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doc-src/Ref/theories.tex

author | paulson |

Wed, 07 Oct 1998 10:31:30 +0200 | |

changeset 5619 | 76a8c72e3fd4 |

parent 5390 | 0c9e6d860485 |

child 6568 | b38bc78d9a9d |

permissions | -rw-r--r-- |

new theorems

%% $Id$ \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{*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|)} %%% Local Variables: %%% mode: latex %%% TeX-master: "ref" %%% End: