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\chapter{Theories, Terms and Types} \label{theories} 
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\index{theories(}\index{signaturesbold} 

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\index{reading!axiomssee{\texttt{assume_ax}}} Theories organize the syntax, 
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declarations and axioms of a mathematical development. They are built, 

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starting from the Pure or CPure theory, by extending and merging existing 

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theories. They have the \ML\ type \mltydx{theory}. Theory operations signal 

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errors by raising exception \xdx{THEORY}, returning a message and a list of 

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theories. 

104  12 

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

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syntax, have the \ML\ type~\mltydx{Sign.sg}. For identification, each 
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signature carries a unique list of \bfindex{stamps}, which are \ML\ 
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references to strings. The strings serve as humanreadable names; the 
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references serve as unique identifiers. Each primitive signature has a 
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single stamp. When two signatures are merged, their lists of stamps are 

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also merged. Every theory carries a unique signature. 

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

275  22 
\ML\ definitions are irrelevant to naive Isabelle users. Programmers who 
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wish to extend Isabelle may need to know such details, say to code a tactic 

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

104  25 
`certified' to be wellformed with respect to a given signature. 
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324  27 

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\section{Defining theories}\label{sec:refdefiningtheories} 

104  29 

6571  30 
Theories are defined via theory files $name$\texttt{.thy} (there are also 
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\MLlevel interfaces which are only intended for people building advanced 

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theory definition packages). Appendix~\ref{app:TheorySyntax} presents the 

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concrete syntax for theory files; here follows an explanation of the 

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constituent parts. 

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\begin{description} 
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\item[{\it theoryDef}] is the full definition. The new theory is called $id$. 
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It is the union of the named \textbf{parent 
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theories}\indexbold{theories!parent}, possibly extended with new 
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components. \thydx{Pure} and \thydx{CPure} are the basic theories, which 
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contain only the metalogic. They differ just in their concrete syntax for 

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function applications. 

6571  42 

43 
The new theory begins as a merge of its parents. 

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\begin{ttbox} 

45 
Attempt to merge different versions of theories: "\(T@1\)", \(\ldots\), "\(T@n\)" 

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\end{ttbox} 

47 
This error may especially occur when a theory is redeclared  say to 

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change an inappropriate definition  and bindings to old versions persist. 

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Isabelle ensures that old and new theories of the same name are not involved 

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in a proof. 

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\item[$classes$] 
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is a series of class declarations. Declaring {\tt$id$ < $id@1$ \dots\ 
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$id@n$} makes $id$ a subclass of the existing classes $id@1\dots 
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id@n$. This rules out cyclic class structures. Isabelle automatically 

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computes the transitive closure of subclass hierarchies; it is not 

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necessary to declare \texttt{c < e} in addition to \texttt{c < d} and \texttt{d < 
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e}. 
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\item[$default$] 
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introduces $sort$ as the new default sort for type variables. This applies 
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to unconstrained type variables in an input string but not to type 
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variables created internally. If omitted, the default sort is the listwise 
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union of the default sorts of the parent theories (i.e.\ their logical 
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intersection). 
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\item[$sort$] is a finite set of classes. A single class $id$ abbreviates the 
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sort $\{id\}$. 

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\item[$types$] 
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is a series of type declarations. Each declares a new type constructor 
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or type synonym. An $n$place type constructor is specified by 

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

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indicate the number~$n$. 

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A \textbf{type synonym}\indexbold{type synonyms} is an abbreviation 
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$(\alpha@1,\dots,\alpha@n)name = \tau$, where $name$ and $\tau$ can 
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be strings. 

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\item[$infix$] 
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declares a type or constant to be an infix operator having priority $nat$ 
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and associating to the left (\texttt{infixl}) or right (\texttt{infixr}). 

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

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('a,'b)~"*"~(infixr~20)}, which may express binary product types. 
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\item[$arities$] is a series of type arity declarations. Each assigns 
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arities to type constructors. The $name$ must be an existing type 

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

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\item[$nonterminals$]\index{*nonterminal symbols} declares purely 
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syntactic types to be used as nonterminal symbols of the context 

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free grammar. 

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\item[$consts$] is a series of constant declarations. Each new 
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constant $name$ is given the specified type. The optional $mixfix$ 

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annotations may attach concrete syntax to the constant. 

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

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of $consts$ which adds just syntax without actually declaring 

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

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free grammar. The optional $mode$ specifies the print mode where the 
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mixfix productions should be added. If there is no \texttt{output} 
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option given, all productions are also added to the input syntax 
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(regardless of the print mode). 

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

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annotations can take three forms: 

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\begin{itemize} 
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\item A mixfix template given as a $string$ of the form 

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

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indicates the position where the $i$th argument should go. The list 
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of numbers gives the priority of each argument. The final number gives 

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the priority of the whole construct. 

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

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\item A constant $f$ of type $(\tau@1\To\tau@2)\To\tau$ can be given {\bf 
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binder} status. The declaration \texttt{binder} $\cal Q$ $p$ causes 
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${\cal Q}\,x.F(x)$ to be treated 
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like $f(F)$, where $p$ is the priority. 

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\end{itemize} 
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\item[$trans$] 
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specifies syntactic translation rules (macros). There are three forms: 
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parse rules (\texttt{=>}), print rules (\texttt{<=}), and parse/print rules ({\tt 
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==}). 
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\item[$rules$] 
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is a series of rule declarations. Each has a name $id$ and the formula is 
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given by the $string$. Rule names must be distinct within any single 
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theory. 
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\item[$defs$] is a series of definitions. They are just like $rules$, except 
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that every $string$ must be a definition (see below for details). 

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

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definition. The first $string$ is the type, the second the definition. 
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\item[$axclass$] \index{*axclass section} defines an \rmindex{axiomatic type 
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class} \cite{Wenzel:1997:TPHOL} as the intersection of existing classes, 

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with additional axioms holding. Class axioms may not contain more than one 

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type variable. The class axioms (with implicit sort constraints added) are 

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bound to the given names. Furthermore a class introduction rule is 

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generated, which is automatically employed by $instance$ to prove 

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instantiations of this class. 

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

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

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type signature. The instantiation is proven and checked properly. 
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The user has to supply sufficient witness information: theorems 
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($longident$), axioms ($string$), or even arbitrary \ML{} tactic 

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code $verbatim$. 

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\item[$oracle$] links the theory to a trusted external reasoner. It is 
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allowed to create theorems, but each theorem carries a proof object 

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

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\item[$local$, $global$] change the current name declaration mode. 
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Initially, theories start in $local$ mode, causing all names of 
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types, constants, axioms etc.\ to be automatically qualified by the 

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

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declared as short base names only. 

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

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all following theory sections until changed again explicitly. Also 

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

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particular, this determines how the names of theorems stored later 

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on are handled. 

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

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the theory. The argument should denote a value of type 

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

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initialized in this way. 

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\item[$ml$] \index{*ML section} 
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consists of \ML\ code, typically for parse and print translation functions. 
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\end{description} 
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Chapters~\ref{DefiningLogics} and \ref{chap:syntax} explain mixfix 
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declarations, translation rules and the \texttt{ML} section in more detail. 
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\subsection{Definitions}\indexbold{definitions} 
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\textbf{Definitions} are intended to express abbreviations. The simplest 
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form of a definition is $f \equiv t$, where $f$ is a constant. 
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Isabelle also allows a derived forms where the arguments of~$f$ appear 

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

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191 
Isabelle makes the following checks on definitions: 

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\begin{itemize} 

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\item Arguments (on the lefthand side) must be distinct variables. 
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\item All variables on the righthand side must also appear on the lefthand 
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side. 

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\item All type variables on the righthand side must also appear on 
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the lefthand side; this prohibits definitions such as {\tt 

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

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\item The definition must not be recursive. Most objectlogics provide 
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definitional principles that can be used to express recursion safely. 

201 
\end{itemize} 

202 
These checks are intended to catch the sort of errors that might be made 

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accidentally. Misspellings, for instance, might result in additional 

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variables appearing on the righthand side. More elaborate checks could be 

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

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207 

275  208 
\subsection{*Classes and arities} 
324  209 
\index{classes!context conditions}\index{arities!context conditions} 
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286  211 
In order to guarantee principal types~\cite{nipkowprehofer}, 
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arity declarations must obey two conditions: 
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\begin{itemize} 
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\item There must not be any two declarations $ty :: (\vec{r})c$ and 
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$ty :: (\vec{s})c$ with $\vec{r} \neq \vec{s}$. For example, this 

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excludes the following: 

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\begin{ttbox} 
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arities 
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foo :: (\{logic{\}}) logic 
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foo :: (\{{\}})logic 

145  221 
\end{ttbox} 
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145  223 
\item If there are two declarations $ty :: (s@1,\dots,s@n)c$ and $ty :: 
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(s@1',\dots,s@n')c'$ such that $c' < c$ then $s@i' \preceq s@i$ must hold 

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

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

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expresses that the set of types represented by $s'$ is a subset of the 
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set of types represented by $s$. Assuming $term \preceq logic$, the 

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following is forbidden: 

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\begin{ttbox} 
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arities 
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foo :: (\{logic{\}})logic 
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foo :: (\{{\}})term 

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\end{ttbox} 
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145  236 
\end{itemize} 
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\section{The theory loader}\label{sec:moretheories} 
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\index{theories!reading}\index{files!reading} 

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Isabelle's theory loader manages dependencies of the internal graph of theory 

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nodes (the \emph{theory database}) and the external view of the file system. 

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See \S\ref{sec:introtheories} for its most basic commands, such as 

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\texttt{use_thy}. There are a few more operations available. 

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\begin{ttbox} 
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use_thy_only : string > unit 
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update_thy_only : string > unit 
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touch_thy : string > unit 
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remove_thy : string > unit 
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delete_tmpfiles : bool ref \hfill\textbf{initially true} 
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\end{ttbox} 
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324  255 
\begin{ttdescription} 
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\item[\ttindexbold{use_thy_only} "$name$";] is similar to \texttt{use_thy}, 
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but processes the actual theory file $name$\texttt{.thy} only, ignoring 

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$name$\texttt{.ML}. This might be useful in replaying proof scripts by hand 
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from the very beginning, starting with the fresh theory. 

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\item[\ttindexbold{update_thy_only} "$name$";] is similar to 
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\texttt{update_thy}, but processes the actual theory file 

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$name$\texttt{.thy} only, ignoring $name$\texttt{.ML}. 

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\item[\ttindexbold{touch_thy} "$name$";] marks theory node $name$ of the 
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internal graph as outdated. While the theory remains usable, subsequent 
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operations such as \texttt{use_thy} may cause a reload. 

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\item[\ttindexbold{remove_thy} "$name$";] deletes theory node $name$, 
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including \emph{all} of its descendants. Beware! This is a quick way to 

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dispose a large number of theories at once. Note that {\ML} bindings to 

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theorems etc.\ of removed theories may still persist. 

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\item[reset \ttindexbold{delete_tmpfiles};] processing theory files usually 
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involves temporary {\ML} files to be created. By default, these are deleted 

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afterwards. Resetting the \texttt{delete_tmpfiles} flag inhibits this, 

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leaving the generated code for debugging purposes. The basic location for 

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temporary files is determined by the \texttt{ISABELLE_TMP} environment 

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variable (which is private to the running Isabelle process and may be 
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retrieved by \ttindex{getenv} from {\ML}). 
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\end{ttdescription} 
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\medskip Theory and {\ML} files are located by skimming through the 
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directories listed in Isabelle's internal load path, which merely contains the 

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current directory ``\texttt{.}'' by default. The load path may be accessed by 

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the following operations. 

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\begin{ttbox} 
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show_path: unit > string list 
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add_path: string > unit 

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del_path: string > unit 

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reset_path: unit > unit 

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with_path: string > ('a > 'b) > 'a > 'b 
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no_document: ('a > 'b) > 'a > 'b 
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\end{ttbox} 
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\begin{ttdescription} 
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\item[\ttindexbold{show_path}();] displays the load path components in 
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canonical string representation (which is always according to Unix rules). 
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\item[\ttindexbold{add_path} "$dir$";] adds component $dir$ to the beginning 
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of the load path. 

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\item[\ttindexbold{del_path} "$dir$";] removes any occurrences of component 
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$dir$ from the load path. 
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\item[\ttindexbold{reset_path}();] resets the load path to ``\texttt{.}'' 

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(current directory) only. 

