%% $Id$

\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[$nonterminals$]\index{*nonterminal symbols} declares purely

  syntactic types to be used as nonterminal symbols of the context

  free grammar.

\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[$local$, $global$] change the current name declaration mode.

  Initially, theories start in $local$ mode, causing all names of

  types, constants, axioms etc.\ to be automatically qualified by the

  theory name.  Changing this to $global$ causes all names to be

  declared as short base names only.

  The $local$ and $global$ declarations act like switches, affecting

  all following theory sections until changed again explicitly.  Also

  note that the final state at the end of the theory will persist.  In

  particular, this determines how the names of theorems stored later

  on are handled.

\item[$setup$]\index{*setup!theory} applies a list of ML functions to

  the theory.  The argument should denote a value of type

  \texttt{(theory -> theory) list}.  Typically, ML packages are

  initialized in this way.

\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

\texttt{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.

\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

\texttt{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{*Theory inclusion}

\begin{ttbox}

subthy      : theory * theory -> bool

eq_thy      : theory * theory -> bool

transfer    : theory -> thm -> thm

transfer_sg : Sign.sg -> thm -> thm

\end{ttbox}

Inclusion and equality of theories is determined by unique

identification stamps that are created when declaring new components.

Theorems contain a reference to the theory (actually to its signature)

they have been derived in.  Transferring theorems to super theories

has no logical significance, but may affect some operations in subtle

ways (e.g.\ implicit merges of signatures when applying rules, or

pretty printing of theorems).

\begin{ttdescription}

\item[\ttindexbold{subthy} ($thy@1$, $thy@2$)] determines if $thy@1$

  is included in $thy@2$ wrt.\ identification stamps.

\item[\ttindexbold{eq_thy} ($thy@1$, $thy@2$)] determines if $thy@1$

  is exactly the same as $thy@2$.

\item[\ttindexbold{transfer} $thy$ $thm$] transfers theorem $thm$ to

  theory $thy$, provided the latter includes the theory of $thm$.

\item[\ttindexbold{transfer_sg} $sign$ $thm$] is similar to

  \texttt{transfer}, but identifies the super theory via its

  signature.

\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

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{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

Sign.certify_term : Sign.sg -> term -> term * typ * 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.

\item[\ttindexbold{Sign.certify_term}] is a more primitive version of

  \texttt{cterm_of}, returning the internal representation instead of

  an abstract \texttt{cterm}.

\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

Sign.certify_typ : Sign.sg -> typ -> typ

\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.

\item[\ttindexbold{Sign.certify_typ}] is a more primitive version of

  \texttt{ctyp_of}, returning the internal representation instead of

  an abstract \texttt{ctyp}.

\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|)}

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