theory Spec

 imports Main

 begin

 chapter {* Theory specifications *}

 text {*

   The Isabelle/Isar theory format integrates specifications and

   proofs, supporting interactive development with unlimited undo

   operation.  There is an integrated document preparation system (see

   \chref{ch:document-prep}), for typesetting formal developments

   together with informal text.  The resulting hyper-linked PDF

   documents can be used both for WWW presentation and printed copies.

   The Isar proof language (see \chref{ch:proofs}) is embedded into the

   theory language as a proper sub-language.  Proof mode is entered by

   stating some @{command theorem} or @{command lemma} at the theory

   level, and left again with the final conclusion (e.g.\ via @{command

   qed}).  Some theory specification mechanisms also require a proof,

   such as @{command typedef} in HOL, which demands non-emptiness of

   the representing sets.

 *}

 section {* Defining theories \label{sec:begin-thy} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "theory"} & : & @{text "toplevel \<rightarrow> theory"} \\

     @{command_def (global) "end"} & : & @{text "theory \<rightarrow> toplevel"} \\

   \end{matharray}

   Isabelle/Isar theories are defined via theory files, which may

   contain both specifications and proofs; occasionally definitional

   mechanisms also require some explicit proof.  The theory body may be

   sub-structured by means of \emph{local theory targets}, such as

   @{command "locale"} and @{command "class"}.

   The first proper command of a theory is @{command "theory"}, which

   indicates imports of previous theories and optional dependencies on

   other source files (usually in ML).  Just preceding the initial

   @{command "theory"} command there may be an optional @{command

   "header"} declaration, which is only relevant to document

   preparation: see also the other section markup commands in

   \secref{sec:markup}.

   A theory is concluded by a final @{command (global) "end"} command,

   one that does not belong to a local theory target.  No further

   commands may follow such a global @{command (global) "end"},

   although some user-interfaces might pretend that trailing input is

   admissible.

   \begin{rail}

     'theory' name 'imports' (name +) uses? 'begin'

     ;

     uses: 'uses' ((name | parname) +);

   \end{rail}

   \begin{description}

   \item @{command "theory"}~@{text "A \<IMPORTS> B\<^sub>1 \<dots> B\<^sub>n \<BEGIN>"}

   starts a new theory @{text A} based on the merge of existing

   theories @{text "B\<^sub>1 \<dots> B\<^sub>n"}.

   Due to the possibility to import more than one ancestor, the

   resulting theory structure of an Isabelle session forms a directed

   acyclic graph (DAG).  Isabelle's theory loader ensures that the

   sources contributing to the development graph are always up-to-date:

   changed files are automatically reloaded whenever a theory header

   specification is processed.

   The optional @{keyword_def "uses"} specification declares additional

   dependencies on extra files (usually ML sources).  Files will be

   loaded immediately (as ML), unless the name is parenthesized.  The

   latter case records a dependency that needs to be resolved later in

   the text, usually via explicit @{command_ref "use"} for ML files;

   other file formats require specific load commands defined by the

   corresponding tools or packages.

   \item @{command (global) "end"} concludes the current theory

   definition.  Note that local theory targets involve a local

   @{command (local) "end"}, which is clear from the nesting.

   \end{description}

 *}

 section {* Local theory targets \label{sec:target} *}

 text {*

   A local theory target is a context managed separately within the

   enclosing theory.  Contexts may introduce parameters (fixed

   variables) and assumptions (hypotheses).  Definitions and theorems

   depending on the context may be added incrementally later on.  Named

   contexts refer to locales (cf.\ \secref{sec:locale}) or type classes

   (cf.\ \secref{sec:class}); the name ``@{text "-"}'' signifies the

   global theory context.

   \begin{matharray}{rcll}

     @{command_def "context"} & : & @{text "theory \<rightarrow> local_theory"} \\

     @{command_def (local) "end"} & : & @{text "local_theory \<rightarrow> theory"} \\

   \end{matharray}

   \indexouternonterm{target}

   \begin{rail}

     'context' name 'begin'

     ;

     target: '(' 'in' name ')'

     ;

   \end{rail}

   \begin{description}

   \item @{command "context"}~@{text "c \<BEGIN>"} recommences an

   existing locale or class context @{text c}.  Note that locale and

   class definitions allow to include the @{keyword "begin"} keyword as

   well, in order to continue the local theory immediately after the

   initial specification.

   \item @{command (local) "end"} concludes the current local theory

   and continues the enclosing global theory.  Note that a global

   @{command (global) "end"} has a different meaning: it concludes the

   theory itself (\secref{sec:begin-thy}).

   \item @{text "("}@{keyword_def "in"}~@{text "c)"} given after any

   local theory command specifies an immediate target, e.g.\

   ``@{command "definition"}~@{text "(\<IN> c) \<dots>"}'' or ``@{command

   "theorem"}~@{text "(\<IN> c) \<dots>"}''.  This works both in a local or

   global theory context; the current target context will be suspended

   for this command only.  Note that ``@{text "(\<IN> -)"}'' will

   always produce a global result independently of the current target

   context.

   \end{description}

   The exact meaning of results produced within a local theory context

   depends on the underlying target infrastructure (locale, type class

   etc.).  The general idea is as follows, considering a context named

   @{text c} with parameter @{text x} and assumption @{text "A[x]"}.

   Definitions are exported by introducing a global version with

   additional arguments; a syntactic abbreviation links the long form

   with the abstract version of the target context.  For example,

   @{text "a \<equiv> t[x]"} becomes @{text "c.a ?x \<equiv> t[?x]"} at the theory

   level (for arbitrary @{text "?x"}), together with a local

   abbreviation @{text "c \<equiv> c.a x"} in the target context (for the

   fixed parameter @{text x}).

