wenzelm@26840: %
wenzelm@26840: \begin{isabellebody}%
wenzelm@40685: \def\isabellecontext{HOL{\isaliteral{5F}{\isacharunderscore}}Specific}%
wenzelm@26840: %
wenzelm@26840: \isadelimtheory
wenzelm@26840: %
wenzelm@26840: \endisadelimtheory
wenzelm@26840: %
wenzelm@26840: \isatagtheory
wenzelm@26840: \isacommand{theory}\isamarkupfalse%
wenzelm@40685: \ HOL{\isaliteral{5F}{\isacharunderscore}}Specific\isanewline
wenzelm@43522: \isakeyword{imports}\ Base\ Main\isanewline
wenzelm@26849: \isakeyword{begin}%
wenzelm@26849: \endisatagtheory
wenzelm@26849: {\isafoldtheory}%
wenzelm@26849: %
wenzelm@26849: \isadelimtheory
wenzelm@26849: %
wenzelm@26849: \endisadelimtheory
wenzelm@26849: %
wenzelm@26852: \isamarkupchapter{Isabelle/HOL \label{ch:hol}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@44118: \isamarkupsection{Higher-Order Logic%
wenzelm@44118: }
wenzelm@44118: \isamarkuptrue%
wenzelm@44118: %
wenzelm@44118: \begin{isamarkuptext}%
wenzelm@44118: Isabelle/HOL is based on Higher-Order Logic, a polymorphic
wenzelm@44118:   version of Church's Simple Theory of Types.  HOL can be best
wenzelm@44118:   understood as a simply-typed version of classical set theory.  The
wenzelm@44118:   logic was first implemented in Gordon's HOL system
wenzelm@44118:   \cite{mgordon-hol}.  It extends Church's original logic
wenzelm@44118:   \cite{church40} by explicit type variables (naive polymorphism) and
wenzelm@44118:   a sound axiomatization scheme for new types based on subsets of
wenzelm@44118:   existing types.
wenzelm@44118: 
wenzelm@44118:   Andrews's book \cite{andrews86} is a full description of the
wenzelm@44118:   original Church-style higher-order logic, with proofs of correctness
wenzelm@44118:   and completeness wrt.\ certain set-theoretic interpretations.  The
wenzelm@44118:   particular extensions of Gordon-style HOL are explained semantically
wenzelm@44118:   in two chapters of the 1993 HOL book \cite{pitts93}.
wenzelm@44118: 
wenzelm@44118:   Experience with HOL over decades has demonstrated that higher-order
wenzelm@44118:   logic is widely applicable in many areas of mathematics and computer
wenzelm@44118:   science.  In a sense, Higher-Order Logic is simpler than First-Order
wenzelm@44118:   Logic, because there are fewer restrictions and special cases.  Note
wenzelm@44118:   that HOL is \emph{weaker} than FOL with axioms for ZF set theory,
wenzelm@44118:   which is traditionally considered the standard foundation of regular
wenzelm@44118:   mathematics, but for most applications this does not matter.  If you
wenzelm@44118:   prefer ML to Lisp, you will probably prefer HOL to ZF.
wenzelm@44118: 
wenzelm@44118:   \medskip The syntax of HOL follows \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-calculus and
wenzelm@44118:   functional programming.  Function application is curried.  To apply
wenzelm@44118:   the function \isa{f} of type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{3}}{\isaliteral{22}{\isachardoublequote}}} to the
wenzelm@44118:   arguments \isa{a} and \isa{b} in HOL, you simply write \isa{{\isaliteral{22}{\isachardoublequote}}f\ a\ b{\isaliteral{22}{\isachardoublequote}}} (as in ML or Haskell).  There is no ``apply'' operator; the
wenzelm@44118:   existing application of the Pure \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-calculus is re-used.
wenzelm@44118:   Note that in HOL \isa{{\isaliteral{22}{\isachardoublequote}}f\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} means ``\isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{22}{\isachardoublequote}}} applied to
wenzelm@44118:   the pair \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} (which is notation for \isa{{\isaliteral{22}{\isachardoublequote}}Pair\ a\ b{\isaliteral{22}{\isachardoublequote}}}).  The latter typically introduces extra formal efforts that can
wenzelm@44118:   be avoided by currying functions by default.  Explicit tuples are as
wenzelm@44118:   infrequent in HOL formalizations as in good ML or Haskell programs.
wenzelm@44118: 
wenzelm@44118:   \medskip Isabelle/HOL has a distinct feel, compared to other
wenzelm@44118:   object-logics like Isabelle/ZF.  It identifies object-level types
wenzelm@44118:   with meta-level types, taking advantage of the default
wenzelm@44118:   type-inference mechanism of Isabelle/Pure.  HOL fully identifies
wenzelm@44118:   object-level functions with meta-level functions, with native
wenzelm@44118:   abstraction and application.
wenzelm@44118: 
wenzelm@44118:   These identifications allow Isabelle to support HOL particularly
wenzelm@44118:   nicely, but they also mean that HOL requires some sophistication
wenzelm@44118:   from the user.  In particular, an understanding of Hindley-Milner
wenzelm@44118:   type-inference with type-classes, which are both used extensively in
wenzelm@44118:   the standard libraries and applications.  Beginners can set
wenzelm@44118:   \hyperlink{attribute.show-types}{\mbox{\isa{show{\isaliteral{5F}{\isacharunderscore}}types}}} or even \hyperlink{attribute.show-sorts}{\mbox{\isa{show{\isaliteral{5F}{\isacharunderscore}}sorts}}} to get more
wenzelm@44118:   explicit information about the result of type-inference.%
wenzelm@44118: \end{isamarkuptext}%
wenzelm@44118: \isamarkuptrue%
wenzelm@44118: %
wenzelm@44112: \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44117: An \emph{inductive definition} specifies the least predicate
wenzelm@44117:   or set \isa{R} closed under given rules: applying a rule to
wenzelm@44117:   elements of \isa{R} yields a result within \isa{R}.  For
wenzelm@44117:   example, a structural operational semantics is an inductive
wenzelm@44117:   definition of an evaluation relation.
wenzelm@44112: 
wenzelm@44117:   Dually, a \emph{coinductive definition} specifies the greatest
wenzelm@44117:   predicate or set \isa{R} that is consistent with given rules:
wenzelm@44117:   every element of \isa{R} can be seen as arising by applying a rule
wenzelm@44117:   to elements of \isa{R}.  An important example is using
wenzelm@44117:   bisimulation relations to formalise equivalence of processes and
wenzelm@44117:   infinite data structures.
wenzelm@44117:   
wenzelm@44117:   Both inductive and coinductive definitions are based on the
wenzelm@44117:   Knaster-Tarski fixed-point theorem for complete lattices.  The
wenzelm@44117:   collection of introduction rules given by the user determines a
wenzelm@44117:   functor on subsets of set-theoretic relations.  The required
wenzelm@44117:   monotonicity of the recursion scheme is proven as a prerequisite to
wenzelm@44117:   the fixed-point definition and the resulting consequences.  This
wenzelm@44117:   works by pushing inclusion through logical connectives and any other
wenzelm@44117:   operator that might be wrapped around recursive occurrences of the
wenzelm@44117:   defined relation: there must be a monotonicity theorem of the form
wenzelm@44117:   \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ B{\isaliteral{22}{\isachardoublequote}}}, for each premise \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4D3E}{\isasymM}}\ R\ t{\isaliteral{22}{\isachardoublequote}}} in an
wenzelm@44117:   introduction rule.  The default rule declarations of Isabelle/HOL
wenzelm@44117:   already take care of most common situations.
wenzelm@44112: 
wenzelm@44112:   \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{inductive\_set}\hypertarget{command.HOL.inductive-set}{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{coinductive\_set}\hypertarget{command.HOL.coinductive-set}{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isaliteral{5F}{\isacharunderscore}}set}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44117: \rail@begin{10}{}
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@term{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}}[]
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}}}}}[]
wenzelm@44112: \rail@nextbar{2}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}}[]
wenzelm@44112: \rail@nextbar{3}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isaliteral{5F}{\isacharunderscore}}set}}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44117: \rail@cr{5}
wenzelm@44112: \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44117: \rail@nextbar{6}
wenzelm@44112: \rail@term{\isa{\isakeyword{for}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{6}
wenzelm@44112: \rail@term{\isa{\isakeyword{where}}}[]
wenzelm@44112: \rail@nont{\isa{clauses}}[]
wenzelm@44112: \rail@endbar
wenzelm@44117: \rail@cr{8}
wenzelm@44112: \rail@bar
wenzelm@44117: \rail@nextbar{9}
wenzelm@44112: \rail@term{\isa{\isakeyword{monos}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{3}{\isa{clauses}}
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
wenzelm@44112: \rail@nextplus{2}
wenzelm@44112: \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{3}{}
wenzelm@44112: \rail@term{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{add}}[]
wenzelm@44112: \rail@nextbar{2}
wenzelm@44112: \rail@term{\isa{del}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44117:   \item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the introduction
wenzelm@44117:   rules.
wenzelm@44117: 
wenzelm@44117:   The propositions given as \isa{{\isaliteral{22}{\isachardoublequote}}clauses{\isaliteral{22}{\isachardoublequote}}} in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part are either rules of the usual \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{2F}{\isacharslash}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} format
wenzelm@44117:   (with arbitrary nesting), or equalities using \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}}.  The
wenzelm@44117:   latter specifies extra-logical abbreviations in the sense of
wenzelm@44117:   \indexref{}{command}{abbreviation}\hyperlink{command.abbreviation}{\mbox{\isa{\isacommand{abbreviation}}}}.  Introducing abstract syntax
wenzelm@44117:   simultaneously with the actual introduction rules is occasionally
wenzelm@44117:   useful for complex specifications.
wenzelm@44117: 
wenzelm@44117:   The optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of
wenzelm@44117:   the (co)inductive predicates that remain fixed throughout the
wenzelm@44117:   definition, in contrast to arguments of the relation that may vary
wenzelm@44117:   in each occurrence within the given \isa{{\isaliteral{22}{\isachardoublequote}}clauses{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44117: 
wenzelm@44117:   The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} declaration contains additional
wenzelm@44112:   \emph{monotonicity theorems}, which are required for each operator
wenzelm@44117:   applied to a recursive set in the introduction rules.
wenzelm@44112: 
wenzelm@44117:   \item \hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}}}} and \hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isaliteral{5F}{\isacharunderscore}}set}}}} are wrappers for to the previous commands for
wenzelm@44117:   native HOL predicates.  This allows to define (co)inductive sets,
wenzelm@44117:   where multiple arguments are simulated via tuples.
wenzelm@44112: 
wenzelm@44117:   \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules in the
wenzelm@44117:   context.  These rule are involved in the automated monotonicity
wenzelm@44117:   proof of the above inductive and coinductive definitions.
wenzelm@44112: 
wenzelm@44112:   \end{description}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Derived rules%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44117: A (co)inductive definition of \isa{R} provides the following
wenzelm@44117:   main theorems:
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \isa{R{\isaliteral{2E}{\isachardot}}intros} is the list of introduction rules as proven
wenzelm@44112:   theorems, for the recursive predicates (or sets).  The rules are
wenzelm@44112:   also available individually, using the names given them in the
wenzelm@44112:   theory file;
wenzelm@44112: 
wenzelm@44112:   \item \isa{R{\isaliteral{2E}{\isachardot}}cases} is the case analysis (or elimination) rule;
wenzelm@44112: 
wenzelm@44112:   \item \isa{R{\isaliteral{2E}{\isachardot}}induct} or \isa{R{\isaliteral{2E}{\isachardot}}coinduct} is the (co)induction
wenzelm@44112:   rule.
wenzelm@44112: 
wenzelm@44112:   \end{description}
wenzelm@44112: 
wenzelm@44112:   When several predicates \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} are
wenzelm@44112:   defined simultaneously, the list of introduction rules is called
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{5F}{\isacharunderscore}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}intros{\isaliteral{22}{\isachardoublequote}}}, the case analysis rules are
wenzelm@44112:   called \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2E}{\isachardot}}cases{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}cases{\isaliteral{22}{\isachardoublequote}}}, and the list
wenzelm@44112:   of mutual induction rules is called \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{5F}{\isacharunderscore}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}inducts{\isaliteral{22}{\isachardoublequote}}}.%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Monotonicity theorems%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44117: The context maintains a default set of theorems that are used
wenzelm@44117:   in monotonicity proofs.  New rules can be declared via the
wenzelm@44117:   \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  See the main Isabelle/HOL
wenzelm@44117:   sources for some examples.  The general format of such monotonicity
wenzelm@44117:   theorems is as follows:
wenzelm@44112: 
wenzelm@44112:   \begin{itemize}
wenzelm@44112: 
wenzelm@44117:   \item Theorems of the form \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ B{\isaliteral{22}{\isachardoublequote}}}, for proving
wenzelm@44112:   monotonicity of inductive definitions whose introduction rules have
wenzelm@44117:   premises involving terms such as \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4D3E}{\isasymM}}\ R\ t{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \item Monotonicity theorems for logical operators, which are of the
wenzelm@44112:   general form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}.  For example, in
wenzelm@44112:   the case of the operator \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6F723E}{\isasymor}}{\isaliteral{22}{\isachardoublequote}}}, the corresponding theorem is
wenzelm@44112:   \[
wenzelm@44112:   \infer{\isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}}
wenzelm@44112:   \]
wenzelm@44112: 
wenzelm@44112:   \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@44112:   of expressions, e.g.
wenzelm@44112:   \[
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ P{\isaliteral{22}{\isachardoublequote}}} \qquad\qquad
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C616E643E}{\isasymand}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q{\isaliteral{22}{\isachardoublequote}}}
wenzelm@44112:   \]
wenzelm@44112: 
wenzelm@44112:   \item Equations for reducing complex operators to more primitive
wenzelm@44112:   ones whose monotonicity can easily be proved, e.g.
wenzelm@44112:   \[
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{22}{\isachardoublequote}}} \qquad\qquad
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}Ball\ A\ P\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ x{\isaliteral{22}{\isachardoublequote}}}
wenzelm@44112:   \]
wenzelm@44112: 
wenzelm@44117:   \end{itemize}%
wenzelm@44117: \end{isamarkuptext}%
wenzelm@44117: \isamarkuptrue%
wenzelm@44117: %
wenzelm@44117: \isamarkupsubsubsection{Examples%
wenzelm@44117: }
wenzelm@44117: \isamarkuptrue%
wenzelm@44117: %
wenzelm@44117: \begin{isamarkuptext}%
wenzelm@44117: The finite powerset operator can be defined inductively like this:%
wenzelm@44117: \end{isamarkuptext}%
wenzelm@44117: \isamarkuptrue%
wenzelm@44117: \isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse%
wenzelm@44117: \ Fin\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ set\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ set\ set{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{for}\ A\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44117: \isakeyword{where}\isanewline
wenzelm@44117: \ \ empty{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ Fin\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44117: {\isaliteral{7C}{\isacharbar}}\ insert{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C696E3E}{\isasymin}}\ Fin\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ insert\ a\ B\ {\isaliteral{5C3C696E3E}{\isasymin}}\ Fin\ A{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44117: \begin{isamarkuptext}%
wenzelm@44117: The accessible part of a relation is defined as follows:%
wenzelm@44117: \end{isamarkuptext}%
wenzelm@44117: \isamarkuptrue%
wenzelm@44117: \isacommand{inductive}\isamarkupfalse%
wenzelm@44117: \ acc\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44117: \ \ \isakeyword{for}\ r\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infix}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C707265633E}{\isasymprec}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
wenzelm@44117: \isakeyword{where}\ acc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}y{\isaliteral{2E}{\isachardot}}\ y\ {\isaliteral{5C3C707265633E}{\isasymprec}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ acc\ r\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ acc\ r\ x{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44117: \begin{isamarkuptext}%
wenzelm@44117: Common logical connectives can be easily characterized as
wenzelm@44117: non-recursive inductive definitions with parameters, but without
wenzelm@44117: arguments.%
wenzelm@44117: \end{isamarkuptext}%
wenzelm@44117: \isamarkuptrue%
wenzelm@44117: \isacommand{inductive}\isamarkupfalse%
wenzelm@44117: \ AND\ \isakeyword{for}\ A\ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ bool\isanewline
wenzelm@44117: \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ AND\ A\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44117: \isanewline
wenzelm@44117: \isacommand{inductive}\isamarkupfalse%
wenzelm@44117: \ OR\ \isakeyword{for}\ A\ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ bool\isanewline
wenzelm@44117: \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ OR\ A\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44117: \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ OR\ A\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44117: \isanewline
wenzelm@44117: \isacommand{inductive}\isamarkupfalse%
wenzelm@44117: \ EXISTS\ \isakeyword{for}\ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44117: \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ a\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ EXISTS\ B{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44117: \begin{isamarkuptext}%
wenzelm@44117: Here the \isa{{\isaliteral{22}{\isachardoublequote}}cases{\isaliteral{22}{\isachardoublequote}}} or \isa{{\isaliteral{22}{\isachardoublequote}}induct{\isaliteral{22}{\isachardoublequote}}} rules produced by
wenzelm@44117:   the \hyperlink{command.inductive}{\mbox{\isa{\isacommand{inductive}}}} package coincide with the expected
wenzelm@44117:   elimination rules for Natural Deduction.  Already in the original
wenzelm@44117:   article by Gerhard Gentzen \cite{Gentzen:1935} there is a hint that
wenzelm@44117:   each connective can be characterized by its introductions, and the
wenzelm@44117:   elimination can be constructed systematically.%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsection{Recursive functions \label{sec:recursion}%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{2}{}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
wenzelm@44112: \rail@term{\isa{\isakeyword{where}}}[]
wenzelm@44112: \rail@nont{\isa{equations}}[]
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{4}{}
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@term{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}}[]
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\isa{functionopts}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
wenzelm@44112: \rail@cr{3}
wenzelm@44112: \rail@term{\isa{\isakeyword{where}}}[]
wenzelm@44112: \rail@nont{\isa{equations}}[]
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{3}{\isa{equations}}
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
wenzelm@44112: \rail@nextplus{2}
wenzelm@44112: \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{3}{\isa{functionopts}}
wenzelm@44112: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@term{\isa{sequential}}[]
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{domintros}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nextplus{2}
wenzelm@44112: \rail@cterm{\isa{{\isaliteral{2C}{\isacharcomma}}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
wenzelm@44116:   functions over datatypes (see also \indexref{HOL}{command}{datatype}\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} and
wenzelm@44116:   \indexref{HOL}{command}{rep\_datatype}\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}}).  The given \isa{equations}
wenzelm@44116:   specify reduction rules that are produced by instantiating the
wenzelm@44116:   generic combinator for primitive recursion that is available for
wenzelm@44116:   each datatype.
wenzelm@44116: 
wenzelm@44116:   Each equation needs to be of the form:
wenzelm@44116: 
wenzelm@44116:   \begin{isabelle}%
wenzelm@44116: {\isaliteral{22}{\isachardoublequote}}f\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{28}{\isacharparenleft}}C\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{29}{\isacharparenright}}\ z\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ z\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3D}{\isacharequal}}\ rhs{\isaliteral{22}{\isachardoublequote}}%
wenzelm@44116: \end{isabelle}
wenzelm@44116: 
wenzelm@44116:   such that \isa{C} is a datatype constructor, \isa{rhs} contains
wenzelm@44116:   only the free variables on the left-hand side (or from the context),
wenzelm@44116:   and all recursive occurrences of \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{22}{\isachardoublequote}}} in \isa{{\isaliteral{22}{\isachardoublequote}}rhs{\isaliteral{22}{\isachardoublequote}}} are of
wenzelm@44116:   the form \isa{{\isaliteral{22}{\isachardoublequote}}f\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} for some \isa{i}.  At most one
wenzelm@44116:   reduction rule for each constructor can be given.  The order does
wenzelm@44116:   not matter.  For missing constructors, the function is defined to
wenzelm@44116:   return a default value, but this equation is made difficult to
wenzelm@44116:   access for users.
wenzelm@44116: 
wenzelm@44116:   The reduction rules are declared as \hyperlink{attribute.simp}{\mbox{\isa{simp}}} by default,
wenzelm@44116:   which enables standard proof methods like \hyperlink{method.simp}{\mbox{\isa{simp}}} and
wenzelm@44116:   \hyperlink{method.auto}{\mbox{\isa{auto}}} to normalize expressions of \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{22}{\isachardoublequote}}} applied to
wenzelm@44116:   datatype constructions, by simulating symbolic computation via
wenzelm@44116:   rewriting.
