%% $Id$

 \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 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 classes, etc.  The basic theory, which contains only

   the meta-logic, is called \thydx{Pure}.

   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 singleton

   set {\tt\{}$id${\tt\}}.

 \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

   $(\alpha@1,\dots,\alpha@n)name = \mbox{\tt"}\tau\mbox{\tt"}$, where

   $name$ can be a string and $\tau$ must be enclosed in quotation marks.

 \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 expresses binary product types.

 \item[$arities$]

   is a series of arity declarations.  Each assigns arities to type

   constructors.  The $name$ must be an existing type constructor, which is

   given the additional arity $arity$.

 \item[$constDecl$]

   is a series of constant declarations.  Each new constant $name$ is given

   the type specified by the $string$.  The optional $mixfix$ annotations may

   attach concrete syntax to the constant. A variant of {\tt consts} is the

   {\tt syntax} section\index{*syntax section}, which adds just syntax without

   declaring logical constants.

 \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[$rule$]

   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 file.

 \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{*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 be no two declarations $ty :: (\vec{r})c$ and $ty ::

   (\vec{s})c$ with $\vec{r} \neq \vec{s}$.  For example, the following is

   forbidden:

 \begin{ttbox}

 types

   'a ty

 arities

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

   ty :: ({\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 $s$.  For example, the following is forbidden:

 \begin{ttbox}

 classes

   term < logic

 types

   'a ty

 arities

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

   ty :: ({\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[\ttindexbold{delete_tmpfiles} := false;]

 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 \ttindexbold{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 to {\tt

 true} and no errors occurred.

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

 begins with the declaration {\tt open~$T$} and contains proofs involving

 the new theory.

 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}

 \goodbreak

 \subsection{Important note for Poly/ML users}\index{Poly/{\ML} compiler}

 The theory mechanism depends upon reference variables.  At the end of a

 Poly/\ML{} session, the contents of references are lost unless they are

 declared in the current database.  In particular, assignments to references

 of the {\tt Pure} database are lost, including all information about loaded

 theories. To avoid losing this information simply call

 \begin{ttbox}

 init_thy_reader();

 \end{ttbox}

 when building the new database.

 \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, consult the

 sources of \ZF{} set theory.

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

 \subsection{Extracting an axiom or theorem from a theory}

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

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

 \begin{ttbox}

 get_axiom : theory -> string -> thm

 get_thm   : theory -> string -> thm

 assume_ax : theory -> string -> thm

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{get_axiom} $thy$ $name$]

   returns an axiom with the given $name$ from $thy$, 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.

 \item[\ttindexbold{get_thm} $thy$ $name$]

   is analogous to {\tt get_axiom}, but looks for a stored theorem. Like

   {\tt get_axiom} it searches all parents of a theory if the theorem

   is not associated with $thy$.

 \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{result}

   complains 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}

 pure_thy       : theory

 merge_theories : theory * theory -> theory

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{pure_thy}] contains just the syntax and signature

   of the meta-logic.  There are no axioms: meta-level inferences are carried

   out by \ML\ functions.

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

   theories $thy@1$ and $thy@2$.  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 incorrect axiom --- 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{ttdescription}

 %% 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_theory  : theory -> unit

 axioms_of     : theory -> (string * thm) list

 thms_of       : theory -> (string * thm) list

 parents_of    : theory -> theory list

 sign_of       : theory -> Sign.sg

 stamps_of_thy : theory -> string ref list

 \end{ttbox}

 These provide means of viewing a theory's components.

 \begin{ttdescription}

 \item[\ttindexbold{print_theory} $thy$]

 prints the contents of $thy$ excluding the syntax related parts (which are

 shown by {\tt print_syntax}).  The output is quite verbose.

 \item[\ttindexbold{axioms_of} $thy$]

 returns the additional axioms of the most recent extend node of~$thy$.

 \item[\ttindexbold{thms_of} $thy$]

 returns all theorems that are associated with $thy$.

 \item[\ttindexbold{parents_of} $thy$]

 returns the direct ancestors of~$thy$.

 \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{stamps_of_thy} $thy$]\index{signatures}

 returns the identification \rmindex{stamps} of the signature associated

 with~$thy$.

 \end{ttdescription}

 \section{Viewing theories as HTML documents}

 \index{HTML|bold} 

 Isabelle is able to make HTML documents that show a theory's

 definition, the theorems proved in its ML file and the relationship

 with its ancestors and descendants. HTML stands for Hypertext Markup

 Language and is used in the World Wide Web to represent text

 containing images and links to other documents. Web browsers like the

 ones from Mosaic or Netscape are used to view these documents.

 Besides the three HTML files that are made for every theory

 (definition and theorems, ancestors, descendants), Isabelle stores

 links to all theories in an index file. These indexes are themself

 linked with other indexes.

 To make HTML files for logics that are part of the Isabelle

 distribution, simply set the environment variable {\tt MAKE_HTML}

 before compiling a logic. The entry point to all logics is the {\tt

 index.html} file located in Isabelle's main directory. You also can

 access a HTML version of the Isabelle distribution package at

 \begin{ttbox}

 http://www4.informatik.tu-muenchen.de/~nipkow/isabelle

 \end{ttbox}

 Internally the HTML generation is controlled by

 \begin{ttbox}

 index_path  : string ref

 gif_path    : string ref

 base_path   : string ref

 init_html   : unit -> unit

 make_html   : bool ref

 finish_html : unit -> unit

 \end{ttbox}

 \begin{ttdescription}

 \item[\ttindexbold{base_path}]

 contains the absolute path of Isabelle's main directory. To make them

 independent from their storage place, the HTML files only contain

 relative paths which are derived from absolute ones like the current

 working directory, {\tt index_path} or {\tt base_path}.

 As {\tt base_path} and {\tt gif_path} are set at the time when the

 {\tt Pure} database is made, they are not valid if you use someone

 else's database to read theories stored in your private directory. In

 that case you first have to set {\tt base_path} to the value of {\em

 your} Isabelle main directory, i.e. the directory that contains the

 subdirectories where logics like {\tt FOL}, {\tt HOL} etc. are

 stored. Besides you have to set the next variable:

 \item[\ttindexbold{gif_path}]

 contains the absolute path of two GIF images used in the HTML

 documents. Normally it points to the {\tt Tools} subdirectory of

 Isabelle's main directory. As with {\tt base_path} you have to set it

 manually if you use someone else's database.

 \item[\ttindexbold{init_html}]

 activates the HTML facility. It stores the current working directory

 in {\tt index_path} which is were the {\tt index.html} file for all

 theories loaded afterwards will be placed.

 \item[\ttindexbold{make_html}]

 is a variable read by {\tt use_thy} to decide whether HTML files

 should be made or not. After you have used {\tt init_html} you can

 manually change {\tt make_html}'s value to temporarily disable HTML

 generation.

 \item[\ttindexbold{finish_html}]

 has to be called after all theories have been read that should be

 contained in the current {\tt index.html} file. It reads a temporary

 file with information about the theories read since the last use of

 {\tt init_html} and makes the {\tt index.html} file. If {\tt

 make_html} is {\tt false} nothing is done.

 The indexes made by this function also contain a link to the {\tt

 README} file if there exists one in the directory were the index is

 stored. If there's a {\tt README.html} file it's used instead of the

 {\tt README} file.

 \end{ttdescription}

 Please note that the HTML facility was developed to make HTML

 documents for a stable version of a logic. It is not intended to make

 these documents for a logic were theories are changed and only a part

 of the logic is reread.

 If you add new subdirectories to Isabelle's logics (or add a completly

 new logic), you would have to call {\tt init_html} at the start of the

 directory's {\tt ROOT.ML} file and {\tt finish_html} at the end of

 it. This is automatically done if you use

 \begin{ttbox}

 use_dir : string -> unit

 \end{ttbox}

 This function takes a path as its parameter, changes the working

 directory, calls {\tt init_html} if {\tt make_html} is {\tt true},

 executes {\tt ROOT.ML}, and calls {\tt finish_html}. The {\tt

 index.html} file written in this directory will be automatically

 linked to the first index found in the (recursively searched)

 superdirectories.

 \section{Terms}

 \index{terms|bold}

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

 with six constructors: there are six kinds of term.

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

 \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 -> \{T: typ, sign: Sign.sg\}

 \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}

 \index{theories|)}