   Theorems are exported by discharging the assumptions and

   generalizing the parameters of the context.  For example, @{text "a:

   B[x]"} becomes @{text "c.a: A[?x] \<Longrightarrow> B[?x]"}, again for arbitrary

   @{text "?x"}.

 *}

 section {* Basic specification elements *}

 text {*

   \begin{matharray}{rcll}

     @{command_def "axiomatization"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!)\\

     @{command_def "definition"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{attribute_def "defn"} & : & @{text attribute} \\

     @{command_def "abbreviation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "print_abbrevs"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow> "} \\

   \end{matharray}

   These specification mechanisms provide a slightly more abstract view

   than the underlying primitives of @{command "consts"}, @{command

   "defs"} (see \secref{sec:consts}), and @{command "axioms"} (see

   \secref{sec:axms-thms}).  In particular, type-inference is commonly

   available, and result names need not be given.

   \begin{rail}

     'axiomatization' target? fixes? ('where' specs)?

     ;

     'definition' target? (decl 'where')? thmdecl? prop

     ;

     'abbreviation' target? mode? (decl 'where')? prop

     ;

     fixes: ((name ('::' type)? mixfix? | vars) + 'and')

     ;

     specs: (thmdecl? props + 'and')

     ;

     decl: name ('::' type)? mixfix?

     ;

   \end{rail}

   \begin{description}

   \item @{command "axiomatization"}~@{text "c\<^sub>1 \<dots> c\<^sub>m \<WHERE> \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}

   introduces several constants simultaneously and states axiomatic

   properties for these.  The constants are marked as being specified

   once and for all, which prevents additional specifications being

   issued later on.

   Note that axiomatic specifications are only appropriate when

   declaring a new logical system; axiomatic specifications are

   restricted to global theory contexts.  Normal applications should

   only use definitional mechanisms!

   \item @{command "definition"}~@{text "c \<WHERE> eq"} produces an

   internal definition @{text "c \<equiv> t"} according to the specification

   given as @{text eq}, which is then turned into a proven fact.  The

   given proposition may deviate from internal meta-level equality

   according to the rewrite rules declared as @{attribute defn} by the

   object-logic.  This usually covers object-level equality @{text "x =

   y"} and equivalence @{text "A \<leftrightarrow> B"}.  End-users normally need not

   change the @{attribute defn} setup.

   Definitions may be presented with explicit arguments on the LHS, as

   well as additional conditions, e.g.\ @{text "f x y = t"} instead of

   @{text "f \<equiv> \<lambda>x y. t"} and @{text "y \<noteq> 0 \<Longrightarrow> g x y = u"} instead of an

   unrestricted @{text "g \<equiv> \<lambda>x y. u"}.

   \item @{command "abbreviation"}~@{text "c \<WHERE> eq"} introduces a

   syntactic constant which is associated with a certain term according

   to the meta-level equality @{text eq}.

   Abbreviations participate in the usual type-inference process, but

   are expanded before the logic ever sees them.  Pretty printing of

   terms involves higher-order rewriting with rules stemming from

   reverted abbreviations.  This needs some care to avoid overlapping

   or looping syntactic replacements!

   The optional @{text mode} specification restricts output to a

   particular print mode; using ``@{text input}'' here achieves the

   effect of one-way abbreviations.  The mode may also include an

   ``@{keyword "output"}'' qualifier that affects the concrete syntax

   declared for abbreviations, cf.\ @{command "syntax"} in

   \secref{sec:syn-trans}.

   \item @{command "print_abbrevs"} prints all constant abbreviations

   of the current context.

   \end{description}

 *}

 section {* Generic declarations *}

 text {*

   Arbitrary operations on the background context may be wrapped-up as

   generic declaration elements.  Since the underlying concept of local

   theories may be subject to later re-interpretation, there is an

   additional dependency on a morphism that tells the difference of the

   original declaration context wrt.\ the application context

   encountered later on.  A fact declaration is an important special

   case: it consists of a theorem which is applied to the context by

   means of an attribute.

   \begin{matharray}{rcl}

     @{command_def "declaration"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "declare"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   \end{matharray}

   \begin{rail}

     'declaration' target? text

     ;

     'declare' target? (thmrefs + 'and')

     ;

   \end{rail}

   \begin{description}

   \item @{command "declaration"}~@{text d} adds the declaration

   function @{text d} of ML type @{ML_type declaration}, to the current

   local theory under construction.  In later application contexts, the

   function is transformed according to the morphisms being involved in

   the interpretation hierarchy.

   \item @{command "declare"}~@{text thms} declares theorems to the

   current local theory context.  No theorem binding is involved here,

   unlike @{command "theorems"} or @{command "lemmas"} (cf.\

   \secref{sec:axms-thms}), so @{command "declare"} only has the effect

   of applying attributes as included in the theorem specification.

   \end{description}

 *}

 section {* Locales \label{sec:locale} *}

 text {*

   Locales are named local contexts, consisting of a list of

   declaration elements that are modeled after the Isar proof context

   commands (cf.\ \secref{sec:proof-context}).

 *}

 subsection {* Locale specifications *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "locale"} & : & @{text "theory \<rightarrow> local_theory"} \\

     @{command_def "print_locale"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{command_def "print_locales"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{method_def intro_locales} & : & @{text method} \\

     @{method_def unfold_locales} & : & @{text method} \\

   \end{matharray}

   \indexouternonterm{contextexpr}\indexouternonterm{contextelem}

   \indexisarelem{fixes}\indexisarelem{constrains}\indexisarelem{assumes}

   \indexisarelem{defines}\indexisarelem{notes}

   \begin{rail}

     'locale' name ('=' localeexpr)? 'begin'?

