theory Spec

 imports Base Main

 begin

 chapter {* 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.

   @{rail \<open>

     @@{command theory} @{syntax name} imports keywords? \<newline> @'begin'

     ;

     imports: @'imports' (@{syntax name} +)

     ;

     keywords: @'keywords' (keyword_decls + @'and')

     ;

     keyword_decls: (@{syntax string} +)

       ('::' @{syntax name} @{syntax tags})? ('==' @{syntax name})?

   \<close>}

   \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 takes care

   that sources contributing to the development graph are always

   up-to-date: changed files are automatically rechecked whenever a

   theory header specification is processed.

   The optional @{keyword_def "keywords"} specification declares outer

   syntax (\chref{ch:outer-syntax}) that is introduced in this theory

   later on (rare in end-user applications).  Both minor keywords and

   major keywords of the Isar command language need to be specified, in

   order to make parsing of proof documents work properly.  Command

   keywords need to be classified according to their structural role in

   the formal text.  Examples may be seen in Isabelle/HOL sources

   itself, such as @{keyword "keywords"}~@{verbatim "\"typedef\""}

   @{text ":: thy_goal"} or @{keyword "keywords"}~@{verbatim

   "\"datatype\""} @{text ":: thy_decl"} for theory-level declarations

   with and without proof, respectively.  Additional @{syntax tags}

   provide defaults for document preparation (\secref{sec:tags}).

   It is possible to specify an alternative completion via @{text "==

   text"}, while the default is the corresponding keyword name.

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

   definition.  Note that some other commands, e.g.\ local theory

   targets @{command locale} or @{command class} may involve a

   @{keyword "begin"} that needs to be matched by @{command (local)

   "end"}, according to the usual rules for nested blocks.

   \end{description}

 *}

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

 text {*

   \begin{matharray}{rcll}

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

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

   \end{matharray}

   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.

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

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

   signifies the global theory context.

   \emph{Unnamed contexts} may introduce additional parameters and

   assumptions, and results produced in the context are generalized

   accordingly.  Such auxiliary contexts may be nested within other

   targets, like @{command "locale"}, @{command "class"}, @{command

   "instantiation"}, @{command "overloading"}.

   @{rail \<open>

     @@{command context} @{syntax nameref} @'begin'

     ;

     @@{command context} @{syntax_ref "includes"}? (@{syntax context_elem} * ) @'begin'

     ;

     @{syntax_def target}: '(' @'in' @{syntax nameref} ')'

   \<close>}

   \begin{description}

   \item @{command "context"}~@{text "c \<BEGIN>"} opens a named

   context, by recommencing an existing locale or class @{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 "context"}~@{text "bundles elements \<BEGIN>"} opens

   an unnamed context, by extending the enclosing global or local

   theory target by the given declaration bundles (\secref{sec:bundle})

   and context elements (@{text "\<FIXES>"}, @{text "\<ASSUMES>"}

   etc.).  This means any results stemming from definitions and proofs

   in the extended context will be exported into the enclosing target

   by lifting over extra parameters and premises.

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

   according to the nesting of contexts.  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"}.

   \medskip The Isabelle/HOL library contains numerous applications of

   locales and classes, e.g.\ see @{file "~~/src/HOL/Algebra"}.  An

   example for an unnamed auxiliary contexts is given in @{file

   "~~/src/HOL/Isar_Examples/Group_Context.thy"}.  *}

 section {* Bundled declarations \label{sec:bundle} *}

 text {*

   \begin{matharray}{rcl}

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

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

     @{command_def "include"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

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

     @{keyword_def "includes"} & : & syntax \\

   \end{matharray}

   The outer syntax of fact expressions (\secref{sec:syn-att}) involves

   theorems and attributes, which are evaluated in the context and

   applied to it.  Attributes may declare theorems to the context, as

   in @{text "this_rule [intro] that_rule [elim]"} for example.

   Configuration options (\secref{sec:config}) are special declaration

   attributes that operate on the context without a theorem, as in

   @{text "[[show_types = false]]"} for example.

   Expressions of this form may be defined as \emph{bundled

   declarations} in the context, and included in other situations later

   on.  Including declaration bundles augments a local context casually

   without logical dependencies, which is in contrast to locales and

   locale interpretation (\secref{sec:locale}).

   @{rail \<open>

     @@{command bundle} @{syntax target}? \<newline>

     @{syntax name} '=' @{syntax thmrefs} (@'for' (@{syntax vars} + @'and'))?

     ;

     (@@{command include} | @@{command including}) (@{syntax nameref}+)

     ;

     @{syntax_def "includes"}: @'includes' (@{syntax nameref}+)

   \<close>}

   \begin{description}

   \item @{command bundle}~@{text "b = decls"} defines a bundle of

   declarations in the current context.  The RHS is similar to the one

   of the @{command declare} command.  Bundles defined in local theory

   targets are subject to transformations via morphisms, when moved

   into different application contexts; this works analogously to any

   other local theory specification.

   \item @{command print_bundles} prints the named bundles that are

   available in the current context.

   \item @{command include}~@{text "b\<^sub>1 \<dots> b\<^sub>n"} includes the declarations

   from the given bundles into the current proof body context.  This is

   analogous to @{command "note"} (\secref{sec:proof-facts}) with the

   expanded bundles.

   \item @{command including} is similar to @{command include}, but

   works in proof refinement (backward mode).  This is analogous to

   @{command "using"} (\secref{sec:proof-facts}) with the expanded

   bundles.

   \item @{keyword "includes"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"} is similar to

   @{command include}, but works in situations where a specification

   context is constructed, notably for @{command context} and long

   statements of @{command theorem} etc.

   \end{description}

   Here is an artificial example of bundling various configuration

   options: *}

 bundle trace = [[simp_trace, linarith_trace, metis_trace, smt_trace]]

 lemma "x = x"

   including trace by metis

 section {* Term definitions \label{sec:term-definitions} *}

 text {*

   \begin{matharray}{rcll}

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

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

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

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

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

   \end{matharray}

   Term definitions may either happen within the logic (as equational axioms

   of a certain form, see also see \secref{sec:consts}), or outside of it as

   rewrite system on abstract syntax. The second form is called

   ``abbreviation''.

   @{rail \<open>

     @@{command definition} @{syntax target}? \<newline>

       (decl @'where')? @{syntax thmdecl}? @{syntax prop}

     ;

     @@{command abbreviation} @{syntax target}? @{syntax mode}? \<newline>

       (decl @'where')? @{syntax prop}

     ;

     decl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}?

   \<close>}

   \begin{description}

   \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 "print_defn_rules"} prints the definitional rewrite rules

   declared via @{attribute defn} in the current context.

   \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 {* Axiomatizations \label{sec:axiomatizations} *}

 text {*

   \begin{matharray}{rcll}

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

   \end{matharray}

   @{rail \<open>

     @@{command axiomatization} @{syntax "fixes"}? (@'where' specs)?

     ;

     @{syntax_def "fixes"}: ((@{syntax name} ('::' @{syntax type})?

       @{syntax mixfix}? | @{syntax vars}) + @'and')

     ;

     specs: (@{syntax thmdecl}? @{syntax props} + @'and')

   \<close>}

   \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 for the same constants

   later on, but it is always possible do emit axiomatizations without

   referring to particular constants. Note that lack of precise dependency

   tracking of axiomatizations may disrupt the well-formedness of an

   otherwise definitional theory.

   Axiomatization is restricted to a global theory context: support for local

   theory targets \secref{sec:target} would introduce an extra dimension of

   uncertainty what the written specifications really are, and make it

   infeasible to argue why they are correct.

   Axiomatic specifications are required when declaring a new logical system

   within Isabelle/Pure, but in an application environment like Isabelle/HOL

   the user normally stays within definitional mechanisms provided by the

   logic and its libraries.

   \end{description}

 *}

 section {* Generic declarations *}

 text {*

   \begin{matharray}{rcl}

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

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

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

   \end{matharray}

   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.

   @{rail \<open>

     (@@{command declaration} | @@{command syntax_declaration})

       ('(' @'pervasive' ')')? \<newline> @{syntax target}? @{syntax text}

     ;

     @@{command declare} @{syntax target}? (@{syntax thmrefs} + @'and')

   \<close>}

   \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 "syntax_declaration"} is similar to @{command

   "declaration"}, but is meant to affect only ``syntactic'' tools by

   convention (such as notation and type-checking information).

   \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:theorems}), so @{command "declare"} only has the effect

   of applying attributes as included in the theorem specification.

   \end{description}

 *}

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

 text {*

   A locale is a functor that maps parameters (including implicit type

   parameters) and a specification to a list of declarations.  The

   syntax of locales is modeled after the Isar proof context commands

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

   Locale hierarchies are supported by maintaining a graph of

   dependencies between locale instances in the global theory.

   Dependencies may be introduced through import (where a locale is

   defined as sublocale of the imported instances) or by proving that

   an existing locale is a sublocale of one or several locale

   instances.

   A locale may be opened with the purpose of appending to its list of

   declarations (cf.\ \secref{sec:target}).  When opening a locale

   declarations from all dependencies are collected and are presented

   as a local theory.  In this process, which is called \emph{roundup},

   redundant locale instances are omitted.  A locale instance is

   redundant if it is subsumed by an instance encountered earlier.  A

   more detailed description of this process is available elsewhere

   \cite{Ballarin2014}.

 *}

 subsection {* Locale expressions \label{sec:locale-expr} *}

 text {*

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

   of existing locales.  The context consists of the declaration

   elements from the locale instances.  Redundant locale instances are

   omitted according to roundup.

   @{rail \<open>

     @{syntax_def locale_expr}: (instance + '+') (@'for' (@{syntax "fixes"} + @'and'))?

     ;

     instance: (qualifier ':')? @{syntax nameref} (pos_insts | named_insts)

     ;

     qualifier: @{syntax name} ('?' | '!')?

     ;

     pos_insts: ('_' | @{syntax term})*

     ;

     named_insts: @'where' (@{syntax name} '=' @{syntax term} + @'and')

   \<close>}

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

   positional or named parameter instantiations.  Identical

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

   may be omitted.  The notation `@{text "_"}' 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 @{keyword "for"} clause.  This is useful for shadowing names

   bound in outer contexts, and for declaring syntax.  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''.  Qualifiers play no role

   in determining whether one locale instance subsumes another.

 *}

 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>"} \\

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

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

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

   \end{matharray}

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

   \indexisarelem{defines}\indexisarelem{notes}

   @{rail \<open>

     @@{command locale} @{syntax name} ('=' @{syntax locale})? @'begin'?

     ;

     @@{command print_locale} '!'? @{syntax nameref}

     ;

     @{syntax_def locale}: @{syntax context_elem}+ |

       @{syntax locale_expr} ('+' (@{syntax context_elem}+))?

     ;

     @{syntax_def context_elem}:

       @'fixes' (@{syntax "fixes"} + @'and') |

       @'constrains' (@{syntax name} '::' @{syntax type} + @'and') |

       @'assumes' (@{syntax props} + @'and') |

       @'defines' (@{syntax thmdecl}? @{syntax prop} @{syntax prop_pat}? + @'and') |

       @'notes' (@{syntax thmdef}? @{syntax thmrefs} + @'and')

   \<close>}

   \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 locale expression; see

   \secref{sec:proof-context} above.  Its @{keyword "for"} clause defines

   the parameters of @{text import}.  These are parameters of

   the defined locale.  Locale parameters whose instantiation is

   omitted automatically extend the (possibly empty) @{keyword "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

   "("}@{keyword_ref "structure"}@{text ")"}'' 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 @{keyword "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.

   \end{description}

   Both @{element "assumes"} and @{element "defines"} elements

   contribute to the locale specification.  When defining an operation

   derived from the parameters, @{command "definition"}

   (\secref{sec:term-definitions}) is usually more appropriate.

   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 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 @{command "locale_deps"} visualizes all locales and their

   relations as a Hasse diagram. This includes locales defined as type

   classes (\secref{sec:class}).  See also @{command

   "print_dependencies"} below.

   \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 descend 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 interpretation *}

 text {*

   \begin{matharray}{rcl}

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

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

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

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

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

   \end{matharray}

   Locales may be instantiated, and the resulting instantiated

   declarations added to the current context.  This requires proof (of

   the instantiated specification) and is called \emph{locale

   interpretation}.  Interpretation is possible in locales (@{command

   "sublocale"}), global and local theories (@{command

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

   "interpret"}).

   @{rail \<open>

     @@{command interpretation} @{syntax locale_expr} equations?

     ;

     @@{command interpret} @{syntax locale_expr} equations?

     ;

     @@{command sublocale} (@{syntax nameref} ('<' | '\<subseteq>'))? @{syntax locale_expr} \<newline>

       equations?

     ;

     @@{command print_dependencies} '!'? @{syntax locale_expr}

     ;

     @@{command print_interps} @{syntax nameref}

     ;

     equations: @'where' (@{syntax thmdecl}? @{syntax prop} + @'and')

   \<close>}

   \begin{description}

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

   interprets @{text expr} in a global or local theory.  The command

   generates proof obligations for the instantiated specifications.

   Once these are discharged by the user, instantiated declarations (in

   particular, facts) are added to the theory in a post-processing

   phase.

   The command is aware of interpretations that are already active.

   Post-processing is achieved through a variant of roundup that takes

   the interpretations of the current global or local theory into

   account.  In order to simplify the proof obligations according to

   existing interpretations use methods @{method intro_locales} or

   @{method unfold_locales}.

   When adding declarations to locales, interpreted versions of these

   declarations are added to the global theory for all interpretations

   in the global theory as well. That is, interpretations in global

   theories dynamically participate in any declarations added to

   locales.

   In contrast, the lifetime of an interpretation in a local theory is

   limited to the current context block.  At the closing @{command end}

   of the block the interpretation and its declarations disappear.

   This enables establishing facts based on interpretations without

   creating permanent links to the interpreted locale instances, as

   would be the case with @{command sublocale}.

   While @{command "interpretation"}~@{text "(\<IN> c)

   \<dots>"} is technically possible, it is not useful since its result is

   discarded immediately.

   Free variables in the interpreted expression are allowed.  They are

   turned into schematic variables in the generated declarations.  In

   order to use a free variable whose name is already bound in the

   context --- for example, because a constant of that name exists ---

   add it to the @{keyword "for"} clause.

   The equations @{text eqns} yield \emph{rewrite morphisms}, which are

   unfolded during post-processing and are useful for interpreting

   concepts introduced through definitions.  The equations must be

   proved.

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

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

   interpretation in local theories.  Note that for @{command

   "interpret"} the @{text eqns} should be

   explicitly universally quantified.

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

   eqns"}

   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 discharged, the locale hierarchy is

   changed as if @{text name} imported @{text expr} (hence the name

   @{command "sublocale"}).  When the context of @{text name} is

   subsequently entered, traversing the locale hierarchy will involve

   the locale instances of @{text expr}, and their declarations will be

   added to the context.  This makes @{command "sublocale"}

   dynamic: extensions of a locale that is instantiated in @{text expr}

   may take place after the @{command "sublocale"} declaration and

   still become available in the context.  Circular @{command

   "sublocale"} declarations are allowed as long as they do not lead to

   infinite chains.

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

   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.

   The equations @{text eqns} amend the morphism through

   which @{text expr} is interpreted.  This enables mapping definitions

   from the interpreted locales to entities of @{text name} and can

   help break infinite chains induced by circular @{command

   "sublocale"} declarations.

   In a named context block the @{command sublocale} command may also

   be used, but the locale argument must be omitted.  The command then

   refers to the locale (or class) target of the context block.

   \item @{command "print_dependencies"}~@{text "expr"} is useful for

   understanding the effect of an interpretation of @{text "expr"} in

   the current context.  It lists all locale instances for which

   interpretations would be added to the current context.  Variant

   @{command "print_dependencies"}@{text "!"} does not generalize

   parameters and assumes an empty context --- that is, it prints all

   locale instances that would be considered for interpretation.  The

   latter is useful for understanding the dependencies of a locale

   expression.

   \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 an @{command

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

   @{command "sublocale"} declarations.

   \end{description}

   \begin{warn}

     If a global theory inherits declarations (body elements) for a

     locale from one parent and an interpretation of that locale from

     another parent, the interpretation will not be applied to the

     declarations.

   \end{warn}

   \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 local 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 {*

   \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 "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}

   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.

   @{rail \<open>

     @@{command class} class_spec @'begin'?

     ;

     class_spec: @{syntax name} '='

       ((@{syntax nameref} '+' (@{syntax context_elem}+)) |

         @{syntax nameref} | (@{syntax context_elem}+))      

     ;

     @@{command instantiation} (@{syntax nameref} + @'and') '::' @{syntax arity} @'begin'

     ;

     @@{command instance} (() | (@{syntax nameref} + @'and') '::' @{syntax arity} |

       @{syntax nameref} ('<' | '\<subseteq>') @{syntax nameref} )

     ;

     @@{command subclass} @{syntax target}? @{syntax nameref}

   \<close>}

   \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 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 weakened 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\<^sub>2"} implies @{text "c\<^sub>1"} without reference

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

   the logical connection are not sufficient 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 {*

   \begin{matharray}{rcl}

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

   \end{matharray}

   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 occasions

   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.

   @{rail \<open>

     @@{command overloading} ( spec + ) @'begin'

     ;

     spec: @{syntax name} ( '==' | '\<equiv>' ) @{syntax term} ( '(' @'unchecked' ')' )?

   \<close>}

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

     @{command_def "ML_file"} & : & @{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{tabular}{rcll}

     @{attribute_def ML_print_depth} & : & @{text attribute} & default 10 \\

     @{attribute_def ML_source_trace} & : & @{text attribute} & default @{text false} \\

     @{attribute_def ML_exception_trace} & : & @{text attribute} & default @{text false} \\

   \end{tabular}

   @{rail \<open>

     (@@{command SML_file} | @@{command ML_file}) @{syntax name}

     ;

     (@@{command ML} | @@{command ML_prf} | @@{command ML_val} |

       @@{command ML_command} | @@{command setup} | @@{command local_setup}) @{syntax text}

     ;

     @@{command attribute_setup} @{syntax name} '=' @{syntax text} @{syntax text}?

   \<close>}

   \begin{description}

   \item @{command "SML_file"}~@{text "name"} reads and evaluates the

   given Standard ML file.  Top-level SML bindings are stored within

   the theory context; the initial environment is restricted to the

   Standard ML implementation of Poly/ML, without the many add-ons of

   Isabelle/ML.  Multiple @{command "SML_file"} commands may be used to

   build larger Standard ML projects, independently of the regular

   Isabelle/ML environment.

   \item @{command "ML_file"}~@{text "name"} reads and evaluates the

   given ML file.  The current theory context is passed down to the ML

   toplevel and may be modified, using @{ML "Context.>>"} or derived ML

   commands.  Top-level ML bindings are stored within the (global or

   local) theory context.

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

   "ML_file"}, but evaluates directly 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_structure Args} and @{ML_structure 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)) *}

   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)) *}

 text {*

   \begin{description}

   \item @{attribute ML_print_depth} controls the printing depth of the ML

   toplevel pretty printer; the precise effect depends on the ML compiler and

   run-time system. Typically the limit should be less than 10. Bigger values

   such as 100--1000 are occasionally useful for debugging.

