%% $Id$

 \chapter{Theories, Terms and Types} \label{theories}

 \index{theories|(}\index{signatures|bold}

 \index{reading!axioms|see{{\tt assume_ax}}} Theories organize the

 syntax, declarations and axioms of a mathematical development.  They

 are built, starting from the {\Pure} or {\CPure} theory, by extending

 and merging existing theories.  They have the \ML\ type

 \mltydx{theory}.  Theory operations signal errors by raising exception

 \xdx{THEORY}, returning a message and a list of theories.

 Signatures, which contain information about sorts, types, constants and

 syntax, have the \ML\ type~\mltydx{Sign.sg}.  For identification, each

 signature carries a unique list of \bfindex{stamps}, which are \ML\

 references to strings.  The strings serve as human-readable names; the

 references serve as unique identifiers.  Each primitive signature has a

 single stamp.  When two signatures are merged, their lists of stamps are

 also merged.  Every theory carries a unique signature.

 Terms and types are the underlying representation of logical syntax.  Their

 \ML\ definitions are irrelevant to naive Isabelle users.  Programmers who

 wish to extend Isabelle may need to know such details, say to code a tactic

 that looks for subgoals of a particular form.  Terms and types may be

 `certified' to be well-formed with respect to a given signature.

 \section{Defining theories}\label{sec:ref-defining-theories}

 Theories are usually defined using theory definition files (which have a name

 suffix {\tt .thy}).  There is also a low level interface provided by certain

 \ML{} functions (see \S\ref{BuildingATheory}).

 Appendix~\ref{app:TheorySyntax} presents the concrete syntax for theory

 definitions; here is an explanation of the constituent parts:

 \begin{description}

 \item[{\it theoryDef}] is the full definition.  The new theory is

   called $id$.  It is the union of the named {\bf parent

     theories}\indexbold{theories!parent}, possibly extended with new

   components.  \thydx{Pure} and \thydx{CPure} are the basic theories,

   which contain only the meta-logic.  They differ just in their

   concrete syntax for function applications.

   Normally each {\it name\/} is an identifier, the name of the parent theory.

   Quoted strings can be used to document additional file dependencies; see

   \S\ref{LoadingTheories} for details.

 \item[$classes$]

   is a series of class declarations.  Declaring {\tt$id$ < $id@1$ \dots\

     $id@n$} makes $id$ a subclass of the existing classes $id@1\dots

   id@n$.  This rules out cyclic class structures.  Isabelle automatically

   computes the transitive closure of subclass hierarchies; it is not

   necessary to declare {\tt c < e} in addition to {\tt c < d} and {\tt d <

     e}.

 \item[$default$]

   introduces $sort$ as the new default sort for type variables.  This applies

   to unconstrained type variables in an input string but not to type

   variables created internally.  If omitted, the default sort is the listwise

   union of the default sorts of the parent theories (i.e.\ their logical

   intersection).

 \item[$sort$] is a finite set of classes.  A single class $id$

   abbreviates the sort $\ttlbrace id\ttrbrace$.

 \item[$types$]

   is a series of type declarations.  Each declares a new type constructor

   or type synonym.  An $n$-place type constructor is specified by

   $(\alpha@1,\dots,\alpha@n)name$, where the type variables serve only to

   indicate the number~$n$.

   A {\bf type synonym}\indexbold{type synonyms} is an abbreviation

   $(\alpha@1,\dots,\alpha@n)name = \tau$, where $name$ and $\tau$ can

   be strings.

 \item[$infix$]

   declares a type or constant to be an infix operator of priority $nat$

   associating to the left ({\tt infixl}) or right ({\tt infixr}).  Only

   2-place type constructors can have infix status; an example is {\tt

   ('a,'b)~"*"~(infixr~20)}, which may express binary product types.

 \item[$arities$] is a series of type arity declarations.  Each assigns

   arities to type constructors.  The $name$ must be an existing type

   constructor, which is given the additional arity $arity$.

 \item[$consts$] is a series of constant declarations.  Each new

   constant $name$ is given the specified type.  The optional $mixfix$

   annotations may attach concrete syntax to the constant.

 \item[$syntax$] \index{*syntax section}\index{print mode} is a variant

   of $consts$ which adds just syntax without actually declaring

   logical constants.  This gives full control over a theory's context

   free grammar.  The optional $mode$ specifies the print mode where the

   mixfix productions should be added.  If there is no \texttt{output}

   option given, all productions are also added to the input syntax

   (regardless of the print mode).

 \item[$mixfix$] \index{mixfix declarations}

   annotations can take three forms:

   \begin{itemize}

   \item A mixfix template given as a $string$ of the form

     {\tt"}\dots{\tt\_}\dots{\tt\_}\dots{\tt"} where the $i$-th underscore

     indicates the position where the $i$-th argument should go.  The list

     of numbers gives the priority of each argument.  The final number gives

     the priority of the whole construct.

   \item A constant $f$ of type $\tau@1\To(\tau@2\To\tau)$ can be given {\bf

     infix} status.

   \item A constant $f$ of type $(\tau@1\To\tau@2)\To\tau$ can be given {\bf

     binder} status.  The declaration {\tt binder} $\cal Q$ $p$ causes

   ${\cal Q}\,x.F(x)$ to be treated

   like $f(F)$, where $p$ is the priority.

   \end{itemize}

 \item[$trans$]

   specifies syntactic translation rules (macros).  There are three forms:

   parse rules ({\tt =>}), print rules ({\tt <=}), and parse/print rules ({\tt

   ==}).

 \item[$rules$]

   is a series of rule declarations.  Each has a name $id$ and the formula is

   given by the $string$.  Rule names must be distinct within any single

   theory.

 \item[$defs$] is a series of definitions.  They are just like $rules$, except

   that every $string$ must be a definition (see below for details).

 \item[$constdefs$] combines the declaration of constants and their

   definition.  The first $string$ is the type, the second the definition.

 \item[$axclass$] \index{*axclass section} defines an

   \rmindex{axiomatic type class} as the intersection of existing

   classes, with additional axioms holding.  Class axioms may not

   contain more than one type variable.  The class axioms (with implicit

   sort constraints added) are bound to the given names.  Furthermore a

   class introduction rule is generated, which is automatically

   employed by $instance$ to prove instantiations of this class.

 \item[$instance$] \index{*instance section} proves class inclusions or

   type arities at the logical level and then transfers these to the

   type signature.  The instantiation is proven and checked properly.

   The user has to supply sufficient witness information: theorems

   ($longident$), axioms ($string$), or even arbitrary \ML{} tactic

   code $verbatim$.

 \item[$oracle$] links the theory to a trusted external reasoner.  It is

   allowed to create theorems, but each theorem carries a proof object

   describing the oracle invocation.  See \S\ref{sec:oracles} for details.

 \item[$ml$] \index{*ML section}

   consists of \ML\ code, typically for parse and print translation functions.

 \end{description}

 %

 Chapters~\ref{Defining-Logics} and \ref{chap:syntax} explain mixfix

 declarations, translation rules and the {\tt ML} section in more detail.

 \subsection{Definitions}\indexbold{definitions}

 {\bf Definitions} are intended to express abbreviations.  The simplest

 form of a definition is $f \equiv t$, where $f$ is a constant.

 Isabelle also allows a derived forms where the arguments of~$f$ appear

 on the left, abbreviating a string of $\lambda$-abstractions.

 Isabelle makes the following checks on definitions:

 \begin{itemize}

 \item Arguments (on the left-hand side) must be distinct variables.

 \item All variables on the right-hand side must also appear on the left-hand

   side. 

 \item All type variables on the right-hand side must also appear on

   the left-hand side; this prohibits definitions such as {\tt

     (zero::nat) == length ([]::'a list)}.

 \item The definition must not be recursive.  Most object-logics provide

   definitional principles that can be used to express recursion safely.

 \end{itemize}

 These checks are intended to catch the sort of errors that might be made

 accidentally.  Misspellings, for instance, might result in additional

 variables appearing on the right-hand side.  More elaborate checks could be

 made, but the cost might be overly strict rules on declaration order, etc.

 \subsection{*Classes and arities}

 \index{classes!context conditions}\index{arities!context conditions}

 In order to guarantee principal types~\cite{nipkow-prehofer},

 arity declarations must obey two conditions:

 \begin{itemize}

 \item There must not be any two declarations $ty :: (\vec{r})c$ and

   $ty :: (\vec{s})c$ with $\vec{r} \neq \vec{s}$.  For example, this

   excludes the following:

 \begin{ttbox}

 arities

   foo :: ({\ttlbrace}logic{\ttrbrace}) logic

   foo :: ({\ttlbrace}{\ttrbrace})logic

 \end{ttbox}

 \item If there are two declarations $ty :: (s@1,\dots,s@n)c$ and $ty ::

   (s@1',\dots,s@n')c'$ such that $c' < c$ then $s@i' \preceq s@i$ must hold

   for $i=1,\dots,n$.  The relationship $\preceq$, defined as

 \[ s' \preceq s \iff \forall c\in s. \exists c'\in s'.~ c'\le c, \]

 expresses that the set of types represented by $s'$ is a subset of the

 set of types represented by $s$.  Assuming $term \preceq logic$, the

 following is forbidden:

 \begin{ttbox}

 arities

   foo :: ({\ttlbrace}logic{\ttrbrace})logic

   foo :: ({\ttlbrace}{\ttrbrace})term

 \end{ttbox}

 \end{itemize}

 \section{Loading a new theory}\label{LoadingTheories}

 \index{theories!loading}\index{files!reading}

 \begin{ttbox}

 use_thy         : string -> unit

 time_use_thy    : string -> unit

 loadpath        : string list ref \hfill{\bf initially {\tt["."]}}

 delete_tmpfiles : bool ref \hfill{\bf initially true}

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{use_thy} $thyname$]

   reads the theory $thyname$ and creates an \ML{} structure as described below.

 \item[\ttindexbold{time_use_thy} $thyname$]

   calls {\tt use_thy} $thyname$ and reports the time taken.

 \item[\ttindexbold{loadpath}]

   contains a list of directories to search when locating the files that

   define a theory.  This list is only used if the theory name in {\tt

     use_thy} does not specify the path explicitly.

 \item[reset \ttindexbold{delete_tmpfiles};]

 suppresses the deletion of temporary files.

 \end{ttdescription}

 %

 Each theory definition must reside in a separate file.  Let the file

 {\it T}{\tt.thy} contain the definition of a theory called~$T$, whose

 parent theories are $TB@1$ \dots $TB@n$.  Calling

 \ttindex{use_thy}~{\tt"{\it T\/}"} reads the file {\it T}{\tt.thy},

 writes a temporary \ML{} file {\tt.{\it T}.thy.ML}, and reads the

 latter file.  Recursive {\tt use_thy} calls load those parent theories

 that have not been loaded previously; the recursive calls may continue

 to any depth.  One {\tt use_thy} call can read an entire logic

 provided all theories are linked appropriately.

 The result is an \ML\ structure~$T$ containing at least a component

 {\tt thy} for the new theory and components for each of the rules.

 The structure also contains the definitions of the {\tt ML} section,

 if present.  The file {\tt.{\it T}.thy.ML} is then deleted if {\tt

   delete_tmpfiles} is set and no errors occurred.

 Finally the file {\it T}{\tt.ML} is read, if it exists.  The structure

 $T$ is automatically open in this context.  Proof scripts typically

 refer to its components by unqualified names.

 Some applications construct theories directly by calling \ML\ functions.  In

 this situation there is no {\tt.thy} file, only an {\tt.ML} file.  The

 {\tt.ML} file must declare an \ML\ structure having the theory's name and a

 component {\tt thy} containing the new theory object.

 Section~\ref{sec:pseudo-theories} below describes a way of linking such

 theories to their parents.

 \begin{warn}

   Temporary files are written to the current directory, so this must be

   writable.  Isabelle inherits the current directory from the operating

   system; you can change it within Isabelle by typing {\tt

   cd"$dir$"}\index{*cd}.

 \end{warn}

 \section{Reloading modified theories}\label{sec:reloading-theories}

 \indexbold{theories!reloading}

 \begin{ttbox}

 update     : unit -> unit

 unlink_thy : string -> unit

 \end{ttbox}

 Changing a theory on disk often makes it necessary to reload all theories

 descended from it.  However, {\tt use_thy} reads only one theory, even if

 some of the parent theories are out of date.  In this case you should call

 {\tt update()}.

 Isabelle keeps track of all loaded theories and their files.  If

 \ttindex{use_thy} finds that the theory to be loaded has been read before,

 it determines whether to reload the theory as follows.  First it looks for

 the theory's files in their previous location.  If it finds them, it

 compares their modification times to the internal data and stops if they

 are equal.  If the files have been moved, {\tt use_thy} searches for them

 as it would for a new theory.  After {\tt use_thy} reloads a theory, it

 marks the children as out-of-date.

 \begin{ttdescription}

 \item[\ttindexbold{update}()]

   reloads all modified theories and their descendants in the correct order.

 \item[\ttindexbold{unlink_thy} $thyname$]\indexbold{theories!removing}

   informs Isabelle that theory $thyname$ no longer exists.  If you delete the

   theory files for $thyname$ then you must execute {\tt unlink_thy};

   otherwise {\tt update} will complain about a missing file.

 \end{ttdescription}

 \subsection{*Pseudo theories}\label{sec:pseudo-theories}

 \indexbold{theories!pseudo}%

 Any automatic reloading facility requires complete knowledge of all

 dependencies.  Sometimes theories depend on objects created in \ML{} files

 with no associated theory definition file.  These objects may be theories but

 they could also be theorems, proof procedures, etc.

 Unless such dependencies are documented, {\tt update} fails to reload these

 \ML{} files and the system is left in a state where some objects, such as

 theorems, still refer to old versions of theories.  This may lead to the

 error

 \begin{ttbox}

 Attempt to merge different versions of theories: \dots

 \end{ttbox}

 Therefore there is a way to link theories and {\bf orphaned} \ML{} files ---

 those not associated with a theory definition.

 Let us assume we have an orphaned \ML{} file named {\tt orphan.ML} and a

 theory~$B$ that depends on {\tt orphan.ML} --- for example, {\tt B.ML} uses

 theorems proved in {\tt orphan.ML}.  Then {\tt B.thy} should

 mention this dependency as follows:

 \begin{ttbox}

 B = \(\ldots\) + "orphan" + \(\ldots\)

 \end{ttbox}

 Quoted strings stand for theories which have to be loaded before the

 current theory is read but which are not used in building the base of

 theory~$B$.  Whenever {\tt orphan} changes and is reloaded, Isabelle

 knows that $B$ has to be updated, too.

 Note that it's necessary for {\tt orphan} to declare a special ML

 object of type {\tt theory} which is present in all theories.  This is

 normally achieved by adding the file {\tt orphan.thy} to make {\tt

 orphan} a {\bf pseudo theory}.  A minimum version of {\tt orphan.thy}

 would be

 \begin{ttbox}

 orphan = Pure

 \end{ttbox}

 which uses {\tt Pure} to make a dummy theory.  Normally though the

 orphaned file has its own dependencies.  If {\tt orphan.ML} depends on

 theories or files $A@1$, \ldots, $A@n$, record this by creating the

 pseudo theory in the following way:

 \begin{ttbox}

 orphan = \(A@1\) + \(\ldots\) + \(A@n\)

 \end{ttbox}

 The resulting theory ensures that {\tt update} reloads {\tt orphan}

 whenever it reloads one of the $A@i$.

 For an extensive example of how this technique can be used to link

 lots of theory files and load them by just a few {\tt use_thy} calls

 see the sources of one of the major object-logics (e.g.\ \ZF).

 \section{Basic operations on theories}\label{BasicOperationsOnTheories}

 \subsection{Retrieving axioms and theorems}

 \index{theories!axioms of}\index{axioms!extracting}

 \index{theories!theorems of}\index{theorems!extracting}

 \begin{ttbox}

 get_axiom : theory -> xstring -> thm

 get_thm   : theory -> xstring -> thm

 get_thms  : theory -> xstring -> thm list

 axioms_of : theory -> (string * thm) list

 thms_of   : theory -> (string * thm) list

 assume_ax : theory -> string -> thm

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{get_axiom} $thy$ $name$] returns an axiom with the

   given $name$ from $thy$ or any of its ancestors, raising exception

   \xdx{THEORY} if none exists.  Merging theories can cause several

   axioms to have the same name; {\tt get_axiom} returns an arbitrary

   one.  Usually, axioms are also stored as theorems and may be

   retrieved via \texttt{get_thm} as well.

 \item[\ttindexbold{get_thm} $thy$ $name$] is analogous to {\tt

     get_axiom}, but looks for a theorem stored in the theory's

   database.  Like {\tt get_axiom} it searches all parents of a theory

   if the theorem is not found directly in $thy$.

 \item[\ttindexbold{get_thms} $thy$ $name$] is like \texttt{get_thm}

   for retrieving theorem lists stored within the theory.  It returns a

   singleton list if $name$ actually refers to a theorem rather than a

   theorem list.

 \item[\ttindexbold{axioms_of} $thy$] returns the axioms of this theory

   node, not including the ones of its ancestors.

 \item[\ttindexbold{thms_of} $thy$] returns all theorems stored within

   the database of this theory node, not including the ones of its

   ancestors.

 \item[\ttindexbold{assume_ax} $thy$ $formula$] reads the {\it formula}

   using the syntax of $thy$, following the same conventions as axioms

   in a theory definition.  You can thus pretend that {\it formula} is

   an axiom and use the resulting theorem like an axiom.  Actually {\tt

     assume_ax} returns an assumption; \ttindex{qed} and

   \ttindex{result} complain about additional assumptions, but

   \ttindex{uresult} does not.

 For example, if {\it formula} is

 \hbox{\tt a=b ==> b=a} then the resulting theorem has the form

 \hbox{\verb'?a=?b ==> ?b=?a  [!!a b. a=b ==> b=a]'}

 \end{ttdescription}

 \subsection{*Building a theory}

 \label{BuildingATheory}

 \index{theories!constructing|bold}

 \begin{ttbox}

 ProtoPure.thy  : theory

 Pure.thy       : theory

 CPure.thy      : theory

 merge_theories : string -> theory * theory -> theory

 \end{ttbox}

 \begin{description}

 \item[\ttindexbold{ProtoPure.thy}, \ttindexbold{Pure.thy},

   \ttindexbold{CPure.thy}] contain the syntax and signature of the

   meta-logic.  There are basically no axioms: meta-level inferences

   are carried out by \ML\ functions.  \texttt{Pure} and \texttt{CPure}

   just differ in their concrete syntax of prefix function application:

   $t(u@1, \ldots, u@n)$ in \texttt{Pure} vs.\ $t\,u@1,\ldots\,u@n$ in

   \texttt{CPure}.  \texttt{ProtoPure} is their common parent,

   containing no syntax for printing prefix applications at all!