     ;

     'print\_locale' '!'? localeexpr

     ;

     localeexpr: ((contextexpr '+' (contextelem+)) | contextexpr | (contextelem+))

     ;

     contextexpr: nameref | '(' contextexpr ')' |

     (contextexpr (name mixfix? +)) | (contextexpr + '+')

     ;

     contextelem: fixes | constrains | assumes | defines | notes

     ;

     fixes: 'fixes' ((name ('::' type)? structmixfix? | vars) + 'and')

     ;

     constrains: 'constrains' (name '::' type + 'and')

     ;

     assumes: 'assumes' (thmdecl? props + 'and')

     ;

     defines: 'defines' (thmdecl? prop proppat? + 'and')

     ;

     notes: 'notes' (thmdef? thmrefs + 'and')

     ;

   \end{rail}

   \begin{description}

   \item @{command "locale"}~@{text "loc = import + body"} defines a

   new locale @{text loc} as a context consisting of a certain view of

   existing locales (@{text import}) plus some additional elements

   (@{text body}).  Both @{text import} and @{text body} are optional;

   the degenerate form @{command "locale"}~@{text loc} defines an empty

   locale, which may still be useful to collect declarations of facts

   later on.  Type-inference on locale expressions automatically takes

   care of the most general typing that the combined context elements

   may acquire.

   The @{text import} consists of a structured context expression,

   consisting of references to existing locales, renamed contexts, or

   merged contexts.  Renaming uses positional notation: @{text "c

   x\<^sub>1 \<dots> x\<^sub>n"} means that (a prefix of) the fixed

   parameters of context @{text c} are named @{text "x\<^sub>1, \<dots>,

   x\<^sub>n"}; a ``@{text _}'' (underscore) means to skip that

   position.  Renaming by default deletes concrete syntax, but new

   syntax may by specified with a mixfix annotation.  An exeption of

   this rule is the special syntax declared with ``@{text

   "(\<STRUCTURE>)"}'' (see below), which is neither deleted nor can it

   be changed.  Merging proceeds from left-to-right, suppressing any

   duplicates stemming from different paths through the import

   hierarchy.

   The @{text body} consists of basic context elements, further context

   expressions may be included as well.

   \begin{description}

   \item @{element "fixes"}~@{text "x :: \<tau> (mx)"} declares a local

   parameter of type @{text \<tau>} and mixfix annotation @{text mx} (both

   are optional).  The special syntax declaration ``@{text

   "(\<STRUCTURE>)"}'' means that @{text x} may be referenced

   implicitly in this context.

   \item @{element "constrains"}~@{text "x :: \<tau>"} introduces a type

   constraint @{text \<tau>} on the local parameter @{text x}.

   \item @{element "assumes"}~@{text "a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}

   introduces local premises, similar to @{command "assume"} within a

   proof (cf.\ \secref{sec:proof-context}).

   \item @{element "defines"}~@{text "a: x \<equiv> t"} defines a previously

   declared parameter.  This is similar to @{command "def"} within a

   proof (cf.\ \secref{sec:proof-context}), but @{element "defines"}

   takes an equational proposition instead of variable-term pair.  The

   left-hand side of the equation may have additional arguments, e.g.\

   ``@{element "defines"}~@{text "f x\<^sub>1 \<dots> x\<^sub>n \<equiv> t"}''.

   \item @{element "notes"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}

   reconsiders facts within a local context.  Most notably, this may

   include arbitrary declarations in any attribute specifications

   included here, e.g.\ a local @{attribute simp} rule.

   The initial @{text import} specification of a locale expression

   maintains a dynamic relation to the locales being referenced

   (benefiting from any later fact declarations in the obvious manner).

   \end{description}

   Note that ``@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}'' patterns given

   in the syntax of @{element "assumes"} and @{element "defines"} above

   are illegal in locale definitions.  In the long goal format of

   \secref{sec:goals}, term bindings may be included as expected,

   though.

   \medskip By default, locale specifications are ``closed up'' by

   turning the given text into a predicate definition @{text

   loc_axioms} and deriving the original assumptions as local lemmas

   (modulo local definitions).  The predicate statement covers only the

   newly specified assumptions, omitting the content of included locale

   expressions.  The full cumulative view is only provided on export,

   involving another predicate @{text loc} that refers to the complete

   specification text.

   In any case, the predicate arguments are those locale parameters

   that actually occur in the respective piece of text.  Also note that

   these predicates operate at the meta-level in theory, but the locale

   packages attempts to internalize statements according to the

   object-logic setup (e.g.\ replacing @{text \<And>} by @{text \<forall>}, and

   @{text "\<Longrightarrow>"} by @{text "\<longrightarrow>"} in HOL; see also

   \secref{sec:object-logic}).  Separate introduction rules @{text

   loc_axioms.intro} and @{text loc.intro} are provided as well.

   \item @{command "print_locale"}~@{text "import + body"} prints the

   specified locale expression in a flattened form.  The notable

   special case @{command "print_locale"}~@{text loc} just prints the

   contents of the named locale, but keep in mind that type-inference

   will normalize type variables according to the usual alphabetical

   order.  The command omits @{element "notes"} elements by default.

   Use @{command "print_locale"}@{text "!"} to get them included.

   \item @{command "print_locales"} prints the names of all locales

   of the current theory.

   \item @{method intro_locales} and @{method unfold_locales}

   repeatedly expand all introduction rules of locale predicates of the

   theory.  While @{method intro_locales} only applies the @{text

   loc.intro} introduction rules and therefore does not decend to

   assumptions, @{method unfold_locales} is more aggressive and applies

   @{text loc_axioms.intro} as well.  Both methods are aware of locale

   specifications entailed by the context, both from target statements,

   and from interpretations (see below).  New goals that are entailed

   by the current context are discharged automatically.

   \end{description}

 *}

 subsection {* Interpretation of locales *}

 text {*

   Locale expressions (more precisely, \emph{context expressions}) may

   be instantiated, and the instantiated facts added to the current

   context.  This requires a proof of the instantiated specification

   and is called \emph{locale interpretation}.  Interpretation is

   possible in theories and locales (command @{command

   "interpretation"}) and also within a proof body (command @{command

   "interpret"}).

   \begin{matharray}{rcl}

     @{command_def "interpretation"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

     @{command_def "interpret"} & : & @{text "proof(state) | proof(chain \<rightarrow> proof(prove)"} \\

   \end{matharray}

   \indexouternonterm{interp}

   \begin{rail}

     'interpretation' (interp | name ('<' | subseteq) contextexpr)

     ;

     'interpret' interp

     ;

     instantiation: ('[' (inst+) ']')?

     ;

     interp: (name ':')? \\ (contextexpr instantiation |

       name instantiation 'where' (thmdecl? prop + 'and'))

     ;

   \end{rail}

   \begin{description}

   \item @{command "interpretation"}~@{text "expr insts \<WHERE> eqns"}

   The first form of @{command "interpretation"} interprets @{text

   expr} in the theory.  The instantiation is given as a list of terms

   @{text insts} and is positional.  All parameters must receive an

   instantiation term --- with the exception of defined parameters.

   These are, if omitted, derived from the defining equation and other

   instantiations.  Use ``@{text _}'' to omit an instantiation term.

   The command generates proof obligations for the instantiated

   specifications (assumes and defines elements).  Once these are

   discharged by the user, instantiated facts are added to the theory

   in a post-processing phase.

   Additional equations, which are unfolded in facts during

   post-processing, may be given after the keyword @{keyword "where"}.

   This is useful for interpreting concepts introduced through

   definition specification elements.  The equations must be proved.

   Note that if equations are present, the context expression is

   restricted to a locale name.

   The command is aware of interpretations already active in the

   theory, but does not simplify the goal automatically.  In order to

   simplify the proof obligations use methods @{method intro_locales}

   or @{method unfold_locales}.  Post-processing is not applied to

   facts of interpretations that are already active.  This avoids

   duplication of interpreted facts, in particular.  Note that, in the

   case of a locale with import, parts of the interpretation may

   already be active.  The command will only process facts for new

   parts.

   The context expression may be preceded by a name, which takes effect

   in the post-processing of facts.  It is used to prefix fact names,

   for example to avoid accidental hiding of other facts.

   Adding facts to locales has the effect of adding interpreted facts

   to the theory for all active interpretations also.  That is,

   interpretations dynamically participate in any facts added to

   locales.

   \item @{command "interpretation"}~@{text "name \<subseteq> expr"}

   This form of the command interprets @{text expr} in the locale

   @{text name}.  It requires a proof that the specification of @{text

   name} implies the specification of @{text expr}.  As in the

   localized version of the theorem command, the proof is in the

   context of @{text name}.  After the proof obligation has been

   dischared, the facts of @{text expr} become part of locale @{text

   name} as \emph{derived} context elements and are available when the

   context @{text name} is subsequently entered.  Note that, like

   import, this is dynamic: facts added to a locale part of @{text

   expr} after interpretation become also available in @{text name}.

   Like facts of renamed context elements, facts obtained by

   interpretation may be accessed by prefixing with the parameter

   renaming (where the parameters are separated by ``@{text _}'').

   Unlike interpretation in theories, instantiation is confined to the

   renaming of parameters, which may be specified as part of the

   context expression @{text expr}.  Using defined parameters in @{text

   name} one may achieve an effect similar to instantiation, though.

   Only specification fragments of @{text expr} that are not already

   part of @{text name} (be it imported, derived or a derived fragment

   of the import) are considered by interpretation.  This enables

   circular interpretations.

   If interpretations of @{text name} exist in the current theory, the

   command adds interpretations for @{text expr} as well, with the same

   prefix and attributes, although only for fragments of @{text expr}

   that are not interpreted in the theory already.

   \item @{command "interpret"}~@{text "expr insts \<WHERE> eqns"}

   interprets @{text expr} in the proof context and is otherwise

   similar to interpretation in theories.

   \end{description}

   \begin{warn}

     Since attributes are applied to interpreted theorems,

     interpretation may modify the context of common proof tools, e.g.\

     the Simplifier or Classical Reasoner.  Since the behavior of such

     automated reasoning tools is \emph{not} stable under

     interpretation morphisms, manual declarations might have to be

     issued.

   \end{warn}

   \begin{warn}

     An interpretation in a theory may subsume previous

     interpretations.  This happens if the same specification fragment

     is interpreted twice and the instantiation of the second

     interpretation is more general than the interpretation of the

     first.  A warning is issued, since it is likely that these could

     have been generalized in the first place.  The locale package does

     not attempt to remove subsumed interpretations.

   \end{warn}

 *}

 section {* Classes \label{sec:class} *}

 text {*

   A class is a particular locale with \emph{exactly one} type variable

   @{text \<alpha>}.  Beyond the underlying locale, a corresponding type class

   is established which is interpreted logically as axiomatic type

   class \cite{Wenzel:1997:TPHOL} whose logical content are the

   assumptions of the locale.  Thus, classes provide the full

   generality of locales combined with the commodity of type classes

   (notably type-inference).  See \cite{isabelle-classes} for a short

   tutorial.

   \begin{matharray}{rcl}

     @{command_def "class"} & : & @{text "theory \<rightarrow> local_theory"} \\

     @{command_def "instantiation"} & : & @{text "theory \<rightarrow> local_theory"} \\

     @{command_def "instance"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "subclass"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "print_classes"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{command_def "class_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{method_def intro_classes} & : & @{text method} \\

   \end{matharray}

   \begin{rail}

     'class' name '=' ((superclassexpr '+' (contextelem+)) | superclassexpr | (contextelem+)) \\

       'begin'?

