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\begin{isabellebody}%
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\def\isabellecontext{Framework}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ Framework\isanewline
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\isakeyword{imports}\ Base\ Main\isanewline
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\isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isamarkupchapter{The Isabelle/Isar Framework \label{ch:isar-framework}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Isabelle/Isar
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\cite{Wenzel:1999:TPHOL,Wenzel-PhD,Nipkow-TYPES02,Wenzel-Paulson:2006,Wenzel:2006:Festschrift}
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is intended as a generic framework for developing formal
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mathematical documents with full proof checking. Definitions and
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proofs are organized as theories. An assembly of theory sources may
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be presented as a printed document; see also
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\chref{ch:document-prep}.
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The main objective of Isar is the design of a human-readable
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structured proof language, which is called the ``primary proof
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format'' in Isar terminology. Such a primary proof language is
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somewhere in the middle between the extremes of primitive proof
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objects and actual natural language. In this respect, Isar is a bit
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more formalistic than Mizar
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\cite{Trybulec:1993:MizarFeatures,Rudnicki:1992:MizarOverview,Wiedijk:1999:Mizar},
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using logical symbols for certain reasoning schemes where Mizar
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would prefer English words; see \cite{Wenzel-Wiedijk:2002} for
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further comparisons of these systems.
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So Isar challenges the traditional way of recording informal proofs
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in mathematical prose, as well as the common tendency to see fully
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formal proofs directly as objects of some logical calculus (e.g.\
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\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-terms in a version of type theory). In fact, Isar is
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better understood as an interpreter of a simple block-structured
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language for describing the data flow of local facts and goals,
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interspersed with occasional invocations of proof methods.
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Everything is reduced to logical inferences internally, but these
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steps are somewhat marginal compared to the overall bookkeeping of
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the interpretation process. Thanks to careful design of the syntax
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and semantics of Isar language elements, a formal record of Isar
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instructions may later appear as an intelligible text to the
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attentive reader.
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The Isar proof language has emerged from careful analysis of some
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inherent virtues of the existing logical framework of Isabelle/Pure
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\cite{paulson-found,paulson700}, notably composition of higher-order
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natural deduction rules, which is a generalization of Gentzen's
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original calculus \cite{Gentzen:1935}. The approach of generic
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inference systems in Pure is continued by Isar towards actual proof
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texts.
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Concrete applications require another intermediate layer: an
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object-logic. Isabelle/HOL \cite{isa-tutorial} (simply-typed
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set-theory) is being used most of the time; Isabelle/ZF
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\cite{isabelle-ZF} is less extensively developed, although it would
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probably fit better for classical mathematics.
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\medskip In order to illustrate natural deduction in Isar, we shall
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refer to the background theory and library of Isabelle/HOL. This
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includes common notions of predicate logic, naive set-theory etc.\
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using fairly standard mathematical notation. From the perspective
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of generic natural deduction there is nothing special about the
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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}}},
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\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}{\isaliteral{22}{\isachardoublequote}}}, etc.), only the resulting reasoning principles are
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relevant to the user. There are similar rules available for
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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
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theory, topology etc.).
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Subsequently we briefly review fragments of Isar proof texts
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corresponding directly to such general deduction schemes. The
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examples shall refer to set-theory, to minimize the danger of
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understanding connectives of predicate logic as something special.
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\medskip The following deduction performs \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C696E7465723E}{\isasyminter}}{\isaliteral{22}{\isachardoublequote}}}-introduction,
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working forwards from assumptions towards the conclusion. We give
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both the Isar text, and depict the primitive rule involved, as
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determined by unification of the problem against rules that are
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declared in the library context.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\medskip\begin{minipage}{0.6\textwidth}
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%
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\isadelimproof
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\ \ \ \ %
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\endisadelimproof
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%
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\isatagproof
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\isacommand{assume}\isamarkupfalse%
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\ {\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
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\ \ \ \ \isacommand{then}\isamarkupfalse%
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\ \isacommand{have}\isamarkupfalse%
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\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\end{minipage}\begin{minipage}{0.4\textwidth}
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%
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\begin{isamarkuptext}%
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\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}}}}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\end{minipage}
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%
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\begin{isamarkuptext}%
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\medskip\noindent Note that \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} augments the proof
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context, \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} indicates that the current fact shall be
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used in the next step, and \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} states an intermediate
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goal. The two dots ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' refer to a complete proof of
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this claim, using the indicated facts and a canonical rule from the
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context. We could have been more explicit here by spelling out the
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final proof step via the \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}} command:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isadelimproof
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\ \ \ \ %
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\endisadelimproof
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%
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\isatagproof
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\isacommand{assume}\isamarkupfalse%
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\ {\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
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\ \ \ \ \isacommand{then}\isamarkupfalse%
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\ \isacommand{have}\isamarkupfalse%
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\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
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\ {\isaliteral{28}{\isacharparenleft}}rule\ IntI{\isaliteral{29}{\isacharparenright}}%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\begin{isamarkuptext}%
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\noindent The format of the \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C696E7465723E}{\isasyminter}}{\isaliteral{22}{\isachardoublequote}}}-introduction rule represents
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the most basic inference, which proceeds from given premises to a
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conclusion, without any nested proof context involved.
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The next example performs backwards introduction on \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}},
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the intersection of all sets within a given set. This requires a
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nested proof of set membership within a local context, where \isa{A} is an arbitrary-but-fixed member of the collection:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\medskip\begin{minipage}{0.6\textwidth}
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%
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\isadelimproof
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\ \ \ \ %
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\endisadelimproof
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%
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\isatagproof
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\isacommand{have}\isamarkupfalse%
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\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
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\ \ \ \ \isacommand{proof}\isamarkupfalse%
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\isanewline
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\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
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\ A\isanewline
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\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
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\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
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\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
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\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\isadelimnoproof
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\ %
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\endisadelimnoproof
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%
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\isatagnoproof
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\isacommand{sorry}\isamarkupfalse%
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%
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\endisatagnoproof
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{\isafoldnoproof}%
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%
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\isadelimnoproof
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%
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\endisadelimnoproof
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\isanewline
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%
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\isadelimproof
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\ \ \ \ %
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\endisadelimproof
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%
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\isatagproof
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\isacommand{qed}\isamarkupfalse%
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%
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\endisatagproof
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{\isafoldproof}%
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%
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\isadelimproof
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%
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\endisadelimproof
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%
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\end{minipage}\begin{minipage}{0.4\textwidth}
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%
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\begin{isamarkuptext}%
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\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}}}}}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\end{minipage}
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%
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\begin{isamarkuptext}%
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\medskip\noindent This Isar reasoning pattern again refers to the
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primitive rule depicted above. The system determines it in the
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``\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}'' step, which could have been spelt out more
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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
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that the rule involves both a local parameter \isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}} and an
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assumption \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}} in the nested reasoning. This kind of
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compound rule typically demands a genuine sub-proof in Isar, working
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backwards rather than forwards as seen before. In the proof body we
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encounter the \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}-\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}-\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}
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outline of nested sub-proofs that is typical for Isar. The final
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\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} is like \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} followed by an additional
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refinement of the enclosing claim, using the rule derived from the
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proof body.
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\medskip The next example involves \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}}, which can be
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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
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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
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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
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inference rule, respectively:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\medskip\begin{minipage}{0.6\textwidth}
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%
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\isadelimproof
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\ \ \ \ %
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\endisadelimproof
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%
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\isatagproof
|
36357
|
258 |
\isacommand{assume}\isamarkupfalse%
|
40406
|
259 |
\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29722
|
260 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
|
261 |
\ \isacommand{have}\isamarkupfalse%
|
|
262 |
\ C\isanewline
|
|
263 |
\ \ \ \ \isacommand{proof}\isamarkupfalse%
|
|
264 |
\isanewline
|
|
265 |
\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
266 |
\ A\isanewline
|
|
267 |
\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
|
40406
|
268 |
\ {\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
|
29722
|
269 |
\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
|
29740
|
270 |
\ C%
|
|
271 |
\endisatagproof
|
|
272 |
{\isafoldproof}%
|
|
273 |
%
|
|
274 |
\isadelimproof
|
|
275 |
%
|
|
276 |
\endisadelimproof
|
|
277 |
%
|
|
278 |
\isadelimnoproof
|
|
279 |
\ %
|
|
280 |
\endisadelimnoproof
|
|
281 |
%
|
|
282 |
\isatagnoproof
|
|
283 |
\isacommand{sorry}\isamarkupfalse%
|
|
284 |
%
|
|
285 |
\endisatagnoproof
|
|
286 |
{\isafoldnoproof}%
|
|
287 |
%
|
|
288 |
\isadelimnoproof
|
|
289 |
%
|
|
290 |
\endisadelimnoproof
|
36357
|
291 |
\isanewline
|
29740
|
292 |
%
|
|
293 |
\isadelimproof
|
|
294 |
\ \ \ \ %
|
|
295 |
\endisadelimproof
|
|
296 |
%
|
|
297 |
\isatagproof
|
|
298 |
\isacommand{qed}\isamarkupfalse%
|
29722
|
299 |
%
|
|
300 |
\endisatagproof
|
|
301 |
{\isafoldproof}%
|
|
302 |
%
|
|
303 |
\isadelimproof
|
|
304 |
%
|
|
305 |
\endisadelimproof
|
|
306 |
%
|
|
307 |
\end{minipage}\begin{minipage}{0.4\textwidth}
|
|
308 |
%
|
|
309 |
\begin{isamarkuptext}%
|
40406
|
310 |
\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}}}}}%
|
29722
|
311 |
\end{isamarkuptext}%
|
|
312 |
\isamarkuptrue%
|
|
313 |
%
|
|
314 |
\end{minipage}
|
|
315 |
%
|
|
316 |
\begin{isamarkuptext}%
|
|
317 |
\medskip\noindent Although the Isar proof follows the natural
|
|
318 |
deduction rule closely, the text reads not as natural as
|
|
319 |
anticipated. There is a double occurrence of an arbitrary
|
40406
|
320 |
conclusion \isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}}, which represents the final result, but is
|
29722
|
321 |
irrelevant for now. This issue arises for any elimination rule
|
|
322 |
involving local parameters. Isar provides the derived language
|
|
323 |
element \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}, which is able to perform the same
|
|
324 |
elimination proof more conveniently:%
|
|
325 |
\end{isamarkuptext}%
|
|
326 |
\isamarkuptrue%
|
|
327 |
%
|
|
328 |
\isadelimproof
|
36357
|
329 |
\ \ \ \ %
|
29722
|
330 |
\endisadelimproof
|
|
331 |
%
|
|
332 |
\isatagproof
|
36357
|
333 |
\isacommand{assume}\isamarkupfalse%
|
40406
|
334 |
\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C556E696F6E3E}{\isasymUnion}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29722
|
335 |
\ \ \ \ \isacommand{then}\isamarkupfalse%
|
|
336 |
\ \isacommand{obtain}\isamarkupfalse%
|
40406
|
337 |
\ 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%
|
29722
|
338 |
%
|
|
339 |
\endisatagproof
|
|
340 |
{\isafoldproof}%
|
|
341 |
%
|
|
342 |
\isadelimproof
|
|
343 |
%
|
|
344 |
\endisadelimproof
|
|
345 |
%
|
|
346 |
\begin{isamarkuptext}%
|
40406
|
347 |
\noindent Here we avoid to mention the final conclusion \isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}}
|
29722
|
348 |
and return to plain forward reasoning. The rule involved in the
|
40406
|
349 |
``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' proof is the same as before.%
|
29717
|
350 |
\end{isamarkuptext}%
|
|
351 |
\isamarkuptrue%
|
|
352 |
%
|
29731
|
353 |
\isamarkupsection{The Pure framework \label{sec:framework-pure}%
|
|
354 |
}
|
|
355 |
\isamarkuptrue%
|
|
356 |
%
|
|
357 |
\begin{isamarkuptext}%
|
|
358 |
The Pure logic \cite{paulson-found,paulson700} is an intuitionistic
|
|
359 |
fragment of higher-order logic \cite{church40}. In type-theoretic
|
40406
|
360 |
parlance, there are three levels of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-calculus with
|
|
361 |
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}}}:
|
29740
|
362 |
|
|
363 |
\medskip
|
|
364 |
\begin{tabular}{ll}
|
40406
|
365 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{22}{\isachardoublequote}}} & syntactic function space (terms depending on terms) \\
|
|
366 |
\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) \\
|
|
367 |
\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}} & implication (proofs depending on proofs) \\
|
29740
|
368 |
\end{tabular}
|
|
369 |
\medskip
|
29731
|
370 |
|
29740
|
371 |
\noindent Here only the types of syntactic terms, and the
|
40406
|
372 |
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
|
29740
|
373 |
the Pure inference kernel \cite{Berghofer-Nipkow:2000:TPHOL}, but
|
|
374 |
the formal system can never depend on them due to \emph{proof
|
|
375 |
irrelevance}.
|
|
376 |
|
|
377 |
On top of this most primitive layer of proofs, Pure implements a
|
|
378 |
generic calculus for nested natural deduction rules, similar to
|
|
379 |
\cite{Schroeder-Heister:1984}. Here object-logic inferences are
|
40406
|
380 |
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}}}.
|
29740
|
381 |
Combining such rule statements may involve higher-order unification
|
|
382 |
\cite{paulson-natural}.%
|
29731
|
383 |
\end{isamarkuptext}%
|
|
384 |
\isamarkuptrue%
|
|
385 |
%
|
|
386 |
\isamarkupsubsection{Primitive inferences%
|
|
387 |
}
|
|
388 |
\isamarkuptrue%
|
|
389 |
%
|
|
390 |
\begin{isamarkuptext}%
|
40406
|
391 |
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
|
|
392 |
implicit thanks to type-inference; terms of type \isa{{\isaliteral{22}{\isachardoublequote}}prop{\isaliteral{22}{\isachardoublequote}}} are
|
|
393 |
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
|
|
394 |
judgments of the form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}}, with standard introduction
|
|
395 |
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
|
|
396 |
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
|
|
397 |
\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}}};
|
29731
|
398 |
the corresponding proof terms are left implicit. The subsequent
|
40406
|
399 |
inference rules define \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} inductively, relative to a
|
29731
|
400 |
collection of axioms:
|
|
401 |
|
|
402 |
\[
|
40406
|
403 |
\infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A{\isaliteral{22}{\isachardoublequote}}}}{(\isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}} \text{~axiom})}
|
29731
|
404 |
\qquad
|
40406
|
405 |
\infer{\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A{\isaliteral{22}{\isachardoublequote}}}}{}
|
29731
|
406 |
\]
|
|
407 |
|
|
408 |
\[
|
40406
|
409 |
\infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}{\isaliteral{22}{\isachardoublequote}}}}
|
29731
|
410 |
\qquad
|
40406
|
411 |
\infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}
|
29731
|
412 |
\]
|
|
413 |
|
|
414 |
\[
|
40406
|
415 |
\infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}
|
29731
|
416 |
\qquad
|
40406
|
417 |
\infer{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ {\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A{\isaliteral{22}{\isachardoublequote}}}}
|
29731
|
418 |
\]
|
|
419 |
|
40406
|
420 |
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,
|
|
421 |
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.
|
29731
|
422 |
|
|
423 |
\medskip An object-logic introduces another layer on top of Pure,
|
40406
|
424 |
e.g.\ with types \isa{{\isaliteral{22}{\isachardoublequote}}i{\isaliteral{22}{\isachardoublequote}}} for individuals and \isa{{\isaliteral{22}{\isachardoublequote}}o{\isaliteral{22}{\isachardoublequote}}} for
|
|
425 |
propositions, term constants \isa{{\isaliteral{22}{\isachardoublequote}}Trueprop\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ o\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ prop{\isaliteral{22}{\isachardoublequote}}} as
|
|
426 |
(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
|
|
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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
|
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|
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Pure. After the initial object-logic setup, further axiomatizations
|
|
429 |
are usually avoided; plain definitions and derived principles are
|
|
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used exclusively.%
|
|
431 |
\end{isamarkuptext}%
|
|
432 |
\isamarkuptrue%
|
|
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%
|
|
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\isamarkupsubsection{Reasoning with rules \label{sec:framework-resolution}%
|
|
435 |
}
|
|
436 |
\isamarkuptrue%
|
|
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%
|
|
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\begin{isamarkuptext}%
|
|
439 |
Primitive inferences mostly serve foundational purposes. The main
|
|
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reasoning mechanisms of Pure operate on nested natural deduction
|
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|
441 |
rules expressed as formulae, using \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{22}{\isachardoublequote}}} to bind local
|
|
442 |
parameters and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} to express entailment. Multiple
|
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|
443 |
parameters and premises are represented by repeating these
|
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|
444 |
connectives in a right-associative manner.
|
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|
445 |
|
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|
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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
|
|
447 |
\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.\
|
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|
448 |
that rule statements always observe the normal form where
|
|
449 |
quantifiers are pulled in front of implications at each level of
|
|
450 |
nesting. This means that any Pure proposition may be presented as a
|
|
451 |
\emph{Hereditary Harrop Formula} \cite{Miller:1991} which is of the
|
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|
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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.
|
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|
453 |
Following the convention that outermost quantifiers are implicit,
|
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|
454 |
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
|
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|
455 |
case of this.
|
|
456 |
|
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|
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For example, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C696E7465723E}{\isasyminter}}{\isaliteral{22}{\isachardoublequote}}}-introduction rule encountered before is
|
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|
458 |
represented as a Pure theorem as follows:
|
|
459 |
\[
|
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|
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\isa{{\isaliteral{22}{\isachardoublequote}}IntI{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B{\isaliteral{22}{\isachardoublequote}}}
|
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|
461 |
\]
|
|
462 |
|
|
463 |
\noindent This is a plain Horn clause, since no further nesting on
|
40406
|
464 |
the left is involved. The general \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{22}{\isachardoublequote}}}-introduction
|
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|
465 |
corresponds to a Hereditary Harrop Formula with one additional level
|
|
466 |
of nesting:
|
|
467 |
\[
|
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|
468 |
\isa{{\isaliteral{22}{\isachardoublequote}}InterI{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}A{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C413E}{\isasymA}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{5C3C496E7465723E}{\isasymInter}}{\isaliteral{5C3C413E}{\isasymA}}{\isaliteral{22}{\isachardoublequote}}}
|
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|
469 |
\]
|
|
470 |
|
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|
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\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
|
|
472 |
goal is finished. To allow \isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{22}{\isachardoublequote}}} being a rule statement
|
|
473 |
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
|
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|
474 |
initialize and finish goal states as follows:
|
|
475 |
|
|
476 |
\[
|
|
477 |
\begin{array}{c@ {\qquad}c}
|
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|
478 |
\infer[(\indexdef{}{inference}{init}\hypertarget{inference.init}{\hyperlink{inference.init}{\mbox{\isa{init}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}C\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{23}{\isacharhash}}C{\isaliteral{22}{\isachardoublequote}}}}{} &
|
|
479 |
\infer[(\indexdef{}{inference}{finish}\hypertarget{inference.finish}{\hyperlink{inference.finish}{\mbox{\isa{finish}}}})]{\isa{C}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}C{\isaliteral{22}{\isachardoublequote}}}}
|
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|
480 |
\end{array}
|
|
481 |
\]
|
|
482 |
|
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|
483 |
\noindent Goal states are refined in intermediate proof steps until
|
|
484 |
a finished form is achieved. Here the two main reasoning principles
|
|
485 |
are \hyperlink{inference.resolution}{\mbox{\isa{resolution}}}, for back-chaining a rule against a
|
|
486 |
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
|
40406
|
487 |
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}}}).
