ \item {\tt prop}, 23, 24
+ \item {\tt prth}, \bold{25}
+ \item {\tt prthq}, \bold{25}, 26
+ \item {\tt prths}, \bold{25}
+ \item {\tt Pure}, \bold{44}
+
+ \indexspace
+
+ \item quantifiers
+ \subitem meta-level, \bold{5}
+ \subitem reasoning about, 31
+
+ \indexspace
+
+ \item {\tt read_instantiate}, 27
+ \item refinement, \bold{15}
+ \subitem with instantiation, \bold{52}
+ \item {\tt refl}, 27
+ \item {\tt REPEAT}, 30, 35, 57, 59
+ \item {\tt res_inst_tac}, \bold{52}, 54, 55
+ \item reserved words, \bold{22}
+ \item resolution, \bold{11}
+ \subitem in backward proof, 14
+ \item {\tt resolve_tac}, \bold{27}, 29, 43, 53
+ \item {\tt result}, \bold{28}, 29, 38, 39, 44
+ \item {\tt rewrite_goals_tac}, 41, 42
+ \item {\tt rewrite_rule}, 42, 43
+ \item rewriting
+ \subitem meta-level, 40, \bold{40}
+ \subitem object-level, 54
+ \item {\tt RL}, 27
+ \item {\tt RLN}, 27
+ \item {\tt RS}, \bold{25}, 27, 43
+ \item {\tt RSN}, \bold{25}, 27
+ \item rules
+ \subitem declaring, \bold{45}
+ \subitem derived, 12, \bold{20}, 38, 40
+ \subitem propositional, \bold{6}
+ \subitem quantifier, \bold{7}
+
+ \indexspace
+
+ \item search
+ \subitem depth-first, 58
+ \item signatures, \bold{8}
+ \item {\tt simp_tac}, 55
+ \item simplification set, \bold{54}
+ \item sorts, \bold{4}
+ \item {\tt spec}, 26, 33, 35
+ \item {\tt standard}, \bold{25}
+ \item subgoals, \bold{14}
+ \item substitution, \bold{7}
+ \item {\tt Suc_inject}, 53
+ \item {\tt Suc_neq_0}, 53
+
+ \indexspace
+
+ \item tacticals, \bold{14}, \bold{17}, 35
+ \item Tactics, \bold{14}
+ \item tactics, \bold{17}
+ \subitem basic, \bold{27}
+ \item terms
+ \subitem syntax, \bold{23}
+ \item theorems
+ \subitem basic operations on, \bold{24}
+ \item theories, \bold{8}
+ \subitem defining, 44--52
+ \item {\tt thm}, 24
+ \item {\tt topthm}, 39
+ \item {$Trueprop$}, 6, 7, 9, 23
+ \item type constructors
+ \subitem declaring, \bold{46}
+ \item types, 1
+ \subitem higher, \bold{4}
+ \subitem polymorphic, \bold{3}
+ \subitem simple, \bold{1}
+ \subitem syntax, \bold{23}
+
+ \indexspace
+
+ \item {\tt undo}, \bold{28}
+ \item unfolding, \bold{40}
+ \item unification
+ \subitem higher-order, \bold{10}, 53
+ \item unknowns, \bold{9}, 31
+ \subitem of function type, \bold{11}, 26
+ \item {\tt use_thy}, \bold{44, 45}
+
+ \indexspace
+
+ \item variables
+ \subitem bound, 7
+ \subitem schematic, \see{unknowns}{9}
+
+\end{theindex}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/intro.tex Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,149 @@
+\documentstyle[a4,12pt,proof,iman,alltt]{article}
+%% $Id$
+%% run bibtex intro to prepare bibliography
+%% run ../sedindex intro to prepare index file
+%prth *(\(.*\)); \1;
+%{\\out \(.*\)} {\\out val it = "\1" : thm}
+
+\title{Introduction to Isabelle}
+\author{{\em Lawrence C. Paulson}\\
+ Computer Laboratory \\ University of Cambridge \\[2ex]
+ {\small{\em Electronic mail\/}: {\tt lcp@cl.cam.ac.uk}}
+}
+\date{}
+\makeindex
+
+\underscoreoff
+
+\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
+
+\sloppy
+\binperiod %%%treat . like a binary operator
+
+\newcommand\qeq{\stackrel{?}{\equiv}} %for disagreement pairs in unification
+\newcommand{\nand}{\mathbin{\lnot\&}}
+\newcommand{\xor}{\mathbin{\#}}
+
+\pagenumbering{roman}
+\begin{document}
+\pagestyle{empty}
+\begin{titlepage}
+\maketitle
+\thispagestyle{empty}
+\vfill
+{\small Copyright \copyright{} \number\year{} by Lawrence C. Paulson}
+\end{titlepage}
+
+\pagestyle{headings}
+\part*{Preface}
+\index{Isabelle!overview}
+\index{Isabelle!object-logics supported}
+Isabelle~\cite{paulson86,paulson89,paulson700} is a generic theorem prover.
