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

--- a/doc-src/IsarRef/Thy/document/Framework.tex	Tue Aug 28 12:31:53 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1518 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Framework}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Framework\isanewline
-\isakeyword{imports}\ Base\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupchapter{The Isabelle/Isar Framework \label{ch:isar-framework}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Isabelle/Isar
-  \cite{Wenzel:1999:TPHOL,Wenzel-PhD,Nipkow-TYPES02,Wenzel-Paulson:2006,Wenzel:2006:Festschrift}
-  is intended as a generic framework for developing formal
-  mathematical documents with full proof checking.  Definitions and
-  proofs are organized as theories.  An assembly of theory sources may
-  be presented as a printed document; see also
-  \chref{ch:document-prep}.
-
-  The main objective of Isar is the design of a human-readable
-  structured proof language, which is called the ``primary proof
-  format'' in Isar terminology.  Such a primary proof language is
-  somewhere in the middle between the extremes of primitive proof
-  objects and actual natural language.  In this respect, Isar is a bit
-  more formalistic than Mizar
-  \cite{Trybulec:1993:MizarFeatures,Rudnicki:1992:MizarOverview,Wiedijk:1999:Mizar},
-  using logical symbols for certain reasoning schemes where Mizar
-  would prefer English words; see \cite{Wenzel-Wiedijk:2002} for
-  further comparisons of these systems.
-
-  So Isar challenges the traditional way of recording informal proofs
-  in mathematical prose, as well as the common tendency to see fully
-  formal proofs directly as objects of some logical calculus (e.g.\
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-terms in a version of type theory).  In fact, Isar is
-  better understood as an interpreter of a simple block-structured
-  language for describing the data flow of local facts and goals,
-  interspersed with occasional invocations of proof methods.
-  Everything is reduced to logical inferences internally, but these
-  steps are somewhat marginal compared to the overall bookkeeping of
-  the interpretation process.  Thanks to careful design of the syntax
-  and semantics of Isar language elements, a formal record of Isar
-  instructions may later appear as an intelligible text to the
-  attentive reader.
-
-  The Isar proof language has emerged from careful analysis of some
-  inherent virtues of the existing logical framework of Isabelle/Pure
-  \cite{paulson-found,paulson700}, notably composition of higher-order
-  natural deduction rules, which is a generalization of Gentzen's
-  original calculus \cite{Gentzen:1935}.  The approach of generic
-  inference systems in Pure is continued by Isar towards actual proof
-  texts.
-
-  Concrete applications require another intermediate layer: an
-  object-logic.  Isabelle/HOL \cite{isa-tutorial} (simply-typed
-  set-theory) is being used most of the time; Isabelle/ZF
-  \cite{isabelle-ZF} is less extensively developed, although it would
-  probably fit better for classical mathematics.
-
-  \medskip In order to illustrate natural deduction in Isar, we shall
-  refer to the background theory and library of Isabelle/HOL.  This
-  includes common notions of predicate logic, naive set-theory etc.\
-  using fairly standard mathematical notation.  From the perspective
-  of generic natural deduction there is nothing special about the
-  logical connectives of HOL (\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616E643E}{\isasymand}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6F723E}{\isasymor}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}{\isaliteral{22}{\isachardoublequote}}},
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}{\isaliteral{22}{\isachardoublequote}}}, etc.), only the resulting reasoning principles are
-  relevant to the user.  There are similar rules available for
-  set-theory operators (\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C696E7465723E}{\isasyminter}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C756E696F6E3E}{\isasymunion}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{22}{\isachardoublequote}}}, etc.), or any other theory developed in the library (lattice
-  theory, topology etc.).
-
-  Subsequently we briefly review fragments of Isar proof texts
-  corresponding directly to such general deduction schemes.  The
-  examples shall refer to set-theory, to minimize the danger of
-  understanding connectives of predicate logic as something special.
-
-  \medskip The following deduction performs \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C696E7465723E}{\isasyminter}}{\isaliteral{22}{\isachardoublequote}}}-introduction,
-  working forwards from assumptions towards the conclusion.  We give
-  both the Isar text, and depict the primitive rule involved, as
-  determined by unification of the problem against rules that are
-  declared in the library context.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\medskip\begin{minipage}{0.6\textwidth}
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\end{minipage}\begin{minipage}{0.4\textwidth}
-%
-\begin{isamarkuptext}%
-\infer{\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{22}{\isachardoublequote}}}}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\end{minipage}
-%
-\begin{isamarkuptext}%
-\medskip\noindent Note that \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} augments the proof
-  context, \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} indicates that the current fact shall be
-  used in the next step, and \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} states an intermediate
-  goal.  The two dots ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' refer to a complete proof of
-  this claim, using the indicated facts and a canonical rule from the
-  context.  We could have been more explicit here by spelling out the
-  final proof step via the \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}} command:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ IntI{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent The format of the \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C696E7465723E}{\isasyminter}}{\isaliteral{22}{\isachardoublequote}}}-introduction rule represents
-  the most basic inference, which proceeds from given premises to a
-  conclusion, without any nested proof context involved.
-
-  The next example performs backwards introduction on \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}},
-  the intersection of all sets within a given set.  This requires a
-  nested proof of set membership within a local context, where \isa{A} is an arbitrary-but-fixed member of the collection:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\medskip\begin{minipage}{0.6\textwidth}
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ A\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-%
-\endisadelimnoproof
-\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\end{minipage}\begin{minipage}{0.4\textwidth}
-%
-\begin{isamarkuptext}%
-\infer{\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}}}{\infer*{\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5B}{\isacharbrackleft}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}}}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\end{minipage}
-%
-\begin{isamarkuptext}%
-\medskip\noindent This Isar reasoning pattern again refers to the
-  primitive rule depicted above.  The system determines it in the
-  ``\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}'' step, which could have been spelt out more
-  explicitly as ``\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}rule\ InterI{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}''.  Note
-  that the rule involves both a local parameter \isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}} and an
-  assumption \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}} in the nested reasoning.  This kind of
-  compound rule typically demands a genuine sub-proof in Isar, working
-  backwards rather than forwards as seen before.  In the proof body we
-  encounter the \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}-\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}-\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}
-  outline of nested sub-proofs that is typical for Isar.  The final
-  \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} is like \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} followed by an additional
-  refinement of the enclosing claim, using the rule derived from the
-  proof body.
-
-  \medskip The next example involves \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}}, which can be
-  characterized as the set of all \isa{{\isaliteral{22}{\isachardoublequote}}x{\isaliteral{22}{\isachardoublequote}}} such that \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}A{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}}.  The elimination rule for \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}} does
-  not mention \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616E643E}{\isasymand}}{\isaliteral{22}{\isachardoublequote}}} at all, but admits to obtain
-  directly a local \isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}} such that \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}} hold.  This corresponds to the following Isar proof and
-  inference rule, respectively:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\medskip\begin{minipage}{0.6\textwidth}
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ C\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ A\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ C%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-%
-\endisadelimnoproof
-\isanewline
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\end{minipage}\begin{minipage}{0.4\textwidth}
-%
-\begin{isamarkuptext}%
-\infer{\isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}} & \infer*{\isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}}~}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}A{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5B}{\isacharbrackleft}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{2C}{\isacharcomma}}\ A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}}}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\end{minipage}
-%
-\begin{isamarkuptext}%
-\medskip\noindent Although the Isar proof follows the natural
-  deduction rule closely, the text reads not as natural as
-  anticipated.  There is a double occurrence of an arbitrary
-  conclusion \isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}}, which represents the final result, but is
-  irrelevant for now.  This issue arises for any elimination rule
-  involving local parameters.  Isar provides the derived language
-  element \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}, which is able to perform the same
-  elimination proof more conveniently:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent Here we avoid to mention the final conclusion \isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}}
-  and return to plain forward reasoning.  The rule involved in the
-  ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' proof is the same as before.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{The Pure framework \label{sec:framework-pure}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The Pure logic \cite{paulson-found,paulson700} is an intuitionistic
-  fragment of higher-order logic \cite{church40}.  In type-theoretic
-  parlance, there are three levels of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-calculus with
-  corresponding arrows \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}{\isaliteral{22}{\isachardoublequote}}}/\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{22}{\isachardoublequote}}}/\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}}:
-
-  \medskip
-  \begin{tabular}{ll}
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{22}{\isachardoublequote}}} & syntactic function space (terms depending on terms) \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} & universal quantification (proofs depending on terms) \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}} & implication (proofs depending on proofs) \\
-  \end{tabular}
-  \medskip
-
-  \noindent Here only the types of syntactic terms, and the
-  propositions of proof terms have been shown.  The \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-structure of proofs can be recorded as an optional feature of
-  the Pure inference kernel \cite{Berghofer-Nipkow:2000:TPHOL}, but
-  the formal system can never depend on them due to \emph{proof
-  irrelevance}.
-
-  On top of this most primitive layer of proofs, Pure implements a
-  generic calculus for nested natural deduction rules, similar to
-  \cite{Schroeder-Heister:1984}.  Here object-logic inferences are
-  internalized as formulae over \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}}.
-  Combining such rule statements may involve higher-order unification
-  \cite{paulson-natural}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Primitive inferences%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Term syntax provides explicit notation for abstraction \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{2E}{\isachardot}}\ b{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and application \isa{{\isaliteral{22}{\isachardoublequote}}b\ a{\isaliteral{22}{\isachardoublequote}}}, while types are usually
-  implicit thanks to type-inference; terms of type \isa{{\isaliteral{22}{\isachardoublequote}}prop{\isaliteral{22}{\isachardoublequote}}} are
-  called propositions.  Logical statements are composed via \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{2E}{\isachardot}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}.  Primitive reasoning operates on
-  judgments of the form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}}, with standard introduction
-  and elimination rules for \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} that refer to
-  fixed parameters \isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub m{\isaliteral{22}{\isachardoublequote}}} and hypotheses
-  \isa{{\isaliteral{22}{\isachardoublequote}}A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} from the context \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{22}{\isachardoublequote}}};
-  the corresponding proof terms are left implicit.  The subsequent
-  inference rules define \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} inductively, relative to a
-  collection of axioms:
-
-  \[
-  \infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A{\isaliteral{22}{\isachardoublequote}}}}{(\isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}} \text{~axiom})}
-  \qquad
-  \infer{\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A{\isaliteral{22}{\isachardoublequote}}}}{}
-  \]
-
-  \[
-  \infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{22}{\isachardoublequote}}}}
-  \qquad
-  \infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}
-  \]
-
-  \[
-  \infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}
-  \qquad
-  \infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A{\isaliteral{22}{\isachardoublequote}}}}
-  \]
-
-  Furthermore, Pure provides a built-in equality \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ prop{\isaliteral{22}{\isachardoublequote}}} with axioms for reflexivity, substitution, extensionality,
-  and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{5C3C6574613E}{\isasymeta}}{\isaliteral{22}{\isachardoublequote}}}-conversion on \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-terms.
-
-  \medskip An object-logic introduces another layer on top of Pure,
-  e.g.\ with types \isa{{\isaliteral{22}{\isachardoublequote}}i{\isaliteral{22}{\isachardoublequote}}} for individuals and \isa{{\isaliteral{22}{\isachardoublequote}}o{\isaliteral{22}{\isachardoublequote}}} for
-  propositions, term constants \isa{{\isaliteral{22}{\isachardoublequote}}Trueprop\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ prop{\isaliteral{22}{\isachardoublequote}}} as
-  (implicit) derivability judgment and connectives like \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequote}}} or \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ o{\isaliteral{22}{\isachardoublequote}}}, and axioms for object-level
-  rules such as \isa{{\isaliteral{22}{\isachardoublequote}}conjI{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{22}{\isachardoublequote}}} or \isa{{\isaliteral{22}{\isachardoublequote}}allI{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{22}{\isachardoublequote}}}.  Derived object rules are represented as theorems of
-  Pure.  After the initial object-logic setup, further axiomatizations
-  are usually avoided; plain definitions and derived principles are
-  used exclusively.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Reasoning with rules \label{sec:framework-resolution}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Primitive inferences mostly serve foundational purposes.  The main
-  reasoning mechanisms of Pure operate on nested natural deduction
-  rules expressed as formulae, using \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{22}{\isachardoublequote}}} to bind local
-  parameters and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} to express entailment.  Multiple
-  parameters and premises are represented by repeating these
-  connectives in a right-associative manner.
-
-  Since \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} commute thanks to the theorem
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, we may assume w.l.o.g.\
-  that rule statements always observe the normal form where
-  quantifiers are pulled in front of implications at each level of
-  nesting.  This means that any Pure proposition may be presented as a
-  \emph{Hereditary Harrop Formula} \cite{Miller:1991} which is of the
-  form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub m{\isaliteral{2E}{\isachardot}}\ H\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ H\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{22}{\isachardoublequote}}} for \isa{{\isaliteral{22}{\isachardoublequote}}m{\isaliteral{2C}{\isacharcomma}}\ n\ {\isaliteral{5C3C67653E}{\isasymge}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}}, and \isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}} atomic, and \isa{{\isaliteral{22}{\isachardoublequote}}H\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ H\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} being recursively of the same format.
-  Following the convention that outermost quantifiers are implicit,
-  Horn clauses \isa{{\isaliteral{22}{\isachardoublequote}}A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{22}{\isachardoublequote}}} are a special
-  case of this.
-
-  For example, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C696E7465723E}{\isasyminter}}{\isaliteral{22}{\isachardoublequote}}}-introduction rule encountered before is
-  represented as a Pure theorem as follows:
-  \[
-  \isa{{\isaliteral{22}{\isachardoublequote}}IntI{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{22}{\isachardoublequote}}}
-  \]
-
-  \noindent This is a plain Horn clause, since no further nesting on
-  the left is involved.  The general \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{22}{\isachardoublequote}}}-introduction
-  corresponds to a Hereditary Harrop Formula with one additional level
-  of nesting:
-  \[
-  \isa{{\isaliteral{22}{\isachardoublequote}}InterI{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}A{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}}
-  \]
-
-  \medskip Goals are also represented as rules: \isa{{\isaliteral{22}{\isachardoublequote}}A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequote}}} states that the sub-goals \isa{{\isaliteral{22}{\isachardoublequote}}A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} entail the result \isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}}; for \isa{{\isaliteral{22}{\isachardoublequote}}n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}} the
-  goal is finished.  To allow \isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}} being a rule statement
-  itself, we introduce the protective marker \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ prop\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ prop{\isaliteral{22}{\isachardoublequote}}}, which is defined as identity and hidden from the user.  We
-  initialize and finish goal states as follows:
-
-  \[
-  \begin{array}{c@ {\qquad}c}
-  \infer[(\indexdef{}{inference}{init}\hypertarget{inference.init}{\hyperlink{inference.init}{\mbox{\isa{init}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}C\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}C{\isaliteral{22}{\isachardoublequote}}}}{} &
-  \infer[(\indexdef{}{inference}{finish}\hypertarget{inference.finish}{\hyperlink{inference.finish}{\mbox{\isa{finish}}}})]{\isa{C}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}C{\isaliteral{22}{\isachardoublequote}}}}
-  \end{array}
-  \]
-
-  \noindent Goal states are refined in intermediate proof steps until
-  a finished form is achieved.  Here the two main reasoning principles
-  are \hyperlink{inference.resolution}{\mbox{\isa{resolution}}}, for back-chaining a rule against a
-  sub-goal (replacing it by zero or more sub-goals), and \hyperlink{inference.assumption}{\mbox{\isa{assumption}}}, for solving a sub-goal (finding a short-circuit with
-  local assumptions).  Below \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{22}{\isachardoublequote}}} stands for \isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} (\isa{{\isaliteral{22}{\isachardoublequote}}n\ {\isaliteral{5C3C67653E}{\isasymge}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}}).
-
-  \[
-  \infer[(\indexdef{}{inference}{resolution}\hypertarget{inference.resolution}{\hyperlink{inference.resolution}{\mbox{\isa{resolution}}}})]
-  {\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec H\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec a\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}}}
-  {\begin{tabular}{rl}
-    \isa{{\isaliteral{22}{\isachardoublequote}}rule{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a{\isaliteral{22}{\isachardoublequote}}} \\
-    \isa{{\isaliteral{22}{\isachardoublequote}}goal{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec H\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{27}{\isacharprime}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequote}}} \\
-    \isa{{\isaliteral{22}{\isachardoublequote}}goal\ unifier{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ B\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec a\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}\ {\isaliteral{3D}{\isacharequal}}\ B{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}} \\
-   \end{tabular}}
-  \]
-
-  \medskip
-
-  \[
-  \infer[(\indexdef{}{inference}{assumption}\hypertarget{inference.assumption}{\hyperlink{inference.assumption}{\mbox{\isa{assumption}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}}}
-  {\begin{tabular}{rl}
-    \isa{{\isaliteral{22}{\isachardoublequote}}goal{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec H\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequote}}} \\
-    \isa{{\isaliteral{22}{\isachardoublequote}}assm\ unifier{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}\ {\isaliteral{3D}{\isacharequal}}\ H\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}}~~\text{(for some~\isa{{\isaliteral{22}{\isachardoublequote}}H\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}})} \\
-   \end{tabular}}
-  \]
-
-  The following trace illustrates goal-oriented reasoning in
-  Isabelle/Pure:
-
-  {\footnotesize
-  \medskip
-  \begin{tabular}{r@ {\quad}l}
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}init{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}resolution\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}resolution\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}assumption{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}resolution\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}assumption{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}finish{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \end{tabular}
-  \medskip
-  }
-
-  Compositions of \hyperlink{inference.assumption}{\mbox{\isa{assumption}}} after \hyperlink{inference.resolution}{\mbox{\isa{resolution}}} occurs quite often, typically in elimination steps.
-  Traditional Isabelle tactics accommodate this by a combined
-  \indexdef{}{inference}{elim\_resolution}\hypertarget{inference.elim-resolution}{\hyperlink{inference.elim-resolution}{\mbox{\isa{elim{\isaliteral{5F}{\isacharunderscore}}resolution}}}} principle.  In contrast, Isar uses
-  a slightly more refined combination, where the assumptions to be
-  closed are marked explicitly, using again the protective marker
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}{\isaliteral{22}{\isachardoublequote}}}:
-
-  \[
-  \infer[(\hyperlink{inference.refinement}{\mbox{\isa{refinement}}})]
-  {\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec H\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec G{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec a\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}}}
-  {\begin{tabular}{rl}
-    \isa{{\isaliteral{22}{\isachardoublequote}}sub{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}proof{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec G\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a{\isaliteral{22}{\isachardoublequote}}} \\
-    \isa{{\isaliteral{22}{\isachardoublequote}}goal{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec H\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{27}{\isacharprime}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequote}}} \\
-    \isa{{\isaliteral{22}{\isachardoublequote}}goal\ unifier{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ B\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec a\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}\ {\isaliteral{3D}{\isacharequal}}\ B{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}} \\
-    \isa{{\isaliteral{22}{\isachardoublequote}}assm\ unifiers{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ G\isaliteral{5C3C5E7375623E}{}\isactrlsub j\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec a\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{23}{\isacharhash}}H\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}} \\
-    & \quad (for each marked \isa{{\isaliteral{22}{\isachardoublequote}}G\isaliteral{5C3C5E7375623E}{}\isactrlsub j{\isaliteral{22}{\isachardoublequote}}} some \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}H\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}}) \\
-   \end{tabular}}
-  \]
-
-  \noindent Here the \isa{{\isaliteral{22}{\isachardoublequote}}sub{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}proof{\isaliteral{22}{\isachardoublequote}}} rule stems from the
-  main \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}-\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}-\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} outline of
-  Isar (cf.\ \secref{sec:framework-subproof}): each assumption
-  indicated in the text results in a marked premise \isa{{\isaliteral{22}{\isachardoublequote}}G{\isaliteral{22}{\isachardoublequote}}} above.
-  The marking enforces resolution against one of the sub-goal's
-  premises.  Consequently, \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}-\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}-\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} enables to fit the result of a sub-proof quite robustly into a
-  pending sub-goal, while maintaining a good measure of flexibility.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{The Isar proof language \label{sec:framework-isar}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Structured proofs are presented as high-level expressions for
-  composing entities of Pure (propositions, facts, and goals).  The
-  Isar proof language allows to organize reasoning within the
-  underlying rule calculus of Pure, but Isar is not another logical
-  calculus!
-
-  Isar is an exercise in sound minimalism.  Approximately half of the
-  language is introduced as primitive, the rest defined as derived
-  concepts.  The following grammar describes the core language
-  (category \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{22}{\isachardoublequote}}}), which is embedded into theory
-  specification elements such as \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}; see also
-  \secref{sec:framework-stmt} for the separate category \isa{{\isaliteral{22}{\isachardoublequote}}statement{\isaliteral{22}{\isachardoublequote}}}.
-
-  \medskip
-  \begin{tabular}{rcl}
-    \isa{{\isaliteral{22}{\isachardoublequote}}theory{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}stmt{\isaliteral{22}{\isachardoublequote}}} & = & \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}statement\ proof\ \ {\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}}~~\hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ \ {\isaliteral{7C}{\isacharbar}}\ \ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\[1ex]
-
-    \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{22}{\isachardoublequote}}} & = & \isa{{\isaliteral{22}{\isachardoublequote}}prfx\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}method\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{3F}{\isacharquery}}\ stmt\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}method\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}} \\[1ex]
-
-    \isa{prfx} & = & \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}facts{\isaliteral{22}{\isachardoublequote}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}facts{\isaliteral{22}{\isachardoublequote}}} \\
-
-    \isa{stmt} & = & \hyperlink{command.braceleft}{\mbox{\isa{\isacommand{{\isaliteral{7B}{\isacharbraceleft}}}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}stmt\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isaliteral{7D}{\isacharbraceright}}}}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}name\ {\isaliteral{3D}{\isacharequal}}\ facts{\isaliteral{22}{\isachardoublequote}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}term\ {\isaliteral{3D}{\isacharequal}}\ term{\isaliteral{22}{\isachardoublequote}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}var\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2B}{\isacharplus}}{\isaliteral{22}{\isachardoublequote}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6775696C6C656D6F746C6566743E}{\isasymguillemotleft}}inference{\isaliteral{5C3C6775696C6C656D6F7472696768743E}{\isasymguillemotright}}\ name{\isaliteral{3A}{\isacharcolon}}\ props{\isaliteral{22}{\isachardoublequote}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}~\isa{goal} \\
-    \isa{goal} & = & \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}name{\isaliteral{3A}{\isacharcolon}}\ props\ proof{\isaliteral{22}{\isachardoublequote}}} \\
-    & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}name{\isaliteral{3A}{\isacharcolon}}\ props\ proof{\isaliteral{22}{\isachardoublequote}}} \\
-  \end{tabular}
-
-  \medskip Simultaneous propositions or facts may be separated by the
-  \hyperlink{keyword.and}{\mbox{\isa{\isakeyword{and}}}} keyword.
-
-  \medskip The syntax for terms and propositions is inherited from
-  Pure (and the object-logic).  A \isa{{\isaliteral{22}{\isachardoublequote}}pattern{\isaliteral{22}{\isachardoublequote}}} is a \isa{{\isaliteral{22}{\isachardoublequote}}term{\isaliteral{22}{\isachardoublequote}}} with schematic variables, to be bound by higher-order
-  matching.
-
-  \medskip Facts may be referenced by name or proposition.  For
-  example, the result of ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ A\ {\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}proof{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}{\isaliteral{22}{\isachardoublequote}}}''
-  becomes available both as \isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{22}{\isachardoublequote}}} and
-  \isacharbackquoteopen\isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}}\isacharbackquoteclose.  Moreover,
-  fact expressions may involve attributes that modify either the
-  theorem or the background context.  For example, the expression
-  ``\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5B}{\isacharbrackleft}}OF\ b{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' refers to the composition of two facts
-  according to the \hyperlink{inference.resolution}{\mbox{\isa{resolution}}} inference of
-  \secref{sec:framework-resolution}, while ``\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}''
-  declares a fact as introduction rule in the context.
-
-  The special fact called ``\hyperlink{fact.this}{\mbox{\isa{this}}}'' always refers to the last
-  result, as produced by \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}, \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}, \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}, or \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}.  Since \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}} occurs
-  frequently together with \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} we provide some
-  abbreviations:
-
-  \medskip
-  \begin{tabular}{rcl}
-    \hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{a} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{a}~\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} \\
-    \hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}}~\isa{a} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ this{\isaliteral{22}{\isachardoublequote}}} \\
-  \end{tabular}
-  \medskip
-
-  The \isa{{\isaliteral{22}{\isachardoublequote}}method{\isaliteral{22}{\isachardoublequote}}} category is essentially a parameter and may be
-  populated later.  Methods use the facts indicated by \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} or \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}, and then operate on the goal state.
-  Some basic methods are predefined: ``\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}}'' leaves the goal
-  unchanged, ``\hyperlink{method.this}{\mbox{\isa{this}}}'' applies the facts as rules to the
-  goal, ``\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}'' applies the facts to another rule and the
-  result to the goal (both ``\hyperlink{method.this}{\mbox{\isa{this}}}'' and ``\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}''
-  refer to \hyperlink{inference.resolution}{\mbox{\isa{resolution}}} of
-  \secref{sec:framework-resolution}).  The secondary arguments to
-  ``\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}'' may be specified explicitly as in ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}rule\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'', or picked from the context.  In the latter case, the system
-  first tries rules declared as \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}} or
-  \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}, followed by those declared as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}.
-
-  The default method for \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} is ``\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}''
-  (arguments picked from the context), for \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} it is
-  ``\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}}''.  Further abbreviations for terminal proof steps
-  are ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}'' for
-  ``\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}method\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}'', and ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' for ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}, and ``\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}'' for ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\hyperlink{method.this}{\mbox{\isa{this}}}''.  The \hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}} element operates
-  directly on the current facts and goal by applying equalities.
-
-  \medskip Block structure can be indicated explicitly by ``\hyperlink{command.braceleft}{\mbox{\isa{\isacommand{{\isaliteral{7B}{\isacharbraceleft}}}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isaliteral{7D}{\isacharbraceright}}}}}}'', although the body of a sub-proof
-  already involves implicit nesting.  In any case, \hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}}
-  jumps into the next section of a block, i.e.\ it acts like closing
-  an implicit block scope and opening another one; there is no direct
-  correspondence to subgoals here.
-
-  The remaining elements \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}} and \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} build up
-  a local context (see \secref{sec:framework-context}), while
-  \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} refines a pending sub-goal by the rule resulting
-  from a nested sub-proof (see \secref{sec:framework-subproof}).
-  Further derived concepts will support calculational reasoning (see
-  \secref{sec:framework-calc}).%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Context elements \label{sec:framework-context}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In judgments \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} of the primitive framework, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{22}{\isachardoublequote}}}
-  essentially acts like a proof context.  Isar elaborates this idea
-  towards a higher-level notion, with additional information for
-  type-inference, term abbreviations, local facts, hypotheses etc.
-
-  The element \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}{\isaliteral{22}{\isachardoublequote}}} declares a local
-  parameter, i.e.\ an arbitrary-but-fixed entity of a given type; in
-  results exported from the context, \isa{{\isaliteral{22}{\isachardoublequote}}x{\isaliteral{22}{\isachardoublequote}}} may become anything.
-  The \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6775696C6C656D6F746C6566743E}{\isasymguillemotleft}}inference{\isaliteral{5C3C6775696C6C656D6F7472696768743E}{\isasymguillemotright}}{\isaliteral{22}{\isachardoublequote}}} element provides a
-  general interface to hypotheses: ``\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6775696C6C656D6F746C6566743E}{\isasymguillemotleft}}inference{\isaliteral{5C3C6775696C6C656D6F7472696768743E}{\isasymguillemotright}}\ A{\isaliteral{22}{\isachardoublequote}}}'' produces \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A{\isaliteral{22}{\isachardoublequote}}} locally, while the
-  included inference tells how to discharge \isa{A} from results
-  \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}} later on.  There is no user-syntax for \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6775696C6C656D6F746C6566743E}{\isasymguillemotleft}}inference{\isaliteral{5C3C6775696C6C656D6F7472696768743E}{\isasymguillemotright}}{\isaliteral{22}{\isachardoublequote}}}, i.e.\ it may only occur internally when derived
-  commands are defined in ML.
-
-  At the user-level, the default inference for \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} is
-  \hyperlink{inference.discharge}{\mbox{\isa{discharge}}} as given below.  The additional variants
-  \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}} and \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}} are defined as follows:
-
-  \medskip
-  \begin{tabular}{rcl}
-    \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}~\isa{A} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6775696C6C656D6F746C6566743E}{\isasymguillemotleft}}weak{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}discharge{\isaliteral{5C3C6775696C6C656D6F7472696768743E}{\isasymguillemotright}}\ A{\isaliteral{22}{\isachardoublequote}}} \\
-    \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ a{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{x}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6775696C6C656D6F746C6566743E}{\isasymguillemotleft}}expansion{\isaliteral{5C3C6775696C6C656D6F7472696768743E}{\isasymguillemotright}}\ x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ a{\isaliteral{22}{\isachardoublequote}}} \\
-  \end{tabular}
-  \medskip
-
-  \[
-  \infer[(\indexdef{}{inference}{discharge}\hypertarget{inference.discharge}{\hyperlink{inference.discharge}{\mbox{\isa{discharge}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{23}{\isacharhash}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}
-  \]
-  \[
-  \infer[(\indexdef{}{inference}{weak-discharge}\hypertarget{inference.weak-discharge}{\hyperlink{inference.weak-discharge}{\mbox{\isa{weak{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}discharge}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}
-  \]
-  \[
-  \infer[(\indexdef{}{inference}{expansion}\hypertarget{inference.expansion}{\hyperlink{inference.expansion}{\mbox{\isa{expansion}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B\ a{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B\ x{\isaliteral{22}{\isachardoublequote}}}}
-  \]
-
-  \medskip Note that \hyperlink{inference.discharge}{\mbox{\isa{discharge}}} and \hyperlink{inference.weak-discharge}{\mbox{\isa{weak{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}discharge}}} differ in the marker for \isa{A}, which is
-  relevant when the result of a \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}-\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}-\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} outline is composed with a pending goal,
-  cf.\ \secref{sec:framework-subproof}.
-
-  The most interesting derived context element in Isar is \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} \cite[\S5.3]{Wenzel-PhD}, which supports generalized
-  elimination steps in a purely forward manner.  The \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}
-  command takes a specification of parameters \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{22}{\isachardoublequote}}} and
-  assumptions \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec A{\isaliteral{22}{\isachardoublequote}}} to be added to the context, together
-  with a proof of a case rule stating that this extension is
-  conservative (i.e.\ may be removed from closed results later on):
-
-  \medskip
-  \begin{tabular}{l}
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}facts{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}{\isaliteral{22}{\isachardoublequote}}}~~\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ \ {\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}proof{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} \\[0.5ex]
-  \quad \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}case{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}thesis{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \quad \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}} \\
-  \qquad \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{thesis} \\
-  \qquad \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\
-  \qquad \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{thesis}~\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}facts{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}\ {\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}proof{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \quad \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} \\
-  \quad \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6775696C6C656D6F746C6566743E}{\isasymguillemotleft}}elimination\ case{\isaliteral{5C3C6775696C6C656D6F7472696768743E}{\isasymguillemotright}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{22}{\isachardoublequote}}} \\
-  \end{tabular}
-  \medskip
-
-  \[
-  \infer[(\hyperlink{inference.elimination}{\mbox{\isa{elimination}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}{
-    \begin{tabular}{rl}
-    \isa{{\isaliteral{22}{\isachardoublequote}}case{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}thesis{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\[0.2ex]
-    \isa{{\isaliteral{22}{\isachardoublequote}}result{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
-    \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec y\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}} \\[0.2ex]
-    \end{tabular}}
-  \]
-
-  \noindent Here the name ``\isa{thesis}'' is a specific convention
-  for an arbitrary-but-fixed proposition; in the primitive natural
-  deduction rules shown before we have occasionally used \isa{C}.
-  The whole statement of ``\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}~\isa{x}~\hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}A\ x{\isaliteral{22}{\isachardoublequote}}}'' may be read as a claim that \isa{{\isaliteral{22}{\isachardoublequote}}A\ x{\isaliteral{22}{\isachardoublequote}}}
-  may be assumed for some arbitrary-but-fixed \isa{{\isaliteral{22}{\isachardoublequote}}x{\isaliteral{22}{\isachardoublequote}}}.  Also note
-  that ``\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ B{\isaliteral{22}{\isachardoublequote}}}'' without parameters
-  is similar to ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ B{\isaliteral{22}{\isachardoublequote}}}'', but the
-  latter involves multiple sub-goals.
-
-  \medskip The subsequent Isar proof texts explain all context
-  elements introduced above using the formal proof language itself.
-  After finishing a local proof within a block, we indicate the
-  exported result via \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{minipage}[t]{0.45\textwidth}
-\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-\isanewline
-%
-\endisadelimnoproof
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{note}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{60}{\isacharbackquoteclose}}%
-\end{minipage}\quad\begin{minipage}[t]{0.45\textwidth}
-\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ A\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ B%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-\isanewline
-%
-\endisadelimnoproof
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{note}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{60}{\isacharbackquoteclose}}%
-\end{minipage}\\[3ex]\begin{minipage}[t]{0.45\textwidth}
-\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{def}\isamarkupfalse%
-\ x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ a\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-\isanewline
-%
-\endisadelimnoproof
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{note}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}B\ a{\isaliteral{60}{\isacharbackquoteclose}}%
-\end{minipage}\quad\begin{minipage}[t]{0.45\textwidth}
-\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ x\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-\isanewline
-%
-\endisadelimnoproof
-%
-\isadelimproof
-\ \ \ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{have}\isamarkupfalse%
-\ B%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-\isanewline
-%
-\endisadelimnoproof
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{note}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}B{\isaliteral{60}{\isacharbackquoteclose}}%
-\end{minipage}
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\bigskip\noindent This illustrates the meaning of Isar context
-  elements without goals getting in between.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Structured statements \label{sec:framework-stmt}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The category \isa{{\isaliteral{22}{\isachardoublequote}}statement{\isaliteral{22}{\isachardoublequote}}} of top-level theorem specifications
-  is defined as follows:
-
-  \medskip
-  \begin{tabular}{rcl}
-  \isa{{\isaliteral{22}{\isachardoublequote}}statement{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}name{\isaliteral{3A}{\isacharcolon}}\ props\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
-  & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}context\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}\ conclusion{\isaliteral{22}{\isachardoublequote}}} \\[0.5ex]
-
-  \isa{{\isaliteral{22}{\isachardoublequote}}context{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C46495845533E}{\isasymFIXES}}\ vars\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
-  & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ name{\isaliteral{3A}{\isacharcolon}}\ props\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
-
-  \isa{{\isaliteral{22}{\isachardoublequote}}conclusion{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ name{\isaliteral{3A}{\isacharcolon}}\ props\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
-  & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ vars\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ name{\isaliteral{3A}{\isacharcolon}}\ props\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
-  & & \quad \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C424241523E}{\isasymBBAR}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \end{tabular}
-
-  \medskip\noindent A simple \isa{{\isaliteral{22}{\isachardoublequote}}statement{\isaliteral{22}{\isachardoublequote}}} consists of named
-  propositions.  The full form admits local context elements followed
-  by the actual conclusions, such as ``\hyperlink{keyword.fixes}{\mbox{\isa{\isakeyword{fixes}}}}~\isa{x}~\hyperlink{keyword.assumes}{\mbox{\isa{\isakeyword{assumes}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}A\ x{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{keyword.shows}{\mbox{\isa{\isakeyword{shows}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}B\ x{\isaliteral{22}{\isachardoublequote}}}''.  The final result emerges as a Pure rule after discharging
-  the context: \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ A\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ x{\isaliteral{22}{\isachardoublequote}}}.
-
-  The \hyperlink{keyword.obtains}{\mbox{\isa{\isakeyword{obtains}}}} variant is another abbreviation defined
-  below; unlike \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} (cf.\
-  \secref{sec:framework-context}) there may be several ``cases''
-  separated by ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C424241523E}{\isasymBBAR}}{\isaliteral{22}{\isachardoublequote}}}'', each consisting of several
-  parameters (\isa{{\isaliteral{22}{\isachardoublequote}}vars{\isaliteral{22}{\isachardoublequote}}}) and several premises (\isa{{\isaliteral{22}{\isachardoublequote}}props{\isaliteral{22}{\isachardoublequote}}}).
-  This specifies multi-branch elimination rules.
-
-  \medskip
-  \begin{tabular}{l}
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ \ \ {\isaliteral{5C3C424241523E}{\isasymBBAR}}\ \ \ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ \ \ {\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} \\[0.5ex]
-  \quad \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C46495845533E}{\isasymFIXES}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\
-  \quad \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C415353554D45533E}{\isasymASSUMES}}\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis\ \ {\isaliteral{5C3C414E443E}{\isasymAND}}\ \ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
-  \quad \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\
-  \end{tabular}
-  \medskip
-
-  Presenting structured statements in such an ``open'' format usually
-  simplifies the subsequent proof, because the outer structure of the
-  problem is already laid out directly.  E.g.\ consider the following
-  canonical patterns for \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}{\isaliteral{22}{\isachardoublequote}}},
-  respectively:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{minipage}{0.5\textwidth}
-\isacommand{theorem}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{fixes}\ x\ \isakeyword{and}\ y\isanewline
-\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}A\ x{\isaliteral{60}{\isacharbackquoteclose}}\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}B\ y{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-\isanewline
-%
-\endisadelimnoproof
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\end{minipage}\begin{minipage}{0.5\textwidth}
-\isacommand{theorem}\isamarkupfalse%
-\isanewline
-\ \ \isakeyword{obtains}\ x\ \isakeyword{and}\ y\isanewline
-\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}A\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ b{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-\isanewline
-%
-\endisadelimnoproof
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\end{minipage}
-%
-\begin{isamarkuptext}%
-\medskip\noindent Here local facts \isacharbackquoteopen\isa{{\isaliteral{22}{\isachardoublequote}}A\ x{\isaliteral{22}{\isachardoublequote}}}\isacharbackquoteclose\ and \isacharbackquoteopen\isa{{\isaliteral{22}{\isachardoublequote}}B\ y{\isaliteral{22}{\isachardoublequote}}}\isacharbackquoteclose\ are referenced immediately; there is no
-  need to decompose the logical rule structure again.  In the second
-  proof the final ``\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{thesis}~\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}''  involves the local rule case \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ A\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}} for the particular instance of terms \isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}b{\isaliteral{22}{\isachardoublequote}}} produced in the body.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Structured proof refinement \label{sec:framework-subproof}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-By breaking up the grammar for the Isar proof language, we may
-  understand a proof text as a linear sequence of individual proof
-  commands.  These are interpreted as transitions of the Isar virtual
-  machine (Isar/VM), which operates on a block-structured
-  configuration in single steps.  This allows users to write proof
-  texts in an incremental manner, and inspect intermediate
-  configurations for debugging.
-
-  The basic idea is analogous to evaluating algebraic expressions on a
-  stack machine: \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C63646F743E}{\isasymcdot}}\ c{\isaliteral{22}{\isachardoublequote}}} then corresponds to a sequence
-  of single transitions for each symbol \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{2C}{\isacharcomma}}\ a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{2B}{\isacharplus}}{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C63646F743E}{\isasymcdot}}{\isaliteral{2C}{\isacharcomma}}\ c{\isaliteral{22}{\isachardoublequote}}}.
-  In Isar the algebraic values are facts or goals, and the operations
-  are inferences.
-
-  \medskip The Isar/VM state maintains a stack of nodes, each node
-  contains the local proof context, the linguistic mode, and a pending
-  goal (optional).  The mode determines the type of transition that
-  may be performed next, it essentially alternates between forward and
-  backward reasoning, with an intermediate stage for chained facts
-  (see \figref{fig:isar-vm}).
-
-  \begin{figure}[htb]
-  \begin{center}
-  \includegraphics[width=0.8\textwidth]{Thy/document/isar-vm}
-  \end{center}
-  \caption{Isar/VM modes}\label{fig:isar-vm}
-  \end{figure}
-
-  For example, in \isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} mode Isar acts like a mathematical
-  scratch-pad, accepting declarations like \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}, \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}, and claims like \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}, \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}.  A goal
-  statement changes the mode to \isa{{\isaliteral{22}{\isachardoublequote}}prove{\isaliteral{22}{\isachardoublequote}}}, which means that we
-  may now refine the problem via \hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}} or \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}.  Then we are again in \isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} mode of a proof body,
-  which may issue \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} statements to solve pending
-  sub-goals.  A concluding \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} will return to the original
-  \isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} mode one level upwards.  The subsequent Isar/VM
-  trace indicates block structure, linguistic mode, goal state, and
-  inferences:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begingroup\footnotesize
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{minipage}[t]{0.18\textwidth}
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ A\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ B%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-%
-\isadelimnoproof
-\ \ \ \ \ \ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-%
-\endisadelimnoproof
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{qed}\isamarkupfalse%
-%
-\end{minipage}\quad
-\begin{minipage}[t]{0.06\textwidth}
-\isa{{\isaliteral{22}{\isachardoublequote}}begin{\isaliteral{22}{\isachardoublequote}}} \\
-\\
-\\
-\isa{{\isaliteral{22}{\isachardoublequote}}begin{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}end{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}end{\isaliteral{22}{\isachardoublequote}}} \\
-\end{minipage}
-\begin{minipage}[t]{0.08\textwidth}
-\isa{{\isaliteral{22}{\isachardoublequote}}prove{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}prove{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
-\end{minipage}\begin{minipage}[t]{0.35\textwidth}
-\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-\\
-\\
-\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}} \\
-\end{minipage}\begin{minipage}[t]{0.4\textwidth}
-\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}init{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}resolution\ impI{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-\\
-\\
-\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}refinement\ {\isaliteral{23}{\isacharhash}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}finish{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
-\end{minipage}
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\endgroup
-%
-\begin{isamarkuptext}%
-\noindent Here the \hyperlink{inference.refinement}{\mbox{\isa{refinement}}} inference from
-  \secref{sec:framework-resolution} mediates composition of Isar
-  sub-proofs nicely.  Observe that this principle incorporates some
-  degree of freedom in proof composition.  In particular, the proof
-  body allows parameters and assumptions to be re-ordered, or commuted
-  according to Hereditary Harrop Form.  Moreover, context elements
-  that are not used in a sub-proof may be omitted altogether.  For
-  example:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{minipage}{0.5\textwidth}
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ A\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\ \isakeyword{and}\ y\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-%
-\endisadelimnoproof
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{qed}\isamarkupfalse%
-%
-\end{minipage}\begin{minipage}{0.5\textwidth}
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ A\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ y\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimnoproof
-\ %
-\endisadelimnoproof
-%
-\isatagnoproof
-\isacommand{sorry}\isamarkupfalse%
-%
-\endisatagnoproof
-{\isafoldnoproof}%
-%
-\isadelimnoproof
-%
-\endisadelimnoproof
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{qed}\isamarkupfalse%
-%
-\end{minipage}\\[3ex]\begin{minipage}{0.5\textwidth}
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ A\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ y\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\end{minipage}\begin{minipage}{0.5\textwidth}
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\ y{\isaliteral{2E}{\isachardot}}\ A\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ y\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ x\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\end{minipage}
-%
-\begin{isamarkuptext}%
-\medskip\noindent Such ``peephole optimizations'' of Isar texts are
-  practically important to improve readability, by rearranging
-  contexts elements according to the natural flow of reasoning in the
-  body, while still observing the overall scoping rules.
-
-  \medskip This illustrates the basic idea of structured proof
-  processing in Isar.  The main mechanisms are based on natural
-  deduction rule composition within the Pure framework.  In
-  particular, there are no direct operations on goal states within the
-  proof body.  Moreover, there is no hidden automated reasoning
-  involved, just plain unification.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Calculational reasoning \label{sec:framework-calc}%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The existing Isar infrastructure is sufficiently flexible to support
-  calculational reasoning (chains of transitivity steps) as derived
-  concept.  The generic proof elements introduced below depend on
-  rules declared as \hyperlink{attribute.trans}{\mbox{\isa{trans}}} in the context.  It is left to
-  the object-logic to provide a suitable rule collection for mixed
-  relations of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3C}{\isacharless}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C653E}{\isasymle}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7375627365743E}{\isasymsubset}}{\isaliteral{22}{\isachardoublequote}}},
-  \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}{\isaliteral{22}{\isachardoublequote}}} etc.  Due to the flexibility of rule composition
-  (\secref{sec:framework-resolution}), substitution of equals by
-  equals is covered as well, even substitution of inequalities
-  involving monotonicity conditions; see also \cite[\S6]{Wenzel-PhD}
-  and \cite{Bauer-Wenzel:2001}.
-
-  The generic calculational mechanism is based on the observation that
-  rules such as \isa{{\isaliteral{22}{\isachardoublequote}}trans{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{3D}{\isacharequal}}\ z{\isaliteral{22}{\isachardoublequote}}}
-  proceed from the premises towards the conclusion in a deterministic
-  fashion.  Thus we may reason in forward mode, feeding intermediate
-  results into rules selected from the context.  The course of
-  reasoning is organized by maintaining a secondary fact called
-  ``\hyperlink{fact.calculation}{\mbox{\isa{calculation}}}'', apart from the primary ``\hyperlink{fact.this}{\mbox{\isa{this}}}''
-  already provided by the Isar primitives.  In the definitions below,
-  \hyperlink{attribute.OF}{\mbox{\isa{OF}}} refers to \hyperlink{inference.resolution}{\mbox{\isa{resolution}}}
-  (\secref{sec:framework-resolution}) with multiple rule arguments,
-  and \isa{{\isaliteral{22}{\isachardoublequote}}trans{\isaliteral{22}{\isachardoublequote}}} represents to a suitable rule from the context:
-
-  \begin{matharray}{rcl}
-    \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}calculation\ {\isaliteral{3D}{\isacharequal}}\ this{\isaliteral{22}{\isachardoublequote}}} \\
-    \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{2B}{\isacharplus}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}calculation\ {\isaliteral{3D}{\isacharequal}}\ trans\ {\isaliteral{5B}{\isacharbrackleft}}OF\ calculation\ this{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}} \\[0.5ex]
-    \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}} & \equiv & \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}~\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{calculation} \\
-  \end{matharray}
-
-  \noindent The start of a calculation is determined implicitly in the
-  text: here \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} sets \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} to the current
-  result; any subsequent occurrence will update \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} by
-  combination with the next result and a transitivity rule.  The
-  calculational sequence is concluded via \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}, where
-  the final result is exposed for use in a concluding claim.
-
-  Here is a canonical proof pattern, using \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} to
-  establish the intermediate results:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ d{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ d{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent The term ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' above is a special abbreviation
-  provided by the Isabelle/Isar syntax layer: it statically refers to
-  the right-hand side argument of the previous statement given in the
-  text.  Thus it happens to coincide with relevant sub-expressions in
-  the calculational chain, but the exact correspondence is dependent
-  on the transitivity rules being involved.
-
-  \medskip Symmetry rules such as \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3D}{\isacharequal}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequote}}} are like
-  transitivities with only one premise.  Isar maintains a separate
-  rule collection declared via the \hyperlink{attribute.sym}{\mbox{\isa{sym}}} attribute, to be
-  used in fact expressions ``\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'', or single-step
-  proofs ``\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}''.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End: