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' ('(pervasive)')? 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.

   If the @{text "(pervasive)"} option is given, the corresponding

   declaration is applied to all possible contexts involved, including

   the global background theory.

   \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 parametric 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 expressions \label{sec:locale-expr} *}

 text {*

   A \emph{locale expression} denotes a structured context composed of

   instances of existing locales.  The context consists of a list of

   instances of declaration elements from the locales.  Two locale

   instances are equal if they are of the same locale and the

   parameters are instantiated with equivalent terms.  Declaration

   elements from equal instances are never repeated, thus avoiding

   duplicate declarations.

   \indexouternonterm{localeexpr}

   \begin{rail}

     localeexpr: (instance + '+') ('for' (fixes + 'and'))?

     ;

     instance: (qualifier ':')? nameref (posinsts | namedinsts)

     ;

     qualifier: name ('?' | '!')?

     ;

     posinsts: (term | '_')*

     ;

     namedinsts: 'where' (name '=' term + 'and')

     ;

   \end{rail}

   A locale instance consists of a reference to a locale and either

   positional or named parameter instantiations.  Identical

   instantiations (that is, those that instante a parameter by itself)

   may be omitted.  The notation `\_' enables to omit the instantiation

   for a parameter inside a positional instantiation.

   Terms in instantiations are from the context the locale expressions

   is declared in.  Local names may be added to this context with the

   optional for clause.  In addition, syntax declarations from one

   instance are effective when parsing subsequent instances of the same

   expression.

   Instances have an optional qualifier which applies to names in

   declarations.  Names include local definitions and theorem names.

   If present, the qualifier itself is either optional

   (``\texttt{?}''), which means that it may be omitted on input of the

   qualified name, or mandatory (``\texttt{!}'').  If neither

   ``\texttt{?}'' nor ``\texttt{!}'' are present, the command's default

   is used.  For @{command "interpretation"} and @{command "interpret"}

   the default is ``mandatory'', for @{command "locale"} and @{command

   "sublocale"} the default is ``optional''.

 *}

 subsection {* Locale declarations *}

 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{contextelem}

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

   \indexisarelem{defines}\indexisarelem{notes}

   \begin{rail}

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

     ;

     'print\_locale' '!'? nameref

     ;

     locale: contextelem+ | localeexpr ('+' (contextelem+))?

     ;

     contextelem:

     'fixes' (fixes + 'and')

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

     | 'assumes' (props + 'and')

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

     | '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 locale expression; see

   \secref{sec:proof-context} above.  Its for clause defines the local

   parameters of the @{text import}.  In addition, locale parameters

   whose instantance is omitted automatically extend the (possibly

   empty) for clause: they are inserted at its beginning.  This means

   that these parameters may be referred to from within the expression

   and also in the subsequent context elements and provides a

   notational convenience for the inheritance of parameters in locale

   declarations.

   The @{text body} consists of context elements.

   \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}.  This

   element is deprecated.  The type constraint should be introduced in

   the for clause or the relevant @{element "fixes"} element.

   \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 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 "locale"} prints the

   contents of the named locale.  The command omits @{element "notes"}

   elements by default.  Use @{command "print_locale"}@{text "!"} to

   have 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 {* Locale interpretations *}

 text {*

   Locale 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 locales @{command

   "sublocale"}, theories (command @{command "interpretation"}) and

   also within a proof body (command @{command "interpret"}).

   \begin{matharray}{rcl}

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

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

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

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

   \end{matharray}

   \indexouternonterm{interp}

   \begin{rail}

     'sublocale' nameref ('<' | subseteq) localeexpr

     ;

     'interpretation' localeepxr equations?

     ;

     'interpret' localeexpr equations?

     ;

     'print\_interps' nameref

     ;

     equations: 'where' (thmdecl? prop + 'and')

     ;

   \end{rail}

   \begin{description}

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

   interprets @{text expr} in the locale @{text name}.  A proof that

   the specification of @{text name} implies the specification of

   @{text expr} is required.  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}.

   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 in this process.  This enables

   circular interpretations to the extent that no infinite chains are

   generated in the locale hierarchy.

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

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

   qualifier, although only for fragments of @{text expr} that are not

   interpreted in the theory already.

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

   interprets @{text expr} in the theory.  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 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.

   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.

   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 "interpret"}~@{text "expr \<WHERE> eqns"} interprets

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

   interpretation in theories.  Note that rewrite rules given to

   @{command "interpret"} should be explicitly universally quantified.