+It has been instantiated to support reasoning in several object-logics:
+\begin{itemize}
+\item first-order logic, constructive and classical versions
+\item higher-order logic, similar to that of Gordon's {\sc
+hol}~\cite{gordon88a}
+\item Zermelo-Fraenkel set theory~\cite{suppes72}
+\item an extensional version of Martin-L\"of's Type Theory~\cite{nordstrom90}
+\item the classical first-order sequent calculus, {\sc lk}
+\item the modal logics $T$, $S4$, and $S43$
+\item the Logic for Computable Functions~\cite{paulson87}
+\end{itemize}
+A logic's syntax and inference rules are specified declaratively; this
+allows single-step proof construction. Isabelle provides control
+structures for expressing search procedures. Isabelle also provides
+several generic tools, such as simplifiers and classical theorem provers,
+which can be applied to object-logics.
+
+\index{ML}
+Isabelle is a large system, but beginners can get by with a small
+repertoire of commands and a basic knowledge of how Isabelle works. Some
+knowledge of Standard~\ML{} is essential, because \ML{} is Isabelle's user
+interface. Advanced Isabelle theorem proving can involve writing \ML{}
+code, possibly with Isabelle's sources at hand. My book
+on~\ML{}~\cite{paulson91} covers much material connected with Isabelle,
+including a simple theorem prover. Users must be familiar with logic as
+used in computer science; there are many good
+texts~\cite{galton90,reeves90}.
+
+\index{LCF}
+{\sc lcf}, developed by Robin Milner and colleagues~\cite{gordon79}, is an
+ancestor of {\sc hol}, Nuprl, and several other systems. Isabelle borrows
+ideas from {\sc lcf}: formulae are~\ML{} values; theorems belong to an
+abstract type; tactics and tacticals support backward proof. But {\sc lcf}
+represents object-level rules by functions, while Isabelle represents them
+by terms. You may find my other writings~\cite{paulson87,paulson-handbook}
+helpful in understanding the relationship between {\sc lcf} and Isabelle.
+
+Isabelle does not keep a record of inference steps. Each step is checked
+at run time to ensure that theorems can only be constructed by applying
+inference rules. An Isabelle proof typically involves hundreds of
+primitive inferences, and would be unintelligible if displayed.
+Discarding proofs saves vast amounts of storage. But can Isabelle be
+trusted? If desired, object-logics can be formalized such that each
+theorem carries a proof term, which could be checked by another program.
+Proofs can also be traced.
+
+\index{Isabelle!release history} Isabelle was first distributed in 1986.
+The 1987 version introduced a higher-order meta-logic with an improved
+treatment of quantifiers. The 1988 version added limited polymorphism and
+support for natural deduction. The 1989 version included a parser and
+pretty printer generator. The 1992 version introduced type classes, to
+support many-sorted and higher-order logics. The current version provides
+greater support for theories and is much faster. Isabelle is still under
+development and will continue to change.
+
+\subsubsection*{Overview}
+This manual consists of three parts.
+Part~I discusses the Isabelle's foundations.
+Part~II, presents simple on-line sessions, starting with forward proof.
+It also covers basic tactics and tacticals, and some commands for invoking
+Part~III contains further examples for users with a bit of experience.
+It explains how to derive rules define theories, and concludes with an
+extended example: a Prolog interpreter.
+
+Isabelle's Reference Manual and Object-Logics manual contain more details.
+They assume familiarity with the concepts presented here.
+
+
+\subsubsection*{Acknowledgements}
+Tobias Nipkow contributed most of the section on ``Defining Theories''.
+Sara Kalvala and Markus Wenzel suggested improvements.
+
+Tobias Nipkow has made immense contributions to Isabelle, including the
+parser generator, type classes, and the simplifier. Carsten Clasohm, Sonia
+Mahjoub, Karin Nimmermann and Markus Wenzel also made improvements.
+Isabelle was developed using Dave Matthews's Standard~{\sc ml} compiler,
+Poly/{\sc ml}. Many people have contributed to Isabelle's standard
+object-logics, including Martin Coen, Philippe de Groote, Philippe No\"el.
+The research has been funded by the SERC (grants GR/G53279, GR/H40570) and
+by ESPRIT (projects 3245: Logical Frameworks, and 6453: Types).
+
+\newpage
+\pagestyle{plain} \tableofcontents
+\newpage
+
+\newfont{\sanssi}{cmssi12}
+\vspace*{2.5cm}
+\begin{quote}
+\raggedleft
+{\sanssi
+You can only find truth with logic\\
+if you have already found truth without it.}\\
+\bigskip
+
+G.K. Chesterton, {\em The Man who was Orthodox}
+\end{quote}
+
+\clearfirst \pagestyle{headings}
+\include{foundations}
+\include{getting}
+\include{advanced}
+
+
+\bibliographystyle{plain} \small\raggedright\frenchspacing
+\bibliography{atp,funprog,general,logicprog,theory}
+
+\input{intro.ind}
+\end{document}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/intro.toc Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,67 @@
+\contentsline {part}{\uppercase {i}\phspace {1em}Foundations}{1}
+\contentsline {section}{\numberline {1}Formalizing logical syntax in Isabelle}{1}
+\contentsline {subsection}{\numberline {1.1}Simple types and constants}{1}
+\contentsline {subsection}{\numberline {1.2}Polymorphic types and constants}{3}
+\contentsline {subsection}{\numberline {1.3}Higher types and quantifiers}{4}
+\contentsline {section}{\numberline {2}Formalizing logical rules in Isabelle}{5}
+\contentsline {subsection}{\numberline {2.1}Expressing propositional rules}{6}
+\contentsline {subsection}{\numberline {2.2}Quantifier rules and substitution}{7}
+\contentsline {subsection}{\numberline {2.3}Signatures and theories}{8}
+\contentsline {section}{\numberline {3}Proof construction in Isabelle}{9}
+\contentsline {subsection}{\numberline {3.1}Higher-order unification}{10}
+\contentsline {subsection}{\numberline {3.2}Joining rules by resolution}{11}
+\contentsline {subsection}{\numberline {3.3}Lifting a rule into a context}{13}
+\contentsline {subsubsection}{Lifting over assumptions}{13}
+\contentsline {subsubsection}{Lifting over parameters}{13}
+\contentsline {section}{\numberline {4}Backward proof by resolution}{14}
+\contentsline {subsection}{\numberline {4.1}Refinement by resolution}{15}
+\contentsline {subsection}{\numberline {4.2}Proof by assumption}{15}
+\contentsline {subsection}{\numberline {4.3}A propositional proof}{16}
+\contentsline {subsection}{\numberline {4.4}A quantifier proof}{17}
+\contentsline {subsection}{\numberline {4.5}Tactics and tacticals}{17}
+\contentsline {section}{\numberline {5}Variations on resolution}{18}
+\contentsline {subsection}{\numberline {5.1}Elim-resolution}{18}
+\contentsline {subsection}{\numberline {5.2}Destruction rules}{20}
+\contentsline {subsection}{\numberline {5.3}Deriving rules by resolution}{20}
+\contentsline {part}{\uppercase {ii}\phspace {1em}Getting started with Isabelle}{22}
+\contentsline {section}{\numberline {6}Forward proof}{22}
+\contentsline {subsection}{\numberline {6.1}Lexical matters}{22}
+\contentsline {subsection}{\numberline {6.2}Syntax of types and terms}{23}
+\contentsline {subsection}{\numberline {6.3}Basic operations on theorems}{24}
+\contentsline {subsection}{\numberline {6.4}Flex-flex equations}{26}
+\contentsline {section}{\numberline {7}Backward proof}{27}
+\contentsline {subsection}{\numberline {7.1}The basic tactics}{27}
+\contentsline {subsection}{\numberline {7.2}Commands for backward proof}{28}
+\contentsline {subsection}{\numberline {7.3}A trivial example in propositional logic}{28}
+\contentsline {subsection}{\numberline {7.4}Proving a distributive law}{30}
+\contentsline {section}{\numberline {8}Quantifier reasoning}{31}
+\contentsline {subsection}{\numberline {8.1}Two quantifier proofs, successful and not}{31}
+\contentsline {subsubsection}{The successful proof}{31}
+\contentsline {subsubsection}{The unsuccessful proof}{32}
+\contentsline {subsection}{\numberline {8.2}Nested quantifiers}{33}
+\contentsline {subsubsection}{The wrong approach}{33}
+\contentsline {subsubsection}{The right approach}{34}
+\contentsline {subsubsection}{A one-step proof using tacticals}{35}
+\contentsline {subsection}{\numberline {8.3}A realistic quantifier proof}{35}
+\contentsline {subsection}{\numberline {8.4}The classical reasoning package}{36}
+\contentsline {part}{\uppercase {iii}\phspace {1em}Advanced methods}{38}
+\contentsline {section}{\numberline {9}Deriving rules in Isabelle}{38}
+\contentsline {subsection}{\numberline {9.1}Deriving a rule using tactics}{38}
+\contentsline {subsection}{\numberline {9.2}Definitions and derived rules}{40}
+\contentsline {subsubsection}{Deriving the introduction rule}{41}
+\contentsline {subsubsection}{Deriving the elimination rule}{42}
+\contentsline {section}{\numberline {10}Defining theories}{44}
+\contentsline {subsection}{\numberline {10.1}Declaring constants and rules}{45}
+\contentsline {subsection}{\numberline {10.2}Declaring type constructors}{46}
+\contentsline {subsection}{\numberline {10.3}Infixes and Mixfixes}{47}
+\contentsline {subsection}{\numberline {10.4}Overloading}{48}
+\contentsline {subsection}{\numberline {10.5}Extending first-order logic with the natural numbers}{50}
+\contentsline {subsection}{\numberline {10.6}Declaring the theory to Isabelle}{51}
+\contentsline {section}{\numberline {11}Refinement with explicit instantiation}{52}
+\contentsline {subsection}{\numberline {11.1}A simple proof by induction}{52}
+\contentsline {subsection}{\numberline {11.2}An example of ambiguity in {\ptt resolve_tac}}{53}
+\contentsline {subsection}{\numberline {11.3}Proving that addition is associative}{54}
+\contentsline {section}{\numberline {12}A {\psc Prolog} interpreter}{55}
+\contentsline {subsection}{\numberline {12.1}Simple executions}{56}
+\contentsline {subsection}{\numberline {12.2}Backtracking}{57}
+\contentsline {subsection}{\numberline {12.3}Depth-first search}{58}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/list.thy Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,6 @@
+List = FOL +
+types list 1
+arities list :: (term)term
+consts Nil :: "'a list"
+ Cons :: "['a, 'a list] => 'a list"
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/prod.thy Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,9 @@
+Prod = FOL +
+types "*" 2 (infixl 20)
+arities "*" :: (term,term)term
+consts fst :: "'a * 'b => 'a"
+ snd :: "'a * 'b => 'b"
+ Pair :: "['a,'b] => 'a * 'b" ("(1<_,/_>)")
+rules fst "fst(<a,b>) = a"
+ snd "snd(<a,b>) = b"
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/quant.txt Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,244 @@
+(**** quantifier examples -- process using Doc/tout quant.txt ****)
+
+Pretty.setmargin 72; (*existing macros just allow this margin*)
+print_depth 0;
+
+
+goal Int_Rule.thy "(ALL x y.P(x,y)) --> (ALL z w.P(w,z))";
+by (resolve_tac [impI] 1);
+by (dresolve_tac [spec] 1);
+by (resolve_tac [allI] 1);
+by (dresolve_tac [spec] 1);
+by (resolve_tac [allI] 1);
+by (assume_tac 1);
+choplev 1;
+by (resolve_tac [allI] 1);
+by (resolve_tac [allI] 1);
+by (dresolve_tac [spec] 1);
+by (dresolve_tac [spec] 1);
+by (assume_tac 1);
+
+choplev 0;
+by (REPEAT (assume_tac 1
+ ORELSE resolve_tac [impI,allI] 1
+ ORELSE dresolve_tac [spec] 1));
+
+
+- goal Int_Rule.thy "(ALL x y.P(x,y)) --> (ALL z w.P(w,z))";
+Level 0
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. (ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+val it = [] : thm list
+- by (resolve_tac [impI] 1);
+Level 1
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. ALL x y. P(x,y) ==> ALL z w. P(w,z)
+val it = () : unit
+- by (dresolve_tac [spec] 1);
+Level 2
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. ALL y. P(?x1,y) ==> ALL z w. P(w,z)
+val it = () : unit
+- by (resolve_tac [allI] 1);
+Level 3
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. !!z. ALL y. P(?x1,y) ==> ALL w. P(w,z)
+val it = () : unit
+- by (dresolve_tac [spec] 1);
+Level 4
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. !!z. P(?x1,?y3(z)) ==> ALL w. P(w,z)
+val it = () : unit
+- by (resolve_tac [allI] 1);
+Level 5
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. !!z w. P(?x1,?y3(z)) ==> P(w,z)
+val it = () : unit
+- by (assume_tac 1);
+by: tactic returned no results
+
+uncaught exception ERROR
+- choplev 1;
+Level 1
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. ALL x y. P(x,y) ==> ALL z w. P(w,z)
+val it = () : unit
+- by (resolve_tac [allI] 1);
+Level 2
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. !!z. ALL x y. P(x,y) ==> ALL w. P(w,z)
+val it = () : unit
+- by (resolve_tac [allI] 1);
+Level 3
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. !!z w. ALL x y. P(x,y) ==> P(w,z)
+val it = () : unit
+- by (dresolve_tac [spec] 1);
+Level 4
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. !!z w. ALL y. P(?x3(z,w),y) ==> P(w,z)
+val it = () : unit
+- by (dresolve_tac [spec] 1);
+Level 5
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. !!z w. P(?x3(z,w),?y4(z,w)) ==> P(w,z)
+val it = () : unit
+- by (assume_tac 1);
+Level 6
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+No subgoals!
+
+> choplev 0;
+Level 0
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+ 1. (ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+> by (REPEAT (assume_tac 1
+# ORELSE resolve_tac [impI,allI] 1
+# ORELSE dresolve_tac [spec] 1));
+Level 1
+(ALL x y. P(x,y)) --> (ALL z w. P(w,z))
+No subgoals!
+
+
+
+goal FOL_thy "ALL x. EX y. x=y";
+by (resolve_tac [allI] 1);
+by (resolve_tac [exI] 1);
+by (resolve_tac [refl] 1);
+
+- goal Int_Rule.thy "ALL x. EX y. x=y";
+Level 0
+ALL x. EX y. x = y
+ 1. ALL x. EX y. x = y
+val it = [] : thm list
+- by (resolve_tac [allI] 1);
+Level 1
+ALL x. EX y. x = y
+ 1. !!x. EX y. x = y
+val it = () : unit
+- by (resolve_tac [exI] 1);
+Level 2
+ALL x. EX y. x = y
+ 1. !!x. x = ?y1(x)
+val it = () : unit
+- by (resolve_tac [refl] 1);
+Level 3
+ALL x. EX y. x = y
+No subgoals!
+val it = () : unit
+-
+
+goal FOL_thy "EX y. ALL x. x=y";
+by (resolve_tac [exI] 1);
+by (resolve_tac [allI] 1);
+by (resolve_tac [refl] 1);
+
+- goal Int_Rule.thy "EX y. ALL x. x=y";
+Level 0
+EX y. ALL x. x = y
+ 1. EX y. ALL x. x = y
+val it = [] : thm list
+- by (resolve_tac [exI] 1);
+Level 1
+EX y. ALL x. x = y
+ 1. ALL x. x = ?y
+val it = () : unit
+- by (resolve_tac [allI] 1);
+Level 2
+EX y. ALL x. x = y
+ 1. !!x. x = ?y
+val it = () : unit
+- by (resolve_tac [refl] 1);
+by: tactic returned no results
+
+uncaught exception ERROR
+
+
+
+goal FOL_thy "EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)";
+by (resolve_tac [exI, allI] 1);
+by (resolve_tac [exI, allI] 1);
+by (resolve_tac [exI, allI] 1);
+by (resolve_tac [exI, allI] 1);
+by (resolve_tac [exI, allI] 1);
+
+- goal Int_Rule.thy "EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)";
+Level 0
+EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)
+ 1. EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)
+val it = [] : thm list
+- by (resolve_tac [exI, allI] 1);
+Level 1
+EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)
+ 1. ALL x. EX v. ALL y. EX w. P(?u,x,v,y,w)
+val it = () : unit
+- by (resolve_tac [exI, allI] 1);
+Level 2
+EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)
+ 1. !!x. EX v. ALL y. EX w. P(?u,x,v,y,w)
+val it = () : unit
+- by (resolve_tac [exI, allI] 1);
+Level 3
+EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)
+ 1. !!x. ALL y. EX w. P(?u,x,?v2(x),y,w)
+val it = () : unit
+- by (resolve_tac [exI, allI] 1);
+Level 4
+EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)
+ 1. !!x y. EX w. P(?u,x,?v2(x),y,w)
+val it = () : unit
+- by (resolve_tac [exI, allI] 1);
+Level 5
+EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)
+ 1. !!x y. P(?u,x,?v2(x),y,?w4(x,y))
+val it = () : unit
+
+
+goal FOL_thy "(ALL x.P(x) --> Q) --> (EX x.P(x))-->Q";
+by (REPEAT (resolve_tac [impI] 1));
+by (eresolve_tac [exE] 1);
+by (dresolve_tac [spec] 1);
+by (eresolve_tac [mp] 1);
+by (assume_tac 1);
+
+- goal Int_Rule.thy "(ALL x.P(x) --> Q) --> (EX x.P(x))-->Q";
+Level 0
+(ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q
+ 1. (ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q
+val it = [] : thm list
+- by (REPEAT (resolve_tac [impI] 1));
+Level 1
+(ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q
+ 1. [| ALL x. P(x) --> Q; EX x. P(x) |] ==> Q
+val it = () : unit
+- by (eresolve_tac [exE] 1);
+Level 2
+(ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q
+ 1. !!x. [| ALL x. P(x) --> Q; P(x) |] ==> Q
+val it = () : unit
+- by (dresolve_tac [spec] 1);
+Level 3
+(ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q
+ 1. !!x. [| P(x); P(?x3(x)) --> Q |] ==> Q
+val it = () : unit
+- by (eresolve_tac [mp] 1);
+Level 4
+(ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q
+ 1. !!x. P(x) ==> P(?x3(x))
+val it = () : unit
+- by (assume_tac 1);
+Level 5
+(ALL x. P(x) --> Q) --> (EX x. P(x)) --> Q
+No subgoals!