     ;

     'instantiation' (nameref + 'and') '::' arity 'begin'

     ;

     'instance'

     ;

     'subclass' target? nameref

     ;

     'print\_classes'

     ;

     'class\_deps'

     ;

     superclassexpr: nameref | (nameref '+' superclassexpr)

     ;

   \end{rail}

   \begin{description}

   \item @{command "class"}~@{text "c = superclasses + body"} defines

   a new class @{text c}, inheriting from @{text superclasses}.  This

   introduces a locale @{text c} with import of all locales @{text

   superclasses}.

   Any @{element "fixes"} in @{text body} are lifted to the global

   theory level (\emph{class operations} @{text "f\<^sub>1, \<dots>,

   f\<^sub>n"} of class @{text c}), mapping the local type parameter

   @{text \<alpha>} to a schematic type variable @{text "?\<alpha> :: c"}.

   Likewise, @{element "assumes"} in @{text body} are also lifted,

   mapping each local parameter @{text "f :: \<tau>[\<alpha>]"} to its

   corresponding global constant @{text "f :: \<tau>[?\<alpha> :: c]"}.  The

   corresponding introduction rule is provided as @{text

   c_class_axioms.intro}.  This rule should be rarely needed directly

   --- the @{method intro_classes} method takes care of the details of

   class membership proofs.

   \item @{command "instantiation"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s

   \<BEGIN>"} opens a theory target (cf.\ \secref{sec:target}) which

   allows to specify class operations @{text "f\<^sub>1, \<dots>, f\<^sub>n"} corresponding

   to sort @{text s} at the particular type instance @{text "(\<alpha>\<^sub>1 :: s\<^sub>1,

   \<dots>, \<alpha>\<^sub>n :: s\<^sub>n) t"}.  A plain @{command "instance"} command in the

   target body poses a goal stating these type arities.  The target is

   concluded by an @{command_ref (local) "end"} command.

   Note that a list of simultaneous type constructors may be given;

   this corresponds nicely to mutual recursive type definitions, e.g.\

   in Isabelle/HOL.

   \item @{command "instance"} in an instantiation target body sets

   up a goal stating the type arities claimed at the opening @{command

   "instantiation"}.  The proof would usually proceed by @{method

   intro_classes}, and then establish the characteristic theorems of

   the type classes involved.  After finishing the proof, the

   background theory will be augmented by the proven type arities.

   \item @{command "subclass"}~@{text c} in a class context for class

   @{text d} sets up a goal stating that class @{text c} is logically

   contained in class @{text d}.  After finishing the proof, class

   @{text d} is proven to be subclass @{text c} and the locale @{text

   c} is interpreted into @{text d} simultaneously.

   \item @{command "print_classes"} prints all classes in the current

   theory.

   \item @{command "class_deps"} visualizes all classes and their

   subclass relations as a Hasse diagram.

   \item @{method intro_classes} repeatedly expands all class

   introduction rules of this theory.  Note that this method usually

   needs not be named explicitly, as it is already included in the

   default proof step (e.g.\ of @{command "proof"}).  In particular,

   instantiation of trivial (syntactic) classes may be performed by a

   single ``@{command ".."}'' proof step.

   \end{description}

 *}

 subsection {* The class target *}

 text {*

   %FIXME check

   A named context may refer to a locale (cf.\ \secref{sec:target}).

   If this locale is also a class @{text c}, apart from the common

   locale target behaviour the following happens.

   \begin{itemize}

   \item Local constant declarations @{text "g[\<alpha>]"} referring to the

   local type parameter @{text \<alpha>} and local parameters @{text "f[\<alpha>]"}

   are accompanied by theory-level constants @{text "g[?\<alpha> :: c]"}

   referring to theory-level class operations @{text "f[?\<alpha> :: c]"}.

   \item Local theorem bindings are lifted as are assumptions.

   \item Local syntax refers to local operations @{text "g[\<alpha>]"} and

   global operations @{text "g[?\<alpha> :: c]"} uniformly.  Type inference

   resolves ambiguities.  In rare cases, manual type annotations are

   needed.

   \end{itemize}

 *}

 subsection {* Old-style axiomatic type classes \label{sec:axclass} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "axclass"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "instance"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

   \end{matharray}

   Axiomatic type classes are Isabelle/Pure's primitive

   \emph{definitional} interface to type classes.  For practical

   applications, you should consider using classes

   (cf.~\secref{sec:classes}) which provide high level interface.

   \begin{rail}

     'axclass' classdecl (axmdecl prop +)

     ;

     'instance' (nameref ('<' | subseteq) nameref | nameref '::' arity)

     ;

   \end{rail}

   \begin{description}

   \item @{command "axclass"}~@{text "c \<subseteq> c\<^sub>1, \<dots>, c\<^sub>n axms"} defines an

   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 (being bound as @{text

   c_class.intro}); this rule is employed by method @{method

   intro_classes} to support instantiation proofs of this class.

   The ``class axioms'' (which are derived from the internal class

   definition) are stored as theorems according to the given name

   specifications; the name space prefix @{text "c_class"} is added

   here.  The full collection of these facts is also stored as @{text

   c_class.axioms}.

   \item @{command "instance"}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} and @{command

   "instance"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s"} setup a goal stating a class

   relation or type arity.  The proof would usually proceed by @{method

   intro_classes}, and then establish the characteristic theorems of

   the type classes involved.  After finishing the proof, the theory

   will be augmented by a type signature declaration corresponding to

   the resulting theorem.

   \end{description}

 *}

 section {* Unrestricted overloading *}

 text {*

   Isabelle/Pure's definitional schemes support certain forms of

   overloading (see \secref{sec:consts}).  At most occassions

   overloading will be used in a Haskell-like fashion together with

   type classes by means of @{command "instantiation"} (see

   \secref{sec:class}).  Sometimes low-level overloading is desirable.

   The @{command "overloading"} target provides a convenient view for

   end-users.

   \begin{matharray}{rcl}

     @{command_def "overloading"} & : & @{text "theory \<rightarrow> local_theory"} \\

   \end{matharray}

   \begin{rail}

     'overloading' \\

     ( string ( '==' | equiv ) term ( '(' 'unchecked' ')' )? + ) 'begin'

   \end{rail}

   \begin{description}

   \item @{command "overloading"}~@{text "x\<^sub>1 \<equiv> c\<^sub>1 :: \<tau>\<^sub>1 \<AND> \<dots> x\<^sub>n \<equiv> c\<^sub>n :: \<tau>\<^sub>n \<BEGIN>"}

   opens a theory target (cf.\ \secref{sec:target}) which allows to

   specify constants with overloaded definitions.  These are identified

   by an explicitly given mapping from variable names @{text "x\<^sub>i"} to

   constants @{text "c\<^sub>i"} at particular type instances.  The

   definitions themselves are established using common specification

   tools, using the names @{text "x\<^sub>i"} as reference to the

   corresponding constants.  The target is concluded by @{command

   (local) "end"}.

   A @{text "(unchecked)"} option disables global dependency checks for

   the corresponding definition, which is occasionally useful for

   exotic overloading.  It is at the discretion of the user to avoid

   malformed theory specifications!

   \end{description}

 *}

 section {* Incorporating ML code \label{sec:ML} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "use"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "ML"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "ML_prf"} & : & @{text "proof \<rightarrow> proof"} \\

     @{command_def "ML_val"} & : & @{text "any \<rightarrow>"} \\

     @{command_def "ML_command"} & : & @{text "any \<rightarrow>"} \\

     @{command_def "setup"} & : & @{text "theory \<rightarrow> theory"} \\

   \end{matharray}

   \begin{mldecls}

     @{index_ML bind_thms: "string * thm list -> unit"} \\

     @{index_ML bind_thm: "string * thm -> unit"} \\

   \end{mldecls}

   \begin{rail}

     'use' name

     ;

     ('ML' | 'ML\_prf' | 'ML\_val' | 'ML\_command' | 'setup') text

     ;

   \end{rail}

   \begin{description}

   \item @{command "use"}~@{text "file"} reads and executes ML

   commands from @{text "file"}.  The current theory context is passed

   down to the ML toplevel and may be modified, using @{ML [source=false]

   "Context.>>"} or derived ML commands.  The file name is checked with

   the @{keyword_ref "uses"} dependency declaration given in the theory

   header (see also \secref{sec:begin-thy}).

   Top-level ML bindings are stored within the (global or local) theory

   context.

   \item @{command "ML"}~@{text "text"} is similar to @{command "use"},

   but executes ML commands directly from the given @{text "text"}.

   Top-level ML bindings are stored within the (global or local) theory

   context.

   \item @{command "ML_prf"} is analogous to @{command "ML"} but works

   within a proof context.

   Top-level ML bindings are stored within the proof context in a

   purely sequential fashion, disregarding the nested proof structure.

   ML bindings introduced by @{command "ML_prf"} are discarded at the

   end of the proof.

   \item @{command "ML_val"} and @{command "ML_command"} are diagnostic

   versions of @{command "ML"}, which means that the context may not be

   updated.  @{command "ML_val"} echos the bindings produced at the ML

   toplevel, but @{command "ML_command"} is silent.

   \item @{command "setup"}~@{text "text"} changes the current theory

   context by applying @{text "text"}, which refers to an ML expression

   of type @{ML_type [source=false] "theory -> theory"}.  This enables

   to initialize any object-logic specific tools and packages written

   in ML, for example.

   \item @{ML bind_thms}~@{text "(name, thms)"} stores a list of

   theorems produced in ML both in the theory context and the ML

   toplevel, associating it with the provided name.  Theorems are put

   into a global ``standard'' format before being stored.

   \item @{ML bind_thm} is similar to @{ML bind_thms} but refers to a

   singleton theorem.

   \end{description}

 *}

 section {* Primitive specification elements *}

 subsection {* Type classes and sorts \label{sec:classes} *}

 text {*

   \begin{matharray}{rcll}

     @{command_def "classes"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "classrel"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\

     @{command_def "defaultsort"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "class_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

   \end{matharray}

   \begin{rail}

     'classes' (classdecl +)

     ;

     'classrel' (nameref ('<' | subseteq) nameref + 'and')

     ;

     'defaultsort' sort

     ;

   \end{rail}

   \begin{description}

   \item @{command "classes"}~@{text "c \<subseteq> c\<^sub>1, \<dots>, c\<^sub>n"} declares class

   @{text c} to be a subclass of existing classes @{text "c\<^sub>1, \<dots>, c\<^sub>n"}.

   Isabelle implicitly maintains the transitive closure of the class

   hierarchy.  Cyclic class structures are not permitted.

   \item @{command "classrel"}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} states subclass

   relations between existing classes @{text "c\<^sub>1"} and @{text "c\<^sub>2"}.

   This is done axiomatically!  The @{command_ref "instance"} command

   (see \secref{sec:axclass}) provides a way to introduce proven class

   relations.

   \item @{command "defaultsort"}~@{text s} makes sort @{text s} the

   new default sort for any type variable that is given explicitly in

   the text, but lacks a sort constraint (wrt.\ the current context).

   Type variables generated by type inference are not affected.

   Usually the default sort is only changed when defining a new

   object-logic.  For example, the default sort in Isabelle/HOL is

   @{text type}, the class of all HOL types.  %FIXME sort antiq?

   When merging theories, the default sorts of the parents are

   logically intersected, i.e.\ the representations as lists of classes

   are joined.

   \item @{command "class_deps"} visualizes the subclass relation,

   using Isabelle's graph browser tool (see also \cite{isabelle-sys}).

   \end{description}

 *}

 subsection {* Types and type abbreviations \label{sec:types-pure} *}

 text {*

   \begin{matharray}{rcll}

     @{command_def "types"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "typedecl"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "arities"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\

   \end{matharray}

   \begin{rail}

     'types' (typespec '=' type infix? +)

     ;

     'typedecl' typespec infix?

     ;

     'arities' (nameref '::' arity +)

     ;

   \end{rail}

   \begin{description}

   \item @{command "types"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = \<tau>"} introduces a

   \emph{type synonym} @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} for the existing type

   @{text "\<tau>"}.  Unlike actual type definitions, as are available in

   Isabelle/HOL for example, type synonyms are merely syntactic

   abbreviations without any logical significance.  Internally, type

   synonyms are fully expanded.

   \item @{command "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} declares a new

   type constructor @{text t}.  If the object-logic defines a base sort

   @{text s}, then the constructor is declared to operate on that, via

   the axiomatic specification @{command arities}~@{text "t :: (s, \<dots>,

   s)s"}.

   \item @{command "arities"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s"} augments

   Isabelle's order-sorted signature of types by new type constructor

   arities.  This is done axiomatically!  The @{command_ref "instance"}

   command (see \secref{sec:axclass}) provides a way to introduce

   proven type arities.

   \end{description}

 *}

 subsection {* Co-regularity of type classes and arities *}

 text {* The class relation together with the collection of

   type-constructor arities must obey the principle of

   \emph{co-regularity} as defined below.

   \medskip For the subsequent formulation of co-regularity we assume

   that the class relation is closed by transitivity and reflexivity.

   Moreover the collection of arities @{text "t :: (\<^vec>s)c"} is

   completed such that @{text "t :: (\<^vec>s)c"} and @{text "c \<subseteq> c'"}

   implies @{text "t :: (\<^vec>s)c'"} for all such declarations.

   Treating sorts as finite sets of classes (meaning the intersection),

   the class relation @{text "c\<^sub>1 \<subseteq> c\<^sub>2"} is extended to sorts as

   follows:

   \[

     @{text "s\<^sub>1 \<subseteq> s\<^sub>2 \<equiv> \<forall>c\<^sub>2 \<in> s\<^sub>2. \<exists>c\<^sub>1 \<in> s\<^sub>1. c\<^sub>1 \<subseteq> c\<^sub>2"}

   \]

   This relation on sorts is further extended to tuples of sorts (of

   the same length) in the component-wise way.

   \smallskip Co-regularity of the class relation together with the

   arities relation means:

   \[

     @{text "t :: (\<^vec>s\<^sub>1)c\<^sub>1 \<Longrightarrow> t :: (\<^vec>s\<^sub>2)c\<^sub>2 \<Longrightarrow> c\<^sub>1 \<subseteq> c\<^sub>2 \<Longrightarrow> \<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2"}

   \]

   \noindent for all such arities.  In other words, whenever the result

   classes of some type-constructor arities are related, then the

   argument sorts need to be related in the same way.

   \medskip Co-regularity is a very fundamental property of the

   order-sorted algebra of types.  For example, it entails principle

   types and most general unifiers, e.g.\ see \cite{nipkow-prehofer}.

 *}

 subsection {* Constants and definitions \label{sec:consts} *}

 text {*

   Definitions essentially express abbreviations within the logic.  The

   simplest form of a definition is @{text "c :: \<sigma> \<equiv> t"}, where @{text

   c} is a newly declared constant.  Isabelle also allows derived forms

   where the arguments of @{text c} appear on the left, abbreviating a

   prefix of @{text \<lambda>}-abstractions, e.g.\ @{text "c \<equiv> \<lambda>x y. t"} may be

   written more conveniently as @{text "c x y \<equiv> t"}.  Moreover,

   definitions may be weakened by adding arbitrary pre-conditions:

   @{text "A \<Longrightarrow> c x y \<equiv> t"}.

   \medskip The built-in well-formedness conditions for definitional

   specifications are:

   \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 @{text "0 :: nat \<equiv> length ([] ::

   \<alpha> list)"} for example.

   \item The definition must not be recursive.  Most object-logics

   provide definitional principles that can be used to express

   recursion safely.

   \end{itemize}

   Overloading means that a constant being declared as @{text "c :: \<alpha>

   decl"} may be defined separately on type instances @{text "c ::

   (\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n) t decl"} for each type constructor @{text

   t}.  The right-hand side may mention overloaded constants

   recursively at type instances corresponding to the immediate

   argument types @{text "\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n"}.  Incomplete

   specification patterns impose global constraints on all occurrences,

   e.g.\ @{text "d :: \<alpha> \<times> \<alpha>"} on the left-hand side means that all

   corresponding occurrences on some right-hand side need to be an

   instance of this, general @{text "d :: \<alpha> \<times> \<beta>"} will be disallowed.

   \begin{matharray}{rcl}

     @{command_def "consts"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "defs"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "constdefs"} & : & @{text "theory \<rightarrow> theory"} \\

   \end{matharray}

   \begin{rail}

     'consts' ((name '::' type mixfix?) +)

     ;

     'defs' ('(' 'unchecked'? 'overloaded'? ')')? \\ (axmdecl prop +)

     ;

   \end{rail}

   \begin{rail}

     'constdefs' structs? (constdecl? constdef +)

     ;

     structs: '(' 'structure' (vars + 'and') ')'

     ;

     constdecl:  ((name '::' type mixfix | name '::' type | name mixfix) 'where'?) | name 'where'

     ;

     constdef: thmdecl? prop

     ;

   \end{rail}

   \begin{description}

   \item @{command "consts"}~@{text "c :: \<sigma>"} declares constant @{text

   c} to have any instance of type scheme @{text \<sigma>}.  The optional

   mixfix annotations may attach concrete syntax to the constants

   declared.

   \item @{command "defs"}~@{text "name: eqn"} introduces @{text eqn}

   as a definitional axiom for some existing constant.

   The @{text "(unchecked)"} option disables global dependency checks

   for this definition, which is occasionally useful for exotic

   overloading.  It is at the discretion of the user to avoid malformed

   theory specifications!

   The @{text "(overloaded)"} option declares definitions to be

   potentially overloaded.  Unless this option is given, a warning

   message would be issued for any definitional equation with a more

   special type than that of the corresponding constant declaration.

   \item @{command "constdefs"} combines constant declarations and

   definitions, with type-inference taking care of the most general

   typing of the given specification (the optional type constraint may

   refer to type-inference dummies ``@{text _}'' as usual).  The

   resulting type declaration needs to agree with that of the

   specification; overloading is \emph{not} supported here!

   The constant name may be omitted altogether, if neither type nor

   syntax declarations are given.  The canonical name of the

   definitional axiom for constant @{text c} will be @{text c_def},

   unless specified otherwise.  Also note that the given list of

   specifications is processed in a strictly sequential manner, with

   type-checking being performed independently.

   An optional initial context of @{text "(structure)"} declarations

   admits use of indexed syntax, using the special symbol @{verbatim

   "\<index>"} (printed as ``@{text "\<index>"}'').  The latter concept is

   particularly useful with locales (see also \secref{sec:locale}).

   \end{description}

 *}

 section {* Axioms and theorems \label{sec:axms-thms} *}

 text {*

   \begin{matharray}{rcll}

     @{command_def "axioms"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\

     @{command_def "lemmas"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "theorems"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   \end{matharray}

   \begin{rail}

     'axioms' (axmdecl prop +)

     ;

     ('lemmas' | 'theorems') target? (thmdef? thmrefs + 'and')

     ;

   \end{rail}

   \begin{description}

   \item @{command "axioms"}~@{text "a: \<phi>"} introduces arbitrary

   statements as axioms of the meta-logic.  In fact, axioms are

   ``axiomatic theorems'', and may be referred later just as any other

   theorem.

   Axioms are usually only introduced when declaring new logical

   systems.  Everyday work is typically done the hard way, with proper

   definitions and proven theorems.

   \item @{command "lemmas"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"} retrieves and stores

   existing facts in the theory context, or the specified target

   context (see also \secref{sec:target}).  Typical applications would

   also involve attributes, to declare Simplifier rules, for example.

   \item @{command "theorems"} is essentially the same as @{command

   "lemmas"}, but marks the result as a different kind of facts.

   \end{description}

 *}

 section {* Oracles *}

 text {* 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 the responsibility of the user to ensure that the external

   reasoner is as trustworthy as the application requires.  Another

   typical source of errors is the linkup between Isabelle and the

   external tool, not just its concrete implementation, but also the

   required translation between two different logical environments.

   Isabelle merely guarantees well-formedness of the propositions being

   asserted, and records within the internal derivation object how

   presumed theorems depend on unproven suppositions.

   \begin{matharray}{rcl}

     @{command_def "oracle"} & : & @{text "theory \<rightarrow> theory"} \\

   \end{matharray}

   \begin{rail}

     'oracle' name '=' text

     ;

   \end{rail}

   \begin{description}

   \item @{command "oracle"}~@{text "name = text"} turns the given ML

   expression @{text "text"} of type @{ML_text "'a -> cterm"} into an

   ML function of type @{ML_text "'a -> thm"}, which is bound to the

   global identifier @{ML_text name}.  This acts like an infinitary

   specification of axioms!  Invoking the oracle only works within the

   scope of the resulting theory.

   \end{description}

   See @{"file" "~~/src/FOL/ex/Iff_Oracle.thy"} for a worked example of

   defining a new primitive rule as oracle, and turning it into a proof

   method.

 *}

 section {* Name spaces *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "global"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "local"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "hide"} & : & @{text "theory \<rightarrow> theory"} \\

   \end{matharray}

   \begin{rail}

     'hide' ('(open)')? name (nameref + )

     ;

   \end{rail}

   Isabelle organizes any kind of name declarations (of types,

   constants, theorems etc.) by separate hierarchically structured name

   spaces.  Normally the user does not have to control the behavior of

   name spaces by hand, yet the following commands provide some way to

   do so.

   \begin{description}

   \item @{command "global"} and @{command "local"} change the current

   name declaration mode.  Initially, theories start in @{command

   "local"} mode, causing all names to be automatically qualified by

   the theory name.  Changing this to @{command "global"} causes all

   names to be declared without the theory prefix, until @{command

   "local"} is declared again.

   Note that global names are prone to get hidden accidently later,

   when qualified names of the same base name are introduced.

   \item @{command "hide"}~@{text "space names"} fully removes

   declarations from a given name space (which may be @{text "class"},

   @{text "type"}, @{text "const"}, or @{text "fact"}); with the @{text

   "(open)"} option, only the base name is hidden.  Global

   (unqualified) names may never be hidden.

   Note that hiding name space accesses has no impact on logical

   declarations --- they remain valid internally.  Entities that are no

   longer accessible to the user are printed with the special qualifier

   ``@{text "??"}'' prefixed to the full internal name.

   \end{description}

 *}

 end