%% $Id$

\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

  $(\alpha@1,\dots,\alpha@n)name = \tau$, 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 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 specified type.  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[$rules$]

  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 file.

\item[$defs$]

  is a series of definitions.  Just like $rules$, except that every $string$

  must be a definition.

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

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

\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{*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{Generating 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

{\tt xmosaic} or {\tt 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 represent the hierarchic structure of

Isabelle's logics.

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

distribution, simply set the shell environment variable {\tt

MAKE_HTML} before compiling a logic. This works for single logics as

well as for the shell script {\tt make-all} (see

\ref{sec:shell-scripts}). To make HTML files for {\tt FOL} using a

{\tt csh} style shell, the following commands can be used:

\begin{ttbox}

cd FOL

setenv MAKE_HTML

make

\end{ttbox}

The databases made this way do not differ from the ones made with an

unset {\tt MAKE_HTML}; in particular no HTML files are written if the

database is used to manually load a theory.

As you will see below, the HTML generation is controlled by a boolean

reference variable. If you want to make databases which define this

variable's value as {\tt true} and where therefore HTML files are

written each time {\tt use_thy} is invoked, you have to set {\tt

MAKE_HTML} to ``{\tt true}'':

\begin{ttbox}

cd FOL

setenv MAKE_HTML true

make

\end{ttbox}

All theories loaded from within the {\tt FOL} database and all

databases derived from it will now cause HTML files to be written.

This behaviour can be changed by assigning a value of {\tt false} to

the boolean reference variable {\tt make_html}. Be careful when making

such databases publicly available since it means that your users will

generate HTML files though they might not intend to do so.

As some of Isabelle's logics are based on others (e.g. {\tt ZF} on

{\tt FOL}) and because the HTML files list a theory's ancestors, you

should not make HTML files for a logic if the HTML files for the base

logic do not exist. Otherwise some of the hypertext links might point

to non-existing documents.

The entry point to all logics is the {\tt index.html} file located in

Isabelle's main directory. You can also access a HTML version of the

distribution package at

\begin{ttbox}

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

\end{ttbox}

\subsection*{Manual HTML generation}

To manually control the generation of HTML files the following

commands and reference variables are used:

\begin{ttbox}

init_html   : unit -> unit

make_html   : bool ref

finish_html : unit -> unit

\end{ttbox}

\begin{ttdescription}

\item[\ttindexbold{init_html}]

activates the HTML facility. It stores the current working directory

as the place where the {\tt index.html} file for all theories loaded

afterwards will be stored.

\item[\ttindexbold{make_html}]

is a boolean reference variable read by {\tt use_thy} and other

functions to decide whether HTML files should be made. 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

listed 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 where the index is

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

{\tt README}.

\end{ttdescription}

The above functions could be used in the following way:

\begin{ttbox}

init_html();

{\out Setting path for index.html to "/home/clasohm/isabelle/HOL"}

use_thy "List";

\dots

finish_html();

\end{ttbox}

Please note that HTML files are made only for those theories that are

read while {\tt make_html} is {\tt true}. These files may contain

links to theories that were read with a {\tt false} {\tt make_html}

and therefore point to non-existing files.

\subsection*{Extending or adding a logic}

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

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

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

it. This is automatically done if you use

\begin{ttbox}\index{use_dir}

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.

Instead of adding something like

\begin{ttbox}

use"ex/ROOT.ML";

\end{ttbox}

to the logic's makefile you have to use this:

\begin{ttbox}

use_dir"ex";

\end{ttbox}

Since {\tt use_dir} calls {\tt init_html} only if {\tt make_html} is

{\tt true} the generation of HTML files depends on the value this

reference variable has. It can either be inherited from the used \ML{}

database or set in the makefile before {\tt use_dir} is invoked,

e.g. to set it's value according to the environment variable {\tt

MAKE_HTML}.

The generated HTML files contain all theorems that were proved in the

theory's \ML{} file with {\tt qed}, {\tt qed_goal} and {\tt qed_goalw},

or stored with {\tt bind_thm} and {\tt store_thm}. Additionally there

is a hypertext link to the whole \ML{} file.

You can add section headings to the list of theorems by using

\begin{ttbox}\index{use_dir}

section: string -> unit

\end{ttbox}

in a theory's ML file, which converts a plain string to a HTML

heading and inserts it before the next theorem proved or stored with

one of the above functions. If {\tt make_html} is {\tt false} nothing

is done.

\subsection*{Using someone else's database}

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 gif_path} or {\tt base_path}. The

latter two are reference variables which are initialized at the time

when the {\tt Pure} database is made. Because you need write access

for the current directory to make HTML files and therefore (probably)

generate them in your home directory, the absolute {\tt base_path} is

not correct if you use someone else's database or a database derived

from it.

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

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

subdirectories where standard logics like {\tt FOL} and {\tt HOL} or

your own logics are stored. If you do not do this, the generated HTML

files will still be usable but may contain incomplete titles and lack

some hypertext links.

It's also a good idea to set {\tt gif_path} which points to the

directory containing two GIF images used in the HTML

documents. Normally this is the {\tt Tools} subdirectory of Isabelle's

main directory. While its value in general is still valid, your HTML

files would depend on files not owned by you. This prevents you from

changing the location of the HTML files (as they contain relative

paths) and also causes trouble if the database's maker (re)moves the

GIFs.

Here's what you should do before invoking {\tt init_html} using

someone else's \ML{} database:

\begin{ttbox}

base_path := "/home/smith/isabelle";

gif_path := "/home/smith/isabelle/Tools";

init_html();

\dots

\end{ttbox}

\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}

\section{Oracles: calling 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 * Sign.sg * exn -> thm

     set_oracle    : Sign.sg -> typ -> string

\end{ttbox}

\begin{ttdescription}

\item[\ttindexbold{invoke_oracle} ($thy$, $sign$, $exn$)] invokes the oracle

  of theory $thy$ passing the information contained in the exception value

  $exn$ and creating a theorem having signature $sign$.  Errors arise if $thy$

  does not have an oracle, if the oracle rejects its arguments or if its

  result is ill-typed.

\item[\ttindexbold{set_oracle} $fn$ $thy$] sets the oracle of theory $thy$ to

  be $fn$.  It is seldom called explicitly, as there is syntax for oracles in

  theory files.  A theory can have at most one oracle.

\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.

\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 on directory {\tt FOL/ex}.  This oracle

generates tautologies of the form $P\bimp\cdots\bimp P$, with an even number

of $P$s. 

File {\tt declIffOracle.ML} begins by declaring a new exception constructor

for the oracle the information it requires: here, just an integer.  It

contains some code (suppressed below) for creating the tautologies, and

finally declares the oracle function itself:

\begin{ttbox}

exception IffOracleExn of int;

\(\vdots\)

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 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.  Errors may