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\item[\ttindexbold{with_path} "$dir$" $f$ $x$;] temporarily adds component 

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$dir$ to the beginning of the load path while executing $(f~x)$. 
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\item[\ttindexbold{no_document} $f$ $x$;] temporarily disables {\LaTeX} 

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document generation while executing $(f~x)$. 

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324  316 
\end{ttdescription} 
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Furthermore, in operations referring indirectly to some file (e.g.\ 
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\texttt{use_dir}) the argument may be prefixed by a directory that will be 

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temporarily appended to the load path, too. 

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\section{Locales} 
324 
\label{Locales} 

325 

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Locales \cite{kammuellerlocales} are a concept of local proof contexts. They 

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are introduced as named syntactic objects within theories and can be 

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opened in any descendant theory. 

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330 
\subsection{Declaring Locales} 

331 

332 
A locale is declared in a theory section that starts with the 

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keyword \texttt{locale}. It consists typically of three parts, the 

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\texttt{fixes} part, the \texttt{assumes} part, and the \texttt{defines} part. 

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Appendix \ref{app:TheorySyntax} presents the full syntax. 

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337 
\subsubsection{Parts of Locales} 

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339 
The subsection introduced by the keyword \texttt{fixes} declares the locale 

340 
constants in a way that closely resembles a global \texttt{consts} 

341 
declaration. In particular, there may be an optional pretty printing syntax 

342 
for the locale constants. 

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The subsequent \texttt{assumes} part specifies the locale rules. They are 

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defined like \texttt{rules}: by an identifier followed by the rule 

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given as a string. Locale rules admit the statement of local assumptions 

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about the locale constants. The \texttt{assumes} part is optional. Nonfixed 

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variables in locale rules are automatically bound by the universal quantifier 

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\texttt{!!} of the metalogic. 

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Finally, the \texttt{defines} part introduces the definitions that are 

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available in the locale. Locale constants declared in the \texttt{fixes} 

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section are defined using the metaequality \texttt{==}. If the 

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locale constant is a functiond then its definition can (as usual) have 

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variables on the lefthand side acting as formal parameters; they are 

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considered as schematic variables and are automatically generalized by 

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universal quantification of the metalogic. The right hand side of a 

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definition must not contain variables that are not already on the left hand 

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side. In so far locale definitions behave like theory level definitions. 

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However, the locale concept realizes \emph{dependent definitions}: any variable 

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that is fixed as a locale constant can occur on the right hand side of 

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definitions. For an illustration of these dependent definitions see the 

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occurrence of the locale constant \texttt{G} on the right hand side of the 

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definitions of the locale \texttt{group} below. Naturally, definitions can 

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already use the syntax of the locale constants in the \texttt{fixes} 

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subsection. The \texttt{defines} part is, as the \texttt{assumes} part, 

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optional. 

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369 
\subsubsection{Example for Definition} 

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The concrete syntax of locale definitions is demonstrated by example below. 

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372 
Locale \texttt{group} assumes the definition of groups in a theory 

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file\footnote{This and other examples are from \texttt{HOL/ex}.}. A locale 

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defining a convenient proof environment for group related proofs may be 

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added to the theory as follows: 

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\begin{ttbox} 

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locale group = 

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fixes 

379 
G :: "'a grouptype" 

380 
e :: "'a" 

381 
binop :: "'a => 'a => 'a" (infixr "#" 80) 

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inv :: "'a => 'a" ("i(_)" [90] 91) 

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assumes 

384 
Group_G "G: Group" 

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defines 

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e_def "e == unit G" 

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binop_def "x # y == bin_op G x y" 

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inv_def "i(x) == inverse G x" 

389 
\end{ttbox} 

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391 
\subsubsection{Polymorphism} 

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In contrast to polymorphic definitions in theories, the use of the 

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same type variable for the declaration of different locale constants in the 

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fixes part means \emph{the same} type. In other words, the scope of the 

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polymorphic variables is extended over all constant declarations of a locale. 

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In the above example \texttt{'a} refers to the same type which is fixed inside 

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the locale. In an exported theorem (see \S\ref{sec:localeexport}) the 

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constructors of locale \texttt{group} are polymorphic, yet only simultaneously 

400 
instantiatable. 

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402 
\subsubsection{Nested Locales} 

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A locale can be defined as the extension of a previously defined 

405 
locale. This operation of extension is optional and is syntactically 

406 
expressed as 

407 
\begin{ttbox} 

408 
locale foo = bar + ... 

409 
\end{ttbox} 

410 
The locale \texttt{foo} builds on the constants and syntax of the locale {\tt 

411 
bar}. That is, all contents of the locale \texttt{bar} can be used in 

412 
definitions and rules of the corresponding parts of the locale {\tt 

413 
foo}. Although locale \texttt{foo} assumes the \texttt{fixes} part of \texttt{bar} it 

414 
does not automatically subsume its rules and definitions. Normally, one 

415 
expects to use locale \texttt{foo} only if locale \texttt{bar} is already 

416 
active. These aspects of use and activation of locales are considered in the 

417 
subsequent section. 

418 

419 

420 
\subsection{Locale Scope} 

421 

422 
Locales are by default inactive, but they can be invoked. The list of 

423 
currently active locales is called \emph{scope}. The process of activating 

424 
them is called \emph{opening}; the reverse is \emph{closing}. 

425 

426 
\subsubsection{Scope} 

427 
The locale scope is part of each theory. It is a dynamic stack containing 

428 
all active locales at a certain point in an interactive session. 

429 
The scope lives until all locales are explicitly closed. At one time there 

430 
can be more than one locale open. The contents of these various active 

431 
locales are all visible in the scope. In case of nested locales for example, 

432 
the nesting is actually reflected to the scope, which contains the nested 

433 
locales as layers. To check the state of the scope during a development the 

434 
function \texttt{Print\_scope} may be used. It displays the names of all open 

435 
locales on the scope. The function \texttt{print\_locales} applied to a theory 

436 
displays all locales contained in that theory and in addition also the 

437 
current scope. 