+
+
+goal FOL_thy "((EX x.P(x)) --> Q) --> (ALL x.P(x)-->Q)";
+by (REPEAT (resolve_tac [impI] 1));
+
+
+goal FOL_thy "(EX x.P(x) --> Q) --> (ALL x.P(x))-->Q";
+by (REPEAT (resolve_tac [impI] 1));
+by (eresolve_tac [exE] 1);
+by (eresolve_tac [mp] 1);
+by (eresolve_tac [spec] 1);
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/theorems-out.txt Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,65 @@
+> goal Nat.thy "(k+m)+n = k+(m+n)";
+Level 0
+k + m + n = k + (m + n)
+ 1. k + m + n = k + (m + n)
+val it = [] : thm list
+> by (resolve_tac [induct] 1);
+Level 1
+k + m + n = k + (m + n)
+ 1. k + m + n = 0
+ 2. !!x. k + m + n = x ==> k + m + n = Suc(x)
+val it = () : unit
+> back();
+Level 1
+k + m + n = k + (m + n)
+ 1. k + m + n = k + 0
+ 2. !!x. k + m + n = k + x ==> k + m + n = k + Suc(x)
+val it = () : unit
+> back();
+Level 1
+k + m + n = k + (m + n)
+ 1. k + m + 0 = k + (m + 0)
+ 2. !!x. k + m + x = k + (m + x) ==> k + m + Suc(x) = k + (m + Suc(x))
+val it = () : unit
+> back();
+Level 1
+k + m + n = k + (m + n)
+ 1. k + m + n = k + (m + 0)
+ 2. !!x. k + m + n = k + (m + x) ==> k + m + n = k + (m + Suc(x))
+val it = () : unit
+
+> val nat_congs = prths (mk_congs Nat.thy ["Suc", "op +"]);
+?Xa = ?Ya ==> Suc(?Xa) = Suc(?Ya)
+
+[| ?Xa = ?Ya; ?Xb = ?Yb |] ==> ?Xa + ?Xb = ?Ya + ?Yb
+
+?Xa = ?Ya ==> Suc(?Xa) = Suc(?Ya)
+[| ?Xa = ?Ya; ?Xb = ?Yb |] ==> ?Xa + ?Xb = ?Ya + ?Yb
+val nat_congs = [, ] : thm list
+> val add_ss = FOL_ss addcongs nat_congs
+# addrews [add_0, add_Suc];
+val add_ss = ? : simpset
+> goal Nat.thy "(k+m)+n = k+(m+n)";
+Level 0
+k + m + n = k + (m + n)
+ 1. k + m + n = k + (m + n)
+val it = [] : thm list
+> by (res_inst_tac [("n","k")] induct 1);
+Level 1
+k + m + n = k + (m + n)
+ 1. 0 + m + n = 0 + (m + n)
+ 2. !!x. x + m + n = x + (m + n) ==> Suc(x) + m + n = Suc(x) + (m + n)
+val it = () : unit
+> by (SIMP_TAC add_ss 1);
+Level 2
+k + m + n = k + (m + n)
+ 1. !!x. x + m + n = x + (m + n) ==> Suc(x) + m + n = Suc(x) + (m + n)
+val it = () : unit
+> by (ASM_SIMP_TAC add_ss 1);
+Level 3
+k + m + n = k + (m + n)
+No subgoals!
+val it = () : unit
+> val add_assoc = result();
+?k + ?m + ?n = ?k + (?m + ?n)
+val add_assoc = : thm
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Intro/theorems.txt Wed Nov 10 05:06:55 1993 +0100
@@ -0,0 +1,122 @@
+Pretty.setmargin 72; (*existing macros just allow this margin*)
+print_depth 0;
+
+(*operations for "thm"*)
+
+prth mp;
+
+prth (mp RS mp);
+
+prth (conjunct1 RS mp);
+prth (conjunct1 RSN (2,mp));
+
+prth (mp RS conjunct1);
+prth (spec RS it);
+prth (standard it);
+
+prth spec;
+prth (it RS mp);
+prth (it RS conjunct1);
+prth (standard it);
+
+- prth spec;
+ALL x. ?P(x) ==> ?P(?x)
+- prth (it RS mp);
+[| ALL x. ?P3(x) --> ?Q2(x); ?P3(?x1) |] ==> ?Q2(?x1)
+- prth (it RS conjunct1);
+[| ALL x. ?P4(x) --> ?P6(x) & ?Q5(x); ?P4(?x2) |] ==> ?P6(?x2)
+- prth (standard it);
+[| ALL x. ?P(x) --> ?Pa(x) & ?Q(x); ?P(?x) |] ==> ?Pa(?x)
+
+(*flexflex pairs*)
+- prth refl;
+?a = ?a
+- prth exI;
+?P(?x) ==> EX x. ?P(x)
+- prth (refl RS exI);
+?a3(?x) == ?a2(?x) ==> EX x. ?a3(x) = ?a2(x)
+- prthq (flexflex_rule it);
+EX x. ?a4 = ?a4
+
+(*Usage of RL*)
+- val reflk = prth (read_instantiate [("a","k")] refl);
+k = k
+val reflk = Thm {hyps=#,maxidx=#,prop=#,sign=#} : thm
+- prth (reflk RS exI);
+
+uncaught exception THM
+- prths ([reflk] RL [exI]);
+EX x. x = x
+
+EX x. k = x
+
+EX x. x = k
+
+EX x. k = k
+
+
+
+(*tactics *)
+
+goal cla_thy "P|P --> P";
+by (resolve_tac [impI] 1);
+by (resolve_tac [disjE] 1);
+by (assume_tac 3);
+by (assume_tac 2);
+by (assume_tac 1);
+val mythm = prth(result());
+
+
+goal cla_thy "(P & Q) | R --> (P | R)";
+by (resolve_tac [impI] 1);
+by (eresolve_tac [disjE] 1);
+by (dresolve_tac [conjunct1] 1);
+by (resolve_tac [disjI1] 1);
+by (resolve_tac [disjI2] 2);
+by (REPEAT (assume_tac 1));
+
+
+- goal cla_thy "(P & Q) | R --> (P | R)";
+Level 0
+P & Q | R --> P | R
+ 1. P & Q | R --> P | R
+- by (resolve_tac [impI] 1);
+Level 1
+P & Q | R --> P | R
+ 1. P & Q | R ==> P | R
+- by (eresolve_tac [disjE] 1);
+Level 2
+P & Q | R --> P | R
+ 1. P & Q ==> P | R
+ 2. R ==> P | R
+- by (dresolve_tac [conjunct1] 1);
+Level 3
+P & Q | R --> P | R
+ 1. P ==> P | R
+ 2. R ==> P | R
+- by (resolve_tac [disjI1] 1);
+Level 4
+P & Q | R --> P | R
+ 1. P ==> P
+ 2. R ==> P | R
+- by (resolve_tac [disjI2] 2);
+Level 5
+P & Q | R --> P | R
+ 1. P ==> P
+ 2. R ==> R
+- by (REPEAT (assume_tac 1));
+Level 6
+P & Q | R --> P | R
+No subgoals!
+
+
+goal cla_thy "(P | Q) | R --> P | (Q | R)";
+by (resolve_tac [impI] 1);
+by (eresolve_tac [disjE] 1);
+by (eresolve_tac [disjE] 1);
+by (resolve_tac [disjI1] 1);
+by (resolve_tac [disjI2] 2);
+by (resolve_tac [disjI1] 2);
+by (resolve_tac [disjI2] 3);
+by (resolve_tac [disjI2] 3);
+by (REPEAT (assume_tac 1));