   \item @{command "print_interps"}~@{text "locale"} lists all

   interpretations of @{text "locale"} in the current theory or proof

   context, including those due to a combination of a @{command

   "interpretation"} or @{command "interpret"} and one or several

   @{command "sublocale"} declarations.

   \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.  As the behavior of such

     tools is \emph{not} stable under interpretation morphisms, manual

     declarations might have to be added to the target context of the

     interpretation to revert such declarations.

   \end{warn}

   \begin{warn}

     An interpretation in a theory or proof context 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.  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 "instance"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

     @{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'

     ;

     'instance' (nameref + 'and') '::' arity

     ;

     'subclass' target? nameref

     ;

     'instance' nameref ('<' | subseteq) 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 mutually 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.

   On the theory level, @{command "instance"}~@{text "t :: (s\<^sub>1, \<dots>,

   s\<^sub>n)s"} provides a convenient way to instantiate a type class with no

   need to specify operations: one can continue with the

   instantiation proof immediately.

   \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.

   A weakend form of this is available through a further variant of

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

   a proof that class @{text "c\<^isub>2"} implies @{text "c\<^isub>1"} without reference

   to the underlying locales;  this is useful if the properties to prove

   the logical connection are not sufficent on the locale level but on

   the theory level.

   \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 {* 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}.

 *}

 section {* Unrestricted overloading *}

 text {*

   Isabelle/Pure's definitional schemes support certain forms of

   overloading (see \secref{sec:consts}).  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}.  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 (see \secref{sec:consts} for a precise description).

   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"} \\

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

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

   \end{matharray}

   \begin{rail}

     'use' name

     ;

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

     ;

     'attribute\_setup' name '=' text 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

   "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 "theory -> theory"}.  This enables to initialize

   any object-logic specific tools and packages written in ML, for

   example.

   \item @{command "local_setup"} is similar to @{command "setup"} for

   a local theory context, and an ML expression of type @{ML_type

   "local_theory -> local_theory"}.  This allows to

   invoke local theory specification packages without going through

   concrete outer syntax, for example.

   \item @{command "attribute_setup"}~@{text "name = text description"}

   defines an attribute in the current theory.  The given @{text

   "text"} has to be an ML expression of type

   @{ML_type "attribute context_parser"}, cf.\ basic parsers defined in

   structure @{ML_struct Args} and @{ML_struct Attrib}.

   In principle, attributes can operate both on a given theorem and the

   implicit context, although in practice only one is modified and the

   other serves as parameter.  Here are examples for these two cases:

   \end{description}

 *}

     attribute_setup my_rule = {*

       Attrib.thms >> (fn ths =>

         Thm.rule_attribute (fn context: Context.generic => fn th: thm =>

           let val th' = th OF ths

           in th' end)) *}  "my rule"

     attribute_setup my_declaration = {*

       Attrib.thms >> (fn ths =>

         Thm.declaration_attribute (fn th: thm => fn context: Context.generic =>

           let val context' = context

           in context' end)) *}  "my declaration"

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

   \end{matharray}

   \begin{rail}

     'classes' (classdecl +)

     ;

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

     ;

     'default\_sort' 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 "subclass"} and

   @{command_ref "instance"} commands (see \secref{sec:class}) provide

   a way to introduce proven class relations.

   \item @{command "default_sort"}~@{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

   @{class type}, the class of all HOL types.

   When merging theories, the default sorts of the parents are

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

   are joined.

   \end{description}

 *}

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

 text {*

   \begin{matharray}{rcll}

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

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

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

   \end{matharray}

   \begin{rail}

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

     ;

     'typedecl' typespec mixfix?

     ;

     '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 "instantiation"}

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

   proven type arities.

   \end{description}

 *}

 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}

   The right-hand side of overloaded definitions 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"} \\

   \end{matharray}

   \begin{rail}

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

     ;

     'defs' ('(' 'unchecked'? 'overloaded'? ')')? \\ (axmdecl 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.

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

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

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

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

   \end{matharray}

   \begin{rail}

     ( 'hide\_class' | 'hide\_type' | 'hide\_const' | 'hide\_fact' ) ('(open)')? (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 "hide_class"}~@{text names} fully removes class

   declarations from a given name space; 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.

   \item @{command "hide_type"}, @{command "hide_const"}, and @{command

   "hide_fact"} are similar to @{command "hide_class"}, but hide types,

   constants, and facts, respectively.

   \end{description}

 *}

 end