|
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|
488 |
|
|
489 |
\[
|
|
490 |
\infer[(\indexdef{}{inference}{resolution}\hypertarget{inference.resolution}{\hyperlink{inference.resolution}{\mbox{\isa{resolution}}}})]
|
40406
|
491 |
{\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}}}}
|
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|
492 |
{\begin{tabular}{rl}
|
40406
|
493 |
\isa{{\isaliteral{22}{\isachardoublequote}}rule{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
494 |
\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a{\isaliteral{22}{\isachardoublequote}}} \\
|
|
495 |
\isa{{\isaliteral{22}{\isachardoublequote}}goal{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
496 |
\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}}} \\
|
|
497 |
\isa{{\isaliteral{22}{\isachardoublequote}}goal\ unifier{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
498 |
\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}}} \\
|
29731
|
499 |
\end{tabular}}
|
|
500 |
\]
|
|
501 |
|
|
502 |
\medskip
|
|
503 |
|
|
504 |
\[
|
40406
|
505 |
\infer[(\indexdef{}{inference}{assumption}\hypertarget{inference.assumption}{\hyperlink{inference.assumption}{\mbox{\isa{assumption}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}C{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}}}
|
29731
|
506 |
{\begin{tabular}{rl}
|
40406
|
507 |
\isa{{\isaliteral{22}{\isachardoublequote}}goal{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
508 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec H\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C{\isaliteral{22}{\isachardoublequote}}} \\
|
|
509 |
\isa{{\isaliteral{22}{\isachardoublequote}}assm\ unifier{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}\ {\isaliteral{3D}{\isacharequal}}\ H\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{5C3C76617274686574613E}{\isasymvartheta}}{\isaliteral{22}{\isachardoublequote}}}~~\text{(for some~\isa{{\isaliteral{22}{\isachardoublequote}}H\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}})} \\
|
29731
|
510 |
\end{tabular}}
|
|
511 |
\]
|
|
512 |
|
|
513 |
The following trace illustrates goal-oriented reasoning in
|
|
514 |
Isabelle/Pure:
|
|
515 |
|
29740
|
516 |
{\footnotesize
|
29731
|
517 |
\medskip
|
29740
|
518 |
\begin{tabular}{r@ {\quad}l}
|
40406
|
519 |
\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}}} \\
|
|
520 |
\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}}} \\
|
|
521 |
\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}}} \\
|
|
522 |
\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}}} \\
|
48279
|
523 |
\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}}} \\
|
40406
|
524 |
\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}}} \\
|
|
525 |
\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}}} \\
|
29731
|
526 |
\end{tabular}
|
|
527 |
\medskip
|
29740
|
528 |
}
|
29731
|
529 |
|
|
530 |
Compositions of \hyperlink{inference.assumption}{\mbox{\isa{assumption}}} after \hyperlink{inference.resolution}{\mbox{\isa{resolution}}} occurs quite often, typically in elimination steps.
|
|
531 |
Traditional Isabelle tactics accommodate this by a combined
|
40406
|
532 |
\indexdef{}{inference}{elim\_resolution}\hypertarget{inference.elim-resolution}{\hyperlink{inference.elim-resolution}{\mbox{\isa{elim{\isaliteral{5F}{\isacharunderscore}}resolution}}}} principle. In contrast, Isar uses
|
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|
533 |
a slightly more refined combination, where the assumptions to be
|
|
534 |
closed are marked explicitly, using again the protective marker
|
40406
|
535 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}{\isaliteral{22}{\isachardoublequote}}}:
|
29731
|
536 |
|
|
537 |
\[
|
|
538 |
\infer[(\hyperlink{inference.refinement}{\mbox{\isa{refinement}}})]
|
40406
|
539 |
{\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}}}}
|
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|
540 |
{\begin{tabular}{rl}
|
42666
|
541 |
\isa{{\isaliteral{22}{\isachardoublequote}}sub{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}proof{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
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|
542 |
\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec G\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ \isaliteral{5C3C5E7665633E}{}\isactrlvec a{\isaliteral{22}{\isachardoublequote}}} \\
|
|
543 |
\isa{{\isaliteral{22}{\isachardoublequote}}goal{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
544 |
\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}}} \\
|
|
545 |
\isa{{\isaliteral{22}{\isachardoublequote}}goal\ unifier{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
546 |
\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}}} \\
|
|
547 |
\isa{{\isaliteral{22}{\isachardoublequote}}assm\ unifiers{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
548 |
\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}}} \\
|
|
549 |
& \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}}}) \\
|
29731
|
550 |
\end{tabular}}
|
|
551 |
\]
|
|
552 |
|
42666
|
553 |
\noindent Here the \isa{{\isaliteral{22}{\isachardoublequote}}sub{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}proof{\isaliteral{22}{\isachardoublequote}}} rule stems from the
|
29740
|
554 |
main \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}-\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}-\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} outline of
|
|
555 |
Isar (cf.\ \secref{sec:framework-subproof}): each assumption
|
40406
|
556 |
indicated in the text results in a marked premise \isa{{\isaliteral{22}{\isachardoublequote}}G{\isaliteral{22}{\isachardoublequote}}} above.
|
29740
|
557 |
The marking enforces resolution against one of the sub-goal's
|
|
558 |
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
|
|
559 |
pending sub-goal, while maintaining a good measure of flexibility.%
|
29731
|
560 |
\end{isamarkuptext}%
|
|
561 |
\isamarkuptrue%
|
|
562 |
%
|
|
563 |
\isamarkupsection{The Isar proof language \label{sec:framework-isar}%
|
|
564 |
}
|
|
565 |
\isamarkuptrue%
|
|
566 |
%
|
|
567 |
\begin{isamarkuptext}%
|
|
568 |
Structured proofs are presented as high-level expressions for
|
|
569 |
composing entities of Pure (propositions, facts, and goals). The
|
|
570 |
Isar proof language allows to organize reasoning within the
|
|
571 |
underlying rule calculus of Pure, but Isar is not another logical
|
|
572 |
calculus!
|
|
573 |
|
|
574 |
Isar is an exercise in sound minimalism. Approximately half of the
|
|
575 |
language is introduced as primitive, the rest defined as derived
|
|
576 |
concepts. The following grammar describes the core language
|
40406
|
577 |
(category \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{22}{\isachardoublequote}}}), which is embedded into theory
|
29731
|
578 |
specification elements such as \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}; see also
|
40406
|
579 |
\secref{sec:framework-stmt} for the separate category \isa{{\isaliteral{22}{\isachardoublequote}}statement{\isaliteral{22}{\isachardoublequote}}}.
|
29731
|
580 |
|
|
581 |
\medskip
|
|
582 |
\begin{tabular}{rcl}
|
42666
|
583 |
\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]
|
29731
|
584 |
|
40406
|
585 |
\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]
|
29731
|
586 |
|
40406
|
587 |
\isa{prfx} & = & \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}facts{\isaliteral{22}{\isachardoublequote}}} \\
|
|
588 |
& \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}}} \\
|
29731
|
589 |
|
40406
|
590 |
\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}}}}}} \\
|
|
591 |
& \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7C}{\isacharbar}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}} \\
|
|
592 |
& \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}}} \\
|
|
593 |
& \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}}} \\
|
|
594 |
& \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}}} \\
|
|
595 |
& \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}}} \\
|
|
596 |
& \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} \\
|
|
597 |
\isa{goal} & = & \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}name{\isaliteral{3A}{\isacharcolon}}\ props\ proof{\isaliteral{22}{\isachardoublequote}}} \\
|
|
598 |
& \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}}} \\
|
29731
|
599 |
\end{tabular}
|
|
600 |
|
|
601 |
\medskip Simultaneous propositions or facts may be separated by the
|
|
602 |
\hyperlink{keyword.and}{\mbox{\isa{\isakeyword{and}}}} keyword.
|
|
603 |
|
|
604 |
\medskip The syntax for terms and propositions is inherited from
|
40406
|
605 |
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
|
29731
|
606 |
matching.
|
|
607 |
|
29740
|
608 |
\medskip Facts may be referenced by name or proposition. For
|
40406
|
609 |
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}}}''
|
|
610 |
becomes available both as \isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{22}{\isachardoublequote}}} and
|
|
611 |
\isacharbackquoteopen\isa{{\isaliteral{22}{\isachardoublequote}}A{\isaliteral{22}{\isachardoublequote}}}\isacharbackquoteclose. Moreover,
|
29740
|
612 |
fact expressions may involve attributes that modify either the
|
|
613 |
theorem or the background context. For example, the expression
|
40406
|
614 |
``\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5B}{\isacharbrackleft}}OF\ b{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' refers to the composition of two facts
|
29740
|
615 |
according to the \hyperlink{inference.resolution}{\mbox{\isa{resolution}}} inference of
|
40406
|
616 |
\secref{sec:framework-resolution}, while ``\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5B}{\isacharbrackleft}}intro{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}''
|
29740
|
617 |
declares a fact as introduction rule in the context.
|
29731
|
618 |
|
29740
|
619 |
The special fact called ``\hyperlink{fact.this}{\mbox{\isa{this}}}'' always refers to the last
|
|
620 |
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
|
29731
|
621 |
frequently together with \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} we provide some
|
29740
|
622 |
abbreviations:
|
29731
|
623 |
|
29740
|
624 |
\medskip
|
|
625 |
\begin{tabular}{rcl}
|
40406
|
626 |
\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}}}} \\
|
|
627 |
\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}}} \\
|
29740
|
628 |
\end{tabular}
|
|
629 |
\medskip
|
|
630 |
|
40406
|
631 |
The \isa{{\isaliteral{22}{\isachardoublequote}}method{\isaliteral{22}{\isachardoublequote}}} category is essentially a parameter and may be
|
29740
|
632 |
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.
|
40406
|
633 |
Some basic methods are predefined: ``\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}}'' leaves the goal
|
29740
|
634 |
unchanged, ``\hyperlink{method.this}{\mbox{\isa{this}}}'' applies the facts as rules to the
|
42626
|
635 |
goal, ``\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}'' applies the facts to another rule and the
|
|
636 |
result to the goal (both ``\hyperlink{method.this}{\mbox{\isa{this}}}'' and ``\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}''
|
29740
|
637 |
refer to \hyperlink{inference.resolution}{\mbox{\isa{resolution}}} of
|
29731
|
638 |
\secref{sec:framework-resolution}). The secondary arguments to
|
42626
|
639 |
``\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
|
29731
|
640 |
first tries rules declared as \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}} or
|
|
641 |
\hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}, followed by those declared as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}.
|
|
642 |
|
42626
|
643 |
The default method for \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} is ``\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}''
|
29731
|
644 |
(arguments picked from the context), for \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} it is
|
40406
|
645 |
``\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}}''. Further abbreviations for terminal proof steps
|
|
646 |
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
|
42626
|
647 |
``\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
|
29740
|
648 |
directly on the current facts and goal by applying equalities.
|
29731
|
649 |
|
40406
|
650 |
\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
|
29740
|
651 |
already involves implicit nesting. In any case, \hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}}
|
|
652 |
jumps into the next section of a block, i.e.\ it acts like closing
|
|
653 |
an implicit block scope and opening another one; there is no direct
|
|
654 |
correspondence to subgoals here.
|
29731
|
655 |
|
29740
|
656 |
The remaining elements \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}} and \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} build up
|
|
657 |
a local context (see \secref{sec:framework-context}), while
|
29731
|
658 |
\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} refines a pending sub-goal by the rule resulting
|
|
659 |
from a nested sub-proof (see \secref{sec:framework-subproof}).
|
|
660 |
Further derived concepts will support calculational reasoning (see
|
|
661 |
\secref{sec:framework-calc}).%
|
|
662 |
\end{isamarkuptext}%
|
|
663 |
\isamarkuptrue%
|
|
664 |
%
|
|
665 |
\isamarkupsubsection{Context elements \label{sec:framework-context}%
|
|
666 |
}
|
|
667 |
\isamarkuptrue%
|
|
668 |
%
|
|
669 |
\begin{isamarkuptext}%
|
40406
|
670 |
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}}}
|
29731
|
671 |
essentially acts like a proof context. Isar elaborates this idea
|
29740
|
672 |
towards a higher-level notion, with additional information for
|
29731
|
673 |
type-inference, term abbreviations, local facts, hypotheses etc.
|
|
674 |
|
40406
|
675 |
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
|
29731
|
676 |
parameter, i.e.\ an arbitrary-but-fixed entity of a given type; in
|
40406
|
677 |
results exported from the context, \isa{{\isaliteral{22}{\isachardoublequote}}x{\isaliteral{22}{\isachardoublequote}}} may become anything.
|
|
678 |
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
|
|
679 |
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
|
29740
|
680 |
included inference tells how to discharge \isa{A} from results
|
40406
|
681 |
\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
|
29740
|
682 |
commands are defined in ML.
|
|
683 |
|
|
684 |
At the user-level, the default inference for \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} is
|
|
685 |
\hyperlink{inference.discharge}{\mbox{\isa{discharge}}} as given below. The additional variants
|
|
686 |
\hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}} and \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}} are defined as follows:
|
|
687 |
|
|
688 |
\medskip
|
|
689 |
\begin{tabular}{rcl}
|
42666
|
690 |
\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}}} \\
|
40406
|
691 |
\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}}} \\
|
29740
|
692 |
\end{tabular}
|
|
693 |
\medskip
|
29731
|
694 |
|
|
695 |
\[
|
40406
|
696 |
\infer[(\indexdef{}{inference}{discharge}\hypertarget{inference.discharge}{\hyperlink{inference.discharge}{\mbox{\isa{discharge}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{23}{\isacharhash}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}
|
29740
|
697 |
\]
|
|
698 |
\[
|
42666
|
699 |
\infer[(\indexdef{}{inference}{weak-discharge}\hypertarget{inference.weak-discharge}{\hyperlink{inference.weak-discharge}{\mbox{\isa{weak{\isaliteral{5C3C68797068656E3E}{\isasymhyphen}}discharge}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ A\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}
|
29740
|
700 |
\]
|
|
701 |
\[
|
40406
|
702 |
\infer[(\indexdef{}{inference}{expansion}\hypertarget{inference.expansion}{\hyperlink{inference.expansion}{\mbox{\isa{expansion}}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B\ a{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73747275743E}{\isasymstrut}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B\ x{\isaliteral{22}{\isachardoublequote}}}}
|
29731
|
703 |
\]
|
|
704 |
|
42666
|
705 |
\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
|
29740
|
706 |
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,
|
|
707 |
cf.\ \secref{sec:framework-subproof}.
|
29731
|
708 |
|
29740
|
709 |
The most interesting derived context element in Isar is \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} \cite[\S5.3]{Wenzel-PhD}, which supports generalized
|
|
710 |
elimination steps in a purely forward manner. The \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}
|
40406
|
711 |
command takes a specification of parameters \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{22}{\isachardoublequote}}} and
|
|
712 |
assumptions \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec A{\isaliteral{22}{\isachardoublequote}}} to be added to the context, together
|
29740
|
713 |
with a proof of a case rule stating that this extension is
|
|
714 |
conservative (i.e.\ may be removed from closed results later on):
|
29731
|
715 |
|
|
716 |
\medskip
|
|
717 |
\begin{tabular}{l}
|
40406
|
718 |
\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]
|
|
719 |
\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}}} \\
|
|
720 |
\quad \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}} \\
|
29731
|
721 |
\qquad \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{thesis} \\
|
40406
|
722 |
\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}}} \\
|
|
723 |
\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}}} \\
|
29731
|
724 |
\quad \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} \\
|
40406
|
725 |
\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}}} \\
|
29731
|
726 |
\end{tabular}
|
|
727 |
\medskip
|
|
728 |
|
|
729 |
\[
|
40406
|
730 |
\infer[(\hyperlink{inference.elimination}{\mbox{\isa{elimination}}})]{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}}}{
|
29731
|
731 |
\begin{tabular}{rl}
|
40406
|
732 |
\isa{{\isaliteral{22}{\isachardoublequote}}case{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
733 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}thesis{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}\isaliteral{5C3C5E7665633E}{}\isactrlvec x{\isaliteral{2E}{\isachardot}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\[0.2ex]
|
|
734 |
\isa{{\isaliteral{22}{\isachardoublequote}}result{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
|
|
735 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec A\ \isaliteral{5C3C5E7665633E}{}\isactrlvec y\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ B{\isaliteral{22}{\isachardoublequote}}} \\[0.2ex]
|
29731
|
736 |
\end{tabular}}
|
|
737 |
\]
|
|
738 |
|
|
739 |
\noindent Here the name ``\isa{thesis}'' is a specific convention
|
|
740 |
for an arbitrary-but-fixed proposition; in the primitive natural
|
|
741 |
deduction rules shown before we have occasionally used \isa{C}.
|
40406
|
742 |
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}}}
|
|
743 |
may be assumed for some arbitrary-but-fixed \isa{{\isaliteral{22}{\isachardoublequote}}x{\isaliteral{22}{\isachardoublequote}}}. Also note
|
|
744 |
that ``\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ B{\isaliteral{22}{\isachardoublequote}}}'' without parameters
|
|
745 |
is similar to ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ B{\isaliteral{22}{\isachardoublequote}}}'', but the
|
29740
|
746 |
latter involves multiple sub-goals.
|
29731
|
747 |
|
|
748 |
\medskip The subsequent Isar proof texts explain all context
|
|
749 |
elements introduced above using the formal proof language itself.
|
|
750 |
After finishing a local proof within a block, we indicate the
|
29740
|
751 |
exported result via \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}.%
|
29731
|
752 |
\end{isamarkuptext}%
|
|
753 |
\isamarkuptrue%
|
|
754 |
%
|
|
755 |
\isadelimproof
|
|
756 |
%
|
|
757 |
\endisadelimproof
|
|
758 |
%
|
|
759 |
\isatagproof
|
|
760 |
%
|
40476
|
761 |
\begin{minipage}[t]{0.45\textwidth}
|
40406
|
762 |
\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
|
29731
|
763 |
\isanewline
|
|
764 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
765 |
\ x\isanewline
|
|
766 |
\ \ \ \ \isacommand{have}\isamarkupfalse%
|
40406
|
767 |
\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}%
|
29740
|
768 |
\endisatagproof
|
|
769 |
{\isafoldproof}%
|
|
770 |
%
|
|
771 |
\isadelimproof
|
|
772 |
%
|
|
773 |
\endisadelimproof
|
|
774 |
%
|
|
775 |
\isadelimnoproof
|
|
776 |
\ %
|
|
777 |
\endisadelimnoproof
|
|
778 |
%
|
|
779 |
\isatagnoproof
|
|
780 |
\isacommand{sorry}\isamarkupfalse%
|
|
781 |
%
|
|
782 |
\endisatagnoproof
|
|
783 |
{\isafoldnoproof}%
|
|
784 |
%
|
|
785 |
\isadelimnoproof
|
29731
|
786 |
\isanewline
|
29740
|
787 |
%
|
|
788 |
\endisadelimnoproof
|
|
789 |
%
|
|
790 |
\isadelimproof
|
|
791 |
\ \ %
|
|
792 |
\endisadelimproof
|
|
793 |
%
|
|
794 |
\isatagproof
|
40406
|
795 |
\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
|
29731
|
796 |
\isanewline
|
|
797 |
\ \ \isacommand{note}\isamarkupfalse%
|
40406
|
798 |
\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ B\ x{\isaliteral{60}{\isacharbackquoteclose}}%
|
40476
|
799 |
\end{minipage}\quad\begin{minipage}[t]{0.45\textwidth}
|
40406
|
800 |
\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
|
29740
|
801 |
\isanewline
|
|
802 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
|
803 |
\ A\isanewline
|
|
804 |
\ \ \ \ \isacommand{have}\isamarkupfalse%
|
|
805 |
\ B%
|
|
806 |
\endisatagproof
|
|
807 |
{\isafoldproof}%
|
|
808 |
%
|
|
809 |
\isadelimproof
|
|
810 |
%
|
|
811 |
\endisadelimproof
|
|
812 |
%
|
|
813 |
\isadelimnoproof
|
|
814 |
\ %
|
|
815 |
\endisadelimnoproof
|
|
816 |
%
|
|
817 |
\isatagnoproof
|
|
818 |
\isacommand{sorry}\isamarkupfalse%
|
|
819 |
%
|
|
820 |
\endisatagnoproof
|
|
821 |
{\isafoldnoproof}%
|
|
822 |
%
|
|
823 |
\isadelimnoproof
|
|
824 |
\isanewline
|
|
825 |
%
|
|
826 |
\endisadelimnoproof
|
|
827 |
%
|
|
828 |
\isadelimproof
|
|
829 |
\ \ %
|
|
830 |
\endisadelimproof
|
|
831 |
%
|
|
832 |
\isatagproof
|
40406
|
833 |
\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
|
29740
|
834 |
\isanewline
|
|
835 |
\ \ \isacommand{note}\isamarkupfalse%
|
40406
|
836 |
\ {\isaliteral{60}{\isacharbackquoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{60}{\isacharbackquoteclose}}%
|
40476
|
837 |
\end{minipage}\\[3ex]\begin{minipage}[t]{0.45\textwidth}
|
40406
|
838 |
\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
|
29731
|
839 |
\isanewline
|
|
840 |
\ \ \ \ \isacommand{def}\isamarkupfalse%
|
40406
|
841 |
\ x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ a\isanewline
|
29731
|
842 |
\ \ \ \ \isacommand{have}\isamarkupfalse%
|
40406
|
843 |
\ {\isaliteral{22}{\isachardoublequoteopen}}B\ x{\isaliteral{22}{\isachardoublequoteclose}}%
|
29740
|
844 |
\endisatagproof
|
|
845 |
{\isafoldproof}%
|
|
846 |
%
|
|
847 |
\isadelimproof
|
|
848 |
%
|
|
849 |
\endisadelimproof
|
|
850 |
%
|
|
851 |
\isadelimnoproof
|
|
852 |
\ %
|
|
853 |
\endisadelimnoproof
|
|
854 |
%
|
|
855 |
\isatagnoproof
|
|
856 |
\isacommand{sorry}\isamarkupfalse%
|
|
857 |
%
|
|
858 |
\endisatagnoproof
|
|
859 |
{\isafoldnoproof}%
|
|
860 |
%
|
|
861 |
\isadelimnoproof
|
29731
|
862 |
\isanewline
|
29740
|
863 |
%
|
|
864 |
\endisadelimnoproof
|
|
865 |
%
|
|
866 |
\isadelimproof
|
|
867 |
\ \ %
|
|
868 |
\endisadelimproof
|
|
869 |
%
|
|
870 |
\isatagproof
|
40406
|
871 |
\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
|
29731
|
872 |
\isanewline
|
|
873 |
\ \ \isacommand{note}\isamarkupfalse%
|
40406
|
874 |
\ {\isaliteral{60}{\isacharbackquoteopen}}B\ a{\isaliteral{60}{\isacharbackquoteclose}}%
|
40476
|
875 |
\end{minipage}\quad\begin{minipage}[t]{0.45\textwidth}
|
40406
|
876 |
\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
|
29731
|
877 |
\isanewline
|
|
878 |
\ \ \ \ \isacommand{obtain}\isamarkupfalse%
|
40406
|
879 |
\ x\ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}%
|
29740
|
880 |
\endisatagproof
|
|
881 |
{\isafoldproof}%
|
|
882 |
%
|
|
883 |
\isadelimproof
|
|
884 |
%
|
|
885 |
\endisadelimproof
|
|
886 |
%
|
|
887 |
\isadelimnoproof
|
|
888 |
\ %
|
|
889 |
\endisadelimnoproof
|
|
890 |
%
|
|
891 |
\isatagnoproof
|
|
892 |
\isacommand{sorry}\isamarkupfalse%
|
|
893 |
%
|
|
894 |
\endisatagnoproof
|
|
895 |
{\isafoldnoproof}%
|
|
896 |
%
|
|
897 |
\isadelimnoproof
|
29731
|
898 |
\isanewline
|
29740
|
899 |
%
|
|
900 |
\endisadelimnoproof
|
|
901 |
%
|
|
902 |
\isadelimproof
|
|
903 |
\ \ \ \ %
|
|
904 |
\endisadelimproof
|
|
905 |
%
|
|
906 |
\isatagproof
|
|
907 |
\isacommand{have}\isamarkupfalse%
|
|
908 |
\ B%
|
|
909 |
\endisatagproof
|
|
910 |
{\isafoldproof}%
|
|
911 |
%
|
|
912 |
\isadelimproof
|
|
913 |
%
|
|
914 |
\endisadelimproof
|
|
915 |
%
|
|
916 |
\isadelimnoproof
|
|
917 |
\ %
|
|
918 |
\endisadelimnoproof
|
|
919 |
%
|
|
920 |
\isatagnoproof
|
|
921 |
\isacommand{sorry}\isamarkupfalse%
|
|
922 |
%
|
|
923 |
\endisatagnoproof
|
|
924 |
{\isafoldnoproof}%
|
|
925 |
%
|
|
926 |
\isadelimnoproof
|
29731
|
927 |
\isanewline
|
29740
|
928 |
%
|
|
929 |
\endisadelimnoproof
|
|
930 |
%
|
|
931 |
\isadelimproof
|
|
932 |
\ \ %
|
|
933 |
\endisadelimproof
|
|
934 |
%
|
|
935 |
\isatagproof
|
40406
|
936 |
\isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
|
29731
|
937 |
\isanewline
|
|
938 |
\ \ \isacommand{note}\isamarkupfalse%
|
40406
|
939 |
\ {\isaliteral{60}{\isacharbackquoteopen}}B{\isaliteral{60}{\isacharbackquoteclose}}%
|
29731
|
940 |
\end{minipage}
|
|
941 |
%
|
|
942 |
\endisatagproof
|
|
943 |
{\isafoldproof}%
|
|
944 |
%
|
|
945 |
\isadelimproof
|
|
946 |
%
|
|
947 |
\endisadelimproof
|
|
948 |
%
|
29740
|
949 |
\begin{isamarkuptext}%
|
|
950 |
\bigskip\noindent This illustrates the meaning of Isar context
|
|
951 |
elements without goals getting in between.%
|
|
952 |
\end{isamarkuptext}%
|
|
953 |
\isamarkuptrue%
|
|
954 |
%
|
29731
|
955 |
\isamarkupsubsection{Structured statements \label{sec:framework-stmt}%
|
|
956 |
}
|
|
957 |
\isamarkuptrue%
|
|
958 |
%
|
|
959 |
\begin{isamarkuptext}%
|
40406
|
960 |
The category \isa{{\isaliteral{22}{\isachardoublequote}}statement{\isaliteral{22}{\isachardoublequote}}} of top-level theorem specifications
|
29731
|
961 |
is defined as follows:
|
|
962 |
|
|
963 |
\medskip
|
|
964 |
\begin{tabular}{rcl}
|
40406
|
965 |
\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}}} \\
|
|
966 |
& \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]
|
29731
|
967 |
|
40406
|
968 |
\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}}} \\
|
|
969 |
& \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}}} \\
|
29731
|
970 |
|
40406
|
971 |
\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}}} \\
|
|
972 |
& \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}}} \\
|
|
973 |
& & \quad \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C424241523E}{\isasymBBAR}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} \\
|
29731
|
974 |
\end{tabular}
|
|
975 |
|
40406
|
976 |
\medskip\noindent A simple \isa{{\isaliteral{22}{\isachardoublequote}}statement{\isaliteral{22}{\isachardoublequote}}} consists of named
|
29731
|
977 |
propositions. The full form admits local context elements followed
|
40406
|
978 |
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
|
|
979 |
the context: \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ A\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ x{\isaliteral{22}{\isachardoublequote}}}.
|
29731
|
980 |
|
|
981 |
The \hyperlink{keyword.obtains}{\mbox{\isa{\isakeyword{obtains}}}} variant is another abbreviation defined
|
|
982 |
below; unlike \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} (cf.\
|
|
983 |
\secref{sec:framework-context}) there may be several ``cases''
|
40406
|
984 |
separated by ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C424241523E}{\isasymBBAR}}{\isaliteral{22}{\isachardoublequote}}}'', each consisting of several
|
|
985 |
parameters (\isa{{\isaliteral{22}{\isachardoublequote}}vars{\isaliteral{22}{\isachardoublequote}}}) and several premises (\isa{{\isaliteral{22}{\isachardoublequote}}props{\isaliteral{22}{\isachardoublequote}}}).
|
29731
|
986 |
This specifies multi-branch elimination rules.
|
|
987 |
|
|
988 |
\medskip
|
|
989 |
\begin{tabular}{l}
|
40406
|
990 |
\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]
|
|
991 |
\quad \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C46495845533E}{\isasymFIXES}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\
|
|
992 |
\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}}} \\
|
|
993 |
\quad \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\
|
29731
|
994 |
\end{tabular}
|
|
995 |
\medskip
|
|
996 |
|
|
997 |
Presenting structured statements in such an ``open'' format usually
|
|
998 |
simplifies the subsequent proof, because the outer structure of the
|
|
999 |
problem is already laid out directly. E.g.\ consider the following
|
40406
|
1000 |
canonical patterns for \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C53484F57533E}{\isasymSHOWS}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4F425441494E533E}{\isasymOBTAINS}}{\isaliteral{22}{\isachardoublequote}}},
|
29731
|
1001 |
respectively:%
|
|
1002 |
\end{isamarkuptext}%
|
|
1003 |
\isamarkuptrue%
|
|
1004 |
%
|
|
1005 |
\begin{minipage}{0.5\textwidth}
|
|
1006 |
\isacommand{theorem}\isamarkupfalse%
|
|
1007 |
\isanewline
|
|
1008 |
\ \ \isakeyword{fixes}\ x\ \isakeyword{and}\ y\isanewline
|
40406
|
1009 |
\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
|
1010 |
\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1011 |
%
|
|
1012 |
\isadelimproof
|
|
1013 |
%
|
|
1014 |
\endisadelimproof
|
|
1015 |
%
|
|
1016 |
\isatagproof
|
|
1017 |
\isacommand{proof}\isamarkupfalse%
|
40406
|
1018 |
\ {\isaliteral{2D}{\isacharminus}}\isanewline
|
29731
|
1019 |
\ \ \isacommand{from}\isamarkupfalse%
|
40406
|
1020 |
\ {\isaliteral{60}{\isacharbackquoteopen}}A\ x{\isaliteral{60}{\isacharbackquoteclose}}\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}B\ y{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
|
29731
|
1021 |
\ \ \isacommand{show}\isamarkupfalse%
|
40406
|
1022 |
\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}%
|
29740
|
1023 |
\endisatagproof
|
|
1024 |
{\isafoldproof}%
|
|
1025 |
%
|
|
1026 |
\isadelimproof
|
|
1027 |
%
|
|
1028 |
\endisadelimproof
|
|
1029 |
%
|
|
1030 |
\isadelimnoproof
|
|
1031 |
\ %
|
|
1032 |
\endisadelimnoproof
|
|
1033 |
%
|
|
1034 |
\isatagnoproof
|
|
1035 |
\isacommand{sorry}\isamarkupfalse%
|
|
1036 |
%
|
|
1037 |
\endisatagnoproof
|
|
1038 |
{\isafoldnoproof}%
|
|
1039 |
%
|
|
1040 |
\isadelimnoproof
|
29731
|
1041 |
\isanewline
|
29740
|
1042 |
%
|
|
1043 |
\endisadelimnoproof
|
|
1044 |
%
|
|
1045 |
\isadelimproof
|
|
1046 |
%
|
|
1047 |
\endisadelimproof
|
|
1048 |
%
|
|
1049 |
\isatagproof
|
29731
|
1050 |
\isacommand{qed}\isamarkupfalse%
|
|
1051 |
%
|
|
1052 |
\endisatagproof
|
|
1053 |
{\isafoldproof}%
|
|
1054 |
%
|
|
1055 |
\isadelimproof
|
|
1056 |
%
|
|
1057 |
\endisadelimproof
|
|
1058 |
%
|
|
1059 |
\end{minipage}\begin{minipage}{0.5\textwidth}
|
|
1060 |
\isacommand{theorem}\isamarkupfalse%
|
|
1061 |
\isanewline
|
|
1062 |
\ \ \isakeyword{obtains}\ x\ \isakeyword{and}\ y\isanewline
|
40406
|
1063 |
\ \ \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1064 |
%
|
|
1065 |
\isadelimproof
|
|
1066 |
%
|
|
1067 |
\endisadelimproof
|
|
1068 |
%
|
|
1069 |
\isatagproof
|
|
1070 |
\isacommand{proof}\isamarkupfalse%
|
40406
|
1071 |
\ {\isaliteral{2D}{\isacharminus}}\isanewline
|
29731
|
1072 |
\ \ \isacommand{have}\isamarkupfalse%
|
40406
|
1073 |
\ {\isaliteral{22}{\isachardoublequoteopen}}A\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ b{\isaliteral{22}{\isachardoublequoteclose}}%
|
29740
|
1074 |
\endisatagproof
|
|
1075 |
{\isafoldproof}%
|
|
1076 |
%
|
|
1077 |
\isadelimproof
|
|
1078 |
%
|
|
1079 |
\endisadelimproof
|
|
1080 |
%
|
|
1081 |
\isadelimnoproof
|
|
1082 |
\ %
|
|
1083 |
\endisadelimnoproof
|
|
1084 |
%
|
|
1085 |
\isatagnoproof
|
|
1086 |
\isacommand{sorry}\isamarkupfalse%
|
|
1087 |
%
|
|
1088 |
\endisatagnoproof
|
|
1089 |
{\isafoldnoproof}%
|
|
1090 |
%
|
|
1091 |
\isadelimnoproof
|
29731
|
1092 |
\isanewline
|
29740
|
1093 |
%
|
|
1094 |
\endisadelimnoproof
|
|
1095 |
%
|
|
1096 |
\isadelimproof
|
|
1097 |
\ \ %
|
|
1098 |
\endisadelimproof
|
|
1099 |
%
|
|
1100 |
\isatagproof
|
|
1101 |
\isacommand{then}\isamarkupfalse%
|
29731
|
1102 |
\ \isacommand{show}\isamarkupfalse%
|
40406
|
1103 |
\ thesis\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
|
29731
|
1104 |
\isanewline
|
|
1105 |
\isacommand{qed}\isamarkupfalse%
|
|
1106 |
%
|
|
1107 |
\endisatagproof
|
|
1108 |
{\isafoldproof}%
|
|
1109 |
%
|
|
1110 |
\isadelimproof
|
|
1111 |
%
|
|
1112 |
\endisadelimproof
|
|
1113 |
%
|
|
1114 |
\end{minipage}
|
|
1115 |
%
|
|
1116 |
\begin{isamarkuptext}%
|
40406
|
1117 |
\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
|
29731
|
1118 |
need to decompose the logical rule structure again. In the second
|
40406
|
1119 |
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.%
|
29731
|
1120 |
\end{isamarkuptext}%
|
|
1121 |
\isamarkuptrue%
|
|
1122 |
%
|
|
1123 |
\isamarkupsubsection{Structured proof refinement \label{sec:framework-subproof}%
|
|
1124 |
}
|
|
1125 |
\isamarkuptrue%
|
|
1126 |
%
|
|
1127 |
\begin{isamarkuptext}%
|
|
1128 |
By breaking up the grammar for the Isar proof language, we may
|
|
1129 |
understand a proof text as a linear sequence of individual proof
|
|
1130 |
commands. These are interpreted as transitions of the Isar virtual
|
|
1131 |
machine (Isar/VM), which operates on a block-structured
|
|
1132 |
configuration in single steps. This allows users to write proof
|
|
1133 |
texts in an incremental manner, and inspect intermediate
|
|
1134 |
configurations for debugging.
|
|
1135 |
|
|
1136 |
The basic idea is analogous to evaluating algebraic expressions on a
|
40406
|
1137 |
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
|
|
1138 |
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}}}.
|
29731
|
1139 |
In Isar the algebraic values are facts or goals, and the operations
|
|
1140 |
are inferences.
|
|
1141 |
|
|
1142 |
\medskip The Isar/VM state maintains a stack of nodes, each node
|
|
1143 |
contains the local proof context, the linguistic mode, and a pending
|
|
1144 |
goal (optional). The mode determines the type of transition that
|
|
1145 |
may be performed next, it essentially alternates between forward and
|
29740
|
1146 |
backward reasoning, with an intermediate stage for chained facts
|
|
1147 |
(see \figref{fig:isar-vm}).
|
|
1148 |
|
|
1149 |
\begin{figure}[htb]
|
|
1150 |
\begin{center}
|
|
1151 |
\includegraphics[width=0.8\textwidth]{Thy/document/isar-vm}
|
|
1152 |
\end{center}
|
|
1153 |
\caption{Isar/VM modes}\label{fig:isar-vm}
|
|
1154 |
\end{figure}
|
|
1155 |
|
40406
|
1156 |
For example, in \isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} mode Isar acts like a mathematical
|
29740
|
1157 |
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
|
40406
|
1158 |
statement changes the mode to \isa{{\isaliteral{22}{\isachardoublequote}}prove{\isaliteral{22}{\isachardoublequote}}}, which means that we
|
|
1159 |
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,
|
29740
|
1160 |
which may issue \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} statements to solve pending
|
|
1161 |
sub-goals. A concluding \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} will return to the original
|
40406
|
1162 |
\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} mode one level upwards. The subsequent Isar/VM
|
29740
|
1163 |
trace indicates block structure, linguistic mode, goal state, and
|
|
1164 |
inferences:%
|
29731
|
1165 |
\end{isamarkuptext}%
|
|
1166 |
\isamarkuptrue%
|
|
1167 |
%
|
29740
|
1168 |
\begingroup\footnotesize
|
|
1169 |
%
|
29731
|
1170 |
\isadelimproof
|
|
1171 |
%
|
|
1172 |
\endisadelimproof
|
|
1173 |
%
|
|
1174 |
\isatagproof
|
|
1175 |
%
|
29740
|
1176 |
\begin{minipage}[t]{0.18\textwidth}
|
29731
|
1177 |
\ \ \isacommand{have}\isamarkupfalse%
|
40406
|
1178 |
\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1179 |
\ \ \isacommand{proof}\isamarkupfalse%
|
|
1180 |
\isanewline
|
|
1181 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
|
1182 |
\ A\isanewline
|
|
1183 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
29740
|
1184 |
\ B%
|
|
1185 |
\endisatagproof
|
|
1186 |
{\isafoldproof}%
|
|
1187 |
%
|
|
1188 |
\isadelimproof
|
|
1189 |
%
|
|
1190 |
\endisadelimproof
|
36357
|
1191 |
\isanewline
|
29740
|
1192 |
%
|
|
1193 |
\isadelimnoproof
|
|
1194 |
\ \ \ \ \ \ %
|
|
1195 |
\endisadelimnoproof
|
|
1196 |
%
|
|
1197 |
\isatagnoproof
|
|
1198 |
\isacommand{sorry}\isamarkupfalse%
|
|
1199 |
%
|
|
1200 |
\endisatagnoproof
|
|
1201 |
{\isafoldnoproof}%
|
|
1202 |
%
|
|
1203 |
\isadelimnoproof
|
|
1204 |
%
|
|
1205 |
\endisadelimnoproof
|
36357
|
1206 |
\isanewline
|
29740
|
1207 |
%
|
|
1208 |
\isadelimproof
|
|
1209 |
\ \ %
|
|
1210 |
\endisadelimproof
|
|
1211 |
%
|
|
1212 |
\isatagproof
|
|
1213 |
\isacommand{qed}\isamarkupfalse%
|
29731
|
1214 |
%
|
|
1215 |
\end{minipage}\quad
|
29740
|
1216 |
\begin{minipage}[t]{0.06\textwidth}
|
40406
|
1217 |
\isa{{\isaliteral{22}{\isachardoublequote}}begin{\isaliteral{22}{\isachardoublequote}}} \\
|
29731
|
1218 |
\\
|
|
1219 |
\\
|
40406
|
1220 |
\isa{{\isaliteral{22}{\isachardoublequote}}begin{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1221 |
\isa{{\isaliteral{22}{\isachardoublequote}}end{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1222 |
\isa{{\isaliteral{22}{\isachardoublequote}}end{\isaliteral{22}{\isachardoublequote}}} \\
|
29731
|
1223 |
\end{minipage}
|
|
1224 |
\begin{minipage}[t]{0.08\textwidth}
|
40406
|
1225 |
\isa{{\isaliteral{22}{\isachardoublequote}}prove{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1226 |
\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1227 |
\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1228 |
\isa{{\isaliteral{22}{\isachardoublequote}}prove{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1229 |
\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1230 |
\isa{{\isaliteral{22}{\isachardoublequote}}state{\isaliteral{22}{\isachardoublequote}}} \\
|
29740
|
1231 |
\end{minipage}\begin{minipage}[t]{0.35\textwidth}
|
40406
|
1232 |
\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}}} \\
|
|
1233 |
\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}}} \\
|
29731
|
1234 |
\\
|
|
1235 |
\\
|
40406
|
1236 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{23}{\isacharhash}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1237 |
\isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ B{\isaliteral{22}{\isachardoublequote}}} \\
|
29740
|
1238 |
\end{minipage}\begin{minipage}[t]{0.4\textwidth}
|
40406
|
1239 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}init{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1240 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}resolution\ impI{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
|
29731
|
1241 |
\\
|
|
1242 |
\\
|
40406
|
1243 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}refinement\ {\isaliteral{23}{\isacharhash}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
|
|
1244 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}finish{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
|
29731
|
1245 |
\end{minipage}
|
|
1246 |
%
|
|
1247 |
\endisatagproof
|
|
1248 |
{\isafoldproof}%
|
|
1249 |
%
|
|
1250 |
\isadelimproof
|
|
1251 |
%
|
|
1252 |
\endisadelimproof
|
|
1253 |
%
|
29740
|
1254 |
\endgroup
|
|
1255 |
%
|
29731
|
1256 |
\begin{isamarkuptext}%
|
29740
|
1257 |
\noindent Here the \hyperlink{inference.refinement}{\mbox{\isa{refinement}}} inference from
|
29731
|
1258 |
\secref{sec:framework-resolution} mediates composition of Isar
|
|
1259 |
sub-proofs nicely. Observe that this principle incorporates some
|
|
1260 |
degree of freedom in proof composition. In particular, the proof
|
|
1261 |
body allows parameters and assumptions to be re-ordered, or commuted
|
|
1262 |
according to Hereditary Harrop Form. Moreover, context elements
|
|
1263 |
that are not used in a sub-proof may be omitted altogether. For
|
|
1264 |
example:%
|
|
1265 |
\end{isamarkuptext}%
|
|
1266 |
\isamarkuptrue%
|
|
1267 |
%
|
|
1268 |
\begin{minipage}{0.5\textwidth}
|
|
1269 |
%
|
|
1270 |
\isadelimproof
|
36357
|
1271 |
\ \ %
|
29731
|
1272 |
\endisadelimproof
|
|
1273 |
%
|
|
1274 |
\isatagproof
|
36357
|
1275 |
\isacommand{have}\isamarkupfalse%
|
40406
|
1276 |
\ {\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
|
29731
|
1277 |
\ \ \isacommand{proof}\isamarkupfalse%
|
40406
|
1278 |
\ {\isaliteral{2D}{\isacharminus}}\isanewline
|
29731
|
1279 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
1280 |
\ x\ \isakeyword{and}\ y\isanewline
|
|
1281 |
\ \ \ \ \isacommand{assume}\isamarkupfalse%
|
40406
|
1282 |
\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1283 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
40406
|
1284 |
\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}%
|
29740
|
1285 |
\endisatagproof
|
|
1286 |
{\isafoldproof}%
|
|
1287 |
%
|
|
1288 |
\isadelimproof
|
|
1289 |
%
|
|
1290 |
\endisadelimproof
|
|
1291 |
%
|
|
1292 |
\isadelimnoproof
|
|
1293 |
\ %
|
|
1294 |
\endisadelimnoproof
|
|
1295 |
%
|
|
1296 |
\isatagnoproof
|
|
1297 |
\isacommand{sorry}\isamarkupfalse%
|
|
1298 |
%
|
|
1299 |
\endisatagnoproof
|
|
1300 |
{\isafoldnoproof}%
|
|
1301 |
%
|
|
1302 |
\isadelimnoproof
|
|
1303 |
%
|
|
1304 |
\endisadelimnoproof
|
36357
|
1305 |
\isanewline
|
29740
|
1306 |
%
|
|
1307 |
\isadelimproof
|
|
1308 |
\ \ %
|
|
1309 |
\endisadelimproof
|
|
1310 |
%
|
|
1311 |
\isatagproof
|
|
1312 |
\isacommand{qed}\isamarkupfalse%
|
29731
|
1313 |
%
|
|
1314 |
\end{minipage}\begin{minipage}{0.5\textwidth}
|
|
1315 |
\ \ \isacommand{have}\isamarkupfalse%
|
40406
|
1316 |
\ {\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
|
29731
|
1317 |
\ \ \isacommand{proof}\isamarkupfalse%
|
40406
|
1318 |
\ {\isaliteral{2D}{\isacharminus}}\isanewline
|
29731
|
1319 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
1320 |
\ x\ \isacommand{assume}\isamarkupfalse%
|
40406
|
1321 |
\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1322 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
1323 |
\ y\ \isacommand{assume}\isamarkupfalse%
|
40406
|
1324 |
\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1325 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
40406
|
1326 |
\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}%
|
29740
|
1327 |
\endisatagproof
|
|
1328 |
{\isafoldproof}%
|
|
1329 |
%
|
|
1330 |
\isadelimproof
|
|
1331 |
%
|
|
1332 |
\endisadelimproof
|
|
1333 |
%
|
|
1334 |
\isadelimnoproof
|
|
1335 |
\ %
|
|
1336 |
\endisadelimnoproof
|
|
1337 |
%
|
|
1338 |
\isatagnoproof
|
|
1339 |
\isacommand{sorry}\isamarkupfalse%
|
|
1340 |
%
|
|
1341 |
\endisatagnoproof
|
|
1342 |
{\isafoldnoproof}%
|
|
1343 |
%
|
|
1344 |
\isadelimnoproof
|
|
1345 |
%
|
|
1346 |
\endisadelimnoproof
|
36357
|
1347 |
\isanewline
|
29731
|
1348 |
%
|
29740
|
1349 |
\isadelimproof
|
|
1350 |
\ \ %
|
|
1351 |
\endisadelimproof
|
|
1352 |
%
|
|
1353 |
\isatagproof
|
|
1354 |
\isacommand{qed}\isamarkupfalse%
|
|
1355 |
%
|
|
1356 |
\end{minipage}\\[3ex]\begin{minipage}{0.5\textwidth}
|
29731
|
1357 |
\ \ \isacommand{have}\isamarkupfalse%
|
40406
|
1358 |
\ {\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
|
29731
|
1359 |
\ \ \isacommand{proof}\isamarkupfalse%
|
40406
|
1360 |
\ {\isaliteral{2D}{\isacharminus}}\isanewline
|
29731
|
1361 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
1362 |
\ y\ \isacommand{assume}\isamarkupfalse%
|
40406
|
1363 |
\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1364 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
1365 |
\ x\ \isacommand{assume}\isamarkupfalse%
|
40406
|
1366 |
\ {\isaliteral{22}{\isachardoublequoteopen}}A\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1367 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
40406
|
1368 |
\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
|
29731
|
1369 |
\isanewline
|
|
1370 |
\ \ \isacommand{qed}\isamarkupfalse%
|
|
1371 |
%
|
|
1372 |
\end{minipage}\begin{minipage}{0.5\textwidth}
|
|
1373 |
\ \ \isacommand{have}\isamarkupfalse%
|
40406
|
1374 |
\ {\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
|
29731
|
1375 |
\ \ \isacommand{proof}\isamarkupfalse%
|
40406
|
1376 |
\ {\isaliteral{2D}{\isacharminus}}\isanewline
|
29731
|
1377 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
1378 |
\ y\ \isacommand{assume}\isamarkupfalse%
|
40406
|
1379 |
\ {\isaliteral{22}{\isachardoublequoteopen}}B\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
|
29731
|
1380 |
\ \ \ \ \isacommand{fix}\isamarkupfalse%
|
|
1381 |
\ x\isanewline
|
|
1382 |
\ \ \ \ \isacommand{show}\isamarkupfalse%
|
40406
|
1383 |
\ {\isaliteral{22}{\isachardoublequoteopen}}C\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
|
29731
|
1384 |
\isanewline
|
|
1385 |
\ \ \isacommand{qed}\isamarkupfalse%
|
|
1386 |
%
|
|
1387 |
\endisatagproof
|
|
1388 |
{\isafoldproof}%
|
|
1389 |
%
|
|
1390 |
\isadelimproof
|
|
1391 |
%
|
|
1392 |
\endisadelimproof
|
|
1393 |
%
|
|
1394 |
\end{minipage}
|
|
1395 |
%
|
|
1396 |
\begin{isamarkuptext}%
|
29740
|
1397 |
\medskip\noindent Such ``peephole optimizations'' of Isar texts are
|
29731
|
1398 |
practically important to improve readability, by rearranging
|
|
1399 |
contexts elements according to the natural flow of reasoning in the
|
|
1400 |
body, while still observing the overall scoping rules.
|
|
1401 |
|
|
1402 |
\medskip This illustrates the basic idea of structured proof
|
|
1403 |
processing in Isar. The main mechanisms are based on natural
|
|
1404 |
deduction rule composition within the Pure framework. In
|
|
1405 |
particular, there are no direct operations on goal states within the
|
|
1406 |
proof body. Moreover, there is no hidden automated reasoning
|
|
1407 |
involved, just plain unification.%
|
|
1408 |
\end{isamarkuptext}%
|
|
1409 |
\isamarkuptrue%
|
|
1410 |
%
|
|
1411 |
\isamarkupsubsection{Calculational reasoning \label{sec:framework-calc}%
|
|
1412 |
}
|
|
1413 |
\isamarkuptrue%
|
|
1414 |
%
|
|
1415 |
\begin{isamarkuptext}%
|
29740
|
1416 |
The existing Isar infrastructure is sufficiently flexible to support
|
29731
|
1417 |
calculational reasoning (chains of transitivity steps) as derived
|
|
1418 |
concept. The generic proof elements introduced below depend on
|
29740
|
1419 |
rules declared as \hyperlink{attribute.trans}{\mbox{\isa{trans}}} in the context. It is left to
|
29731
|
1420 |
the object-logic to provide a suitable rule collection for mixed
|
40406
|
1421 |
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}}},
|
|
1422 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}{\isaliteral{22}{\isachardoublequote}}} etc. Due to the flexibility of rule composition
|
29731
|
1423 |
(\secref{sec:framework-resolution}), substitution of equals by
|
|
1424 |
equals is covered as well, even substitution of inequalities
|
|
1425 |
involving monotonicity conditions; see also \cite[\S6]{Wenzel-PhD}
|
|
1426 |
and \cite{Bauer-Wenzel:2001}.
|
|
1427 |
|
|
1428 |
The generic calculational mechanism is based on the observation that
|
40406
|
1429 |
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}}}
|
29740
|
1430 |
proceed from the premises towards the conclusion in a deterministic
|
|
1431 |
fashion. Thus we may reason in forward mode, feeding intermediate
|
|
1432 |
results into rules selected from the context. The course of
|
|
1433 |
reasoning is organized by maintaining a secondary fact called
|
|
1434 |
``\hyperlink{fact.calculation}{\mbox{\isa{calculation}}}'', apart from the primary ``\hyperlink{fact.this}{\mbox{\isa{this}}}''
|
|
1435 |
already provided by the Isar primitives. In the definitions below,
|
|
1436 |
\hyperlink{attribute.OF}{\mbox{\isa{OF}}} refers to \hyperlink{inference.resolution}{\mbox{\isa{resolution}}}
|
|
1437 |
(\secref{sec:framework-resolution}) with multiple rule arguments,
|
40406
|
1438 |
and \isa{{\isaliteral{22}{\isachardoublequote}}trans{\isaliteral{22}{\isachardoublequote}}} represents to a suitable rule from the context:
|
29731
|
1439 |
|
|
1440 |
\begin{matharray}{rcl}
|
40406
|
1441 |
\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}}} \\
|
|
1442 |
\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]
|
29731
|
1443 |
\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}} & \equiv & \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}~\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{calculation} \\
|
|
1444 |
\end{matharray}
|
|
1445 |
|
|
1446 |
\noindent The start of a calculation is determined implicitly in the
|
|
1447 |
text: here \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} sets \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} to the current
|
|
1448 |
result; any subsequent occurrence will update \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} by
|
|
1449 |
combination with the next result and a transitivity rule. The
|
|
1450 |
calculational sequence is concluded via \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}, where
|
|
1451 |
the final result is exposed for use in a concluding claim.
|
|
1452 |
|
|
1453 |
Here is a canonical proof pattern, using \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} to
|
|
1454 |
establish the intermediate results:%
|
|
1455 |
\end{isamarkuptext}%
|
|
1456 |
\isamarkuptrue%
|
|
1457 |
%
|
|
1458 |
\isadelimproof
|
36357
|
1459 |
\ \ %
|
29731
|
1460 |
\endisadelimproof
|
|
1461 |
%
|
|
1462 |
\isatagproof
|
36357
|
1463 |
\isacommand{have}\isamarkupfalse%
|
40406
|
1464 |
\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
|
29731
|
1465 |
\isanewline
|
|
1466 |
\ \ \isacommand{also}\isamarkupfalse%
|
|
1467 |
\ \isacommand{have}\isamarkupfalse%
|
40406
|
1468 |
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
|
29731
|
1469 |
\isanewline
|
|
1470 |
\ \ \isacommand{also}\isamarkupfalse%
|
|
1471 |
\ \isacommand{have}\isamarkupfalse%
|
40406
|
1472 |
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ d{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{sorry}\isamarkupfalse%
|
29731
|
1473 |
\isanewline
|
|
1474 |
\ \ \isacommand{finally}\isamarkupfalse%
|
|
1475 |
\ \isacommand{have}\isamarkupfalse%
|
40406
|
1476 |
\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ d{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
|
29731
|
1477 |
%
|
|
1478 |
\endisatagproof
|
|
1479 |
{\isafoldproof}%
|
|
1480 |
%
|
|
1481 |
\isadelimproof
|
|
1482 |
%
|
|
1483 |
\endisadelimproof
|
|
1484 |
%
|
|
1485 |
\begin{isamarkuptext}%
|
40406
|
1486 |
\noindent The term ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' above is a special abbreviation
|
29731
|
1487 |
provided by the Isabelle/Isar syntax layer: it statically refers to
|
|
1488 |
the right-hand side argument of the previous statement given in the
|
|
1489 |
text. Thus it happens to coincide with relevant sub-expressions in
|
|
1490 |
the calculational chain, but the exact correspondence is dependent
|
|
1491 |
on the transitivity rules being involved.
|
|
1492 |
|
40406
|
1493 |
\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
|
29731
|
1494 |
transitivities with only one premise. Isar maintains a separate
|
|
1495 |
rule collection declared via the \hyperlink{attribute.sym}{\mbox{\isa{sym}}} attribute, to be
|
40406
|
1496 |
used in fact expressions ``\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'', or single-step
|
|
1497 |
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}}}}}}''.%
|
29731
|
1498 |
\end{isamarkuptext}%
|
|
1499 |
\isamarkuptrue%
|
|
1500 |
%
|
29717
|
1501 |
\isadelimtheory
|
|
1502 |
%
|
|
1503 |
\endisadelimtheory
|
|
1504 |
%
|
|
1505 |
\isatagtheory
|
|
1506 |
\isacommand{end}\isamarkupfalse%
|
|
1507 |
%
|
|
1508 |
\endisatagtheory
|
|
1509 |
{\isafoldtheory}%
|
|
1510 |
%
|
|
1511 |
\isadelimtheory
|
|
1512 |
%
|
|
1513 |
\endisadelimtheory
|
|
1514 |
\end{isabellebody}%
|
|
1515 |
%%% Local Variables:
|
|
1516 |
%%% mode: latex
|
|
1517 |
%%% TeX-master: "root"
|
|
1518 |
%%% End:
|