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
wenzelm@44112:   wellfounded recursion. A detailed description with examples can be
wenzelm@44112:   found in \cite{isabelle-function}. The function is specified by a
wenzelm@44112:   set of (possibly conditional) recursive equations with arbitrary
wenzelm@44112:   pattern matching. The command generates proof obligations for the
wenzelm@44112:   completeness and the compatibility of patterns.
wenzelm@44112: 
wenzelm@44112:   The defined function is considered partial, and the resulting
wenzelm@44112:   simplification rules (named \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}psimps{\isaliteral{22}{\isachardoublequote}}}) and induction rule
wenzelm@44112:   (named \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}pinduct{\isaliteral{22}{\isachardoublequote}}}) are guarded by a generated domain
wenzelm@44112:   predicate \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{5F}{\isacharunderscore}}dom{\isaliteral{22}{\isachardoublequote}}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
wenzelm@44112:   command can then be used to establish that the function is total.
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} is a shorthand notation for ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}sequential{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, followed by automated
wenzelm@44112:   proof attempts regarding pattern matching and termination.  See
wenzelm@44112:   \cite{isabelle-function} for further details.
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
wenzelm@44112:   termination proof for the previously defined function \isa{f}.  If
wenzelm@44112:   this is omitted, the command refers to the most recent function
wenzelm@44112:   definition.  After the proof is closed, the recursive equations and
wenzelm@44112:   the induction principle is established.
wenzelm@44112: 
wenzelm@44112:   \end{description}
wenzelm@44112: 
wenzelm@44112:   Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
wenzelm@44116:   command accommodate reasoning by induction (cf.\ \hyperlink{method.induct}{\mbox{\isa{induct}}}):
wenzelm@44116:   rule \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}induct{\isaliteral{22}{\isachardoublequote}}} refers to a specific induction rule, with
wenzelm@44116:   parameters named according to the user-specified equations. Cases
wenzelm@44116:   are numbered starting from 1.  For \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}, the
wenzelm@44116:   induction principle coincides with structural recursion on the
wenzelm@44116:   datatype where the recursion is carried out.
wenzelm@44112: 
wenzelm@44112:   The equations provided by these packages may be referred later as
wenzelm@44112:   theorem list \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}simps{\isaliteral{22}{\isachardoublequote}}}, where \isa{f} is the (collective)
wenzelm@44112:   name of the functions defined.  Individual equations may be named
wenzelm@44112:   explicitly as well.
wenzelm@44112: 
wenzelm@44112:   The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
wenzelm@44112:   options.
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \isa{sequential} enables a preprocessor which disambiguates
wenzelm@44112:   overlapping patterns by making them mutually disjoint.  Earlier
wenzelm@44112:   equations take precedence over later ones.  This allows to give the
wenzelm@44112:   specification in a format very similar to functional programming.
wenzelm@44112:   Note that the resulting simplification and induction rules
wenzelm@44112:   correspond to the transformed specification, not the one given
wenzelm@44112:   originally. This usually means that each equation given by the user
wenzelm@44112:   may result in several theorems.  Also note that this automatic
wenzelm@44112:   transformation only works for ML-style datatype patterns.
wenzelm@44112: 
wenzelm@44112:   \item \isa{domintros} enables the automated generation of
wenzelm@44112:   introduction rules for the domain predicate. While mostly not
wenzelm@44112:   needed, they can be helpful in some proofs about partial functions.
wenzelm@44112: 
wenzelm@44112:   \end{description}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44116: \isamarkupsubsubsection{Example: evaluation of expressions%
wenzelm@44116: }
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: %
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: Subsequently, we define mutual datatypes for arithmetic and
wenzelm@44116:   boolean expressions, and use \hyperlink{command.primrec}{\mbox{\isa{\isacommand{primrec}}}} for evaluation
wenzelm@44116:   functions that follow the same recursive structure.%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{datatype}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{27}{\isacharprime}}a\ aexp\ {\isaliteral{3D}{\isacharequal}}\isanewline
wenzelm@44116: \ \ \ \ IF\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ {\isaliteral{7C}{\isacharbar}}\ Sum\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ {\isaliteral{7C}{\isacharbar}}\ Diff\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ {\isaliteral{7C}{\isacharbar}}\ Var\ {\isaliteral{27}{\isacharprime}}a\isanewline
wenzelm@44116: \ \ {\isaliteral{7C}{\isacharbar}}\ Num\ nat\isanewline
wenzelm@44116: \isakeyword{and}\ {\isaliteral{27}{\isacharprime}}a\ bexp\ {\isaliteral{3D}{\isacharequal}}\isanewline
wenzelm@44116: \ \ \ \ Less\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ {\isaliteral{7C}{\isacharbar}}\ And\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ {\isaliteral{7C}{\isacharbar}}\ Neg\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: \medskip Evaluation of arithmetic and boolean expressions%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{primrec}\isamarkupfalse%
wenzelm@44116: \ evala\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ aexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ \isakeyword{and}\ evalb\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ bexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \isakeyword{where}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ evalb\ env\ b\ then\ evala\ env\ a{\isadigit{1}}\ else\ evala\ env\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isadigit{1}}\ {\isaliteral{2B}{\isacharplus}}\ evala\ env\ a{\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ evala\ env\ a{\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Var\ v{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ env\ v{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Num\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}evala\ env\ a{\isadigit{1}}\ {\isaliteral{3C}{\isacharless}}\ evala\ env\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}evalb\ env\ b{\isadigit{1}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ evalb\ env\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}Neg\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ evalb\ env\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: Since the value of an expression depends on the value of its
wenzelm@44116:   variables, the functions \isa{evala} and \isa{evalb} take an
wenzelm@44116:   additional parameter, an \emph{environment} that maps variables to
wenzelm@44116:   their values.
wenzelm@44116: 
wenzelm@44116:   \medskip Substitution on expressions can be defined similarly.  The
wenzelm@44116:   mapping \isa{f} of type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequote}}} given as a
wenzelm@44116:   parameter is lifted canonically on the types \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequote}}} and
wenzelm@44116:   \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequote}}}, respectively.%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{primrec}\isamarkupfalse%
wenzelm@44116: \ substa\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ aexp{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ aexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ \isakeyword{and}\ substb\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ aexp{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ bexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \isakeyword{where}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ IF\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Sum\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Diff\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Var\ v{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ v{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Num\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Num\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substb\ f\ {\isaliteral{28}{\isacharparenleft}}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Less\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substb\ f\ {\isaliteral{28}{\isacharparenleft}}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ And\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substb\ f\ {\isaliteral{28}{\isacharparenleft}}Neg\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Neg\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: In textbooks about semantics one often finds substitution
wenzelm@44116:   theorems, which express the relationship between substitution and
wenzelm@44116:   evaluation.  For \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequote}}}, we can prove
wenzelm@44116:   such a theorem by mutual induction, followed by simplification.%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{lemma}\isamarkupfalse%
wenzelm@44116: \ subst{\isaliteral{5F}{\isacharunderscore}}one{\isaliteral{3A}{\isacharcolon}}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}substa\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ {\isaliteral{28}{\isacharparenleft}}env\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}substb\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evalb\ {\isaliteral{28}{\isacharparenleft}}env\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: \ \ %
wenzelm@44116: \endisadelimproof
wenzelm@44116: %
wenzelm@44116: \isatagproof
wenzelm@44116: \isacommand{by}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{28}{\isacharparenleft}}induct\ a\ \isakeyword{and}\ b{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
wenzelm@44116: \endisatagproof
wenzelm@44116: {\isafoldproof}%
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: \isanewline
wenzelm@44116: %
wenzelm@44116: \endisadelimproof
wenzelm@44116: \isanewline
wenzelm@44116: \isacommand{lemma}\isamarkupfalse%
wenzelm@44116: \ subst{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{3A}{\isacharcolon}}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}substa\ s\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ evala\ env\ {\isaliteral{28}{\isacharparenleft}}s\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}substb\ s\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evalb\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ evala\ env\ {\isaliteral{28}{\isacharparenleft}}s\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: \ \ %
wenzelm@44116: \endisadelimproof
wenzelm@44116: %
wenzelm@44116: \isatagproof
wenzelm@44116: \isacommand{by}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{28}{\isacharparenleft}}induct\ a\ \isakeyword{and}\ b{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
wenzelm@44116: \endisatagproof
wenzelm@44116: {\isafoldproof}%
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: %
wenzelm@44116: \endisadelimproof
wenzelm@44116: %
wenzelm@44116: \isamarkupsubsubsection{Example: a substitution function for terms%
wenzelm@44116: }
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: %
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: Functions on datatypes with nested recursion are also defined
wenzelm@44116:   by mutual primitive recursion.%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{datatype}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{22}{\isachardoublequoteopen}}term{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{3D}{\isacharequal}}\ Var\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{7C}{\isacharbar}}\ App\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: A substitution function on type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{22}{\isachardoublequote}}} can be
wenzelm@44116:   defined as follows, by working simultaneously on \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list{\isaliteral{22}{\isachardoublequote}}}:%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{primrec}\isamarkupfalse%
wenzelm@44116: \ subst{\isaliteral{5F}{\isacharunderscore}}term\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline
wenzelm@44116: \ \ subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \isakeyword{where}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f\ {\isaliteral{28}{\isacharparenleft}}Var\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f\ {\isaliteral{28}{\isacharparenleft}}App\ b\ ts{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ App\ b\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ ts{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ {\isaliteral{28}{\isacharparenleft}}t\ {\isaliteral{23}{\isacharhash}}\ ts{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ subst{\isaliteral{5F}{\isacharunderscore}}term\ f\ t\ {\isaliteral{23}{\isacharhash}}\ subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ ts{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: The recursion scheme follows the structure of the unfolded
wenzelm@44116:   definition of type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{22}{\isachardoublequote}}}.  To prove properties of this
wenzelm@44116:   substitution function, mutual induction is needed:%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{lemma}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{3D}{\isacharequal}}\ subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{2}}\ t{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ ts\ {\isaliteral{3D}{\isacharequal}}\ subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f{\isadigit{2}}\ ts{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: \ \ %
wenzelm@44116: \endisadelimproof
wenzelm@44116: %
wenzelm@44116: \isatagproof
wenzelm@44116: \isacommand{by}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{28}{\isacharparenleft}}induct\ t\ \isakeyword{and}\ ts{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
wenzelm@44116: \endisatagproof
wenzelm@44116: {\isafoldproof}%
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: %
wenzelm@44116: \endisadelimproof
wenzelm@44116: %
wenzelm@44116: \isamarkupsubsubsection{Example: a map function for infinitely branching trees%
wenzelm@44116: }
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: %
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: Defining functions on infinitely branching datatypes by
wenzelm@44116:   primitive recursion is just as easy.%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{datatype}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{27}{\isacharprime}}a\ tree\ {\isaliteral{3D}{\isacharequal}}\ Atom\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{7C}{\isacharbar}}\ Branch\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ tree{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \isanewline
wenzelm@44116: \isacommand{primrec}\isamarkupfalse%
wenzelm@44116: \ map{\isaliteral{5F}{\isacharunderscore}}tree\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ tree\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ tree{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: \isakeyword{where}\isanewline
wenzelm@44116: \ \ {\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{28}{\isacharparenleft}}Atom\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Atom\ {\isaliteral{28}{\isacharparenleft}}f\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{28}{\isacharparenleft}}Branch\ ts{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Branch\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{28}{\isacharparenleft}}ts\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: Note that all occurrences of functions such as \isa{ts}
wenzelm@44116:   above must be applied to an argument.  In particular, \isa{{\isaliteral{22}{\isachardoublequote}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ ts{\isaliteral{22}{\isachardoublequote}}} is not allowed here.%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: %
wenzelm@44116: \begin{isamarkuptext}%
wenzelm@44116: Here is a simple composition lemma for \isa{map{\isaliteral{5F}{\isacharunderscore}}tree}:%
wenzelm@44116: \end{isamarkuptext}%
wenzelm@44116: \isamarkuptrue%
wenzelm@44116: \isacommand{lemma}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}tree\ g\ {\isaliteral{28}{\isacharparenleft}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ map{\isaliteral{5F}{\isacharunderscore}}tree\ {\isaliteral{28}{\isacharparenleft}}g\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: \ \ %
wenzelm@44116: \endisadelimproof
wenzelm@44116: %
wenzelm@44116: \isatagproof
wenzelm@44116: \isacommand{by}\isamarkupfalse%
wenzelm@44116: \ {\isaliteral{28}{\isacharparenleft}}induct\ t{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
wenzelm@44116: \endisatagproof
wenzelm@44116: {\isafoldproof}%
wenzelm@44116: %
wenzelm@44116: \isadelimproof
wenzelm@44116: %
wenzelm@44116: \endisadelimproof
wenzelm@44116: %
wenzelm@44112: \isamarkupsubsection{Proof methods related to recursive definitions%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isaliteral{5F}{\isacharunderscore}}completeness}}}} & : & \isa{method} \\
wenzelm@44112:     \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
wenzelm@44112:     \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}}}} & : & \isa{method} \\
wenzelm@44112:     \indexdef{HOL}{method}{size\_change}\hypertarget{method.HOL.size-change}{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isaliteral{5F}{\isacharunderscore}}change}}}} & : & \isa{method} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{1}{}
wenzelm@44112: \rail@term{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{}
wenzelm@44112: \rail@term{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@cnont{\hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{}
wenzelm@44112: \rail@term{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isaliteral{5F}{\isacharunderscore}}change}}}}[]
wenzelm@44112: \rail@nont{\isa{orders}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@cnont{\hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{4}{\isa{orders}}
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@term{\isa{max}}[]
wenzelm@44112: \rail@nextbar{2}
wenzelm@44112: \rail@term{\isa{min}}[]
wenzelm@44112: \rail@nextbar{3}
wenzelm@44112: \rail@term{\isa{ms}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isaliteral{5F}{\isacharunderscore}}completeness}}} is a specialized method to
wenzelm@44112:   solve goals regarding the completeness of pattern matching, as
wenzelm@44112:   required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
wenzelm@44112:   \cite{isabelle-function}).
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
wenzelm@44112:   proof using the relation \isa{R}.  The resulting proof state will
wenzelm@44112:   contain goals expressing that \isa{R} is wellfounded, and that the
wenzelm@44112:   arguments of recursive calls decrease with respect to \isa{R}.
wenzelm@44112:   Usually, this method is used as the initial proof step of manual
wenzelm@44112:   termination proofs.
