doc-src/IsarRef/Thy/document/HOL_Specific.tex

 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
 }
 \isamarkuptrue%
 %
 \begin{isamarkuptext}%
-An \emph{inductive definition} specifies the least predicate
-  or set \isa{R} closed under given rules: applying a rule to
-  elements of \isa{R} yields a result within \isa{R}.  For
-  example, a structural operational semantics is an inductive
-  definition of an evaluation relation.
+\begin{matharray}{rcl}
+    \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}}} \\
+    \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}}} \\
+    \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}}} \\
+    \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}}} \\
+    \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
+  \end{matharray}
+
+  An \emph{inductive definition} specifies the least predicate or set
+  \isa{R} closed under given rules: applying a rule to elements of
+  \isa{R} yields a result within \isa{R}.  For example, a
+  structural operational semantics is an inductive definition of an
+  evaluation relation.
 
   Dually, a \emph{coinductive definition} specifies the greatest
   predicate or set \isa{R} that is consistent with given rules:
   every element of \isa{R} can be seen as arising by applying a rule
   to elements of \isa{R}.  An important example is using

   defined relation: there must be a monotonicity theorem 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 each premise \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4D3E}{\isasymM}}\ R\ t{\isaliteral{22}{\isachardoublequote}}} in an
   introduction rule.  The default rule declarations of Isabelle/HOL
   already take care of most common situations.
 
-  \begin{matharray}{rcl}
-    \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}}} \\
-    \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}}} \\
-    \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}}} \\
-    \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}}} \\
-    \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
-  \end{matharray}
-
   \begin{railoutput}
 \rail@begin{10}{}
 \rail@bar
 \rail@term{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}}[]
 \rail@nextbar{1}

 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
 }
 \isamarkuptrue%
 %
 \begin{isamarkuptext}%
-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.
-
-  \begin{matharray}{rcl}
+\begin{matharray}{rcl}
     \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}}} \\
     \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}}} \\
   \end{matharray}
+
+  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.
 
   \begin{railoutput}
 \rail@begin{5}{}
 \rail@term{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}}[]
 \rail@bar

 \isamarkupsection{Typedef axiomatization \label{sec:hol-typedef}%
 }
 \isamarkuptrue%
 %
 \begin{isamarkuptext}%
-A Gordon/HOL-style type definition is a certain axiom scheme
-  that identifies a new type with a subset of an existing type.  More
+\begin{matharray}{rcl}
+    \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}}} \\
+  \end{matharray}
+
+  A Gordon/HOL-style type definition is a certain axiom scheme that
+  identifies a new type with a subset of an existing type.  More
   precisely, the new type is defined by exhibiting an existing type
   \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
   \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
   postulated that establish an isomorphism between the new type and
   the subset.  In general, the type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} may involve type

   downwards-closed wrt.\ arbitrary non-empty subsets.  Thus genuinely
   new types introduced by \hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}} stay within the range
   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
   abbreviations, without any logical significance.
   
-  \begin{matharray}{rcl}
-    \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}}} \\
-  \end{matharray}
-
   \begin{railoutput}
 \rail@begin{2}{}
 \rail@term{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}}[]
 \rail@bar
 \rail@nextbar{1}

 \isamarkupsection{Quotient types%
 }
 \isamarkuptrue%
 %
 \begin{isamarkuptext}%
-The quotient package defines a new quotient type given a raw type
-  and a partial equivalence relation.
-  It also includes automation for transporting definitions and theorems.
-  It can automatically produce definitions and theorems on the quotient type,
-  given the corresponding constants and facts on the raw type.
-
-  \begin{matharray}{rcl}
+\begin{matharray}{rcl}
     \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}}}\\
     \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}}}\\
     \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}}}\\
     \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}}}\\
     \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}}}\\
   \end{matharray}
 
+  The quotient package defines a new quotient type given a raw type
+  and a partial equivalence relation.  It also includes automation for
+  transporting definitions and theorems.  It can automatically produce
+  definitions and theorems on the quotient type, given the
+  corresponding constants and facts on the raw type.
+
   \begin{railoutput}
 \rail@begin{2}{}
 \rail@term{\hyperlink{command.HOL.quotient-type}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}type}}}}}[]
 \rail@plus
 \rail@nont{\isa{spec}}[]

 \end{railoutput}
 
 
   \begin{description}
   
-  \item \hyperlink{command.HOL.quotient-type}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}type}}}} defines quotient types. The injection from a quotient type 
-  to a raw type is called \isa{rep{\isaliteral{5F}{\isacharunderscore}}t}, its inverse \isa{abs{\isaliteral{5F}{\isacharunderscore}}t} unless explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} specification provides alternative names.
-
-  \item \hyperlink{command.HOL.quotient-definition}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}definition}}}} defines a constant on the quotient type.
-
-  \item \hyperlink{command.HOL.print-quotmaps}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotmaps}}}} prints quotient map functions.
+  \item \hyperlink{command.HOL.quotient-type}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}type}}}} defines quotient types.  The
+  injection from a quotient type to a raw type is called \isa{rep{\isaliteral{5F}{\isacharunderscore}}t}, its inverse \isa{abs{\isaliteral{5F}{\isacharunderscore}}t} unless explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} specification provides alternative names.
+
+  \item \hyperlink{command.HOL.quotient-definition}{\mbox{\isa{\isacommand{quotient{\isaliteral{5F}{\isacharunderscore}}definition}}}} defines a constant on
+  the quotient type.
+
+  \item \hyperlink{command.HOL.print-quotmaps}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotmaps}}}} prints quotient map
+  functions.
 
   \item \hyperlink{command.HOL.print-quotients}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotients}}}} prints quotients.
 
   \item \hyperlink{command.HOL.print-quotconsts}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}quotconsts}}}} prints quotient constants.
 

     \indexdef{HOL}{attribute}{coercion}\hypertarget{attribute.HOL.coercion}{\hyperlink{attribute.HOL.coercion}{\mbox{\isa{coercion}}}} & : & \isa{attribute} \\
     \indexdef{HOL}{attribute}{coercion\_enabled}\hypertarget{attribute.HOL.coercion-enabled}{\hyperlink{attribute.HOL.coercion-enabled}{\mbox{\isa{coercion{\isaliteral{5F}{\isacharunderscore}}enabled}}}} & : & \isa{attribute} \\
     \indexdef{HOL}{attribute}{coercion\_map}\hypertarget{attribute.HOL.coercion-map}{\hyperlink{attribute.HOL.coercion-map}{\mbox{\isa{coercion{\isaliteral{5F}{\isacharunderscore}}map}}}} & : & \isa{attribute} \\
   \end{matharray}
 
+  Coercive subtyping allows the user to omit explicit type
+  conversions, also called \emph{coercions}.  Type inference will add
+  them as necessary when parsing a term. See
+  \cite{traytel-berghofer-nipkow-2011} for details.
+
   \begin{railoutput}
 \rail@begin{2}{}
 \rail@term{\hyperlink{attribute.HOL.coercion}{\mbox{\isa{coercion}}}}[]
 \rail@bar
 \rail@nextbar{1}

 \rail@endbar
 \rail@end
 \end{railoutput}
 
 
-  Coercive subtyping allows the user to omit explicit type conversions,
-  also called \emph{coercions}.  Type inference will add them as
-  necessary when parsing a term. See
-  \cite{traytel-berghofer-nipkow-2011} for details.
-
   \begin{description}
 
   \item \hyperlink{attribute.HOL.coercion}{\mbox{\isa{coercion}}}~\isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{22}{\isachardoublequote}}} registers a new
-  coercion function \isa{{\isaliteral{22}{\isachardoublequote}}f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} where \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} are nullary type constructors. Coercions are
-  composed by the inference algorithm if needed. Note that the type
-  inference algorithm is complete only if the registered coercions form
-  a lattice.
-
-
-  \item \hyperlink{attribute.HOL.coercion-map}{\mbox{\isa{coercion{\isaliteral{5F}{\isacharunderscore}}map}}}~\isa{{\isaliteral{22}{\isachardoublequote}}map{\isaliteral{22}{\isachardoublequote}}} registers a new
-  map function to lift coercions through type constructors. The function
-  \isa{{\isaliteral{22}{\isachardoublequote}}map{\isaliteral{22}{\isachardoublequote}}} must conform to the following type pattern
+  coercion function \isa{{\isaliteral{22}{\isachardoublequote}}f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} where \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} and
+  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} are type constructors without arguments.  Coercions are
+  composed by the inference algorithm if needed.  Note that the type
+  inference algorithm is complete only if the registered coercions
+  form a lattice.
+
+  \item \hyperlink{attribute.HOL.coercion-map}{\mbox{\isa{coercion{\isaliteral{5F}{\isacharunderscore}}map}}}~\isa{{\isaliteral{22}{\isachardoublequote}}map{\isaliteral{22}{\isachardoublequote}}} registers a
+  new map function to lift coercions through type constructors. The
+  function \isa{{\isaliteral{22}{\isachardoublequote}}map{\isaliteral{22}{\isachardoublequote}}} must conform to the following type pattern
 
   \begin{matharray}{lll}
     \isa{{\isaliteral{22}{\isachardoublequote}}map{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
       \isa{{\isaliteral{22}{\isachardoublequote}}f\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ f\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} \\
   \end{matharray}
 
-  where \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{22}{\isachardoublequote}}} is a type constructor and \isa{{\isaliteral{22}{\isachardoublequote}}f\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}} is of
-  type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}} or
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}}.
-  Registering a map function overwrites any existing map function for
-  this particular type constructor.
-
+  where \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{22}{\isachardoublequote}}} is a type constructor and \isa{{\isaliteral{22}{\isachardoublequote}}f\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}} is of type
+  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}} or \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}}.  Registering a map function
+  overwrites any existing map function for this particular type
+  constructor.
 
   \item \hyperlink{attribute.HOL.coercion-enabled}{\mbox{\isa{coercion{\isaliteral{5F}{\isacharunderscore}}enabled}}} enables the coercion
   inference algorithm.
 
   \end{description}%

     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
     \indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isaliteral{5F}{\isacharunderscore}}split}}}} & : & \isa{attribute} \\
   \end{matharray}
 
-  The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
-  (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
-  facts are inserted into the goal before running the procedure.
-
-  The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
-  always supplied to the arithmetic provers implicitly.
-
-  The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isaliteral{5F}{\isacharunderscore}}split}}} attribute declares case split
+  \begin{description}
+
+  \item \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} decides linear arithmetic problems (on
+  types \isa{nat}, \isa{int}, \isa{real}).  Any current facts
+  are inserted into the goal before running the procedure.
+
+  \item \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} declares facts that are supplied to
+  the arithmetic provers implicitly.
+
+  \item \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isaliteral{5F}{\isacharunderscore}}split}}} attribute declares case split
   rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked.
 
-  Note that a simpler (but faster) arithmetic prover is
-  already invoked by the Simplifier.%
+  \end{description}
+
+  Note that a simpler (but faster) arithmetic prover is already
+  invoked by the Simplifier.%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
 \isamarkupsection{Intuitionistic proof search%
 }

 \rail@endplus
 \rail@end
 \end{railoutput}
 
 
-  The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
-  search, depending on specifically declared rules from the context,
-  or given as explicit arguments.  Chained facts are inserted into the
-  goal before commencing proof search.
+  \begin{description}
+
+  \item \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} performs intuitionistic proof search,
+  depending on specifically declared rules from the context, or given
+  as explicit arguments.  Chained facts are inserted into the goal
+  before commencing proof search.
 
   Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
   \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
   ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{21}{\isacharbang}}{\isaliteral{22}{\isachardoublequote}}}'' indicator refers to ``safe'' rules, which may be
   applied aggressively (without considering back-tracking later).
   Rules declared with ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}'' are ignored in proof search (the
   single-step \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} method still observes these).  An
   explicit weight annotation may be given as well; otherwise the
-  number of rule premises will be taken into account here.%
+  number of rule premises will be taken into account here.
+
+  \end{description}%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
 \isamarkupsection{Model Elimination and Resolution%
 }

 \rail@endbar
 \rail@end
 \end{railoutput}
 
 
-  The \hyperlink{method.HOL.meson}{\mbox{\isa{meson}}} method implements Loveland's model elimination
-  procedure \cite{loveland-78}. See \verb|~~/src/HOL/ex/Meson_Test.thy| for
-  examples.
-
-  The \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}} method combines ordered resolution and ordered
-  paramodulation to find first-order (or mildly higher-order) proofs. The first
-  optional argument specifies a type encoding; see the Sledgehammer manual
-  \cite{isabelle-sledgehammer} for details. The \verb|~~/src/HOL/Metis_Examples| directory contains several small theories
-  developed to a large extent using Metis.%
+  \begin{description}
+
+  \item \hyperlink{method.HOL.meson}{\mbox{\isa{meson}}} implements Loveland's model elimination
+  procedure \cite{loveland-78}.  See \verb|~~/src/HOL/ex/Meson_Test.thy| for examples.
+
+  \item \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}} combines ordered resolution and ordered
+  paramodulation to find first-order (or mildly higher-order) proofs.
+  The first optional argument specifies a type encoding; see the
+  Sledgehammer manual \cite{isabelle-sledgehammer} for details.  The
+  directory \verb|~~/src/HOL/Metis_Examples| contains several small
+  theories developed to a large extent using \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}.
+
+  \end{description}%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
 \isamarkupsection{Coherent Logic%
 }

 \rail@endbar
 \rail@end
 \end{railoutput}
 
 
-  The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
-  \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
-  applications in confluence theory, lattice theory and projective
-  geometry.  See \verb|~~/src/HOL/ex/Coherent.thy| for some
-  examples.%
+  \begin{description}
+
+  \item \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} solves problems of \emph{Coherent
+  Logic} \cite{Bezem-Coquand:2005}, which covers applications in
+  confluence theory, lattice theory and projective geometry.  See
+  \verb|~~/src/HOL/ex/Coherent.thy| for some examples.
+
+  \end{description}%
 \end{isamarkuptext}%
 \isamarkuptrue%
 %
 \isamarkupsection{Proving propositions%
 }