438 

439 
The scope is manipulated by the commands for opening and closing of locales. 

440 

441 
\subsubsection{Opening} 

442 
Locales can be \emph{opened} at any point during a session where 

443 
we want to prove theorems concerning the locale. Opening a locale means 

444 
making its contents visible by pushing it onto the scope of the current 

445 
theory. Inside a scope of opened locales, theorems can use all definitions and 

446 
rules contained in the locales on the scope. The rules and definitions may 

447 
be accessed individually using the function \ttindex{thm}. This function is 

448 
applied to the names assigned to locale rules and definitions as 

449 
strings. The opening command is called \texttt{Open\_locale} and takes the 

450 
name of the locale to be opened as its argument. 

451 

452 
If one opens a locale \texttt{foo} that is defined by extension from locale 

453 
\texttt{bar}, the function \texttt{Open\_locale} checks if locale \texttt{bar} 

454 
is open. If so, then it just opens \texttt{foo}, if not, then it prints a 

455 
message and opens \texttt{bar} before opening \texttt{foo}. Naturally, this 

456 
carries on, if \texttt{bar} is again an extension. 

457 

458 
\subsubsection{Closing} 

459 

460 
\emph{Closing} means to cancel the last opened locale, pushing it out of the 

461 
scope. Theorems proved during the life cycle of this locale will be disabled, 

462 
unless they have been explicitly exported, as described below. However, when 

463 
the same locale is opened again these theorems may be used again as well, 

464 
provided that they were saved as theorems in the first place, using 

465 
\texttt{qed} or ML assignment. The command \texttt{Close\_locale} takes a 

466 
locale name as a string and checks if this locale is actually the topmost 

467 
locale on the scope. If this is the case, it removes this locale, otherwise 

468 
it prints a warning message and does not change the scope. 

469 

470 
\subsubsection{Export of Theorems} 

471 
\label{sec:localeexport} 

472 

473 
Export of theorems transports theorems out of the scope of locales. Locale 

474 
rules that have been used in the proof of an exported theorem inside the 

475 
locale are carried by the exported form of the theorem as its individual 

476 
metaassumptions. The locale constants are universally quantified variables 

477 
in these theorems, hence such theorems can be instantiated individually. 

478 
Definitions become unfolded; locale constants that were merely used for 

479 
definitions vanish. Logically, exporting corresponds to a combined 

480 
application of introduction rules for implication and universal 

481 
quantification. Exporting forms a kind of normalization of theorems in a 

482 
locale scope. 

483 

484 
According to the possibility of nested locales there are two different forms 

485 
of export. The first one is realized by the function \texttt{export} that 

486 
exports theorems through all layers of opened locales of the scope. Hence, 

487 
the application of export to a theorem yields a theorem of the global level, 

488 
that is, the current theory context without any local assumptions or 

489 
definitions. 

490 

491 
When locales are nested we might want to export a theorem, not to the global 

492 
level of the current theory but just to the previous level. The other export 

493 
function, \texttt{Export}, transports theorems one level up in the scope; the 

494 
theorem still uses locale constants, definitions and rules of the locales 

495 
underneath. 

496 

497 
\subsection{Functions for Locales} 

498 
\label{Syntax} 

499 
\index{locales!functions} 

500 

501 
Here is a quick reference list of locale functions. 

502 
\begin{ttbox} 

503 
Open_locale : xstring > unit 

504 
Close_locale : xstring > unit 

505 
export : thm > thm 

506 
Export : thm > thm 

507 
thm : xstring > thm 

508 
Print_scope : unit > unit 

509 
print_locales: theory > unit 

510 
\end{ttbox} 

511 
\begin{ttdescription} 

512 
\item[\ttindexbold{Open_locale} $xstring$] 

513 
opens the locale {\it xstring}, adding it to the scope of the theory of the 

514 
current context. If the opened locale is built by extension, the ancestors 

515 
are opened automatically. 

516 

517 
\item[\ttindexbold{Close_locale} $xstring$] eliminates the locale {\it 

518 
xstring} from the scope if it is the topmost item on it, otherwise it does 

519 
not change the scope and produces a warning. 

520 

521 
\item[\ttindexbold{export} $thm$] locale definitions become expanded in {\it 

522 
thm} and locale rules that were used in the proof of {\it thm} become part 

523 
of its individual assumptions. This normalization happens with respect to 

524 
\emph{all open locales} on the scope. 

525 

526 
\item[\ttindexbold{Export} $thm$] works like \texttt{export} but normalizes 

527 
theorems only up to the previous level of locales on the scope. 

528 

529 
\item[\ttindexbold{thm} $xstring$] applied to the name of a locale definition 

530 
or rule it returns the definition as a theorem. 

531 

532 
\item[\ttindexbold{Print_scope}()] prints the names of the locales in the 

533 
current scope of the current theory context. 

534 

535 
\item[\ttindexbold{print_locale} $theory$] prints all locales that are 

536 
contained in {\it theory} directly or indirectly. It also displays the 

537 
current scope similar to \texttt{Print\_scope}. 

538 
\end{ttdescription} 

539 

540 

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541 
\section{Basic operations on theories}\label{BasicOperationsOnTheories} 
4384  542 

543 
\subsection{*Theory inclusion} 

544 
\begin{ttbox} 

545 
subthy : theory * theory > bool 

546 
eq_thy : theory * theory > bool 

547 
transfer : theory > thm > thm 

548 
transfer_sg : Sign.sg > thm > thm 

549 
\end{ttbox} 

550 

551 
Inclusion and equality of theories is determined by unique 

552 
identification stamps that are created when declaring new components. 

553 
Theorems contain a reference to the theory (actually to its signature) 

554 
they have been derived in. Transferring theorems to super theories 

555 
has no logical significance, but may affect some operations in subtle 

556 
ways (e.g.\ implicit merges of signatures when applying rules, or 

557 
pretty printing of theorems). 

558 

559 
\begin{ttdescription} 

560 

561 
\item[\ttindexbold{subthy} ($thy@1$, $thy@2$)] determines if $thy@1$ 

562 
is included in $thy@2$ wrt.\ identification stamps. 

563 

564 
\item[\ttindexbold{eq_thy} ($thy@1$, $thy@2$)] determines if $thy@1$ 

565 
is exactly the same as $thy@2$. 

566 

567 
\item[\ttindexbold{transfer} $thy$ $thm$] transfers theorem $thm$ to 

568 
theory $thy$, provided the latter includes the theory of $thm$. 

569 

570 
\item[\ttindexbold{transfer_sg} $sign$ $thm$] is similar to 

571 
\texttt{transfer}, but identifies the super theory via its 

572 
signature. 

573 

574 
\end{ttdescription} 

575 

576 

6571  577 
\subsection{*Primitive theories} 
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578 
\begin{ttbox} 
4317  579 
ProtoPure.thy : theory 
3108  580 
Pure.thy : theory 
581 
CPure.thy : theory 

286  582 
\end{ttbox} 
3108  583 
\begin{description} 
4317  584 
\item[\ttindexbold{ProtoPure.thy}, \ttindexbold{Pure.thy}, 
585 
\ttindexbold{CPure.thy}] contain the syntax and signature of the 

586 
metalogic. There are basically no axioms: metalevel inferences 

587 
are carried out by \ML\ functions. \texttt{Pure} and \texttt{CPure} 

588 
just differ in their concrete syntax of prefix function application: 

589 
$t(u@1, \ldots, u@n)$ in \texttt{Pure} vs.\ $t\,u@1,\ldots\,u@n$ in 

590 
\texttt{CPure}. \texttt{ProtoPure} is their common parent, 

591 
containing no syntax for printing prefix applications at all! 

6571  592 

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593 
%% FIXME 
478  594 
%\item [\ttindexbold{extend_theory} $thy$ {\tt"}$T${\tt"} $\cdots$] extends 
595 
% the theory $thy$ with new types, constants, etc. $T$ identifies the theory 

596 
% internally. When a theory is redeclared, say to change an incorrect axiom, 

597 
% bindings to the old axiom may persist. Isabelle ensures that the old and 

598 
% new theories are not involved in the same proof. Attempting to combine 

599 
% different theories having the same name $T$ yields the fatal error 

600 
%extend_theory : theory > string > \(\cdots\) > theory 

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601 
%\begin{ttbox} 
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602 
%Attempt to merge different versions of theory: \(T\) 
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603 
%\end{ttbox} 
3108  604 
\end{description} 
286  605 

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606 
%% FIXME 
275  607 
%\item [\ttindexbold{extend_theory} $thy$ {\tt"}$T${\tt"} 
608 
% ($classes$, $default$, $types$, $arities$, $consts$, $sextopt$) $rules$] 

609 
%\hfill\break %%% include if line is just too short 

286  610 
%is the \ML{} equivalent of the following theory definition: 
275  611 
%\begin{ttbox} 
612 
%\(T\) = \(thy\) + 

613 
%classes \(c\) < \(c@1\),\(\dots\),\(c@m\) 

614 
% \dots 

615 
%default {\(d@1,\dots,d@r\)} 

616 
%types \(tycon@1\),\dots,\(tycon@i\) \(n\) 

617 
% \dots 

618 
%arities \(tycon@1'\),\dots,\(tycon@j'\) :: (\(s@1\),\dots,\(s@n\))\(c\) 

619 
% \dots 

620 
%consts \(b@1\),\dots,\(b@k\) :: \(\tau\) 

621 
% \dots 

622 
%rules \(name\) \(rule\) 

623 
% \dots 

624 
%end 

625 
%\end{ttbox} 

626 
%where 

627 
%\begin{tabular}[t]{l@{~=~}l} 

628 
%$classes$ & \tt[("$c$",["$c@1$",\dots,"$c@m$"]),\dots] \\ 

629 
%$default$ & \tt["$d@1$",\dots,"$d@r$"]\\ 

630 
%$types$ & \tt[([$tycon@1$,\dots,$tycon@i$], $n$),\dots] \\ 

631 
%$arities$ & \tt[([$tycon'@1$,\dots,$tycon'@j$], ([$s@1$,\dots,$s@n$],$c$)),\dots] 

632 
%\\ 

633 
%$consts$ & \tt[([$b@1$,\dots,$b@k$],$\tau$),\dots] \\ 

634 
%$rules$ & \tt[("$name$",$rule$),\dots] 

635 
%\end{tabular} 

104  636 

637 

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638 
\subsection{Inspecting a theory}\label{sec:inspctthy} 
104  639 
\index{theories!inspectingbold} 
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640 
\begin{ttbox} 
4317  641 
print_syntax : theory > unit 
642 
print_theory : theory > unit 

643 
parents_of : theory > theory list 

644 
ancestors_of : theory > theory list 

645 
sign_of : theory > Sign.sg 

646 
Sign.stamp_names_of : Sign.sg > string list 

104  647 
\end{ttbox} 
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648 
These provide means of viewing a theory's components. 
324  649 
\begin{ttdescription} 
3108  650 
\item[\ttindexbold{print_syntax} $thy$] prints the syntax of $thy$ 
651 
(grammar, macros, translation functions etc., see 

652 
page~\pageref{pg:print_syn} for more details). 

653 

654 
\item[\ttindexbold{print_theory} $thy$] prints the logical parts of 

655 
$thy$, excluding the syntax. 

4317  656 

657 
\item[\ttindexbold{parents_of} $thy$] returns the direct ancestors 

658 
of~$thy$. 

659 

660 
\item[\ttindexbold{ancestors_of} $thy$] returns all ancestors of~$thy$ 

661 
(not including $thy$ itself). 

662 

663 
\item[\ttindexbold{sign_of} $thy$] returns the signature associated 

664 
with~$thy$. It is useful with functions like {\tt 

665 
read_instantiate_sg}, which take a signature as an argument. 

666 

667 
\item[\ttindexbold{Sign.stamp_names_of} $sg$]\index{signatures} 

668 
returns the names of the identification \rmindex{stamps} of ax 

669 
signature. These coincide with the names of its full ancestry 

670 
including that of $sg$ itself. 