 \item[\ttindexbold{merge_theories} $name$ ($thy@1$, $thy@2$)] merges

   the two theories $thy@1$ and $thy@2$, creating a new named theory

   node.  The resulting theory contains all of the syntax, signature

   and axioms of the constituent theories.  Merging theories that

   contain different identification stamps of the same name fails with

   the following message

 \begin{ttbox}

 Attempt to merge different versions of theories: "\(T@1\)", \(\ldots\), "\(T@n\)"

 \end{ttbox}

 This error may especially occur when a theory is redeclared --- say to

 change an inappropriate definition --- and bindings to old versions

 persist.  Isabelle ensures that old and new theories of the same name

 are not involved in a proof.

 %% FIXME

 %\item [\ttindexbold{extend_theory} $thy$ {\tt"}$T${\tt"} $\cdots$] extends

 %  the theory $thy$ with new types, constants, etc.  $T$ identifies the theory

 %  internally.  When a theory is redeclared, say to change an incorrect axiom,

 %  bindings to the old axiom may persist.  Isabelle ensures that the old and

 %  new theories are not involved in the same proof.  Attempting to combine

 %  different theories having the same name $T$ yields the fatal error

 %extend_theory  : theory -> string -> \(\cdots\) -> theory

 %\begin{ttbox}

 %Attempt to merge different versions of theory: \(T\)

 %\end{ttbox}

 \end{description}

 %% FIXME

 %\item [\ttindexbold{extend_theory} $thy$ {\tt"}$T${\tt"}

 %      ($classes$, $default$, $types$, $arities$, $consts$, $sextopt$) $rules$]

 %\hfill\break   %%% include if line is just too short

 %is the \ML{} equivalent of the following theory definition:

 %\begin{ttbox}

 %\(T\) = \(thy\) +

 %classes \(c\) < \(c@1\),\(\dots\),\(c@m\)

 %        \dots

 %default {\(d@1,\dots,d@r\)}

 %types   \(tycon@1\),\dots,\(tycon@i\) \(n\)

 %        \dots

 %arities \(tycon@1'\),\dots,\(tycon@j'\) :: (\(s@1\),\dots,\(s@n\))\(c\)

 %        \dots

 %consts  \(b@1\),\dots,\(b@k\) :: \(\tau\)

 %        \dots

 %rules   \(name\) \(rule\)

 %        \dots

 %end

 %\end{ttbox}

 %where

 %\begin{tabular}[t]{l@{~=~}l}

 %$classes$ & \tt[("$c$",["$c@1$",\dots,"$c@m$"]),\dots] \\

 %$default$ & \tt["$d@1$",\dots,"$d@r$"]\\

 %$types$   & \tt[([$tycon@1$,\dots,$tycon@i$], $n$),\dots] \\

 %$arities$ & \tt[([$tycon'@1$,\dots,$tycon'@j$], ([$s@1$,\dots,$s@n$],$c$)),\dots]

 %\\

 %$consts$  & \tt[([$b@1$,\dots,$b@k$],$\tau$),\dots] \\

 %$rules$   & \tt[("$name$",$rule$),\dots]

 %\end{tabular}

 \subsection{Inspecting a theory}\label{sec:inspct-thy}

 \index{theories!inspecting|bold}

 \begin{ttbox}

 print_syntax        : theory -> unit

 print_theory        : theory -> unit

 print_data          : theory -> string -> unit

 parents_of          : theory -> theory list

 ancestors_of        : theory -> theory list

 sign_of             : theory -> Sign.sg

 Sign.stamp_names_of : Sign.sg -> string list

 \end{ttbox}

 These provide means of viewing a theory's components.

 \begin{ttdescription}

 \item[\ttindexbold{print_syntax} $thy$] prints the syntax of $thy$

   (grammar, macros, translation functions etc., see

   page~\pageref{pg:print_syn} for more details).

 \item[\ttindexbold{print_theory} $thy$] prints the logical parts of

   $thy$, excluding the syntax.

 \item[\ttindexbold{print_data} $thy$ $kind$] prints generic data of

   $thy$.  This invokes the print method associated with $kind$.  Refer

   to the output of \texttt{print_theory} for a list of available data

   kinds in $thy$.

 \item[\ttindexbold{parents_of} $thy$] returns the direct ancestors

   of~$thy$.

 \item[\ttindexbold{ancestors_of} $thy$] returns all ancestors of~$thy$

   (not including $thy$ itself).

 \item[\ttindexbold{sign_of} $thy$] returns the signature associated

   with~$thy$.  It is useful with functions like {\tt

     read_instantiate_sg}, which take a signature as an argument.

 \item[\ttindexbold{Sign.stamp_names_of} $sg$]\index{signatures}

   returns the names of the identification \rmindex{stamps} of ax

   signature.  These coincide with the names of its full ancestry

   including that of $sg$ itself.

 \end{ttdescription}

 \section{Terms}

 \index{terms|bold}

 Terms belong to the \ML\ type \mltydx{term}, which is a concrete datatype

 with six constructors:

 \begin{ttbox}

 type indexname = string * int;

 infix 9 $;

 datatype term = Const of string * typ

               | Free  of string * typ

               | Var   of indexname * typ

               | Bound of int

               | Abs   of string * typ * term

               | op $  of term * term;

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{Const} ($a$, $T$)] \index{constants|bold}

   is the {\bf constant} with name~$a$ and type~$T$.  Constants include

   connectives like $\land$ and $\forall$ as well as constants like~0

   and~$Suc$.  Other constants may be required to define a logic's concrete

   syntax.

 \item[\ttindexbold{Free} ($a$, $T$)] \index{variables!free|bold}

   is the {\bf free variable} with name~$a$ and type~$T$.

 \item[\ttindexbold{Var} ($v$, $T$)] \index{unknowns|bold}

   is the {\bf scheme variable} with indexname~$v$ and type~$T$.  An

   \mltydx{indexname} is a string paired with a non-negative index, or

   subscript; a term's scheme variables can be systematically renamed by

   incrementing their subscripts.  Scheme variables are essentially free

   variables, but may be instantiated during unification.

 \item[\ttindexbold{Bound} $i$] \index{variables!bound|bold}

   is the {\bf bound variable} with de Bruijn index~$i$, which counts the

   number of lambdas, starting from zero, between a variable's occurrence

   and its binding.  The representation prevents capture of variables.  For

   more information see de Bruijn \cite{debruijn72} or

   Paulson~\cite[page~336]{paulson91}.

 \item[\ttindexbold{Abs} ($a$, $T$, $u$)]

   \index{lambda abs@$\lambda$-abstractions|bold}

   is the $\lambda$-{\bf abstraction} with body~$u$, and whose bound

   variable has name~$a$ and type~$T$.  The name is used only for parsing

   and printing; it has no logical significance.

 \item[$t$ \$ $u$] \index{$@{\tt\$}|bold} \index{function applications|bold}

 is the {\bf application} of~$t$ to~$u$.

 \end{ttdescription}

 Application is written as an infix operator to aid readability.

 Here is an \ML\ pattern to recognize \FOL{} formulae of

 the form~$A\imp B$, binding the subformulae to~$A$ and~$B$:

 \begin{ttbox}

 Const("Trueprop",_) $ (Const("op -->",_) $ A $ B)

 \end{ttbox}

 \section{*Variable binding}

 \begin{ttbox}

 loose_bnos     : term -> int list

 incr_boundvars : int -> term -> term

 abstract_over  : term*term -> term

 variant_abs    : string * typ * term -> string * term

 aconv          : term * term -> bool\hfill{\bf infix}

 \end{ttbox}

 These functions are all concerned with the de Bruijn representation of

 bound variables.

 \begin{ttdescription}

 \item[\ttindexbold{loose_bnos} $t$]

   returns the list of all dangling bound variable references.  In

   particular, {\tt Bound~0} is loose unless it is enclosed in an

   abstraction.  Similarly {\tt Bound~1} is loose unless it is enclosed in

   at least two abstractions; if enclosed in just one, the list will contain

   the number 0.  A well-formed term does not contain any loose variables.

 \item[\ttindexbold{incr_boundvars} $j$]

   increases a term's dangling bound variables by the offset~$j$.  This is

   required when moving a subterm into a context where it is enclosed by a

   different number of abstractions.  Bound variables with a matching

   abstraction are unaffected.

 \item[\ttindexbold{abstract_over} $(v,t)$]

   forms the abstraction of~$t$ over~$v$, which may be any well-formed term.

   It replaces every occurrence of \(v\) by a {\tt Bound} variable with the

   correct index.

 \item[\ttindexbold{variant_abs} $(a,T,u)$]

   substitutes into $u$, which should be the body of an abstraction.

   It replaces each occurrence of the outermost bound variable by a free

   variable.  The free variable has type~$T$ and its name is a variant

   of~$a$ chosen to be distinct from all constants and from all variables

   free in~$u$.

 \item[$t$ \ttindexbold{aconv} $u$]

   tests whether terms~$t$ and~$u$ are \(\alpha\)-convertible: identical up

   to renaming of bound variables.

 \begin{itemize}

   \item

     Two constants, {\tt Free}s, or {\tt Var}s are \(\alpha\)-convertible

     if their names and types are equal.

     (Variables having the same name but different types are thus distinct.

     This confusing situation should be avoided!)

   \item

     Two bound variables are \(\alpha\)-convertible

     if they have the same number.

   \item

     Two abstractions are \(\alpha\)-convertible

     if their bodies are, and their bound variables have the same type.

   \item

     Two applications are \(\alpha\)-convertible

     if the corresponding subterms are.

 \end{itemize}

 \end{ttdescription}

 \section{Certified terms}\index{terms!certified|bold}\index{signatures}

 A term $t$ can be {\bf certified} under a signature to ensure that every type

 in~$t$ is well-formed and every constant in~$t$ is a type instance of a

 constant declared in the signature.  The term must be well-typed and its use

 of bound variables must be well-formed.  Meta-rules such as {\tt forall_elim}

 take certified terms as arguments.

 Certified terms belong to the abstract type \mltydx{cterm}.

 Elements of the type can only be created through the certification process.

 In case of error, Isabelle raises exception~\ttindex{TERM}\@.

 \subsection{Printing terms}

 \index{terms!printing of}

 \begin{ttbox}

      string_of_cterm :           cterm -> string

 Sign.string_of_term  : Sign.sg -> term -> string

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{string_of_cterm} $ct$]

 displays $ct$ as a string.

 \item[\ttindexbold{Sign.string_of_term} $sign$ $t$]

 displays $t$ as a string, using the syntax of~$sign$.

 \end{ttdescription}

 \subsection{Making and inspecting certified terms}

 \begin{ttbox}

 cterm_of   : Sign.sg -> term -> cterm

 read_cterm : Sign.sg -> string * typ -> cterm

 cert_axm   : Sign.sg -> string * term -> string * term

 read_axm   : Sign.sg -> string * string -> string * term

 rep_cterm  : cterm -> {\ttlbrace}T:typ, t:term, sign:Sign.sg, maxidx:int\ttrbrace

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{cterm_of} $sign$ $t$] \index{signatures}

 certifies $t$ with respect to signature~$sign$.

 \item[\ttindexbold{read_cterm} $sign$ ($s$, $T$)]

 reads the string~$s$ using the syntax of~$sign$, creating a certified term.

 The term is checked to have type~$T$; this type also tells the parser what

 kind of phrase to parse.

 \item[\ttindexbold{cert_axm} $sign$ ($name$, $t$)]

 certifies $t$ with respect to $sign$ as a meta-proposition and converts all

 exceptions to an error, including the final message

 \begin{ttbox}

 The error(s) above occurred in axiom "\(name\)"

 \end{ttbox}

 \item[\ttindexbold{read_axm} $sign$ ($name$, $s$)]

 similar to {\tt cert_axm}, but first reads the string $s$ using the syntax of

 $sign$.

 \item[\ttindexbold{rep_cterm} $ct$]

 decomposes $ct$ as a record containing its type, the term itself, its

 signature, and the maximum subscript of its unknowns.  The type and maximum

 subscript are computed during certification.

 \end{ttdescription}

 \section{Types}\index{types|bold}

 Types belong to the \ML\ type \mltydx{typ}, which is a concrete datatype with

 three constructor functions.  These correspond to type constructors, free

 type variables and schematic type variables.  Types are classified by sorts,

 which are lists of classes (representing an intersection).  A class is

 represented by a string.

 \begin{ttbox}

 type class = string;

 type sort  = class list;

 datatype typ = Type  of string * typ list

              | TFree of string * sort

              | TVar  of indexname * sort;

 infixr 5 -->;

 fun S --> T = Type ("fun", [S, T]);

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{Type} ($a$, $Ts$)] \index{type constructors|bold}

   applies the {\bf type constructor} named~$a$ to the type operands~$Ts$.

   Type constructors include~\tydx{fun}, the binary function space

   constructor, as well as nullary type constructors such as~\tydx{prop}.

   Other type constructors may be introduced.  In expressions, but not in

   patterns, \hbox{\tt$S$-->$T$} is a convenient shorthand for function

   types.

 \item[\ttindexbold{TFree} ($a$, $s$)] \index{type variables|bold}

   is the {\bf type variable} with name~$a$ and sort~$s$.

 \item[\ttindexbold{TVar} ($v$, $s$)] \index{type unknowns|bold}

   is the {\bf type unknown} with indexname~$v$ and sort~$s$.

   Type unknowns are essentially free type variables, but may be

   instantiated during unification.

 \end{ttdescription}

 \section{Certified types}

 \index{types!certified|bold}

 Certified types, which are analogous to certified terms, have type

 \ttindexbold{ctyp}.

 \subsection{Printing types}

 \index{types!printing of}

 \begin{ttbox}

      string_of_ctyp :           ctyp -> string

 Sign.string_of_typ  : Sign.sg -> typ -> string

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{string_of_ctyp} $cT$]

 displays $cT$ as a string.

 \item[\ttindexbold{Sign.string_of_typ} $sign$ $T$]

 displays $T$ as a string, using the syntax of~$sign$.

 \end{ttdescription}

 \subsection{Making and inspecting certified types}

 \begin{ttbox}

 ctyp_of  : Sign.sg -> typ -> ctyp

 rep_ctyp : ctyp -> {\ttlbrace}T: typ, sign: Sign.sg\ttrbrace

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{ctyp_of} $sign$ $T$] \index{signatures}

 certifies $T$ with respect to signature~$sign$.

 \item[\ttindexbold{rep_ctyp} $cT$]

 decomposes $cT$ as a record containing the type itself and its signature.

 \end{ttdescription}

 \section{Oracles: calling trusted external reasoners}

 \label{sec:oracles}

 \index{oracles|(}

 Oracles allow Isabelle to take advantage of external reasoners such as

 arithmetic decision procedures, model checkers, fast tautology checkers or

 computer algebra systems.  Invoked as an oracle, an external reasoner can

 create arbitrary Isabelle theorems.  It is your responsibility to ensure that

 the external reasoner is as trustworthy as your application requires.

 Isabelle's proof objects~(\S\ref{sec:proofObjects}) record how each theorem

 depends upon oracle calls.

 \begin{ttbox}

 invoke_oracle     : theory -> xstring -> Sign.sg * object -> thm

 Theory.add_oracle : bstring * (Sign.sg * object -> term) -> theory -> theory

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{invoke_oracle} $thy$ $name$ ($sign$, $data$)]

   invokes the oracle $name$ of theory $thy$ passing the information

   contained in the exception value $data$ and creating a theorem

   having signature $sign$.  Note that type \ttindex{object} is just an

   abbreviation for \texttt{exn}.  Errors arise if $thy$ does not have

   an oracle called $name$, if the oracle rejects its arguments or if

   its result is ill-typed.

 \item[\ttindexbold{Theory.add_oracle} $name$ $fun$ $thy$] extends

   $thy$ by oracle $fun$ called $name$.  It is seldom called

   explicitly, as there is concrete syntax for oracles in theory files.

 \end{ttdescription}

 A curious feature of {\ML} exceptions is that they are ordinary constructors.

 The {\ML} type {\tt exn} is a datatype that can be extended at any time.  (See

 my {\em {ML} for the Working Programmer}~\cite{paulson-ml2}, especially

 page~136.)  The oracle mechanism takes advantage of this to allow an oracle to

 take any information whatever.

 There must be some way of invoking the external reasoner from \ML, either

 because it is coded in {\ML} or via an operating system interface.  Isabelle

 expects the {\ML} function to take two arguments: a signature and an

 exception object.

 \begin{itemize}

 \item The signature will typically be that of a desendant of the theory

   declaring the oracle.  The oracle will use it to distinguish constants from

   variables, etc., and it will be attached to the generated theorems.

 \item The exception is used to pass arbitrary information to the oracle.  This

   information must contain a full description of the problem to be solved by

   the external reasoner, including any additional information that might be

   required.  The oracle may raise the exception to indicate that it cannot

   solve the specified problem.

 \end{itemize}

 A trivial example is provided in theory {\tt FOL/ex/IffOracle}.  This

 oracle generates tautologies of the form $P\bimp\cdots\bimp P$, with

 an even number of $P$s.

 The \texttt{ML} section of \texttt{IffOracle.thy} begins by declaring

 a few auxiliary functions (suppressed below) for creating the

 tautologies.  Then it declares a new exception constructor for the

 information required by the oracle: here, just an integer. It finally

 defines the oracle function itself.

 \begin{ttbox}

 exception IffOracleExn of int;\medskip

 fun mk_iff_oracle (sign, IffOracleExn n) =

   if n > 0 andalso n mod 2 = 0

   then Trueprop $ mk_iff n

   else raise IffOracleExn n;

 \end{ttbox}

 Observe the function's two arguments, the signature {\tt sign} and the

 exception given as a pattern.  The function checks its argument for

 validity.  If $n$ is positive and even then it creates a tautology

 containing $n$ occurrences of~$P$.  Otherwise it signals error by

 raising its own exception (just by happy coincidence).  Errors may be

 signalled by other means, such as returning the theorem {\tt True}.

 Please ensure that the oracle's result is correctly typed; Isabelle

 will reject ill-typed theorems by raising a cryptic exception at top

 level.

 The \texttt{oracle} section of {\tt IffOracle.thy} installs above

 \texttt{ML} function as follows:

 \begin{ttbox}

 IffOracle = FOL +\medskip

 oracle

   iff = mk_iff_oracle\medskip

 end

 \end{ttbox}

 Now in \texttt{IffOracle.ML} we first define a wrapper for invoking

 the oracle:

 \begin{ttbox}

 fun iff_oracle n =

   invoke_oracle IffOracle.thy "iff" (sign_of IffOracle.thy, IffOracleExn n);

 \end{ttbox}

 Here are some example applications of the \texttt{iff} oracle.  An

 argument of 10 is allowed, but one of 5 is forbidden:

 \begin{ttbox}

 iff_oracle 10;

 {\out  "P <-> P <-> P <-> P <-> P <-> P <-> P <-> P <-> P <-> P" : thm}

 iff_oracle 5;

 {\out Exception- IffOracleExn 5 raised}

 \end{ttbox}

 \index{oracles|)}

 \index{theories|)}