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}}} attempts a fully
wenzelm@44112:   automated termination proof by searching for a lexicographic
wenzelm@44112:   combination of size measures on the arguments of the function. The
wenzelm@44112:   method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
wenzelm@44134:   which it uses internally to prove local descents.  The \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}} modifiers are accepted (as for \hyperlink{method.auto}{\mbox{\isa{auto}}}).
wenzelm@44112: 
wenzelm@44112:   In case of failure, extensive information is printed, which can help
wenzelm@44112:   to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isaliteral{5F}{\isacharunderscore}}change}}} also works on termination goals,
wenzelm@44112:   using a variation of the size-change principle, together with a
wenzelm@44112:   graph decomposition technique (see \cite{krauss_phd} for details).
wenzelm@44112:   Three kinds of orders are used internally: \isa{max}, \isa{min},
wenzelm@44112:   and \isa{ms} (multiset), which is only available when the theory
wenzelm@44112:   \isa{Multiset} is loaded. When no order kinds are given, they are
wenzelm@44112:   tried in order. The search for a termination proof uses SAT solving
wenzelm@44112:   internally.
wenzelm@44112: 
wenzelm@44134:   For local descent proofs, the \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}} modifiers are
wenzelm@44134:   accepted (as for \hyperlink{method.auto}{\mbox{\isa{auto}}}).
wenzelm@44112: 
wenzelm@44112:   \end{description}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Functions with explicit partiality%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{command}{partial\_function}\hypertarget{command.HOL.partial-function}{\hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isaliteral{5F}{\isacharunderscore}}function}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{attribute}{partial\_function\_mono}\hypertarget{attribute.HOL.partial-function-mono}{\hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}mono}}}} & : & \isa{attribute} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{5}{}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isaliteral{5F}{\isacharunderscore}}function}}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
wenzelm@44112: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
wenzelm@44112: \rail@cr{3}
wenzelm@44112: \rail@term{\isa{\isakeyword{where}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{4}
wenzelm@44112: \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isaliteral{5F}{\isacharunderscore}}function}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}mode{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} defines
wenzelm@44112:   recursive functions based on fixpoints in complete partial
wenzelm@44112:   orders. No termination proof is required from the user or
wenzelm@44112:   constructed internally. Instead, the possibility of non-termination
wenzelm@44112:   is modelled explicitly in the result type, which contains an
wenzelm@44112:   explicit bottom element.
wenzelm@44112: 
wenzelm@44112:   Pattern matching and mutual recursion are currently not supported.
wenzelm@44112:   Thus, the specification consists of a single function described by a
wenzelm@44112:   single recursive equation.
wenzelm@44112: 
wenzelm@44112:   There are no fixed syntactic restrictions on the body of the
wenzelm@44112:   function, but the induced functional must be provably monotonic
wenzelm@44112:   wrt.\ the underlying order.  The monotonicitity proof is performed
wenzelm@44112:   internally, and the definition is rejected when it fails. The proof
wenzelm@44112:   can be influenced by declaring hints using the
wenzelm@44112:   \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}mono}}} attribute.
wenzelm@44112: 
wenzelm@44112:   The mandatory \isa{mode} argument specifies the mode of operation
wenzelm@44112:   of the command, which directly corresponds to a complete partial
wenzelm@44112:   order on the result type. By default, the following modes are
wenzelm@44112:   defined:
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112:   \item \isa{option} defines functions that map into the \isa{option} type. Here, the value \isa{None} is used to model a
wenzelm@44112:   non-terminating computation. Monotonicity requires that if \isa{None} is returned by a recursive call, then the overall result
wenzelm@44112:   must also be \isa{None}. This is best achieved through the use of
wenzelm@44112:   the monadic operator \isa{{\isaliteral{22}{\isachardoublequote}}Option{\isaliteral{2E}{\isachardot}}bind{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \item \isa{tailrec} defines functions with an arbitrary result
wenzelm@44112:   type and uses the slightly degenerated partial order where \isa{{\isaliteral{22}{\isachardoublequote}}undefined{\isaliteral{22}{\isachardoublequote}}} is the bottom element.  Now, monotonicity requires that
wenzelm@44112:   if \isa{undefined} is returned by a recursive call, then the
wenzelm@44112:   overall result must also be \isa{undefined}. In practice, this is
wenzelm@44112:   only satisfied when each recursive call is a tail call, whose result
wenzelm@44112:   is directly returned. Thus, this mode of operation allows the
wenzelm@44112:   definition of arbitrary tail-recursive functions.
wenzelm@44112:   \end{description}
wenzelm@44112: 
wenzelm@44112:   Experienced users may define new modes by instantiating the locale
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}definitions{\isaliteral{22}{\isachardoublequote}}} appropriately.
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}mono}}} declares rules for
wenzelm@44112:   use in the internal monononicity proofs of partial function
wenzelm@44112:   definitions.
wenzelm@44112: 
wenzelm@44112:   \end{description}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Old-style recursive function definitions (TFL)%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: The old TFL commands \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} and \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isaliteral{5F}{\isacharunderscore}}tc}}}} for defining recursive are mostly obsolete; \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} or \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} should be used instead.
wenzelm@44112: 
wenzelm@44112:   \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{recdef\_tc}\hypertarget{command.HOL.recdef-tc}{\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isaliteral{5F}{\isacharunderscore}}tc}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{5}{}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@44112: \rail@term{\isa{\isakeyword{permissive}}}[]
wenzelm@44112: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@cr{3}
wenzelm@44112: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
wenzelm@44112: \rail@nextplus{4}
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{4}
wenzelm@44112: \rail@nont{\isa{hints}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{}
wenzelm@44112: \rail@nont{\isa{recdeftc}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nont{\isa{tc}}[]
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{\isa{hints}}
wenzelm@44112: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@44112: \rail@term{\isa{\isakeyword{hints}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@cnont{\isa{recdefmod}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{4}{\isa{recdefmod}}
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@term{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}}[]
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}}[]
wenzelm@44112: \rail@nextbar{2}
wenzelm@44112: \rail@term{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{add}}[]
wenzelm@44112: \rail@nextbar{2}
wenzelm@44112: \rail@term{\isa{del}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
wenzelm@44112: \rail@nextbar{3}
wenzelm@44112: \rail@nont{\hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{\isa{tc}}
wenzelm@44112: \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
wenzelm@44112: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
wenzelm@44112:   recursive functions (using the TFL package), see also
wenzelm@44112:   \cite{isabelle-HOL}.  The ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}permissive{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' option tells
wenzelm@44112:   TFL to recover from failed proof attempts, returning unfinished
wenzelm@44112:   results.  The \isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}, \isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}, and \isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf} hints refer to auxiliary rules to be used in the internal
wenzelm@44112:   automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
wenzelm@44134:   declarations may be given to tune the context of the Simplifier
wenzelm@44134:   (cf.\ \secref{sec:simplifier}) and Classical reasoner (cf.\
wenzelm@44134:   \secref{sec:classical}).
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isaliteral{5F}{\isacharunderscore}}tc}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} recommences the
wenzelm@44112:   proof for leftover termination condition number \isa{i} (default
wenzelm@44112:   1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
wenzelm@44112:   constant \isa{c}.
wenzelm@44112: 
wenzelm@44112:   Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
wenzelm@44112:   its internal proofs without manual intervention.
wenzelm@44112: 
wenzelm@44112:   \end{description}
wenzelm@44112: 
wenzelm@44112:   \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
wenzelm@44112:   globally, using the following attributes.
wenzelm@44112: 
wenzelm@44112:   \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}}}} & : & \isa{attribute} \\
wenzelm@44112:     \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}}}} & : & \isa{attribute} \\
wenzelm@44112:     \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf}}}} & : & \isa{attribute} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{3}{}
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@term{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}}}}[]
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}}}}[]
wenzelm@44112: \rail@nextbar{2}
wenzelm@44112: \rail@term{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{add}}[]
wenzelm@44112: \rail@nextbar{2}
wenzelm@44112: \rail@term{\isa{del}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsection{Datatypes \label{sec:hol-datatype}%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \indexdef{HOL}{command}{rep\_datatype}\hypertarget{command.HOL.rep-datatype}{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{2}{}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nont{\isa{spec}}[]
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{3}{}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@44112: \rail@nextplus{2}
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{\isa{spec}}
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.parname}{\mbox{\isa{parname}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@nont{\hyperlink{syntax.typespec}{\mbox{\isa{typespec}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nont{\isa{cons}}[]
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \rail@begin{2}{\isa{cons}}
wenzelm@44112: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nextplus{1}
wenzelm@44112: \rail@cnont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in
wenzelm@44112:   HOL.
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}} represents existing types as
wenzelm@44113:   datatypes.
wenzelm@44113: 
wenzelm@44113:   For foundational reasons, some basic types such as \isa{nat}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{22}{\isachardoublequote}}}, \isa{bool} and \isa{unit} are
wenzelm@44113:   introduced by more primitive means using \indexref{}{command}{typedef}\hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}}.  To
wenzelm@44113:   recover the rich infrastructure of \hyperlink{command.datatype}{\mbox{\isa{\isacommand{datatype}}}} (e.g.\ rules
wenzelm@44113:   for \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} and the primitive recursion
wenzelm@44113:   combinators), such types may be represented as actual datatypes
wenzelm@44113:   later.  This is done by specifying the constructors of the desired
wenzelm@44113:   type, and giving a proof of the induction rule, distinctness and
wenzelm@44113:   injectivity of constructors.
wenzelm@44113: 
wenzelm@44113:   For example, see \verb|~~/src/HOL/Sum_Type.thy| for the
wenzelm@44113:   representation of the primitive sum type as fully-featured datatype.
wenzelm@44112: 
wenzelm@44112:   \end{description}
wenzelm@44112: 
wenzelm@44113:   The generated rules for \hyperlink{method.induct}{\mbox{\isa{induct}}} and \hyperlink{method.cases}{\mbox{\isa{cases}}} provide
wenzelm@44113:   case names according to the given constructors, while parameters are
wenzelm@44113:   named after the types (see also \secref{sec:cases-induct}).
wenzelm@44112: 
wenzelm@44112:   See \cite{isabelle-HOL} for more details on datatypes, but beware of
wenzelm@44112:   the old-style theory syntax being used there!  Apart from proper
wenzelm@44112:   proof methods for case-analysis and induction, there are also
wenzelm@44112:   emulations of ML tactics \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}} available, see \secref{sec:hol-induct-tac}; these admit
wenzelm@44112:   to refer directly to the internal structure of subgoals (including
wenzelm@44112:   internally bound parameters).%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44114: \isamarkupsubsubsection{Examples%
wenzelm@44114: }
wenzelm@44114: \isamarkuptrue%
wenzelm@44114: %
wenzelm@44114: \begin{isamarkuptext}%
wenzelm@44114: We define a type of finite sequences, with slightly different
wenzelm@44114:   names than the existing \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{22}{\isachardoublequote}}} that is already in \hyperlink{theory.Main}{\mbox{\isa{Main}}}:%
wenzelm@44114: \end{isamarkuptext}%
wenzelm@44114: \isamarkuptrue%
wenzelm@44114: \isacommand{datatype}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{27}{\isacharprime}}a\ seq\ {\isaliteral{3D}{\isacharequal}}\ Empty\ {\isaliteral{7C}{\isacharbar}}\ Seq\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ seq{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@44114: \begin{isamarkuptext}%
wenzelm@44114: We can now prove some simple lemma by structural induction:%
wenzelm@44114: \end{isamarkuptext}%
wenzelm@44114: \isamarkuptrue%
wenzelm@44114: \isacommand{lemma}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ xs\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44114: %
wenzelm@44114: \isadelimproof
wenzelm@44114: %
wenzelm@44114: \endisadelimproof
wenzelm@44114: %
wenzelm@44114: \isatagproof
wenzelm@44114: \isacommand{proof}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{28}{\isacharparenleft}}induct\ xs\ arbitrary{\isaliteral{3A}{\isacharcolon}}\ x{\isaliteral{29}{\isacharparenright}}\isanewline
wenzelm@44114: \ \ \isacommand{case}\isamarkupfalse%
wenzelm@44114: \ Empty%
wenzelm@44114: \begin{isamarkuptxt}%
wenzelm@44114: This case can be proved using the simplifier: the freeness
wenzelm@44114:     properties of the datatype are already declared as \hyperlink{attribute.simp}{\mbox{\isa{simp}}} rules.%
wenzelm@44114: \end{isamarkuptxt}%
wenzelm@44114: \isamarkuptrue%
wenzelm@44114: \ \ \isacommand{show}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ Empty\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Empty{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44114: \ \ \ \ \isacommand{by}\isamarkupfalse%
wenzelm@44114: \ simp\isanewline
wenzelm@44114: \isacommand{next}\isamarkupfalse%
wenzelm@44114: \isanewline
wenzelm@44114: \ \ \isacommand{case}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{28}{\isacharparenleft}}Seq\ y\ ys{\isaliteral{29}{\isacharparenright}}%
wenzelm@44114: \begin{isamarkuptxt}%
wenzelm@44114: The step case is proved similarly.%
wenzelm@44114: \end{isamarkuptxt}%
wenzelm@44114: \isamarkuptrue%
wenzelm@44114: \ \ \isacommand{show}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ {\isaliteral{28}{\isacharparenleft}}Seq\ y\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Seq\ y\ ys{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44114: \ \ \ \ \isacommand{using}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{60}{\isacharbackquoteopen}}Seq\ y\ ys\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ ys{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{by}\isamarkupfalse%
wenzelm@44114: \ simp\isanewline
wenzelm@44114: \isacommand{qed}\isamarkupfalse%
wenzelm@44114: %
wenzelm@44114: \endisatagproof
wenzelm@44114: {\isafoldproof}%
wenzelm@44114: %
wenzelm@44114: \isadelimproof
wenzelm@44114: %
wenzelm@44114: \endisadelimproof
wenzelm@44114: %
wenzelm@44114: \begin{isamarkuptext}%
wenzelm@44114: Here is a more succinct version of the same proof:%
wenzelm@44114: \end{isamarkuptext}%
wenzelm@44114: \isamarkuptrue%
wenzelm@44114: \isacommand{lemma}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ xs\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44114: %
wenzelm@44114: \isadelimproof
wenzelm@44114: \ \ %
wenzelm@44114: \endisadelimproof
wenzelm@44114: %
wenzelm@44114: \isatagproof
wenzelm@44114: \isacommand{by}\isamarkupfalse%
wenzelm@44114: \ {\isaliteral{28}{\isacharparenleft}}induct\ xs\ arbitrary{\isaliteral{3A}{\isacharcolon}}\ x{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
wenzelm@44114: \endisatagproof
wenzelm@44114: {\isafoldproof}%
wenzelm@44114: %
wenzelm@44114: \isadelimproof
wenzelm@44114: %
wenzelm@44114: \endisadelimproof
wenzelm@44114: %
wenzelm@44112: \isamarkupsection{Records \label{sec:hol-record}%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: In principle, records merely generalize the concept of tuples, where
wenzelm@44112:   components may be addressed by labels instead of just position.  The
wenzelm@44112:   logical infrastructure of records in Isabelle/HOL is slightly more
wenzelm@44112:   advanced, though, supporting truly extensible record schemes.  This
wenzelm@44112:   admits operations that are polymorphic with respect to record
wenzelm@44112:   extension, yielding ``object-oriented'' effects like (single)
wenzelm@44112:   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
wenzelm@44112:   details on object-oriented verification and record subtyping in HOL.%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Basic concepts%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
wenzelm@44112:   at the level of terms and types.  The notation is as follows:
wenzelm@44112: 
wenzelm@44112:   \begin{center}
wenzelm@44112:   \begin{tabular}{l|l|l}
wenzelm@44112:     & record terms & record types \\ \hline
wenzelm@44112:     fixed & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     schematic & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ m{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} &
wenzelm@44112:       \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ M{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{tabular}
wenzelm@44112:   \end{center}
wenzelm@44112: 
wenzelm@44112:   \noindent The ASCII representation of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} is \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{7C}{\isacharbar}}\ x\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{7C}{\isacharbar}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   A fixed record \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} has field \isa{x} of value
wenzelm@44112:   \isa{a} and field \isa{y} of value \isa{b}.  The corresponding
wenzelm@44112:   type is \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, assuming that \isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{22}{\isachardoublequote}}}
wenzelm@44112:   and \isa{{\isaliteral{22}{\isachardoublequote}}b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   A record scheme like \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ m{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} contains fields
wenzelm@44112:   \isa{x} and \isa{y} as before, but also possibly further fields
wenzelm@44112:   as indicated by the ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' notation (which is actually part
wenzelm@44112:   of the syntax).  The improper field ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' of a record
wenzelm@44112:   scheme is called the \emph{more part}.  Logically it is just a free
wenzelm@44112:   variable, which is occasionally referred to as ``row variable'' in
wenzelm@44112:   the literature.  The more part of a record scheme may be
wenzelm@44112:   instantiated by zero or more further components.  For example, the
wenzelm@44112:   previous scheme may get instantiated to \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ z\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ m{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, where \isa{m{\isaliteral{27}{\isacharprime}}} refers to a different more part.
wenzelm@44112:   Fixed records are special instances of record schemes, where
wenzelm@44112:   ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' is properly terminated by the \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ unit{\isaliteral{22}{\isachardoublequote}}}
wenzelm@44112:   element.  In fact, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} is just an abbreviation
wenzelm@44112:   for \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \medskip Two key observations make extensible records in a simply
wenzelm@44112:   typed language like HOL work out:
wenzelm@44112: 
wenzelm@44112:   \begin{enumerate}
wenzelm@44112: 
wenzelm@44112:   \item the more part is internalized, as a free term or type
wenzelm@44112:   variable,
wenzelm@44112: 
wenzelm@44112:   \item field names are externalized, they cannot be accessed within
wenzelm@44112:   the logic as first-class values.
wenzelm@44112: 
wenzelm@44112:   \end{enumerate}
wenzelm@44112: 
wenzelm@44112:   \medskip In Isabelle/HOL record types have to be defined explicitly,
wenzelm@44112:   fixing their field names and types, and their (optional) parent
wenzelm@44112:   record.  Afterwards, records may be formed using above syntax, while
wenzelm@44112:   obeying the canonical order of fields as given by their declaration.
wenzelm@44112:   The record package provides several standard operations like
wenzelm@44112:   selectors and updates.  The common setup for various generic proof
wenzelm@44112:   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
wenzelm@44112:   tutorial \cite{isabelle-hol-book} for further instructions on using
wenzelm@44112:   records in practice.%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Record specifications%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{4}{}
wenzelm@44112: \rail@term{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}}[]
wenzelm@44112: \rail@nont{\hyperlink{syntax.typespec-sorts}{\mbox{\isa{typespec{\isaliteral{5F}{\isacharunderscore}}sorts}}}}[]
wenzelm@44112: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
wenzelm@44112: \rail@cr{2}
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{3}
wenzelm@44112: \rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]
wenzelm@44112: \rail@term{\isa{{\isaliteral{2B}{\isacharplus}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@plus
wenzelm@44112: \rail@nont{\hyperlink{syntax.constdecl}{\mbox{\isa{constdecl}}}}[]
wenzelm@44112: \rail@nextplus{3}
wenzelm@44112: \rail@endplus
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{2B}{\isacharplus}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} defines extensible record type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}},
wenzelm@44112:   derived from the optional parent record \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7461753E}{\isasymtau}}{\isaliteral{22}{\isachardoublequote}}} by adding new
wenzelm@44112:   field components \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} etc.
wenzelm@44112: 
wenzelm@44112:   The type variables of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7461753E}{\isasymtau}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} need to be
wenzelm@44112:   covered by the (distinct) parameters \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}}.  Type constructor \isa{t} has to be new, while \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} needs to specify an instance of an existing record type.  At
wenzelm@44112:   least one new field \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} has to be specified.
wenzelm@44112:   Basically, field names need to belong to a unique record.  This is
wenzelm@44112:   not a real restriction in practice, since fields are qualified by
wenzelm@44112:   the record name internally.
wenzelm@44112: 
wenzelm@44112:   The parent record specification \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} is optional; if omitted
wenzelm@44112:   \isa{t} becomes a root record.  The hierarchy of all records
wenzelm@44112:   declared within a theory context forms a forest structure, i.e.\ a
wenzelm@44112:   set of trees starting with a root record each.  There is no way to
wenzelm@44112:   merge multiple parent records!
wenzelm@44112: 
wenzelm@44112:   For convenience, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} is made a
wenzelm@44112:   type abbreviation for the fixed record type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, likewise is \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{5F}{\isacharunderscore}}scheme{\isaliteral{22}{\isachardoublequote}}} made an abbreviation for
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \end{description}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Record operations%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: Any record definition of the form presented above produces certain
wenzelm@44112:   standard operations.  Selectors and updates are provided for any
wenzelm@44112:   field, including the improper one ``\isa{more}''.  There are also
wenzelm@44112:   cumulative record constructor functions.  To simplify the
wenzelm@44112:   presentation below, we assume for now that \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} is a root record with fields \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \medskip \textbf{Selectors} and \textbf{updates} are available for
wenzelm@44112:   any field (including ``\isa{more}''):
wenzelm@44112: 
wenzelm@44112:   \begin{matharray}{lll}
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{5F}{\isacharunderscore}}update{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   There is special syntax for application of updates: \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} abbreviates term \isa{{\isaliteral{22}{\isachardoublequote}}x{\isaliteral{5F}{\isacharunderscore}}update\ a\ r{\isaliteral{22}{\isachardoublequote}}}.  Further notation for
wenzelm@44112:   repeated updates is also available: \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} may be written \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}.  Note that
wenzelm@44112:   because of postfix notation the order of fields shown here is
wenzelm@44112:   reverse than in the actual term.  Since repeated updates are just
wenzelm@44112:   function applications, fields may be freely permuted in \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, as far as logical equality is concerned.
wenzelm@44112:   Thus commutativity of independent updates can be proven within the
wenzelm@44112:   logic for any two fields, but not as a general theorem.
wenzelm@44112: 
wenzelm@44112:   \medskip The \textbf{make} operation provides a cumulative record
wenzelm@44112:   constructor function:
wenzelm@44112: 
wenzelm@44112:   \begin{matharray}{lll}
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \medskip We now reconsider the case of non-root records, which are
wenzelm@44112:   derived of some parent.  In general, the latter may depend on
wenzelm@44112:   another parent as well, resulting in a list of \emph{ancestor
wenzelm@44112:   records}.  Appending the lists of fields of all ancestors results in
wenzelm@44112:   a certain field prefix.  The record package automatically takes care
wenzelm@44112:   of this by lifting operations over this context of ancestor fields.
wenzelm@44112:   Assuming that \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} has ancestor
wenzelm@44112:   fields \isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub k\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{22}{\isachardoublequote}}},
wenzelm@44112:   the above record operations will get the following types:
wenzelm@44112: 
wenzelm@44112:   \medskip
wenzelm@44112:   \begin{tabular}{lll}
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{5F}{\isacharunderscore}}update{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub k\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{tabular}
wenzelm@44112:   \medskip
wenzelm@44112: 
wenzelm@44112:   \noindent Some further operations address the extension aspect of a
wenzelm@44112:   derived record scheme specifically: \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}} produces a
wenzelm@44112:   record fragment consisting of exactly the new fields introduced here
wenzelm@44112:   (the result may serve as a more part elsewhere); \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}extend{\isaliteral{22}{\isachardoublequote}}}
wenzelm@44112:   takes a fixed record and adds a given more part; \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}truncate{\isaliteral{22}{\isachardoublequote}}} restricts a record scheme to a fixed record.
wenzelm@44112: 
wenzelm@44112:   \medskip
wenzelm@44112:   \begin{tabular}{lll}
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}extend{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:     \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}truncate{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@44112:   \end{tabular}
wenzelm@44112:   \medskip
wenzelm@44112: 
wenzelm@44112:   \noindent Note that \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}} coincide
wenzelm@44112:   for root records.%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \isamarkupsubsection{Derived rules and proof tools%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: The record package proves several results internally, declaring
wenzelm@44112:   these facts to appropriate proof tools.  This enables users to
wenzelm@44112:   reason about record structures quite conveniently.  Assume that
wenzelm@44112:   \isa{t} is a record type as specified above.
wenzelm@44112: 
wenzelm@44112:   \begin{enumerate}
wenzelm@44112: 
wenzelm@44112:   \item Standard conversions for selectors or updates applied to
wenzelm@44112:   record constructor terms are made part of the default Simplifier
wenzelm@44112:   context; thus proofs by reduction of basic operations merely require
wenzelm@44112:   the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments.  These rules
wenzelm@44112:   are available as \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}simps{\isaliteral{22}{\isachardoublequote}}}, too.
wenzelm@44112: 
wenzelm@44112:   \item Selectors applied to updated records are automatically reduced
wenzelm@44112:   by an internal simplification procedure, which is also part of the
wenzelm@44112:   standard Simplifier setup.
wenzelm@44112: 
wenzelm@44112:   \item Inject equations of a form analogous to \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequote}}} are declared to the Simplifier and Classical
wenzelm@44112:   Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules.  These rules are available as
wenzelm@44112:   \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}iffs{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \item The introduction rule for record equality analogous to \isa{{\isaliteral{22}{\isachardoublequote}}x\ r\ {\isaliteral{3D}{\isacharequal}}\ x\ r{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ r\ {\isaliteral{3D}{\isacharequal}}\ y\ r{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ r\ {\isaliteral{3D}{\isacharequal}}\ r{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequote}}} is declared to the Simplifier,
wenzelm@44112:   and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}''.
wenzelm@44112:   The rule is called \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}equality{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \item Representations of arbitrary record expressions as canonical
wenzelm@44112:   constructor terms are provided both in \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} format (cf.\ the generic proof methods of the same name,
wenzelm@44112:   \secref{sec:cases-induct}).  Several variations are available, for
wenzelm@44112:   fixed records, record schemes, more parts etc.
wenzelm@44112: 
wenzelm@44112:   The generic proof methods are sufficiently smart to pick the most
wenzelm@44112:   sensible rule according to the type of the indicated record
wenzelm@44112:   expression: users just need to apply something like ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}cases\ r{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' to a certain proof problem.
wenzelm@44112: 
wenzelm@44112:   \item The derived record operations \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}extend{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}truncate{\isaliteral{22}{\isachardoublequote}}} are \emph{not}
wenzelm@44112:   treated automatically, but usually need to be expanded by hand,
wenzelm@44112:   using the collective fact \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}defs{\isaliteral{22}{\isachardoublequote}}}.
wenzelm@44112: 
wenzelm@44112:   \end{enumerate}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44115: \isamarkupsubsubsection{Examples%
wenzelm@44115: }
wenzelm@44115: \isamarkuptrue%
wenzelm@44115: %
wenzelm@44115: \begin{isamarkuptext}%
wenzelm@44115: See \verb|~~/src/HOL/ex/Records.thy|, for example.%
wenzelm@44115: \end{isamarkuptext}%
wenzelm@44115: \isamarkuptrue%
wenzelm@44115: %
wenzelm@44112: \isamarkupsection{Adhoc tuples%
wenzelm@44112: }
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@44112: \begin{isamarkuptext}%
wenzelm@44112: \begin{matharray}{rcl}
wenzelm@44112:     \indexdef{HOL}{attribute}{split\_format}\hypertarget{attribute.HOL.split-format}{\hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isaliteral{5F}{\isacharunderscore}}format}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{attribute} \\
wenzelm@44112:   \end{matharray}
wenzelm@44112: 
wenzelm@44112:   \begin{railoutput}
wenzelm@44112: \rail@begin{2}{}
wenzelm@44112: \rail@term{\hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isaliteral{5F}{\isacharunderscore}}format}}}}[]
wenzelm@44112: \rail@bar
wenzelm@44112: \rail@nextbar{1}
wenzelm@44112: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@44112: \rail@term{\isa{complete}}[]
wenzelm@44112: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@44112: \rail@endbar
wenzelm@44112: \rail@end
wenzelm@44112: \end{railoutput}
wenzelm@44112: 
wenzelm@44112: 
wenzelm@44112:   \begin{description}
wenzelm@44112: 
wenzelm@44112:   \item \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isaliteral{5F}{\isacharunderscore}}format}}}\ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}complete{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} causes
wenzelm@44112:   arguments in function applications to be represented canonically
wenzelm@44112:   according to their tuple type structure.
wenzelm@44112: 
wenzelm@44112:   Note that this operation tends to invent funny names for new local
wenzelm@44112:   parameters introduced.
wenzelm@44112: 
wenzelm@44112:   \end{description}%
wenzelm@44112: \end{isamarkuptext}%
wenzelm@44112: \isamarkuptrue%
wenzelm@44112: %
wenzelm@35757: \isamarkupsection{Typedef axiomatization \label{sec:hol-typedef}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@44111: A Gordon/HOL-style type definition is a certain axiom scheme
wenzelm@44111:   that identifies a new type with a subset of an existing type.  More
wenzelm@44111:   precisely, the new type is defined by exhibiting an existing type
wenzelm@44111:   \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}}, a set \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\ set{\isaliteral{22}{\isachardoublequote}}}, and a theorem that proves
wenzelm@44111:   \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequote}}}.  Thus \isa{A} is a non-empty subset of \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}}, and the new type denotes this subset.  New functions are
wenzelm@44111:   postulated that establish an isomorphism between the new type and
wenzelm@44111:   the subset.  In general, the type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} may involve type
wenzelm@44111:   variables \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} which means that the type definition
wenzelm@44111:   produces a type constructor \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} depending on
wenzelm@44111:   those type arguments.
wenzelm@44111: 
wenzelm@44111:   The axiomatization can be considered a ``definition'' in the sense
wenzelm@44111:   of the particular set-theoretic interpretation of HOL
wenzelm@44111:   \cite{pitts93}, where the universe of types is required to be
wenzelm@44111:   downwards-closed wrt.\ arbitrary non-empty subsets.  Thus genuinely
wenzelm@44111:   new types introduced by \hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}} stay within the range
wenzelm@44111:   of HOL models by construction.  Note that \indexref{}{command}{type\_synonym}\hyperlink{command.type-synonym}{\mbox{\isa{\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}}}} from Isabelle/Pure merely introduces syntactic
wenzelm@44111:   abbreviations, without any logical significance.
wenzelm@44111:   
wenzelm@44111:   \begin{matharray}{rcl}
wenzelm@40685:     \indexdef{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@44111: \rail@nont{\isa{alt{\isaliteral{5F}{\isacharunderscore}}name}}[]
wenzelm@43467: \rail@endbar
wenzelm@44111: \rail@nont{\isa{abs{\isaliteral{5F}{\isacharunderscore}}type}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
wenzelm@44111: \rail@nont{\isa{rep{\isaliteral{5F}{\isacharunderscore}}set}}[]
wenzelm@43467: \rail@end
wenzelm@44111: \rail@begin{3}{\isa{alt{\isaliteral{5F}{\isacharunderscore}}name}}
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{\isakeyword{open}}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\isa{\isakeyword{open}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@end
wenzelm@44111: \rail@begin{2}{\isa{abs{\isaliteral{5F}{\isacharunderscore}}type}}
wenzelm@43576: \rail@nont{\hyperlink{syntax.typespec-sorts}{\mbox{\isa{typespec{\isaliteral{5F}{\isacharunderscore}}sorts}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@44111: \rail@begin{2}{\isa{rep{\isaliteral{5F}{\isacharunderscore}}set}}
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{\isakeyword{morphisms}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@28788:   \begin{description}
wenzelm@42994: 
wenzelm@40685:   \item \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{3D}{\isacharequal}}\ A{\isaliteral{22}{\isachardoublequote}}}
wenzelm@44111:   axiomatizes a type definition in the background theory of the
wenzelm@44111:   current context, depending on a non-emptiness result of the set
wenzelm@44111:   \isa{A} that needs to be proven here.  The set \isa{A} may
wenzelm@44111:   contain type variables \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} as specified on the LHS,
wenzelm@44111:   but no term variables.
wenzelm@35757: 
wenzelm@44111:   Even though a local theory specification, the newly introduced type
wenzelm@44111:   constructor cannot depend on parameters or assumptions of the
wenzelm@44111:   context: this is structurally impossible in HOL.  In contrast, the
wenzelm@44111:   non-emptiness proof may use local assumptions in unusual situations,
wenzelm@44111:   which could result in different interpretations in target contexts:
wenzelm@44111:   the meaning of the bijection between the representing set \isa{A}
wenzelm@44111:   and the new type \isa{t} may then change in different application
wenzelm@44111:   contexts.
wenzelm@42994: 
wenzelm@44111:   By default, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type
wenzelm@44111:   constructor \isa{t} for the new type, and a term constant \isa{t} for the representing set within the old type.  Use the ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}open{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' option to suppress a separate constant definition
wenzelm@40685:   altogether.  The injection from type to set is called \isa{Rep{\isaliteral{5F}{\isacharunderscore}}t},
wenzelm@44111:   its inverse \isa{Abs{\isaliteral{5F}{\isacharunderscore}}t}, unless explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} specification provides alternative names.
wenzelm@42994: 
wenzelm@44111:   The core axiomatization uses the locale predicate \isa{type{\isaliteral{5F}{\isacharunderscore}}definition} as defined in Isabelle/HOL.  Various basic
wenzelm@44111:   consequences of that are instantiated accordingly, re-using the
wenzelm@44111:   locale facts with names derived from the new type constructor.  Thus
wenzelm@44111:   the generic \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep} is turned into the specific
wenzelm@44111:   \isa{{\isaliteral{22}{\isachardoublequote}}Rep{\isaliteral{5F}{\isacharunderscore}}t{\isaliteral{22}{\isachardoublequote}}}, for example.
wenzelm@44111: 
wenzelm@44111:   Theorems \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep}, \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}inverse}, and \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}inverse}
wenzelm@44111:   provide the most basic characterization as a corresponding
wenzelm@44111:   injection/surjection pair (in both directions).  The derived rules
wenzelm@44111:   \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}inject} and \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}inject} provide a more convenient version of
wenzelm@44111:   injectivity, suitable for automated proof tools (e.g.\ in
wenzelm@44111:   declarations involving \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}).
wenzelm@44111:   Furthermore, the rules \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}cases}~/ \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}induct}, and \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}cases}~/
wenzelm@44111:   \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}induct} provide alternative views on
wenzelm@44111:   surjectivity.  These rules are already declared as set or type rules
wenzelm@44111:   for the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods,
wenzelm@44111:   respectively.
wenzelm@42994: 
wenzelm@35757:   An alternative name for the set definition (and other derived
wenzelm@35757:   entities) may be specified in parentheses; the default is to use
wenzelm@44111:   \isa{t} directly.
wenzelm@26849: 
wenzelm@44111:   \end{description}
wenzelm@44111: 
wenzelm@44111:   \begin{warn}
wenzelm@44111:   If you introduce a new type axiomatically, i.e.\ via \indexref{}{command}{typedecl}\hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}} and \indexref{}{command}{axiomatization}\hyperlink{command.axiomatization}{\mbox{\isa{\isacommand{axiomatization}}}}, the minimum requirement
wenzelm@44111:   is that it has a non-empty model, to avoid immediate collapse of the
wenzelm@44111:   HOL logic.  Moreover, one needs to demonstrate that the
wenzelm@44111:   interpretation of such free-form axiomatizations can coexist with
wenzelm@44111:   that of the regular \indexdef{}{command}{typedef}\hypertarget{command.typedef}{\hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}}} scheme, and any extension
wenzelm@44111:   that other people might have introduced elsewhere (e.g.\ in HOLCF
wenzelm@44111:   \cite{MuellerNvOS99}).
wenzelm@44111:   \end{warn}%
wenzelm@44111: \end{isamarkuptext}%
wenzelm@44111: \isamarkuptrue%
wenzelm@44111: %
wenzelm@44111: \isamarkupsubsubsection{Examples%
wenzelm@44111: }
wenzelm@44111: \isamarkuptrue%
wenzelm@44111: %
wenzelm@44111: \begin{isamarkuptext}%
wenzelm@44111: Type definitions permit the introduction of abstract data
wenzelm@44111:   types in a safe way, namely by providing models based on already
wenzelm@44111:   existing types.  Given some abstract axiomatic description \isa{P}
wenzelm@44111:   of a type, this involves two steps:
wenzelm@44111: 
wenzelm@44111:   \begin{enumerate}
wenzelm@44111: 
wenzelm@44111:   \item Find an appropriate type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} and subset \isa{A} which
wenzelm@44111:   has the desired properties \isa{P}, and make a type definition
wenzelm@44111:   based on this representation.
wenzelm@44111: 
wenzelm@44111:   \item Prove that \isa{P} holds for \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} by lifting \isa{P}
wenzelm@44111:   from the representation.
wenzelm@44111: 
wenzelm@44111:   \end{enumerate}
wenzelm@44111: 
wenzelm@44111:   You can later forget about the representation and work solely in
wenzelm@44111:   terms of the abstract properties \isa{P}.
wenzelm@44111: 
wenzelm@44111:   \medskip The following trivial example pulls a three-element type
wenzelm@44111:   into existence within the formal logical environment of HOL.%
wenzelm@44111: \end{isamarkuptext}%
wenzelm@44111: \isamarkuptrue%
wenzelm@44111: \isacommand{typedef}\isamarkupfalse%
wenzelm@44111: \ three\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ False{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}False{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44111: %
wenzelm@44111: \isadelimproof
wenzelm@44111: \ \ %
wenzelm@44111: \endisadelimproof
wenzelm@44111: %
wenzelm@44111: \isatagproof
wenzelm@44111: \isacommand{by}\isamarkupfalse%
wenzelm@44111: \ blast%
wenzelm@44111: \endisatagproof
wenzelm@44111: {\isafoldproof}%
wenzelm@44111: %
wenzelm@44111: \isadelimproof
wenzelm@44111: \isanewline
wenzelm@44111: %
wenzelm@44111: \endisadelimproof
wenzelm@44111: \isanewline
wenzelm@44111: \isacommand{definition}\isamarkupfalse%
wenzelm@44111: \ {\isaliteral{22}{\isachardoublequoteopen}}One\ {\isaliteral{3D}{\isacharequal}}\ Abs{\isaliteral{5F}{\isacharunderscore}}three\ {\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44111: \isacommand{definition}\isamarkupfalse%
wenzelm@44111: \ {\isaliteral{22}{\isachardoublequoteopen}}Two\ {\isaliteral{3D}{\isacharequal}}\ Abs{\isaliteral{5F}{\isacharunderscore}}three\ {\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ False{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44111: \isacommand{definition}\isamarkupfalse%
wenzelm@44111: \ {\isaliteral{22}{\isachardoublequoteopen}}Three\ {\isaliteral{3D}{\isacharequal}}\ Abs{\isaliteral{5F}{\isacharunderscore}}three\ {\isaliteral{28}{\isacharparenleft}}False{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44111: \isanewline
wenzelm@44111: \isacommand{lemma}\isamarkupfalse%
wenzelm@44111: \ three{\isaliteral{5F}{\isacharunderscore}}distinct{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}One\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Two{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}One\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Three{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}Two\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Three{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44111: %
wenzelm@44111: \isadelimproof
wenzelm@44111: \ \ %
wenzelm@44111: \endisadelimproof
wenzelm@44111: %
wenzelm@44111: \isatagproof
wenzelm@44111: \isacommand{by}\isamarkupfalse%
wenzelm@44111: \ {\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{5F}{\isacharunderscore}}all\ add{\isaliteral{3A}{\isacharcolon}}\ One{\isaliteral{5F}{\isacharunderscore}}def\ Two{\isaliteral{5F}{\isacharunderscore}}def\ Three{\isaliteral{5F}{\isacharunderscore}}def\ Abs{\isaliteral{5F}{\isacharunderscore}}three{\isaliteral{5F}{\isacharunderscore}}inject\ three{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
wenzelm@44111: \endisatagproof
wenzelm@44111: {\isafoldproof}%
wenzelm@44111: %
wenzelm@44111: \isadelimproof
wenzelm@44111: \isanewline
wenzelm@44111: %
wenzelm@44111: \endisadelimproof
wenzelm@44111: \isanewline
wenzelm@44111: \isacommand{lemma}\isamarkupfalse%
wenzelm@44111: \ three{\isaliteral{5F}{\isacharunderscore}}cases{\isaliteral{3A}{\isacharcolon}}\isanewline
wenzelm@44111: \ \ \isakeyword{fixes}\ x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ three\ \isakeyword{obtains}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ One{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ Two{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ Three{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@44111: %
wenzelm@44111: \isadelimproof
wenzelm@44111: \ \ %
wenzelm@44111: \endisadelimproof
wenzelm@44111: %
wenzelm@44111: \isatagproof
wenzelm@44111: \isacommand{by}\isamarkupfalse%
wenzelm@44111: \ {\isaliteral{28}{\isacharparenleft}}cases\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ One{\isaliteral{5F}{\isacharunderscore}}def\ Two{\isaliteral{5F}{\isacharunderscore}}def\ Three{\isaliteral{5F}{\isacharunderscore}}def\ Abs{\isaliteral{5F}{\isacharunderscore}}three{\isaliteral{5F}{\isacharunderscore}}inject\ three{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
wenzelm@44111: \endisatagproof
wenzelm@44111: {\isafoldproof}%
wenzelm@44111: %
wenzelm@44111: \isadelimproof
wenzelm@44111: %
wenzelm@44111: \endisadelimproof
wenzelm@44111: %
wenzelm@44111: \begin{isamarkuptext}%
wenzelm@44111: Note that such trivial constructions are better done with
wenzelm@44111:   derived specification mechanisms such as \hyperlink{command.datatype}{\mbox{\isa{\isacommand{datatype}}}}:%
wenzelm@44111: \end{isamarkuptext}%
wenzelm@44111: \isamarkuptrue%
wenzelm@44111: \isacommand{datatype}\isamarkupfalse%
wenzelm@44111: \ three{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ One{\isaliteral{27}{\isacharprime}}\ {\isaliteral{7C}{\isacharbar}}\ Two{\isaliteral{27}{\isacharprime}}\ {\isaliteral{7C}{\isacharbar}}\ Three{\isaliteral{27}{\isacharprime}}%
wenzelm@44111: \begin{isamarkuptext}%
wenzelm@44111: This avoids re-doing basic definitions and proofs from the
wenzelm@44111:   primitive \hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}} above.%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
haftmann@41644: \isamarkupsection{Functorial structure of types%
haftmann@41644: }
haftmann@41644: \isamarkuptrue%
haftmann@41644: %
haftmann@41644: \begin{isamarkuptext}%
haftmann@41644: \begin{matharray}{rcl}
haftmann@41753:     \indexdef{HOL}{command}{enriched\_type}\hypertarget{command.HOL.enriched-type}{\hyperlink{command.HOL.enriched-type}{\mbox{\isa{\isacommand{enriched{\isaliteral{5F}{\isacharunderscore}}type}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}
haftmann@41644:   \end{matharray}
haftmann@41644: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.enriched-type}{\mbox{\isa{\isacommand{enriched{\isaliteral{5F}{\isacharunderscore}}type}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43488: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
wenzelm@43488: 
haftmann@41644: 
haftmann@41644:   \begin{description}
haftmann@41644: 
wenzelm@43488:   \item \hyperlink{command.HOL.enriched-type}{\mbox{\isa{\isacommand{enriched{\isaliteral{5F}{\isacharunderscore}}type}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}prefix{\isaliteral{3A}{\isacharcolon}}\ m{\isaliteral{22}{\isachardoublequote}}} allows to
wenzelm@43488:   prove and register properties about the functorial structure of type
wenzelm@43488:   constructors.  These properties then can be used by other packages
wenzelm@43488:   to deal with those type constructors in certain type constructions.
wenzelm@43488:   Characteristic theorems are noted in the current local theory.  By
wenzelm@43488:   default, they are prefixed with the base name of the type
wenzelm@43488:   constructor, an explicit prefix can be given alternatively.
haftmann@41644: 
haftmann@41644:   The given term \isa{{\isaliteral{22}{\isachardoublequote}}m{\isaliteral{22}{\isachardoublequote}}} is considered as \emph{mapper} for the
haftmann@41644:   corresponding type constructor and must conform to the following
haftmann@41644:   type pattern:
haftmann@41644: 
haftmann@41644:   \begin{matharray}{lll}
haftmann@41644:     \isa{{\isaliteral{22}{\isachardoublequote}}m{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
haftmann@41644:       \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub k\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} \\
haftmann@41644:   \end{matharray}
haftmann@41644: 
haftmann@41644:   \noindent where \isa{t} is the type constructor, \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} are distinct
haftmann@41644:   type variables free in the local theory and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}},
haftmann@41644:   \ldots, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub k{\isaliteral{22}{\isachardoublequote}}} is a subsequence of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}, \ldots,
haftmann@41644:   \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}}.
haftmann@41644: 
haftmann@41644:   \end{description}%
haftmann@41644: \end{isamarkuptext}%
haftmann@41644: \isamarkuptrue%
haftmann@41644: %
bulwahn@44864: \isamarkupsection{Quotient types%
bulwahn@44864: }
bulwahn@44864: \isamarkuptrue%
bulwahn@44864: %
bulwahn@44864: \begin{isamarkuptext}%
bulwahn@44864: The quotient package defines a new quotient type given a raw type
bulwahn@44864:   and a partial equivalence relation.
bulwahn@44864:   It also includes automation for transporting definitions and theorems.
bulwahn@44864:   It can automatically produce definitions and theorems on the quotient type,
bulwahn@44864:   given the corresponding constants and facts on the raw type.
bulwahn@44864: 
bulwahn@44864:   \begin{matharray}{rcl}
bulwahn@44864:     \indexdef{HOL}{command}{quotient\_type}\hypertarget{command.HOL.quotient-type}{\hyperlink{command.HOL.quotient-type}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}type}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}\\
bulwahn@44864:     \indexdef{HOL}{command}{quotient\_definition}\hypertarget{command.HOL.quotient-definition}{\hyperlink{command.HOL.quotient-definition}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}definition}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}\\
bulwahn@44864:     \indexdef{HOL}{command}{print\_quotmaps}\hypertarget{command.HOL.print-quotmaps}{\hyperlink{command.HOL.print-quotmaps}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotmaps}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}}\\
bulwahn@44864:     \indexdef{HOL}{command}{print\_quotients}\hypertarget{command.HOL.print-quotients}{\hyperlink{command.HOL.print-quotients}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotients}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}}\\
bulwahn@44864:     \indexdef{HOL}{command}{print\_quotconsts}\hypertarget{command.HOL.print-quotconsts}{\hyperlink{command.HOL.print-quotconsts}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotconsts}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}}\\
bulwahn@44864:   \end{matharray}
bulwahn@44864: 
bulwahn@44864:   \begin{railoutput}
bulwahn@44864: \rail@begin{2}{}
bulwahn@44864: \rail@term{\hyperlink{command.HOL.quotient-type}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}type}}}}}[]
bulwahn@44864: \rail@plus
bulwahn@44864: \rail@nont{\isa{spec}}[]
bulwahn@44864: \rail@nextplus{1}
bulwahn@44864: \rail@cterm{\isa{\isakeyword{and}}}[]
bulwahn@44864: \rail@endplus
bulwahn@44864: \rail@end
bulwahn@44864: \rail@begin{5}{\isa{spec}}
bulwahn@44864: \rail@nont{\hyperlink{syntax.typespec}{\mbox{\isa{typespec}}}}[]
bulwahn@44864: \rail@bar
bulwahn@44864: \rail@nextbar{1}
bulwahn@44864: \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
bulwahn@44864: \rail@endbar
bulwahn@44864: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
bulwahn@44864: \rail@cr{3}
bulwahn@44864: \rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]
bulwahn@44864: \rail@term{\isa{{\isaliteral{2F}{\isacharslash}}}}[]
bulwahn@44864: \rail@bar
bulwahn@44864: \rail@nextbar{4}
bulwahn@44864: \rail@term{\isa{partial}}[]
bulwahn@44864: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
bulwahn@44864: \rail@endbar
bulwahn@44864: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
bulwahn@44864: \rail@end
bulwahn@44864: \end{railoutput}
bulwahn@44864: 
bulwahn@44864: 
bulwahn@44864:   \begin{railoutput}
bulwahn@44864: \rail@begin{4}{}
bulwahn@44864: \rail@term{\hyperlink{command.HOL.quotient-definition}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}definition}}}}}[]
bulwahn@44864: \rail@bar
bulwahn@44864: \rail@nextbar{1}
bulwahn@44864: \rail@nont{\isa{constdecl}}[]
bulwahn@44864: \rail@endbar
bulwahn@44864: \rail@bar
bulwahn@44864: \rail@nextbar{1}
bulwahn@44864: \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
bulwahn@44864: \rail@endbar
bulwahn@44864: \rail@cr{3}
bulwahn@44864: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
bulwahn@44864: \rail@term{\isa{is}}[]
bulwahn@44864: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
bulwahn@44864: \rail@end
bulwahn@44864: \rail@begin{2}{\isa{constdecl}}
bulwahn@44864: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
bulwahn@44864: \rail@bar
bulwahn@44864: \rail@nextbar{1}
bulwahn@44864: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}}}[]
bulwahn@44864: \rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]
bulwahn@44864: \rail@endbar
bulwahn@44864: \rail@bar
bulwahn@44864: \rail@nextbar{1}
bulwahn@44864: \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
bulwahn@44864: \rail@endbar
bulwahn@44864: \rail@end
bulwahn@44864: \end{railoutput}
bulwahn@44864: 
bulwahn@44864: 
bulwahn@44864:   \begin{description}
bulwahn@44864:   
bulwahn@44864:   \item \hyperlink{command.HOL.quotient-type}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}type}}}} defines quotient types.
bulwahn@44864: 
bulwahn@44864:   \item \hyperlink{command.HOL.quotient-definition}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}definition}}}} defines a constant on the quotient type.
bulwahn@44864: 
bulwahn@44864:   \item \hyperlink{command.HOL.print-quotmaps}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotmaps}}}} prints quotient map functions.
bulwahn@44864: 
bulwahn@44864:   \item \hyperlink{command.HOL.print-quotients}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotients}}}} prints quotients.
bulwahn@44864: 
bulwahn@44864:   \item \hyperlink{command.HOL.print-quotconsts}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotconsts}}}} prints quotient constants.
bulwahn@44864: 
bulwahn@44864:   \end{description}%
bulwahn@44864: \end{isamarkuptext}%
bulwahn@44864: \isamarkuptrue%
bulwahn@44864: %
wenzelm@26849: \isamarkupsection{Arithmetic proof support%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@28788:     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
nipkow@30863:     \indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
wenzelm@40685:     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isaliteral{5F}{\isacharunderscore}}split}}}} & : & \isa{attribute} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26902:   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
wenzelm@26849:   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
wenzelm@26849:   facts are inserted into the goal before running the procedure.
wenzelm@26849: 
nipkow@30863:   The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
nipkow@30863:   always supplied to the arithmetic provers implicitly.
nipkow@30863: 
wenzelm@40685:   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isaliteral{5F}{\isacharunderscore}}split}}} attribute declares case split
wenzelm@30865:   rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked.
wenzelm@26849: 
nipkow@30863:   Note that a simpler (but faster) arithmetic prover is
nipkow@30863:   already invoked by the Simplifier.%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@30172: \isamarkupsection{Intuitionistic proof search%
wenzelm@30172: }
wenzelm@30172: \isamarkuptrue%
wenzelm@30172: %
wenzelm@30172: \begin{isamarkuptext}%
wenzelm@30172: \begin{matharray}{rcl}
wenzelm@30172:     \indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\
wenzelm@30172:   \end{matharray}
wenzelm@30172: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cnont{\hyperlink{syntax.rulemod}{\mbox{\isa{rulemod}}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
wenzelm@43467: 
wenzelm@30172: 
wenzelm@30172:   The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
wenzelm@30172:   search, depending on specifically declared rules from the context,
wenzelm@30172:   or given as explicit arguments.  Chained facts are inserted into the
wenzelm@35613:   goal before commencing proof search.
wenzelm@35613: 
wenzelm@30172:   Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
wenzelm@30172:   \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
wenzelm@40685:   ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{21}{\isacharbang}}{\isaliteral{22}{\isachardoublequote}}}'' indicator refers to ``safe'' rules, which may be
wenzelm@30172:   applied aggressively (without considering back-tracking later).
wenzelm@40685:   Rules declared with ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}'' are ignored in proof search (the
wenzelm@43497:   single-step \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} method still observes these).  An
wenzelm@30172:   explicit weight annotation may be given as well; otherwise the
wenzelm@30172:   number of rule premises will be taken into account here.%
wenzelm@30172: \end{isamarkuptext}%
wenzelm@30172: \isamarkuptrue%
wenzelm@30172: %
blanchet@44440: \isamarkupsection{Model Elimination and Resolution%
blanchet@44440: }
blanchet@44440: \isamarkuptrue%
blanchet@44440: %
blanchet@44440: \begin{isamarkuptext}%
blanchet@44440: \begin{matharray}{rcl}
blanchet@44440:     \indexdef{HOL}{method}{meson}\hypertarget{method.HOL.meson}{\hyperlink{method.HOL.meson}{\mbox{\isa{meson}}}} & : & \isa{method} \\
blanchet@44440:     \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isa{method} \\
blanchet@44440:   \end{matharray}
blanchet@44440: 
blanchet@44440:   \begin{railoutput}
blanchet@44440: \rail@begin{2}{}
blanchet@44440: \rail@term{\hyperlink{method.HOL.meson}{\mbox{\isa{meson}}}}[]
blanchet@44440: \rail@bar
blanchet@44440: \rail@nextbar{1}
blanchet@44440: \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
blanchet@44440: \rail@endbar
blanchet@44440: \rail@end
blanchet@44440: \rail@begin{5}{}
blanchet@44440: \rail@term{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}}[]
blanchet@44440: \rail@bar
blanchet@44440: \rail@nextbar{1}
blanchet@44440: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
blanchet@44440: \rail@bar
blanchet@44440: \rail@term{\isa{partial{\isaliteral{5F}{\isacharunderscore}}types}}[]
blanchet@44440: \rail@nextbar{2}
blanchet@44440: \rail@term{\isa{full{\isaliteral{5F}{\isacharunderscore}}types}}[]
blanchet@44440: \rail@nextbar{3}
blanchet@44440: \rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}types}}[]
blanchet@44440: \rail@nextbar{4}
blanchet@44440: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
blanchet@44440: \rail@endbar
blanchet@44440: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
blanchet@44440: \rail@endbar
blanchet@44440: \rail@bar
blanchet@44440: \rail@nextbar{1}
blanchet@44440: \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
blanchet@44440: \rail@endbar
blanchet@44440: \rail@end
blanchet@44440: \end{railoutput}
blanchet@44440: 
blanchet@44440: 
blanchet@44440:   The \hyperlink{method.HOL.meson}{\mbox{\isa{meson}}} method implements Loveland's model elimination
blanchet@44440:   procedure \cite{loveland-78}. See \verb|~~/src/HOL/ex/Meson_Test.thy| for
blanchet@44440:   examples.
blanchet@44440: 
blanchet@44440:   The \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}} method combines ordered resolution and ordered
blanchet@44440:   paramodulation to find first-order (or mildly higher-order) proofs. The first
blanchet@44440:   optional argument specifies a type encoding; see the Sledgehammer manual
blanchet@44440:   \cite{isabelle-sledgehammer} for details. The \verb|~~/src/HOL/Metis_Examples| directory contains several small theories
blanchet@44440:   developed to a large extent using Metis.%
blanchet@44440: \end{isamarkuptext}%
blanchet@44440: \isamarkuptrue%
blanchet@44440: %
wenzelm@30172: \isamarkupsection{Coherent Logic%
wenzelm@30172: }
wenzelm@30172: \isamarkuptrue%
wenzelm@30172: %
wenzelm@30172: \begin{isamarkuptext}%
wenzelm@30172: \begin{matharray}{rcl}
wenzelm@30172:     \indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\
wenzelm@30172:   \end{matharray}
wenzelm@30172: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
wenzelm@43467: 
wenzelm@30172: 
wenzelm@30172:   The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
wenzelm@30172:   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
wenzelm@30172:   applications in confluence theory, lattice theory and projective
wenzelm@41052:   geometry.  See \verb|~~/src/HOL/ex/Coherent.thy| for some
wenzelm@30172:   examples.%
wenzelm@30172: \end{isamarkuptext}%
wenzelm@30172: \isamarkuptrue%
wenzelm@30172: %
blanchet@43082: \isamarkupsection{Proving propositions%
blanchet@43082: }
blanchet@43082: \isamarkuptrue%
blanchet@43082: %
blanchet@43082: \begin{isamarkuptext}%
blanchet@43082: In addition to the standard proof methods, a number of diagnosis
blanchet@43082:   tools search for proofs and provide an Isar proof snippet on success.
blanchet@43082:   These tools are available via the following commands.
blanchet@43082: 
blanchet@43082:   \begin{matharray}{rcl}
blanchet@43082:     \indexdef{HOL}{command}{solve\_direct}\hypertarget{command.HOL.solve-direct}{\hyperlink{command.HOL.solve-direct}{\mbox{\isa{\isacommand{solve{\isaliteral{5F}{\isacharunderscore}}direct}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{try}\hypertarget{command.HOL.try}{\hyperlink{command.HOL.try}{\mbox{\isa{\isacommand{try}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43881:     \indexdef{HOL}{command}{try\_methods}\hypertarget{command.HOL.try-methods}{\hyperlink{command.HOL.try-methods}{\mbox{\isa{\isacommand{try{\isaliteral{5F}{\isacharunderscore}}methods}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{sledgehammer}\hypertarget{command.HOL.sledgehammer}{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{sledgehammer\_params}\hypertarget{command.HOL.sledgehammer-params}{\hyperlink{command.HOL.sledgehammer-params}{\mbox{\isa{\isacommand{sledgehammer{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}}
blanchet@43082:   \end{matharray}
blanchet@43082: 
wenzelm@43467:   \begin{railoutput}
blanchet@43881: \rail@begin{1}{}
blanchet@43881: \rail@term{\hyperlink{command.HOL.try}{\mbox{\isa{\isacommand{try}}}}}[]
blanchet@43881: \rail@end
wenzelm@43535: \rail@begin{6}{}
blanchet@43881: \rail@term{\hyperlink{command.HOL.try-methods}{\mbox{\isa{\isacommand{try{\isaliteral{5F}{\isacharunderscore}}methods}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\isa{simp}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\isa{intro}}[]
wenzelm@43467: \rail@nextbar{3}
wenzelm@43467: \rail@term{\isa{elim}}[]
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@term{\isa{dest}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
wenzelm@43467: \rail@nextplus{5}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
wenzelm@43467: \rail@nont{\isa{args}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{facts}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.sledgehammer-params}{\mbox{\isa{\isacommand{sledgehammer{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
wenzelm@43467: \rail@nont{\isa{args}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{args}}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
wenzelm@43467: \rail@nont{\isa{value}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cterm{\isa{{\isaliteral{2C}{\isacharcomma}}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{5}{\isa{facts}}
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\isa{add}}[]
wenzelm@43467: \rail@nextbar{3}
wenzelm@43467: \rail@term{\isa{del}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
wenzelm@43467: \rail@nextplus{4}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
blanchet@43881:  % FIXME check args "value"
blanchet@43082: 
blanchet@43082:   \begin{description}
blanchet@43082: 
blanchet@43082:   \item \hyperlink{command.HOL.solve-direct}{\mbox{\isa{\isacommand{solve{\isaliteral{5F}{\isacharunderscore}}direct}}}} checks whether the current subgoals can
blanchet@43082:     be solved directly by an existing theorem. Duplicate lemmas can be detected
blanchet@43082:     in this way.
blanchet@43082: 
blanchet@43881:   \item \hyperlink{command.HOL.try-methods}{\mbox{\isa{\isacommand{try{\isaliteral{5F}{\isacharunderscore}}methods}}}} attempts to prove a subgoal using a combination
blanchet@43082:     of standard proof methods (\isa{auto}, \isa{simp}, \isa{blast}, etc.).
blanchet@43082:     Additional facts supplied via \isa{{\isaliteral{22}{\isachardoublequote}}simp{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}intro{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}},
blanchet@43082:     \isa{{\isaliteral{22}{\isachardoublequote}}elim{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}, and \isa{{\isaliteral{22}{\isachardoublequote}}dest{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} are passed to the appropriate proof
blanchet@43082:     methods.
blanchet@43082: 
bulwahn@44785:   \item \hyperlink{command.HOL.try}{\mbox{\isa{\isacommand{try}}}} attempts to prove or disprove a subgoal
bulwahn@44785:     using a combination of provers and disprovers (\isa{{\isaliteral{22}{\isachardoublequote}}solve{\isaliteral{5F}{\isacharunderscore}}direct{\isaliteral{22}{\isachardoublequote}}},
bulwahn@44785:     \isa{{\isaliteral{22}{\isachardoublequote}}quickcheck{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}try{\isaliteral{5F}{\isacharunderscore}}methods{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}sledgehammer{\isaliteral{22}{\isachardoublequote}}},
bulwahn@44785:     \isa{{\isaliteral{22}{\isachardoublequote}}nitpick{\isaliteral{22}{\isachardoublequote}}}).
bulwahn@44785: 
blanchet@43082:   \item \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} attempts to prove a subgoal using external
blanchet@43082:     automatic provers (resolution provers and SMT solvers). See the Sledgehammer
blanchet@43082:     manual \cite{isabelle-sledgehammer} for details.
blanchet@43082: 
blanchet@43082:   \item \hyperlink{command.HOL.sledgehammer-params}{\mbox{\isa{\isacommand{sledgehammer{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
blanchet@43082:     \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} configuration options persistently.
blanchet@43082: 
blanchet@43082:   \end{description}%
blanchet@43082: \end{isamarkuptext}%
blanchet@43082: \isamarkuptrue%
blanchet@43082: %
haftmann@31907: \isamarkupsection{Checking and refuting propositions%
haftmann@31907: }
haftmann@31907: \isamarkuptrue%
haftmann@31907: %
haftmann@31907: \begin{isamarkuptext}%
haftmann@31907: Identifying incorrect propositions usually involves evaluation of
blanchet@43082:   particular assignments and systematic counterexample search.  This
haftmann@31907:   is supported by the following commands.
haftmann@31907: 
haftmann@31907:   \begin{matharray}{rcl}
wenzelm@40685:     \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{quickcheck}\hypertarget{command.HOL.quickcheck}{\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{refute}\hypertarget{command.HOL.refute}{\hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{nitpick}\hypertarget{command.HOL.nitpick}{\hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{quickcheck\_params}\hypertarget{command.HOL.quickcheck-params}{\hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{refute\_params}\hypertarget{command.HOL.refute-params}{\hyperlink{command.HOL.refute-params}{\mbox{\isa{\isacommand{refute{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
blanchet@43082:     \indexdef{HOL}{command}{nitpick\_params}\hypertarget{command.HOL.nitpick-params}{\hyperlink{command.HOL.nitpick-params}{\mbox{\isa{\isacommand{nitpick{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}}
haftmann@31907:   \end{matharray}
haftmann@31907: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
wenzelm@43467: \rail@nont{\isa{name}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{modes}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{3}{}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
wenzelm@43467: \rail@nont{\isa{args}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{3}{}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.refute-params}{\mbox{\isa{\isacommand{refute{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.nitpick-params}{\mbox{\isa{\isacommand{nitpick{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
wenzelm@43467: \rail@nont{\isa{args}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{modes}}
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{args}}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
wenzelm@43467: \rail@nont{\isa{value}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cterm{\isa{{\isaliteral{2C}{\isacharcomma}}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
wenzelm@43467:  % FIXME check "value"
haftmann@31907: 
haftmann@31907:   \begin{description}
haftmann@31907: 
haftmann@31907:   \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a
haftmann@31907:     term; optionally \isa{modes} can be specified, which are
wenzelm@44130:     appended to the current print mode; see \secref{sec:print-modes}.
haftmann@31907:     Internally, the evaluation is performed by registered evaluators,
haftmann@31907:     which are invoked sequentially until a result is returned.
haftmann@31907:     Alternatively a specific evaluator can be selected using square
haftmann@37419:     brackets; typical evaluators use the current set of code equations
wenzelm@44130:     to normalize and include \isa{simp} for fully symbolic
wenzelm@44130:     evaluation using the simplifier, \isa{nbe} for
wenzelm@44130:     \emph{normalization by evaluation} and \emph{code} for code
wenzelm@44130:     generation in SML.
haftmann@31907: 
haftmann@31907:   \item \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} tests the current goal for
blanchet@43082:     counterexamples using a series of assignments for its
haftmann@31907:     free variables; by default the first subgoal is tested, an other
haftmann@31907:     can be selected explicitly using an optional goal index.
wenzelm@41185:     Assignments can be chosen exhausting the search space upto a given
bulwahn@44785:     size, or using a fixed number of random assignments in the search space,
bulwahn@44785:     or exploring the search space symbolically using narrowing.
wenzelm@41185:     By default, quickcheck uses exhaustive testing.
haftmann@31907:     A number of configuration options are supported for
haftmann@31907:     \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}, notably:
haftmann@31907: 
haftmann@31907:     \begin{description}
haftmann@31907: 
bulwahn@44785:     \item[\isa{tester}] specifies which testing approach to apply.
bulwahn@44785:       There are three testers, \isa{exhaustive},
bulwahn@44785:       \isa{random}, and \isa{narrowing}.
wenzelm@41185:       An unknown configuration option is treated as an argument to tester,
wenzelm@41185:       making \isa{{\isaliteral{22}{\isachardoublequote}}tester\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{22}{\isachardoublequote}}} optional.
bulwahn@44785:       When multiple testers are given, these are applied in parallel. 
bulwahn@44785:       If no tester is specified, quickcheck uses the testers that are
bulwahn@44785:       set active, i.e., configurations
bulwahn@44785:       \isa{quickcheck{\isaliteral{5F}{\isacharunderscore}}exhaustive{\isaliteral{5F}{\isacharunderscore}}active}, \isa{quickcheck{\isaliteral{5F}{\isacharunderscore}}random{\isaliteral{5F}{\isacharunderscore}}active},
bulwahn@44785:       \isa{quickcheck{\isaliteral{5F}{\isacharunderscore}}narrowing{\isaliteral{5F}{\isacharunderscore}}active} are set to true.
wenzelm@40515:     \item[\isa{size}] specifies the maximum size of the search space
wenzelm@40515:     for assignment values.
haftmann@31907: 
wenzelm@42994:     \item[\isa{eval}] takes a term or a list of terms and evaluates
wenzelm@42994:       these terms under the variable assignment found by quickcheck.
wenzelm@42994: 
wenzelm@40515:     \item[\isa{iterations}] sets how many sets of assignments are
wenzelm@40515:     generated for each particular size.
haftmann@31907: 
wenzelm@40685:     \item[\isa{no{\isaliteral{5F}{\isacharunderscore}}assms}] specifies whether assumptions in
wenzelm@40515:     structured proofs should be ignored.
wenzelm@40515: 
wenzelm@40515:     \item[\isa{timeout}] sets the time limit in seconds.
wenzelm@40515: 
wenzelm@40685:     \item[\isa{default{\isaliteral{5F}{\isacharunderscore}}type}] sets the type(s) generally used to
wenzelm@40515:     instantiate type variables.
wenzelm@40515: 
wenzelm@40515:     \item[\isa{report}] if set quickcheck reports how many tests
wenzelm@40515:     fulfilled the preconditions.
wenzelm@40515: 
wenzelm@40515:     \item[\isa{quiet}] if not set quickcheck informs about the
wenzelm@40515:     current size for assignment values.
wenzelm@40515: 
wenzelm@40515:     \item[\isa{expect}] can be used to check if the user's
wenzelm@40685:     expectation was met (\isa{no{\isaliteral{5F}{\isacharunderscore}}expectation}, \isa{no{\isaliteral{5F}{\isacharunderscore}}counterexample}, or \isa{counterexample}).
wenzelm@35352: 
haftmann@31907:     \end{description}
haftmann@31907: 
haftmann@31907:     These option can be given within square brackets.
haftmann@31907: 
blanchet@43082:   \item \hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
blanchet@43082:     \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} configuration options persistently.
blanchet@43082: 
blanchet@43082:   \item \hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}} tests the current goal for
blanchet@43082:     counterexamples using a reduction to SAT. The following configuration
blanchet@43082:     options are supported:
blanchet@43082: 
blanchet@43082:     \begin{description}
blanchet@43082: 
blanchet@43082:     \item[\isa{minsize}] specifies the minimum size (cardinality) of the
blanchet@43082:       models to search for.
blanchet@43082: 
blanchet@43082:     \item[\isa{maxsize}] specifies the maximum size (cardinality) of the
blanchet@43082:       models to search for. Nonpositive values mean $\infty$.
blanchet@43082: 
blanchet@43082:     \item[\isa{maxvars}] specifies the maximum number of Boolean variables
blanchet@43082:     to use when transforming the term into a propositional formula.
blanchet@43082:     Nonpositive values mean $\infty$.
blanchet@43082: 
blanchet@43082:     \item[\isa{satsolver}] specifies the SAT solver to use.
blanchet@43082: 
blanchet@43082:     \item[\isa{no{\isaliteral{5F}{\isacharunderscore}}assms}] specifies whether assumptions in
blanchet@43082:     structured proofs should be ignored.
blanchet@43082: 
blanchet@43082:     \item[\isa{maxtime}] sets the time limit in seconds.
blanchet@43082: 
blanchet@43082:     \item[\isa{expect}] can be used to check if the user's
blanchet@43082:     expectation was met (\isa{genuine}, \isa{potential},
blanchet@43082:     \isa{none}, or \isa{unknown}).
blanchet@43082: 
blanchet@43082:     \end{description}
blanchet@43082: 
blanchet@43082:     These option can be given within square brackets.
blanchet@43082: 
blanchet@43082:   \item \hyperlink{command.HOL.refute-params}{\mbox{\isa{\isacommand{refute{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
blanchet@43082:     \hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}} configuration options persistently.
blanchet@43082: 
blanchet@43082:   \item \hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}} tests the current goal for counterexamples
blanchet@43082:     using a reduction to first-order relational logic. See the Nitpick manual
blanchet@43082:     \cite{isabelle-nitpick} for details.
blanchet@43082: 
blanchet@43082:   \item \hyperlink{command.HOL.nitpick-params}{\mbox{\isa{\isacommand{nitpick{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
blanchet@43082:     \hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}} configuration options persistently.
haftmann@31907: 
haftmann@31907:   \end{description}%
haftmann@31907: \end{isamarkuptext}%
haftmann@31907: \isamarkuptrue%
haftmann@31907: %
wenzelm@28788: \isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@27124: The following tools of Isabelle/HOL support cases analysis and
wenzelm@27124:   induction in unstructured tactic scripts; see also
wenzelm@27124:   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@40685:     \indexdef{HOL}{method}{case\_tac}\hypertarget{method.HOL.case-tac}{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
wenzelm@40685:     \indexdef{HOL}{method}{induct\_tac}\hypertarget{method.HOL.induct-tac}{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
wenzelm@40685:     \indexdef{HOL}{method}{ind\_cases}\hypertarget{method.HOL.ind-cases}{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
wenzelm@40685:     \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43576: \rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{rule}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{3}{}
wenzelm@43467: \rail@term{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43576: \rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.insts}{\mbox{\isa{insts}}}}[]
wenzelm@43467: \rail@nextplus{2}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{rule}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{3}{}
wenzelm@43467: \rail@term{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{\isakeyword{for}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@nextplus{2}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{3}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@nextplus{2}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{1}{\isa{rule}}
wenzelm@43467: \rail@term{\isa{rule}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.thmref}{\mbox{\isa{thmref}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
wenzelm@26849: 
wenzelm@26849: 
wenzelm@28788:   \begin{description}
wenzelm@26849: 
wenzelm@40685:   \item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}} admit
wenzelm@28788:   to reason about inductive types.  Rules are selected according to
wenzelm@28788:   the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
wenzelm@28788:   attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
wenzelm@27124: 
wenzelm@27124:   These unstructured tactics feature both goal addressing and dynamic
wenzelm@26849:   instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@27124:   as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
wenzelm@40685:   methods (see \secref{sec:cases-induct}).  Unlike the \hyperlink{method.induct}{\mbox{\isa{induct}}} method, \hyperlink{method.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}} does not handle structured rule
wenzelm@27124:   statements, only the compact object-logic conclusion of the subgoal
wenzelm@27124:   being addressed.
wenzelm@42994: 
wenzelm@40685:   \item \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}} and \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}} provide an interface to the internal \verb|mk_cases| operation.  Rules are simplified in an unrestricted
wenzelm@26861:   forward manner.
wenzelm@26849: 
wenzelm@40685:   While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}} is a proof method to apply the
wenzelm@40685:   result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}} provides case split theorems at the theory level
wenzelm@40685:   for later use.  The \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} argument of the \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}} method allows to specify a list of variables that should
wenzelm@26849:   be generalized before applying the resulting rule.
wenzelm@26849: 
wenzelm@28788:   \end{description}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Executable code%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@43498: For validation purposes, it is often useful to \emph{execute}
wenzelm@43498:   specifications.  In principle, execution could be simulated by
wenzelm@43498:   Isabelle's inference kernel, i.e. by a combination of resolution and
wenzelm@43498:   simplification.  Unfortunately, this approach is rather inefficient.
wenzelm@43498:   A more efficient way of executing specifications is to translate
wenzelm@43498:   them into a functional programming language such as ML.
wenzelm@26849: 
wenzelm@43498:   Isabelle provides two generic frameworks to support code generation
wenzelm@43498:   from executable specifications.  Isabelle/HOL instantiates these
wenzelm@43498:   mechanisms in a way that is amenable to end-user applications.%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \isamarkupsubsection{The new code generator (F. Haftmann)%
wenzelm@43498: }
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: This framework generates code from functional programs
haftmann@37397:   (including overloading using type classes) to SML \cite{SML}, OCaml
haftmann@39048:   \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
wenzelm@43498:   \cite{scala-overview-tech-report}.  Conceptually, code generation is
wenzelm@43498:   split up in three steps: \emph{selection} of code theorems,
wenzelm@43498:   \emph{translation} into an abstract executable view and
wenzelm@43498:   \emph{serialization} to a specific \emph{target language}.
wenzelm@43498:   Inductive specifications can be executed using the predicate
wenzelm@43498:   compiler which operates within HOL.  See \cite{isabelle-codegen} for
wenzelm@43498:   an introduction.
haftmann@37397: 
haftmann@37397:   \begin{matharray}{rcl}
wenzelm@40685:     \indexdef{HOL}{command}{export\_code}\hypertarget{command.HOL.export-code}{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isaliteral{5F}{\isacharunderscore}}code}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
haftmann@37397:     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_abort}\hypertarget{command.HOL.code-abort}{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}abort}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}datatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{print\_codesetup}\hypertarget{command.HOL.print-codesetup}{\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codesetup}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{attribute}{code\_inline}\hypertarget{attribute.HOL.code-inline}{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}inline}}}} & : & \isa{attribute} \\
wenzelm@40685:     \indexdef{HOL}{attribute}{code\_post}\hypertarget{attribute.HOL.code-post}{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}post}}}} & : & \isa{attribute} \\
wenzelm@40685:     \indexdef{HOL}{command}{print\_codeproc}\hypertarget{command.HOL.print-codeproc}{\hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codeproc}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_thms}\hypertarget{command.HOL.code-thms}{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}thms}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_deps}\hypertarget{command.HOL.code-deps}{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}deps}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}const}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}type}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}class}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}instance}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reserved}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}monad}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}include}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}modulename}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{code\_reflect}\hypertarget{command.HOL.code-reflect}{\hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reflect}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}}
haftmann@37397:   \end{matharray}
haftmann@37397: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{11}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isaliteral{5F}{\isacharunderscore}}code}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{constexpr}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@cr{3}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@term{\isa{\isakeyword{in}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{5}
wenzelm@43467: \rail@term{\isa{\isakeyword{module{\isaliteral{5F}{\isacharunderscore}}name}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@cr{7}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{8}
wenzelm@43467: \rail@term{\isa{\isakeyword{file}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nextbar{9}
wenzelm@43467: \rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{8}
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@nont{\isa{args}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nextplus{10}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{1}{\isa{const}}
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{3}{\isa{constexpr}}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nont{\isa{const}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{name{\isaliteral{2E}{\isachardot}}{\isaliteral{5F}{\isacharunderscore}}}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\isa{{\isaliteral{5F}{\isacharunderscore}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{1}{\isa{typeconstructor}}
wenzelm@43467: \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{1}{\isa{class}}
wenzelm@43467: \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{4}{\isa{target}}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\isa{SML}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{OCaml}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\isa{Haskell}}[]
wenzelm@43467: \rail@nextbar{3}
wenzelm@43467: \rail@term{\isa{Scala}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{4}{}
wenzelm@43467: \rail@term{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\isa{del}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\isa{abstype}}[]
wenzelm@43467: \rail@nextbar{3}
wenzelm@43467: \rail@term{\isa{abstract}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}abort}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{const}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}datatype}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{const}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}inline}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{del}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}post}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{del}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{3}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}thms}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{constexpr}}[]
wenzelm@43467: \rail@nextplus{2}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{3}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}deps}}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{constexpr}}[]
wenzelm@43467: \rail@nextplus{2}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{7}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}const}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{const}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@cr{3}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@nont{\isa{syntax}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nextplus{5}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@nextplus{6}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{7}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}type}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{typeconstructor}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@cr{3}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@nont{\isa{syntax}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nextplus{5}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@nextplus{6}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{9}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}class}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{class}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@cr{3}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@cr{5}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{6}
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nextplus{7}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@nextplus{8}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{7}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}instance}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{typeconstructor}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}}}[]
wenzelm@43467: \rail@nont{\isa{class}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@cr{3}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nextplus{5}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@nextplus{6}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reserved}}}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{1}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}monad}}}}}[]
wenzelm@43467: \rail@nont{\isa{const}}[]
wenzelm@43467: \rail@nont{\isa{const}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}include}}}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}modulename}}}}}[]
wenzelm@43467: \rail@nont{\isa{target}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{11}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reflect}}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@cr{2}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{3}
wenzelm@43467: \rail@term{\isa{\isakeyword{datatypes}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\isa{{\isaliteral{5F}{\isacharunderscore}}}}[]
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nextplus{5}
wenzelm@43467: \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@nextplus{6}
wenzelm@43467: \rail@cterm{\isa{\isakeyword{and}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@cr{8}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{9}
wenzelm@43467: \rail@term{\isa{\isakeyword{functions}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nextplus{10}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{9}
wenzelm@43467: \rail@term{\isa{\isakeyword{file}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{4}{\isa{syntax}}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\isa{\isakeyword{infix}}}[]
wenzelm@43467: \rail@nextbar{2}
wenzelm@43467: \rail@term{\isa{\isakeyword{infixl}}}[]
wenzelm@43467: \rail@nextbar{3}
wenzelm@43467: \rail@term{\isa{\isakeyword{infixr}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
haftmann@37397: 
haftmann@37397: 
haftmann@37397:   \begin{description}
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isaliteral{5F}{\isacharunderscore}}code}}}} generates code for a given list
haftmann@39832:   of constants in the specified target language(s).  If no
haftmann@39832:   serialization instruction is given, only abstract code is generated
haftmann@39832:   internally.
haftmann@37397: 
haftmann@37397:   Constants may be specified by giving them literally, referring to
wenzelm@40685:   all executable contants within a certain theory by giving \isa{{\isaliteral{22}{\isachardoublequote}}name{\isaliteral{2E}{\isachardot}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}}, or referring to \emph{all} executable constants currently
wenzelm@40685:   available by giving \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}}.
haftmann@37397: 
haftmann@37397:   By default, for each involved theory one corresponding name space
haftmann@37397:   module is generated.  Alternativly, a module name may be specified
wenzelm@40685:   after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isaliteral{5F}{\isacharunderscore}}name}}}} keyword; then \emph{all} code is
haftmann@37397:   placed in this module.
haftmann@37397: 
haftmann@39832:   For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
haftmann@39832:   refers to a single file; for \emph{Haskell}, it refers to a whole
haftmann@39832:   directory, where code is generated in multiple files reflecting the
haftmann@39832:   module hierarchy.  Omitting the file specification denotes standard
haftmann@37748:   output.
haftmann@37397: 
haftmann@37397:   Serializers take an optional list of arguments in parentheses.  For
wenzelm@40685:   \emph{SML} and \emph{OCaml}, ``\isa{no{\isaliteral{5F}{\isacharunderscore}}signatures}`` omits
haftmann@37397:   explicit module signatures.
wenzelm@42994: 
haftmann@39832:   For \emph{Haskell} a module name prefix may be given using the
wenzelm@40685:   ``\isa{{\isaliteral{22}{\isachardoublequote}}root{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}'' argument; ``\isa{string{\isaliteral{5F}{\isacharunderscore}}classes}'' adds a
haftmann@39832:   ``\verb|deriving (Read, Show)|'' clause to each appropriate
haftmann@39832:   datatype declaration.
haftmann@37397: 
haftmann@37397:   \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
wenzelm@40685:   ``\isa{{\isaliteral{22}{\isachardoublequote}}del{\isaliteral{22}{\isachardoublequote}}}'' deselects) a code equation for code generation.
haftmann@38706:   Usually packages introducing code equations provide a reasonable
wenzelm@40685:   default setup for selection.  Variants \isa{{\isaliteral{22}{\isachardoublequote}}code\ abstype{\isaliteral{22}{\isachardoublequote}}} and
wenzelm@40685:   \isa{{\isaliteral{22}{\isachardoublequote}}code\ abstract{\isaliteral{22}{\isachardoublequote}}} declare abstract datatype certificates or
haftmann@38706:   code equations on abstract datatype representations respectively.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}abort}}}} declares constants which are not
haftmann@39832:   required to have a definition by means of code equations; if needed
haftmann@39832:   these are implemented by program abort instead.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}datatype}}}} specifies a constructor set
haftmann@37397:   for a logical type.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codesetup}}}} gives an overview on
haftmann@37397:   selected code equations and code generator datatypes.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}inline}}} declares (or with option
wenzelm@40685:   ``\isa{{\isaliteral{22}{\isachardoublequote}}del{\isaliteral{22}{\isachardoublequote}}}'' removes) inlining theorems which are applied as
haftmann@39832:   rewrite rules to any code equation during preprocessing.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}post}}} declares (or with option ``\isa{{\isaliteral{22}{\isachardoublequote}}del{\isaliteral{22}{\isachardoublequote}}}'' removes) theorems which are applied as rewrite rules to any
haftmann@39832:   result of an evaluation.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codeproc}}}} prints the setup of the code
haftmann@39832:   generator preprocessor.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}thms}}}} prints a list of theorems
haftmann@37397:   representing the corresponding program containing all given
haftmann@37397:   constants after preprocessing.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}deps}}}} visualizes dependencies of
haftmann@37397:   theorems representing the corresponding program containing all given
haftmann@37397:   constants after preprocessing.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}const}}}} associates a list of constants
haftmann@37397:   with target-specific serializations; omitting a serialization
haftmann@37397:   deletes an existing serialization.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}type}}}} associates a list of type
haftmann@37397:   constructors with target-specific serializations; omitting a
haftmann@37397:   serialization deletes an existing serialization.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}class}}}} associates a list of classes
haftmann@37397:   with target-specific class names; omitting a serialization deletes
haftmann@37397:   an existing serialization.  This applies only to \emph{Haskell}.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}instance}}}} declares a list of type
haftmann@37397:   constructor / class instance relations as ``already present'' for a
wenzelm@40685:   given target.  Omitting a ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{22}{\isachardoublequote}}}'' deletes an existing
haftmann@37397:   ``already present'' declaration.  This applies only to
haftmann@37397:   \emph{Haskell}.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reserved}}}} declares a list of names as
haftmann@37397:   reserved for a given target, preventing it to be shadowed by any
haftmann@37397:   generated code.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}monad}}}} provides an auxiliary mechanism
haftmann@37397:   to generate monadic code for Haskell.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}include}}}} adds arbitrary named content
wenzelm@40685:   (``include'') to generated code.  A ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{22}{\isachardoublequote}}}'' as last argument
haftmann@37397:   will remove an already added ``include''.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}modulename}}}} declares aliasings from one
haftmann@37397:   module name onto another.
haftmann@37397: 
wenzelm@40685:   \item \hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reflect}}}} without a ``\isa{{\isaliteral{22}{\isachardoublequote}}file{\isaliteral{22}{\isachardoublequote}}}''
haftmann@39832:   argument compiles code into the system runtime environment and
haftmann@39832:   modifies the code generator setup that future invocations of system
wenzelm@40685:   runtime code generation referring to one of the ``\isa{{\isaliteral{22}{\isachardoublequote}}datatypes{\isaliteral{22}{\isachardoublequote}}}'' or ``\isa{{\isaliteral{22}{\isachardoublequote}}functions{\isaliteral{22}{\isachardoublequote}}}'' entities use these precompiled
wenzelm@40685:   entities.  With a ``\isa{{\isaliteral{22}{\isachardoublequote}}file{\isaliteral{22}{\isachardoublequote}}}'' argument, the corresponding code
haftmann@39832:   is generated into that specified file without modifying the code
haftmann@39832:   generator setup.
haftmann@39832: 
wenzelm@43498:   \end{description}%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \isamarkupsubsection{The old code generator (S. Berghofer)%
wenzelm@43498: }
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: This framework generates code from both functional and
wenzelm@43498:   relational programs to SML, as explained below.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@43498:     \indexdef{}{command}{code\_module}\hypertarget{command.code-module}{\hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@43498:     \indexdef{}{command}{code\_library}\hypertarget{command.code-library}{\hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@43498:     \indexdef{}{command}{consts\_code}\hypertarget{command.consts-code}{\hyperlink{command.consts-code}{\mbox{\isa{\isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@43498:     \indexdef{}{command}{types\_code}\hypertarget{command.types-code}{\hyperlink{command.types-code}{\mbox{\isa{\isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@43497:     \indexdef{}{attribute}{code}\hypertarget{attribute.code}{\hyperlink{attribute.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{11}{}
wenzelm@43467: \rail@bar
wenzelm@43498: \rail@term{\hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43498: \rail@term{\hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{modespec}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@cr{3}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@term{\isa{\isakeyword{file}}}[]
wenzelm@43467: \rail@nont{\isa{name}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@term{\isa{\isakeyword{imports}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@nextplus{5}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@cr{7}
wenzelm@43467: \rail@term{\isa{\isakeyword{contains}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@nextplus{8}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@nextbar{9}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@nextplus{10}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{modespec}}
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@cnont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{codespec}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{codespec}}
wenzelm@43467: \rail@nont{\isa{const}}[]
wenzelm@43467: \rail@nont{\isa{template}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{attachment}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{tycodespec}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{tycodespec}}
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@nont{\isa{template}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{attachment}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{1}{\isa{const}}
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{1}{\isa{template}}
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{attachment}}
wenzelm@43467: \rail@term{\isa{attach}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{modespec}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@term{\isa{{\isaliteral{7B}{\isacharbraceleft}}}}[]
wenzelm@43467: \rail@nont{\hyperlink{syntax.text}{\mbox{\isa{text}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{7D}{\isacharbraceright}}}}[]
wenzelm@43467: \rail@end
wenzelm@43535: \rail@begin{2}{}
wenzelm@43497: \rail@term{\hyperlink{attribute.code}{\mbox{\isa{code}}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\isa{name}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@43498: \isamarkupsubsubsection{Invoking the code generator%
wenzelm@43498: }
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: The code generator is invoked via the \hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}}
wenzelm@43498:   and \hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}} commands, which correspond to
wenzelm@43498:   \emph{incremental} and \emph{modular} code generation, respectively.
wenzelm@43498: 
wenzelm@43498:   \begin{description}
wenzelm@43498: 
wenzelm@43498:   \item [Modular] For each theory, an ML structure is generated,
wenzelm@43498:   containing the code generated from the constants defined in this
wenzelm@43498:   theory.
wenzelm@43498: 
wenzelm@43498:   \item [Incremental] All the generated code is emitted into the same
wenzelm@43498:   structure.  This structure may import code from previously generated
wenzelm@43498:   structures, which can be specified via \hyperlink{keyword.imports}{\mbox{\isa{\isakeyword{imports}}}}.
wenzelm@43498:   Moreover, the generated structure may also be referred to in later
wenzelm@43498:   invocations of the code generator.
wenzelm@43498: 
wenzelm@43498:   \end{description}
wenzelm@43498: 
wenzelm@43498:   After the \hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}} and \hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}}
wenzelm@43498:   keywords, the user may specify an optional list of ``modes'' in
wenzelm@43498:   parentheses. These can be used to instruct the code generator to
wenzelm@43498:   emit additional code for special purposes, e.g.\ functions for
wenzelm@43498:   converting elements of generated datatypes to Isabelle terms, or
wenzelm@43498:   test data generators. The list of modes is followed by a module
wenzelm@43498:   name.  The module name is optional for modular code generation, but
wenzelm@43498:   must be specified for incremental code generation.
wenzelm@43498: 
wenzelm@43498:   The code can either be written to a file, in which case a file name
wenzelm@43498:   has to be specified after the \hyperlink{keyword.file}{\mbox{\isa{\isakeyword{file}}}} keyword, or be loaded
wenzelm@43498:   directly into Isabelle's ML environment. In the latter case, the
wenzelm@43498:   \hyperlink{command.ML}{\mbox{\isa{\isacommand{ML}}}} theory command can be used to inspect the results
wenzelm@43498:   interactively, for example.
wenzelm@43498: 
wenzelm@43498:   The terms from which to generate code can be specified after the
wenzelm@43498:   \hyperlink{keyword.contains}{\mbox{\isa{\isakeyword{contains}}}} keyword, either as a list of bindings, or just
wenzelm@43498:   as a list of terms. In the latter case, the code generator just
wenzelm@43498:   produces code for all constants and types occuring in the term, but
wenzelm@43498:   does not bind the compiled terms to ML identifiers.
wenzelm@43498: 
wenzelm@43498:   Here is an example:%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: \isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}\isamarkupfalse%
wenzelm@43498: \ Test\isanewline
wenzelm@43523: \isakeyword{contains}\ test\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}foldl\ op\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{3}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{4}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{5}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: \noindent This binds the result of compiling the given term to
wenzelm@43498:   the ML identifier \verb|Test.test|.%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \isadelimML
wenzelm@43498: %
wenzelm@43498: \endisadelimML
wenzelm@43498: %
wenzelm@43498: \isatagML
wenzelm@43498: \isacommand{ML}\isamarkupfalse%
wenzelm@43498: \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ %
wenzelm@43498: \isaantiq
wenzelm@43498: assert{}%
wenzelm@43498: \endisaantiq
wenzelm@43498: \ {\isaliteral{28}{\isacharparenleft}}Test{\isaliteral{2E}{\isachardot}}test\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}%
wenzelm@43498: \endisatagML
wenzelm@43498: {\isafoldML}%
wenzelm@43498: %
wenzelm@43498: \isadelimML
wenzelm@43498: %
wenzelm@43498: \endisadelimML
wenzelm@43498: %
wenzelm@43498: \isamarkupsubsubsection{Configuring the code generator%
wenzelm@43498: }
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: When generating code for a complex term, the code generator
wenzelm@43498:   recursively calls itself for all subterms.  When it arrives at a
wenzelm@43498:   constant, the default strategy of the code generator is to look up
wenzelm@43498:   its definition and try to generate code for it.  Constants which
wenzelm@43498:   have no definitions that are immediately executable, may be
wenzelm@43498:   associated with a piece of ML code manually using the \indexref{}{command}{consts\_code}\hyperlink{command.consts-code}{\mbox{\isa{\isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}}}} command.  It takes a list whose elements consist of a
wenzelm@43498:   constant (given in usual term syntax -- an explicit type constraint
wenzelm@43498:   accounts for overloading), and a mixfix template describing the ML
wenzelm@43498:   code. The latter is very much the same as the mixfix templates used
wenzelm@43498:   when declaring new constants.  The most notable difference is that
wenzelm@43498:   terms may be included in the ML template using antiquotation
wenzelm@43498:   brackets \verb|{|\verb|*|~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{22}{\isachardoublequote}}}~\verb|*|\verb|}|.
wenzelm@43498: 
wenzelm@43498:   A similar mechanism is available for types: \indexref{}{command}{types\_code}\hyperlink{command.types-code}{\mbox{\isa{\isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}}}} associates type constructors with specific ML code.
wenzelm@43498: 
wenzelm@43498:   For example, the following declarations copied from \verb|~~/src/HOL/Product_Type.thy| describe how the product type of
wenzelm@43498:   Isabelle/HOL should be compiled to ML.%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: \isacommand{typedecl}\isamarkupfalse%
wenzelm@43498: \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ prod\isanewline
wenzelm@43498: \isacommand{consts}\isamarkupfalse%
wenzelm@43498: \ Pair\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ prod{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43498: \isanewline
wenzelm@43498: \isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}\isamarkupfalse%
wenzelm@43498: \ prod\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{2F}{\isacharslash}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
wenzelm@43498: \isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}\isamarkupfalse%
wenzelm@43498: \ Pair\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{2F}{\isacharslash}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}%
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: Sometimes, the code associated with a constant or type may
wenzelm@43498:   need to refer to auxiliary functions, which have to be emitted when
wenzelm@43498:   the constant is used. Code for such auxiliary functions can be
wenzelm@43498:   declared using \hyperlink{keyword.attach}{\mbox{\isa{\isakeyword{attach}}}}. For example, the \isa{wfrec}
wenzelm@43498:   function can be implemented as follows:%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: \isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}\isamarkupfalse%
wenzelm@43498: \ wfrec\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6D6F64756C653E}{\isasymmodule}}wfrec{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ \ \isanewline
wenzelm@43523: \isakeyword{attach}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\ fun\ wfrec\ f\ x\ {\isaliteral{3D}{\isacharequal}}\ f\ {\isaliteral{28}{\isacharparenleft}}wfrec\ f{\isaliteral{29}{\isacharparenright}}\ x\ {\isaliteral{2A7D}{\isacharverbatimclose}}%
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: If the code containing a call to \isa{wfrec} resides in an
wenzelm@43498:   ML structure different from the one containing the function
wenzelm@43498:   definition attached to \isa{wfrec}, the name of the ML structure
wenzelm@43498:   (followed by a ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2E}{\isachardot}}{\isaliteral{22}{\isachardoublequote}}}'')  is inserted in place of ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6D6F64756C653E}{\isasymmodule}}{\isaliteral{22}{\isachardoublequote}}}'' in the above template.  The ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}''  means that
wenzelm@43498:   the code generator should ignore the first argument of \isa{wfrec}, i.e.\ the termination relation, which is usually not
wenzelm@43498:   executable.
wenzelm@43498: 
wenzelm@43498:   \medskip Another possibility of configuring the code generator is to
wenzelm@43498:   register theorems to be used for code generation. Theorems can be
wenzelm@43498:   registered via the \hyperlink{attribute.code}{\mbox{\isa{code}}} attribute. It takes an optional
wenzelm@43498:   name as an argument, which indicates the format of the
wenzelm@43498:   theorem. Currently supported formats are equations (this is the
wenzelm@43498:   default when no name is specified) and horn clauses (this is
wenzelm@43498:   indicated by the name \texttt{ind}). The left-hand sides of
wenzelm@43498:   equations may only contain constructors and distinct variables,
wenzelm@43498:   whereas horn clauses must have the same format as introduction rules
wenzelm@43498:   of inductive definitions.
wenzelm@43498: 
wenzelm@43498:   The following example specifies three equations from which to
wenzelm@43498:   generate code for \isa{{\isaliteral{22}{\isachardoublequote}}op\ {\isaliteral{3C}{\isacharless}}{\isaliteral{22}{\isachardoublequote}}} on natural numbers (see also
wenzelm@43498:   \verb|~~/src/HOL/Nat.thy|).%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: \isacommand{lemma}\isamarkupfalse%
wenzelm@43498: \ {\isaliteral{5B}{\isacharbrackleft}}code{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}Suc\ m\ {\isaliteral{3C}{\isacharless}}\ Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{3C}{\isacharless}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43498: \ \ \isakeyword{and}\ {\isaliteral{5B}{\isacharbrackleft}}code{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43498: \ \ \isakeyword{and}\ {\isaliteral{5B}{\isacharbrackleft}}code{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@43498: \isadelimproof
wenzelm@43498: \ %
wenzelm@43498: \endisadelimproof
wenzelm@43498: %
wenzelm@43498: \isatagproof
wenzelm@43498: \isacommand{by}\isamarkupfalse%
wenzelm@43498: \ simp{\isaliteral{5F}{\isacharunderscore}}all%
wenzelm@43498: \endisatagproof
wenzelm@43498: {\isafoldproof}%
wenzelm@43498: %
wenzelm@43498: \isadelimproof
wenzelm@43498: %
wenzelm@43498: \endisadelimproof
wenzelm@43498: %
wenzelm@43498: \isamarkupsubsubsection{Specific HOL code generators%
wenzelm@43498: }
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@43498: \begin{isamarkuptext}%
wenzelm@43498: The basic code generator framework offered by Isabelle/Pure
wenzelm@43498:   has already been extended with additional code generators for
wenzelm@43498:   specific HOL constructs. These include datatypes, recursive
wenzelm@43498:   functions and inductive relations. The code generator for inductive
wenzelm@43498:   relations can handle expressions of the form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ r{\isaliteral{22}{\isachardoublequote}}}, where \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{22}{\isachardoublequote}}} is an inductively defined relation. If at
wenzelm@43498:   least one of the \isa{{\isaliteral{22}{\isachardoublequote}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} is a dummy pattern ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequote}}}'',
wenzelm@43498:   the above expression evaluates to a sequence of possible answers. If
wenzelm@43498:   all of the \isa{{\isaliteral{22}{\isachardoublequote}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} are proper terms, the expression evaluates
wenzelm@43498:   to a boolean value.
wenzelm@43498: 
wenzelm@43523:   The following example demonstrates this for beta-reduction on lambda
wenzelm@43523:   terms (see also \verb|~~/src/HOL/Proofs/Lambda/Lambda.thy|).%
wenzelm@43523: \end{isamarkuptext}%
wenzelm@43523: \isamarkuptrue%
wenzelm@43523: \isacommand{datatype}\isamarkupfalse%
wenzelm@43523: \ dB\ {\isaliteral{3D}{\isacharequal}}\isanewline
wenzelm@43523: \ \ \ \ Var\ nat\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ App\ dB\ dB\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6465677265653E}{\isasymdegree}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ Abs\ dB\isanewline
wenzelm@43523: \isanewline
wenzelm@43523: \isacommand{primrec}\isamarkupfalse%
wenzelm@43523: \ lift\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \isakeyword{where}\isanewline
wenzelm@43523: \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}lift\ {\isaliteral{28}{\isacharparenleft}}Var\ i{\isaliteral{29}{\isacharparenright}}\ k\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3C}{\isacharless}}\ k\ then\ Var\ i\ else\ Var\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}lift\ {\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ t{\isaliteral{29}{\isacharparenright}}\ k\ {\isaliteral{3D}{\isacharequal}}\ lift\ s\ k\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ lift\ t\ k{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}lift\ {\isaliteral{28}{\isacharparenleft}}Abs\ s{\isaliteral{29}{\isacharparenright}}\ k\ {\isaliteral{3D}{\isacharequal}}\ Abs\ {\isaliteral{28}{\isacharparenleft}}lift\ s\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \isanewline
wenzelm@43523: \isacommand{primrec}\isamarkupfalse%
wenzelm@43523: \ subst\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{27}{\isacharprime}}{\isaliteral{2F}{\isacharslash}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}\ {\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
wenzelm@43523: \isakeyword{where}\isanewline
wenzelm@43523: \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}Var\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
wenzelm@43523: \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ k\ {\isaliteral{3C}{\isacharless}}\ i\ then\ Var\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ else\ if\ i\ {\isaliteral{3D}{\isacharequal}}\ k\ then\ s\ else\ Var\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}t\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u{\isaliteral{29}{\isacharparenright}}{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ t{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}Abs\ t{\isaliteral{29}{\isacharparenright}}{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ Abs\ {\isaliteral{28}{\isacharparenleft}}t{\isaliteral{5B}{\isacharbrackleft}}lift\ s\ {\isadigit{0}}\ {\isaliteral{2F}{\isacharslash}}\ k{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \isanewline
wenzelm@43523: \isacommand{inductive}\isamarkupfalse%
wenzelm@43523: \ beta\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
wenzelm@43523: \isakeyword{where}\isanewline
wenzelm@43523: \ \ \ \ beta{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}Abs\ s\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ t\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ s{\isaliteral{5B}{\isacharbrackleft}}t{\isaliteral{2F}{\isacharslash}}{\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ appL{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ s\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ appR{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ u\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ u\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ t{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ {\isaliteral{7C}{\isacharbar}}\ abs{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Abs\ s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ Abs\ t{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \isanewline
wenzelm@43523: \isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}\isamarkupfalse%
wenzelm@43523: \ Test\isanewline
wenzelm@43523: \isakeyword{contains}\isanewline
wenzelm@43523: \ \ test{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}Abs\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ Var\ {\isadigit{0}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ Var\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
wenzelm@43523: \ \ test{\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}Abs\ {\isaliteral{28}{\isacharparenleft}}Abs\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isadigit{0}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ {\isaliteral{28}{\isacharparenleft}}Abs\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequoteclose}}%
wenzelm@43523: \begin{isamarkuptext}%
wenzelm@43523: In the above example, \verb|Test.test1| evaluates to a boolean,
wenzelm@43523:   whereas \verb|Test.test2| is a lazy sequence whose elements can be
wenzelm@43523:   inspected separately.%
wenzelm@43523: \end{isamarkuptext}%
wenzelm@43523: \isamarkuptrue%
wenzelm@43523: %
wenzelm@43523: \isadelimML
wenzelm@43523: %
wenzelm@43523: \endisadelimML
wenzelm@43523: %
wenzelm@43523: \isatagML
wenzelm@43523: \isacommand{ML}\isamarkupfalse%
wenzelm@43523: \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ %
wenzelm@43523: \isaantiq
wenzelm@43523: assert{}%
wenzelm@43523: \endisaantiq
wenzelm@43523: \ Test{\isaliteral{2E}{\isachardot}}test{\isadigit{1}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}\isanewline
wenzelm@43523: \isacommand{ML}\isamarkupfalse%
wenzelm@43523: \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ val\ results\ {\isaliteral{3D}{\isacharequal}}\ DSeq{\isaliteral{2E}{\isachardot}}list{\isaliteral{5F}{\isacharunderscore}}of\ Test{\isaliteral{2E}{\isachardot}}test{\isadigit{2}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}\isanewline
wenzelm@43523: \isacommand{ML}\isamarkupfalse%
wenzelm@43523: \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ %
wenzelm@43523: \isaantiq
wenzelm@43523: assert{}%
wenzelm@43523: \endisaantiq
wenzelm@43523: \ {\isaliteral{28}{\isacharparenleft}}length\ results\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}%
wenzelm@43523: \endisatagML
wenzelm@43523: {\isafoldML}%
wenzelm@43523: %
wenzelm@43523: \isadelimML
wenzelm@43523: %
wenzelm@43523: \endisadelimML
wenzelm@43523: %
wenzelm@43523: \begin{isamarkuptext}%
wenzelm@43523: \medskip The theory underlying the HOL code generator is described
wenzelm@43498:   more detailed in \cite{Berghofer-Nipkow:2002}. More examples that
wenzelm@43498:   illustrate the usage of the code generator can be found e.g.\ in
wenzelm@43498:   \verb|~~/src/HOL/MicroJava/J/JListExample.thy| and \verb|~~/src/HOL/MicroJava/JVM/JVMListExample.thy|.%
wenzelm@43498: \end{isamarkuptext}%
wenzelm@43498: \isamarkuptrue%
wenzelm@43498: %
wenzelm@27047: \isamarkupsection{Definition by specification \label{sec:hol-specification}%
wenzelm@27047: }
wenzelm@27047: \isamarkuptrue%
wenzelm@27047: %
wenzelm@27047: \begin{isamarkuptext}%
wenzelm@27047: \begin{matharray}{rcl}
wenzelm@40685:     \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@40685:     \indexdef{HOL}{command}{ax\_specification}\hypertarget{command.HOL.ax-specification}{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
wenzelm@27047:   \end{matharray}
wenzelm@27047: 
wenzelm@43467:   \begin{railoutput}
wenzelm@43535: \rail@begin{6}{}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@term{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}}[]
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@nont{\isa{decl}}[]
wenzelm@43467: \rail@nextplus{1}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@cr{3}
wenzelm@43467: \rail@plus
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{4}
wenzelm@43467: \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
wenzelm@43467: \rail@nextplus{5}
wenzelm@43467: \rail@endplus
wenzelm@43467: \rail@end
wenzelm@43467: \rail@begin{2}{\isa{decl}}
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
wenzelm@43467: \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
wenzelm@43467: \rail@term{\isa{\isakeyword{overloaded}}}[]
wenzelm@43467: \rail@bar
wenzelm@43467: \rail@nextbar{1}
wenzelm@43467: \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
wenzelm@43467: \rail@endbar
wenzelm@43467: \rail@end
wenzelm@43467: \end{railoutput}
wenzelm@43467: 
wenzelm@27047: 
wenzelm@28788:   \begin{description}
wenzelm@27047: 
wenzelm@40685:   \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}decls\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} sets up a
wenzelm@27047:   goal stating the existence of terms with the properties specified to
wenzelm@27047:   hold for the constants given in \isa{decls}.  After finishing the
wenzelm@27047:   proof, the theory will be augmented with definitions for the given
wenzelm@27047:   constants, as well as with theorems stating the properties for these
wenzelm@27047:   constants.
wenzelm@27047: 
wenzelm@40685:   \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}decls\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} sets up
wenzelm@28788:   a goal stating the existence of terms with the properties specified
wenzelm@28788:   to hold for the constants given in \isa{decls}.  After finishing
wenzelm@28788:   the proof, the theory will be augmented with axioms expressing the
wenzelm@28788:   properties given in the first place.
wenzelm@27047: 
wenzelm@28788:   \item \isa{decl} declares a constant to be defined by the
wenzelm@27047:   specification given.  The definition for the constant \isa{c} is
wenzelm@40685:   bound to the name \isa{c{\isaliteral{5F}{\isacharunderscore}}def} unless a theorem name is given in
wenzelm@27047:   the declaration.  Overloaded constants should be declared as such.
wenzelm@27047: 
wenzelm@28788:   \end{description}
wenzelm@27047: 
wenzelm@40685:   Whether to use \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} or \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}} is to some extent a matter of style.  \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} introduces no new axioms, and so by
wenzelm@40685:   construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}} does introduce axioms, but only after the
wenzelm@27047:   user has explicitly proven it to be safe.  A practical issue must be
wenzelm@27047:   considered, though: After introducing two constants with the same
wenzelm@27047:   properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
wenzelm@27047:   that the two constants are, in fact, equal.  If this might be a
wenzelm@40685:   problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}.%
wenzelm@27047: \end{isamarkuptext}%
wenzelm@27047: \isamarkuptrue%
wenzelm@27047: %
wenzelm@26849: \isadelimtheory
wenzelm@26849: %
wenzelm@26849: \endisadelimtheory
wenzelm@26849: %
wenzelm@26849: \isatagtheory
wenzelm@26840: \isacommand{end}\isamarkupfalse%
wenzelm@26840: %
wenzelm@26840: \endisatagtheory
wenzelm@26840: {\isafoldtheory}%
wenzelm@26840: %
wenzelm@26840: \isadelimtheory
wenzelm@26840: %
wenzelm@26840: \endisadelimtheory
wenzelm@26849: \isanewline
wenzelm@26840: \end{isabellebody}%
wenzelm@26840: %%% Local Variables:
wenzelm@26840: %%% mode: latex
wenzelm@26840: %%% TeX-master: "root"
wenzelm@26840: %%% End: