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"} \\

    @{command_def "local_setup"} & : & @{text "local_theory \<rightarrow> local_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' | 'local\_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

  "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 @{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"}