   \item @{attribute ML_source_trace} indicates whether the source text that

   is given to the ML compiler should be output: it shows the raw Standard ML

   after expansion of Isabelle/ML antiquotations.

   \item @{attribute ML_exception_trace} indicates whether the ML run-time

   system should print a detailed stack trace on exceptions. The result is

   dependent on the particular ML compiler version. Note that after Poly/ML

   5.3 some optimizations in the run-time systems may hinder exception

   traces.

   The boundary for the exception trace is the current Isar command

   transactions. It is occasionally better to insert the combinator @{ML

   Runtime.exn_trace} into ML code for debugging

   \cite{isabelle-implementation}, closer to the point where it actually

   happens.

   \end{description}

 *}

 section {* Primitive specification elements *}

 subsection {* Sorts *}

 text {*

   \begin{matharray}{rcll}

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

   \end{matharray}

   @{rail \<open>

     @@{command default_sort} @{syntax sort}

   \<close>}

   \begin{description}

   \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 \label{sec:types-pure} *}

 text {*

   \begin{matharray}{rcll}

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

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

   \end{matharray}

   @{rail \<open>

     @@{command type_synonym} (@{syntax typespec} '=' @{syntax type} @{syntax mixfix}?)

     ;

     @@{command typedecl} @{syntax typespec} @{syntax mixfix}?

   \<close>}

   \begin{description}

   \item @{command "type_synonym"}~@{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 the semantic type definitions in Isabelle/HOL, 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 type-class instance @{text "t :: (s, \<dots>, s)s"}.

   \end{description}

   \begin{warn}

   If you introduce a new type axiomatically, i.e.\ via @{command_ref

   typedecl} and @{command_ref axiomatization}

   (\secref{sec:axiomatizations}), the minimum requirement is that it has a

   non-empty model, to avoid immediate collapse of the logical environment.

   Moreover, one needs to demonstrate that the interpretation of such

   free-form axiomatizations can coexist with other axiomatization schemes

   for types, notably @{command_def typedef} in Isabelle/HOL

   (\secref{sec:hol-typedef}), or any other extension that people might have

   introduced elsewhere.

   \end{warn}

 *}

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

 text {*

   \begin{matharray}{rcl}

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

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

   \end{matharray}

   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.

   @{rail \<open>

     @@{command consts} ((@{syntax name} '::' @{syntax type} @{syntax mixfix}?) +)

     ;

     @@{command defs} opt? (@{syntax axmdecl} @{syntax prop} +)

     ;

     opt: '(' @'unchecked'? @'overloaded'? ')'

   \<close>}

   \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 {* Naming existing theorems \label{sec:theorems} *}

 text {*

   \begin{matharray}{rcll}

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

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

   \end{matharray}

   @{rail \<open>

     (@@{command lemmas} | @@{command theorems}) @{syntax target}? \<newline>

       (@{syntax thmdef}? @{syntax thmrefs} + @'and')

       (@'for' (@{syntax vars} + @'and'))?

   \<close>}

   \begin{description}

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

   "for"}~@{text "x\<^sub>1 \<dots> x\<^sub>m"} evaluates given facts (with attributes) in

   the current context, which may be augmented by local variables.

   Results are standardized before being stored, i.e.\ schematic

   variables are renamed to enforce index @{text "0"} uniformly.

   \item @{command "theorems"} is the same as @{command "lemmas"}, but

   marks the result as a different kind of facts.

   \end{description}

 *}

 section {* Oracles *}

 text {*

   \begin{matharray}{rcll}

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

   \end{matharray}

   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.

   @{rail \<open>

     @@{command oracle} @{syntax name} '=' @{syntax text}

   \<close>}

   \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/HOL/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}

   @{rail \<open>

     ( @{command hide_class} | @{command hide_type} |

       @{command hide_const} | @{command hide_fact} ) ('(' @'open' ')')? (@{syntax nameref} + )

   \<close>}

   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.

   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