104  671 

324  672 
\end{ttdescription} 
104  673 

1369  674 

11623  675 
\section{Terms}\label{sec:terms} 
104  676 
\index{termsbold} 
324  677 
Terms belong to the \ML\ type \mltydx{term}, which is a concrete datatype 
3108  678 
with six constructors: 
104  679 
\begin{ttbox} 
680 
type indexname = string * int; 

681 
infix 9 $; 

682 
datatype term = Const of string * typ 

683 
 Free of string * typ 

684 
 Var of indexname * typ 

685 
 Bound of int 

686 
 Abs of string * typ * term 

687 
 op $ of term * term; 

688 
\end{ttbox} 

324  689 
\begin{ttdescription} 
4317  690 
\item[\ttindexbold{Const} ($a$, $T$)] \index{constantsbold} 
8136  691 
is the \textbf{constant} with name~$a$ and type~$T$. Constants include 
286  692 
connectives like $\land$ and $\forall$ as well as constants like~0 
693 
and~$Suc$. Other constants may be required to define a logic's concrete 

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694 
syntax. 
104  695 

4317  696 
\item[\ttindexbold{Free} ($a$, $T$)] \index{variables!freebold} 
8136  697 
is the \textbf{free variable} with name~$a$ and type~$T$. 
104  698 

4317  699 
\item[\ttindexbold{Var} ($v$, $T$)] \index{unknownsbold} 
8136  700 
is the \textbf{scheme variable} with indexname~$v$ and type~$T$. An 
324  701 
\mltydx{indexname} is a string paired with a nonnegative index, or 
702 
subscript; a term's scheme variables can be systematically renamed by 

703 
incrementing their subscripts. Scheme variables are essentially free 

704 
variables, but may be instantiated during unification. 

104  705 

324  706 
\item[\ttindexbold{Bound} $i$] \index{variables!boundbold} 
8136  707 
is the \textbf{bound variable} with de Bruijn index~$i$, which counts the 
324  708 
number of lambdas, starting from zero, between a variable's occurrence 
709 
and its binding. The representation prevents capture of variables. For 

710 
more information see de Bruijn \cite{debruijn72} or 

6592  711 
Paulson~\cite[page~376]{paulsonml2}. 
104  712 

4317  713 
\item[\ttindexbold{Abs} ($a$, $T$, $u$)] 
324  714 
\index{lambda abs@$\lambda$abstractionsbold} 
8136  715 
is the $\lambda$\textbf{abstraction} with body~$u$, and whose bound 
324  716 
variable has name~$a$ and type~$T$. The name is used only for parsing 
717 
and printing; it has no logical significance. 

104  718 

324  719 
\item[$t$ \$ $u$] \index{$@{\tt\$}bold} \index{function applicationsbold} 
8136  720 
is the \textbf{application} of~$t$ to~$u$. 
324  721 
\end{ttdescription} 
9695  722 
Application is written as an infix operator to aid readability. Here is an 
723 
\ML\ pattern to recognize FOL formulae of the form~$A\imp B$, binding the 

724 
subformulae to~$A$ and~$B$: 

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725 
\begin{ttbox} 
104  726 
Const("Trueprop",_) $ (Const("op >",_) $ A $ B) 
727 
\end{ttbox} 

728 

729 

4317  730 
\section{*Variable binding} 
286  731 
\begin{ttbox} 
732 
loose_bnos : term > int list 

733 
incr_boundvars : int > term > term 

734 
abstract_over : term*term > term 

735 
variant_abs : string * typ * term > string * term 

8136  736 
aconv : term * term > bool\hfill\textbf{infix} 
286  737 
\end{ttbox} 
738 
These functions are all concerned with the de Bruijn representation of 

739 
bound variables. 

324  740 
\begin{ttdescription} 
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741 
\item[\ttindexbold{loose_bnos} $t$] 
286  742 
returns the list of all dangling bound variable references. In 
6669  743 
particular, \texttt{Bound~0} is loose unless it is enclosed in an 
744 
abstraction. Similarly \texttt{Bound~1} is loose unless it is enclosed in 

286  745 
at least two abstractions; if enclosed in just one, the list will contain 
746 
the number 0. A wellformed term does not contain any loose variables. 

747 

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748 
\item[\ttindexbold{incr_boundvars} $j$] 
332  749 
increases a term's dangling bound variables by the offset~$j$. This is 
286  750 
required when moving a subterm into a context where it is enclosed by a 
751 
different number of abstractions. Bound variables with a matching 

752 
abstraction are unaffected. 

753 

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754 
\item[\ttindexbold{abstract_over} $(v,t)$] 
286  755 
forms the abstraction of~$t$ over~$v$, which may be any wellformed term. 
6669  756 
It replaces every occurrence of \(v\) by a \texttt{Bound} variable with the 
286  757 
correct index. 
758 

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759 
\item[\ttindexbold{variant_abs} $(a,T,u)$] 
286  760 
substitutes into $u$, which should be the body of an abstraction. 
761 
It replaces each occurrence of the outermost bound variable by a free 

762 
variable. The free variable has type~$T$ and its name is a variant 

332  763 
of~$a$ chosen to be distinct from all constants and from all variables 
286  764 
free in~$u$. 
765 

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766 
\item[$t$ \ttindexbold{aconv} $u$] 
286  767 
tests whether terms~$t$ and~$u$ are \(\alpha\)convertible: identical up 
768 
to renaming of bound variables. 

769 
\begin{itemize} 

770 
\item 

6669  771 
Two constants, \texttt{Free}s, or \texttt{Var}s are \(\alpha\)convertible 
286  772 
if their names and types are equal. 
773 
(Variables having the same name but different types are thus distinct. 

774 
This confusing situation should be avoided!) 

775 
\item 

776 
Two bound variables are \(\alpha\)convertible 

777 
if they have the same number. 

778 
\item 

779 
Two abstractions are \(\alpha\)convertible 

780 
if their bodies are, and their bound variables have the same type. 

781 
\item 

782 
Two applications are \(\alpha\)convertible 

783 
if the corresponding subterms are. 

784 
\end{itemize} 

785 

324  786 
\end{ttdescription} 
286  787 

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788 
\section{Certified terms}\index{terms!certifiedbold}\index{signatures} 
8136  789 
A term $t$ can be \textbf{certified} under a signature to ensure that every type 
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790 
in~$t$ is wellformed and every constant in~$t$ is a type instance of a 
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791 
constant declared in the signature. The term must be welltyped and its use 
6669  792 
of bound variables must be wellformed. Metarules such as \texttt{forall_elim} 
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793 
take certified terms as arguments. 
104  794 

324  795 
Certified terms belong to the abstract type \mltydx{cterm}. 
104  796 
Elements of the type can only be created through the certification process. 
797 
In case of error, Isabelle raises exception~\ttindex{TERM}\@. 

798 

799 
\subsection{Printing terms} 

324  800 
\index{terms!printing of} 
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801 
\begin{ttbox} 
275  802 
string_of_cterm : cterm > string 
104  803 
Sign.string_of_term : Sign.sg > term > string 
804 
\end{ttbox} 

324  805 
\begin{ttdescription} 
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806 
\item[\ttindexbold{string_of_cterm} $ct$] 
104  807 
displays $ct$ as a string. 
808 

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809 
\item[\ttindexbold{Sign.string_of_term} $sign$ $t$] 
104  810 
displays $t$ as a string, using the syntax of~$sign$. 
324  811 
\end{ttdescription} 
104  812 

813 
\subsection{Making and inspecting certified terms} 

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814 
\begin{ttbox} 
8136  815 
cterm_of : Sign.sg > term > cterm 
816 
read_cterm : Sign.sg > string * typ > cterm 

817 
cert_axm : Sign.sg > string * term > string * term 

818 
read_axm : Sign.sg > string * string > string * term 

819 
rep_cterm : cterm > \{T:typ, t:term, sign:Sign.sg, maxidx:int\} 

4543  820 
Sign.certify_term : Sign.sg > term > term * typ * int 
104  821 
\end{ttbox} 
324  822 
\begin{ttdescription} 
4543  823 

824 
\item[\ttindexbold{cterm_of} $sign$ $t$] \index{signatures} certifies 

825 
$t$ with respect to signature~$sign$. 

826 

827 
\item[\ttindexbold{read_cterm} $sign$ ($s$, $T$)] reads the string~$s$ 

828 
using the syntax of~$sign$, creating a certified term. The term is 

829 
checked to have type~$T$; this type also tells the parser what kind 

830 
of phrase to parse. 

831 

832 
\item[\ttindexbold{cert_axm} $sign$ ($name$, $t$)] certifies $t$ with 

833 
respect to $sign$ as a metaproposition and converts all exceptions 

834 
to an error, including the final message 

864
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835 
\begin{ttbox} 
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836 
The error(s) above occurred in axiom "\(name\)" 
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837 
\end{ttbox} 
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838 

4543  839 
\item[\ttindexbold{read_axm} $sign$ ($name$, $s$)] similar to {\tt 
840 
cert_axm}, but first reads the string $s$ using the syntax of 

841 
$sign$. 

842 

843 
\item[\ttindexbold{rep_cterm} $ct$] decomposes $ct$ as a record 

844 
containing its type, the term itself, its signature, and the maximum 

845 
subscript of its unknowns. The type and maximum subscript are 

846 
computed during certification. 

847 

848 
\item[\ttindexbold{Sign.certify_term}] is a more primitive version of 

849 
\texttt{cterm_of}, returning the internal representation instead of 

850 
an abstract \texttt{cterm}. 

864
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851 

324  852 
\end{ttdescription} 
104  853 

854 

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855 
\section{Types}\index{typesbold} 
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856 
Types belong to the \ML\ type \mltydx{typ}, which is a concrete datatype with 
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857 
three constructor functions. These correspond to type constructors, free 
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858 
type variables and schematic type variables. Types are classified by sorts, 
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859 
which are lists of classes (representing an intersection). A class is 
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860 
represented by a string. 
104  861 
\begin{ttbox} 
862 
type class = string; 

863 
type sort = class list; 

864 

865 
datatype typ = Type of string * typ list 

866 
 TFree of string * sort 

867 
 TVar of indexname * sort; 

868 

869 
infixr 5 >; 

864
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870 
fun S > T = Type ("fun", [S, T]); 
104  871 
\end{ttbox} 
324  872 
\begin{ttdescription} 
4317  873 
\item[\ttindexbold{Type} ($a$, $Ts$)] \index{type constructorsbold} 
8136  874 
applies the \textbf{type constructor} named~$a$ to the type operand list~$Ts$. 
324  875 
Type constructors include~\tydx{fun}, the binary function space 
876 
constructor, as well as nullary type constructors such as~\tydx{prop}. 

877 
Other type constructors may be introduced. In expressions, but not in 

878 
patterns, \hbox{\tt$S$>$T$} is a convenient shorthand for function 

879 
types. 

104  880 

4317  881 
\item[\ttindexbold{TFree} ($a$, $s$)] \index{type variablesbold} 
8136  882 
is the \textbf{type variable} with name~$a$ and sort~$s$. 
104  883 

4317  884 
\item[\ttindexbold{TVar} ($v$, $s$)] \index{type unknownsbold} 
8136  885 
is the \textbf{type unknown} with indexname~$v$ and sort~$s$. 
324  886 
Type unknowns are essentially free type variables, but may be 
887 
instantiated during unification. 

888 
\end{ttdescription} 

104  889 

890 

891 
\section{Certified types} 

892 
\index{types!certifiedbold} 

864
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893 
Certified types, which are analogous to certified terms, have type 
275  894 
\ttindexbold{ctyp}. 
104  895 

896 
\subsection{Printing types} 

324  897 
\index{types!printing of} 
864
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898 
\begin{ttbox} 
275  899 
string_of_ctyp : ctyp > string 
104  900 
Sign.string_of_typ : Sign.sg > typ > string 
901 
\end{ttbox} 

324  902 
\begin{ttdescription} 
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903 
\item[\ttindexbold{string_of_ctyp} $cT$] 
104  904 
displays $cT$ as a string. 
905 

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906 
\item[\ttindexbold{Sign.string_of_typ} $sign$ $T$] 
104  907 
displays $T$ as a string, using the syntax of~$sign$. 
324  908 
\end{ttdescription} 
104  909 

910 

911 
\subsection{Making and inspecting certified types} 

864
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912 
\begin{ttbox} 
4543  913 
ctyp_of : Sign.sg > typ > ctyp 
8136  914 
rep_ctyp : ctyp > \{T: typ, sign: Sign.sg\} 
4543  915 
Sign.certify_typ : Sign.sg > typ > typ 
104  916 
\end{ttbox} 
324  917 
\begin{ttdescription} 
4543  918 

919 
\item[\ttindexbold{ctyp_of} $sign$ $T$] \index{signatures} certifies 

920 
$T$ with respect to signature~$sign$. 

921 

922 
\item[\ttindexbold{rep_ctyp} $cT$] decomposes $cT$ as a record 

923 
containing the type itself and its signature. 

924 

925 
\item[\ttindexbold{Sign.certify_typ}] is a more primitive version of 

926 
\texttt{ctyp_of}, returning the internal representation instead of 

927 
an abstract \texttt{ctyp}. 

104  928 

324  929 
\end{ttdescription} 
104  930 

1846  931 

4317  932 
\section{Oracles: calling trusted external reasoners} 
1846  933 
\label{sec:oracles} 
934 
\index{oracles(} 

935 

936 
Oracles allow Isabelle to take advantage of external reasoners such as 

937 
arithmetic decision procedures, model checkers, fast tautology checkers or 

938 
computer algebra systems. Invoked as an oracle, an external reasoner can 

939 
create arbitrary Isabelle theorems. It is your responsibility to ensure that 

940 
the external reasoner is as trustworthy as your application requires. 

941 
Isabelle's proof objects~(\S\ref{sec:proofObjects}) record how each theorem 

942 
depends upon oracle calls. 

943 

944 
\begin{ttbox} 

4317  945 
invoke_oracle : theory > xstring > Sign.sg * object > thm 
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946 
Theory.add_oracle : bstring * (Sign.sg * object > term) > theory 
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947 
> theory 
1846  948 
\end{ttbox} 
949 
\begin{ttdescription} 

4317  950 
\item[\ttindexbold{invoke_oracle} $thy$ $name$ ($sign$, $data$)] 
951 
invokes the oracle $name$ of theory $thy$ passing the information 

952 
contained in the exception value $data$ and creating a theorem 

953 
having signature $sign$. Note that type \ttindex{object} is just an 

954 
abbreviation for \texttt{exn}. Errors arise if $thy$ does not have 

955 
an oracle called $name$, if the oracle rejects its arguments or if 

956 
its result is illtyped. 

957 

958 
\item[\ttindexbold{Theory.add_oracle} $name$ $fun$ $thy$] extends 

959 
$thy$ by oracle $fun$ called $name$. It is seldom called 

960 
explicitly, as there is concrete syntax for oracles in theory files. 

1846  961 
\end{ttdescription} 
962 

963 
A curious feature of {\ML} exceptions is that they are ordinary constructors. 

6669  964 
The {\ML} type \texttt{exn} is a datatype that can be extended at any time. (See 
1846  965 
my {\em {ML} for the Working Programmer}~\cite{paulsonml2}, especially 
966 
page~136.) The oracle mechanism takes advantage of this to allow an oracle to 

967 
take any information whatever. 

968 

969 
There must be some way of invoking the external reasoner from \ML, either 

970 
because it is coded in {\ML} or via an operating system interface. Isabelle 

971 
expects the {\ML} function to take two arguments: a signature and an 

4317  972 
exception object. 
1846  973 
\begin{itemize} 
974 
\item The signature will typically be that of a desendant of the theory 

975 
declaring the oracle. The oracle will use it to distinguish constants from 

976 
variables, etc., and it will be attached to the generated theorems. 

977 

978 
\item The exception is used to pass arbitrary information to the oracle. This 

979 
information must contain a full description of the problem to be solved by 

980 
the external reasoner, including any additional information that might be 

981 
required. The oracle may raise the exception to indicate that it cannot 

982 
solve the specified problem. 

983 
\end{itemize} 

984 

6669  985 
A trivial example is provided in theory \texttt{FOL/ex/IffOracle}. This 
4317  986 
oracle generates tautologies of the form $P\bimp\cdots\bimp P$, with 
987 
an even number of $P$s. 

1846  988 

4317  989 
The \texttt{ML} section of \texttt{IffOracle.thy} begins by declaring 
990 
a few auxiliary functions (suppressed below) for creating the 

991 
tautologies. Then it declares a new exception constructor for the 

992 
information required by the oracle: here, just an integer. It finally 

993 
defines the oracle function itself. 

1846  994 
\begin{ttbox} 
4317  995 
exception IffOracleExn of int;\medskip 
996 
fun mk_iff_oracle (sign, IffOracleExn n) = 

997 
if n > 0 andalso n mod 2 = 0 

6669  998 
then Trueprop \$ mk_iff n 
4317  999 
else raise IffOracleExn n; 
1846  1000 
\end{ttbox} 
6669  1001 
Observe the function's two arguments, the signature \texttt{sign} and the 
4317  1002 
exception given as a pattern. The function checks its argument for 
1003 
validity. If $n$ is positive and even then it creates a tautology 

1004 
containing $n$ occurrences of~$P$. Otherwise it signals error by 

1005 
raising its own exception (just by happy coincidence). Errors may be 

6669  1006 
signalled by other means, such as returning the theorem \texttt{True}. 
4317  1007 
Please ensure that the oracle's result is correctly typed; Isabelle 
1008 
will reject illtyped theorems by raising a cryptic exception at top 

1009 
level. 

1846  1010 

6669  1011 
The \texttt{oracle} section of \texttt{IffOracle.thy} installs above 
4317  1012 
\texttt{ML} function as follows: 
1846  1013 
\begin{ttbox} 
4317  1014 
IffOracle = FOL +\medskip 
1015 
oracle 

1016 
iff = mk_iff_oracle\medskip 

1846  1017 
end 
1018 
\end{ttbox} 

1019 

4317  1020 
Now in \texttt{IffOracle.ML} we first define a wrapper for invoking 
1021 
the oracle: 

1846  1022 
\begin{ttbox} 
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1023 
fun iff_oracle n = invoke_oracle IffOracle.thy "iff" 
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1024 
(sign_of IffOracle.thy, IffOracleExn n); 
4317  1025 
\end{ttbox} 
1026 

1027 
Here are some example applications of the \texttt{iff} oracle. An 

1028 
argument of 10 is allowed, but one of 5 is forbidden: 

1029 
\begin{ttbox} 

1030 
iff_oracle 10; 

1846  1031 
{\out "P <> P <> P <> P <> P <> P <> P <> P <> P <> P" : thm} 
4317  1032 
iff_oracle 5; 
1846  1033 
{\out Exception IffOracleExn 5 raised} 
1034 
\end{ttbox} 

1035 

1036 
\index{oracles)} 

104  1037 
\index{theories)} 
5369  1038 

1039 

1040 
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1041 
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1042 
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1043 
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