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theory Framework
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imports Main
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begin
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chapter {* The Isabelle/Isar Framework \label{ch:isar-framework} *}
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text {*
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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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@{text "\<lambda>"}-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 (@{text "\<and>"}, @{text "\<or>"}, @{text "\<forall>"},
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@{text "\<exists>"}, 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 (@{text "\<inter>"}, @{text "\<union>"}, @{text "\<Inter>"}, @{text
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"\<Union>"}, 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 @{text "\<inter>"}-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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*}
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text_raw {*\medskip\begin{minipage}{0.6\textwidth}*}
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(*<*)
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example_proof
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(*>*)
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assume "x \<in> A" and "x \<in> B"
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then have "x \<in> A \<inter> B" ..
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(*<*)
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qed
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(*>*)
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text_raw {*\end{minipage}\begin{minipage}{0.4\textwidth}*}
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text {*
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\infer{@{prop "x \<in> A \<inter> B"}}{@{prop "x \<in> A"} & @{prop "x \<in> B"}}
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*}
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text_raw {*\end{minipage}*}
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text {*
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\medskip\noindent Note that @{command assume} augments the proof
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context, @{command then} indicates that the current fact shall be
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used in the next step, and @{command have} states an intermediate
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goal. The two dots ``@{command ".."}'' 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 @{command "by"} command:
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*}
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(*<*)
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example_proof
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(*>*)
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assume "x \<in> A" and "x \<in> B"
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then have "x \<in> A \<inter> B" by (rule IntI)
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(*<*)
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qed
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(*>*)
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text {*
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\noindent The format of the @{text "\<inter>"}-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 @{term "\<Inter>\<A>"},
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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 @{term
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A} is an arbitrary-but-fixed member of the collection:
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*}
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text_raw {*\medskip\begin{minipage}{0.6\textwidth}*}
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(*<*)
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example_proof
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(*>*)
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have "x \<in> \<Inter>\<A>"
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proof
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fix A
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assume "A \<in> \<A>"
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show "x \<in> A" sorry %noproof
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qed
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(*<*)
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qed
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(*>*)
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text_raw {*\end{minipage}\begin{minipage}{0.4\textwidth}*}
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text {*
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\infer{@{prop "x \<in> \<Inter>\<A>"}}{\infer*{@{prop "x \<in> A"}}{@{text "[A][A \<in> \<A>]"}}}
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*}
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text_raw {*\end{minipage}*}
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text {*
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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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``@{command proof}'' step, which could have been spelt out more
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explicitly as ``@{command proof}~@{text "(rule InterI)"}''. Note
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that the rule involves both a local parameter @{term "A"} and an
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assumption @{prop "A \<in> \<A>"} 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 @{command fix}-@{command assume}-@{command show}
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outline of nested sub-proofs that is typical for Isar. The final
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@{command show} is like @{command 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 @{term "\<Union>\<A>"}, which can be
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characterized as the set of all @{term "x"} such that @{prop "\<exists>A. x
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\<in> A \<and> A \<in> \<A>"}. The elimination rule for @{prop "x \<in> \<Union>\<A>"} does
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not mention @{text "\<exists>"} and @{text "\<and>"} at all, but admits to obtain
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directly a local @{term "A"} such that @{prop "x \<in> A"} and @{prop "A
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\<in> \<A>"} hold. This corresponds to the following Isar proof and
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inference rule, respectively:
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*}
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text_raw {*\medskip\begin{minipage}{0.6\textwidth}*}
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(*<*)
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example_proof
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(*>*)
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assume "x \<in> \<Union>\<A>"
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then have C
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proof
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fix A
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assume "x \<in> A" and "A \<in> \<A>"
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show C sorry %noproof
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qed
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(*<*)
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qed
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(*>*)
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text_raw {*\end{minipage}\begin{minipage}{0.4\textwidth}*}
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text {*
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\infer{@{prop "C"}}{@{prop "x \<in> \<Union>\<A>"} & \infer*{@{prop "C"}~}{@{text "[A][x \<in> A, A \<in> \<A>]"}}}
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*}
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text_raw {*\end{minipage}*}
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text {*
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\medskip\noindent Although the Isar proof follows the natural
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deduction rule closely, the text reads not as natural as
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anticipated. There is a double occurrence of an arbitrary
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conclusion @{prop "C"}, which represents the final result, but is
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irrelevant for now. This issue arises for any elimination rule
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involving local parameters. Isar provides the derived language
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element @{command obtain}, which is able to perform the same
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elimination proof more conveniently:
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*}
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(*<*)
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example_proof
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(*>*)
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assume "x \<in> \<Union>\<A>"
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then obtain A where "x \<in> A" and "A \<in> \<A>" ..
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(*<*)
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qed
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(*>*)
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text {*
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\noindent Here we avoid to mention the final conclusion @{prop "C"}
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and return to plain forward reasoning. The rule involved in the
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``@{command ".."}'' proof is the same as before.
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*}
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section {* The Pure framework \label{sec:framework-pure} *}
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text {*
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The Pure logic \cite{paulson-found,paulson700} is an intuitionistic
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fragment of higher-order logic \cite{church40}. In type-theoretic
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parlance, there are three levels of @{text "\<lambda>"}-calculus with
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corresponding arrows @{text "\<Rightarrow>"}/@{text "\<And>"}/@{text "\<Longrightarrow>"}:
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\medskip
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\begin{tabular}{ll}
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@{text "\<alpha> \<Rightarrow> \<beta>"} & syntactic function space (terms depending on terms) \\
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@{text "\<And>x. B(x)"} & universal quantification (proofs depending on terms) \\
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@{text "A \<Longrightarrow> B"} & implication (proofs depending on proofs) \\
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\end{tabular}
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\medskip
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\noindent Here only the types of syntactic terms, and the
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propositions of proof terms have been shown. The @{text
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"\<lambda>"}-structure of proofs can be recorded as an optional feature of
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the Pure inference kernel \cite{Berghofer-Nipkow:2000:TPHOL}, but
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the formal system can never depend on them due to \emph{proof
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irrelevance}.
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On top of this most primitive layer of proofs, Pure implements a
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generic calculus for nested natural deduction rules, similar to
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\cite{Schroeder-Heister:1984}. Here object-logic inferences are
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internalized as formulae over @{text "\<And>"} and @{text "\<Longrightarrow>"}.
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Combining such rule statements may involve higher-order unification
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\cite{paulson-natural}.
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*}
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subsection {* Primitive inferences *}
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text {*
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Term syntax provides explicit notation for abstraction @{text "\<lambda>x ::
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\<alpha>. b(x)"} and application @{text "b a"}, while types are usually
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implicit thanks to type-inference; terms of type @{text "prop"} are
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called propositions. Logical statements are composed via @{text "\<And>x
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:: \<alpha>. B(x)"} and @{text "A \<Longrightarrow> B"}. Primitive reasoning operates on
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judgments of the form @{text "\<Gamma> \<turnstile> \<phi>"}, with standard introduction
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and elimination rules for @{text "\<And>"} and @{text "\<Longrightarrow>"} that refer to
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fixed parameters @{text "x\<^isub>1, \<dots>, x\<^isub>m"} and hypotheses
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@{text "A\<^isub>1, \<dots>, A\<^isub>n"} from the context @{text "\<Gamma>"};
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the corresponding proof terms are left implicit. The subsequent
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inference rules define @{text "\<Gamma> \<turnstile> \<phi>"} inductively, relative to a
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collection of axioms:
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wenzelm@30055
|
274 |
\[
|
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|
275 |
\infer{@{text "\<turnstile> A"}}{(@{text "A"} \text{~axiom})}
|
wenzelm@30055
|
276 |
\qquad
|
wenzelm@30055
|
277 |
\infer{@{text "A \<turnstile> A"}}{}
|
wenzelm@30055
|
278 |
\]
|
wenzelm@30055
|
279 |
|
wenzelm@30055
|
280 |
\[
|
wenzelm@30055
|
281 |
\infer{@{text "\<Gamma> \<turnstile> \<And>x. B(x)"}}{@{text "\<Gamma> \<turnstile> B(x)"} & @{text "x \<notin> \<Gamma>"}}
|
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|
282 |
\qquad
|
wenzelm@30055
|
283 |
\infer{@{text "\<Gamma> \<turnstile> B(a)"}}{@{text "\<Gamma> \<turnstile> \<And>x. B(x)"}}
|
wenzelm@30055
|
284 |
\]
|
wenzelm@30055
|
285 |
|
wenzelm@30055
|
286 |
\[
|
wenzelm@30055
|
287 |
\infer{@{text "\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<Gamma> \<turnstile> B"}}
|
wenzelm@30055
|
288 |
\qquad
|
wenzelm@30055
|
289 |
\infer{@{text "\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B"}}{@{text "\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B"} & @{text "\<Gamma>\<^sub>2 \<turnstile> A"}}
|
wenzelm@30055
|
290 |
\]
|
wenzelm@30055
|
291 |
|
wenzelm@30055
|
292 |
Furthermore, Pure provides a built-in equality @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow>
|
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|
293 |
prop"} with axioms for reflexivity, substitution, extensionality,
|
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|
294 |
and @{text "\<alpha>\<beta>\<eta>"}-conversion on @{text "\<lambda>"}-terms.
|
wenzelm@30055
|
295 |
|
wenzelm@30055
|
296 |
\medskip An object-logic introduces another layer on top of Pure,
|
wenzelm@30055
|
297 |
e.g.\ with types @{text "i"} for individuals and @{text "o"} for
|
wenzelm@30055
|
298 |
propositions, term constants @{text "Trueprop :: o \<Rightarrow> prop"} as
|
wenzelm@30055
|
299 |
(implicit) derivability judgment and connectives like @{text "\<and> :: o
|
wenzelm@30055
|
300 |
\<Rightarrow> o \<Rightarrow> o"} or @{text "\<forall> :: (i \<Rightarrow> o) \<Rightarrow> o"}, and axioms for object-level
|
wenzelm@30055
|
301 |
rules such as @{text "conjI: A \<Longrightarrow> B \<Longrightarrow> A \<and> B"} or @{text "allI: (\<And>x. B
|
wenzelm@30055
|
302 |
x) \<Longrightarrow> \<forall>x. B x"}. Derived object rules are represented as theorems of
|
wenzelm@30055
|
303 |
Pure. After the initial object-logic setup, further axiomatizations
|
wenzelm@30055
|
304 |
are usually avoided; plain definitions and derived principles are
|
wenzelm@30055
|
305 |
used exclusively.
|
wenzelm@30055
|
306 |
*}
|
wenzelm@30055
|
307 |
|
wenzelm@30055
|
308 |
|
wenzelm@30055
|
309 |
subsection {* Reasoning with rules \label{sec:framework-resolution} *}
|
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|
310 |
|
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|
311 |
text {*
|
wenzelm@30055
|
312 |
Primitive inferences mostly serve foundational purposes. The main
|
wenzelm@30055
|
313 |
reasoning mechanisms of Pure operate on nested natural deduction
|
wenzelm@30055
|
314 |
rules expressed as formulae, using @{text "\<And>"} to bind local
|
wenzelm@30055
|
315 |
parameters and @{text "\<Longrightarrow>"} to express entailment. Multiple
|
wenzelm@30055
|
316 |
parameters and premises are represented by repeating these
|
wenzelm@30061
|
317 |
connectives in a right-associative manner.
|
wenzelm@30055
|
318 |
|
wenzelm@30055
|
319 |
Since @{text "\<And>"} and @{text "\<Longrightarrow>"} commute thanks to the theorem
|
wenzelm@30055
|
320 |
@{prop "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}, we may assume w.l.o.g.\
|
wenzelm@30055
|
321 |
that rule statements always observe the normal form where
|
wenzelm@30055
|
322 |
quantifiers are pulled in front of implications at each level of
|
wenzelm@30055
|
323 |
nesting. This means that any Pure proposition may be presented as a
|
wenzelm@30055
|
324 |
\emph{Hereditary Harrop Formula} \cite{Miller:1991} which is of the
|
wenzelm@30055
|
325 |
form @{text "\<And>x\<^isub>1 \<dots> x\<^isub>m. H\<^isub>1 \<Longrightarrow> \<dots> H\<^isub>n \<Longrightarrow>
|
wenzelm@30061
|
326 |
A"} for @{text "m, n \<ge> 0"}, and @{text "A"} atomic, and @{text
|
wenzelm@30061
|
327 |
"H\<^isub>1, \<dots>, H\<^isub>n"} being recursively of the same format.
|
wenzelm@30055
|
328 |
Following the convention that outermost quantifiers are implicit,
|
wenzelm@30055
|
329 |
Horn clauses @{text "A\<^isub>1 \<Longrightarrow> \<dots> A\<^isub>n \<Longrightarrow> A"} are a special
|
wenzelm@30055
|
330 |
case of this.
|
wenzelm@30055
|
331 |
|
wenzelm@30061
|
332 |
For example, @{text "\<inter>"}-introduction rule encountered before is
|
wenzelm@30061
|
333 |
represented as a Pure theorem as follows:
|
wenzelm@30061
|
334 |
\[
|
wenzelm@30061
|
335 |
@{text "IntI:"}~@{prop "x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> x \<in> A \<inter> B"}
|
wenzelm@30061
|
336 |
\]
|
wenzelm@30061
|
337 |
|
wenzelm@30061
|
338 |
\noindent This is a plain Horn clause, since no further nesting on
|
wenzelm@30061
|
339 |
the left is involved. The general @{text "\<Inter>"}-introduction
|
wenzelm@30061
|
340 |
corresponds to a Hereditary Harrop Formula with one additional level
|
wenzelm@30061
|
341 |
of nesting:
|
wenzelm@30061
|
342 |
\[
|
wenzelm@30061
|
343 |
@{text "InterI:"}~@{prop "(\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A) \<Longrightarrow> x \<in> \<Inter>\<A>"}
|
wenzelm@30061
|
344 |
\]
|
wenzelm@30061
|
345 |
|
wenzelm@30055
|
346 |
\medskip Goals are also represented as rules: @{text "A\<^isub>1 \<Longrightarrow>
|
wenzelm@30055
|
347 |
\<dots> A\<^isub>n \<Longrightarrow> C"} states that the sub-goals @{text "A\<^isub>1, \<dots>,
|
wenzelm@30055
|
348 |
A\<^isub>n"} entail the result @{text "C"}; for @{text "n = 0"} the
|
wenzelm@30055
|
349 |
goal is finished. To allow @{text "C"} being a rule statement
|
wenzelm@30055
|
350 |
itself, we introduce the protective marker @{text "# :: prop \<Rightarrow>
|
wenzelm@30055
|
351 |
prop"}, which is defined as identity and hidden from the user. We
|
wenzelm@30055
|
352 |
initialize and finish goal states as follows:
|
wenzelm@30055
|
353 |
|
wenzelm@30055
|
354 |
\[
|
wenzelm@30055
|
355 |
\begin{array}{c@ {\qquad}c}
|
wenzelm@30055
|
356 |
\infer[(@{inference_def init})]{@{text "C \<Longrightarrow> #C"}}{} &
|
wenzelm@30055
|
357 |
\infer[(@{inference_def finish})]{@{text C}}{@{text "#C"}}
|
wenzelm@30055
|
358 |
\end{array}
|
wenzelm@30055
|
359 |
\]
|
wenzelm@30055
|
360 |
|
wenzelm@30061
|
361 |
\noindent Goal states are refined in intermediate proof steps until
|
wenzelm@30061
|
362 |
a finished form is achieved. Here the two main reasoning principles
|
wenzelm@30061
|
363 |
are @{inference resolution}, for back-chaining a rule against a
|
wenzelm@30061
|
364 |
sub-goal (replacing it by zero or more sub-goals), and @{inference
|
wenzelm@30055
|
365 |
assumption}, for solving a sub-goal (finding a short-circuit with
|
wenzelm@30055
|
366 |
local assumptions). Below @{text "\<^vec>x"} stands for @{text
|
wenzelm@30055
|
367 |
"x\<^isub>1, \<dots>, x\<^isub>n"} (@{text "n \<ge> 0"}).
|
wenzelm@30055
|
368 |
|
wenzelm@30055
|
369 |
\[
|
wenzelm@30055
|
370 |
\infer[(@{inference_def resolution})]
|
wenzelm@30055
|
371 |
{@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}}
|
wenzelm@30055
|
372 |
{\begin{tabular}{rl}
|
wenzelm@30055
|
373 |
@{text "rule:"} &
|
wenzelm@30055
|
374 |
@{text "\<^vec>A \<^vec>a \<Longrightarrow> B \<^vec>a"} \\
|
wenzelm@30055
|
375 |
@{text "goal:"} &
|
wenzelm@30055
|
376 |
@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\
|
wenzelm@30055
|
377 |
@{text "goal unifier:"} &
|
wenzelm@30055
|
378 |
@{text "(\<lambda>\<^vec>x. B (\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\
|
wenzelm@30055
|
379 |
\end{tabular}}
|
wenzelm@30055
|
380 |
\]
|
wenzelm@30055
|
381 |
|
wenzelm@30055
|
382 |
\medskip
|
wenzelm@30055
|
383 |
|
wenzelm@30055
|
384 |
\[
|
wenzelm@30055
|
385 |
\infer[(@{inference_def assumption})]{@{text "C\<vartheta>"}}
|
wenzelm@30055
|
386 |
{\begin{tabular}{rl}
|
wenzelm@30055
|
387 |
@{text "goal:"} &
|
wenzelm@30055
|
388 |
@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C"} \\
|
wenzelm@30055
|
389 |
@{text "assm unifier:"} & @{text "A\<vartheta> = H\<^sub>i\<vartheta>"}~~\text{(for some~@{text "H\<^sub>i"})} \\
|
wenzelm@30055
|
390 |
\end{tabular}}
|
wenzelm@30055
|
391 |
\]
|
wenzelm@30055
|
392 |
|
wenzelm@30055
|
393 |
The following trace illustrates goal-oriented reasoning in
|
wenzelm@30055
|
394 |
Isabelle/Pure:
|
wenzelm@30055
|
395 |
|
wenzelm@30061
|
396 |
{\footnotesize
|
wenzelm@30055
|
397 |
\medskip
|
wenzelm@30061
|
398 |
\begin{tabular}{r@ {\quad}l}
|
wenzelm@30055
|
399 |
@{text "(A \<and> B \<Longrightarrow> B \<and> A) \<Longrightarrow> #(A \<and> B \<Longrightarrow> B \<and> A)"} & @{text "(init)"} \\
|
wenzelm@30055
|
400 |
@{text "(A \<and> B \<Longrightarrow> B) \<Longrightarrow> (A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>"} & @{text "(resolution B \<Longrightarrow> A \<Longrightarrow> B \<and> A)"} \\
|
wenzelm@30055
|
401 |
@{text "(A \<and> B \<Longrightarrow> A \<and> B) \<Longrightarrow> (A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>"} & @{text "(resolution A \<and> B \<Longrightarrow> B)"} \\
|
wenzelm@30055
|
402 |
@{text "(A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>"} & @{text "(assumption)"} \\
|
wenzelm@30055
|
403 |
@{text "(A \<and> B \<Longrightarrow> B \<and> A) \<Longrightarrow> #\<dots>"} & @{text "(resolution A \<and> B \<Longrightarrow> A)"} \\
|
wenzelm@30055
|
404 |
@{text "#\<dots>"} & @{text "(assumption)"} \\
|
wenzelm@30055
|
405 |
@{text "A \<and> B \<Longrightarrow> B \<and> A"} & @{text "(finish)"} \\
|
wenzelm@30055
|
406 |
\end{tabular}
|
wenzelm@30055
|
407 |
\medskip
|
wenzelm@30061
|
408 |
}
|
wenzelm@30055
|
409 |
|
wenzelm@30055
|
410 |
Compositions of @{inference assumption} after @{inference
|
wenzelm@30055
|
411 |
resolution} occurs quite often, typically in elimination steps.
|
wenzelm@30055
|
412 |
Traditional Isabelle tactics accommodate this by a combined
|
wenzelm@30055
|
413 |
@{inference_def elim_resolution} principle. In contrast, Isar uses
|
wenzelm@30055
|
414 |
a slightly more refined combination, where the assumptions to be
|
wenzelm@30055
|
415 |
closed are marked explicitly, using again the protective marker
|
wenzelm@30055
|
416 |
@{text "#"}:
|
wenzelm@30055
|
417 |
|
wenzelm@30055
|
418 |
\[
|
wenzelm@30055
|
419 |
\infer[(@{inference refinement})]
|
wenzelm@30055
|
420 |
{@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>G' (\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>"}}
|
wenzelm@30055
|
421 |
{\begin{tabular}{rl}
|
wenzelm@30055
|
422 |
@{text "sub\<dash>proof:"} &
|
wenzelm@30055
|
423 |
@{text "\<^vec>G \<^vec>a \<Longrightarrow> B \<^vec>a"} \\
|
wenzelm@30055
|
424 |
@{text "goal:"} &
|
wenzelm@30055
|
425 |
@{text "(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C"} \\
|
wenzelm@30055
|
426 |
@{text "goal unifier:"} &
|
wenzelm@30055
|
427 |
@{text "(\<lambda>\<^vec>x. B (\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>"} \\
|
wenzelm@30055
|
428 |
@{text "assm unifiers:"} &
|
wenzelm@30055
|
429 |
@{text "(\<lambda>\<^vec>x. G\<^sub>j (\<^vec>a \<^vec>x))\<vartheta> = #H\<^sub>i\<vartheta>"} \\
|
wenzelm@30055
|
430 |
& \quad (for each marked @{text "G\<^sub>j"} some @{text "#H\<^sub>i"}) \\
|
wenzelm@30055
|
431 |
\end{tabular}}
|
wenzelm@30055
|
432 |
\]
|
wenzelm@30055
|
433 |
|
wenzelm@30055
|
434 |
\noindent Here the @{text "sub\<dash>proof"} rule stems from the
|
wenzelm@30063
|
435 |
main @{command fix}-@{command assume}-@{command show} outline of
|
wenzelm@30063
|
436 |
Isar (cf.\ \secref{sec:framework-subproof}): each assumption
|
wenzelm@30055
|
437 |
indicated in the text results in a marked premise @{text "G"} above.
|
wenzelm@30061
|
438 |
The marking enforces resolution against one of the sub-goal's
|
wenzelm@30063
|
439 |
premises. Consequently, @{command fix}-@{command assume}-@{command
|
wenzelm@30063
|
440 |
show} enables to fit the result of a sub-proof quite robustly into a
|
wenzelm@30063
|
441 |
pending sub-goal, while maintaining a good measure of flexibility.
|
wenzelm@30055
|
442 |
*}
|
wenzelm@30055
|
443 |
|
wenzelm@30055
|
444 |
|
wenzelm@30055
|
445 |
section {* The Isar proof language \label{sec:framework-isar} *}
|
wenzelm@30055
|
446 |
|
wenzelm@30055
|
447 |
text {*
|
wenzelm@30055
|
448 |
Structured proofs are presented as high-level expressions for
|
wenzelm@30055
|
449 |
composing entities of Pure (propositions, facts, and goals). The
|
wenzelm@30055
|
450 |
Isar proof language allows to organize reasoning within the
|
wenzelm@30055
|
451 |
underlying rule calculus of Pure, but Isar is not another logical
|
wenzelm@30055
|
452 |
calculus!
|
wenzelm@30055
|
453 |
|
wenzelm@30055
|
454 |
Isar is an exercise in sound minimalism. Approximately half of the
|
wenzelm@30055
|
455 |
language is introduced as primitive, the rest defined as derived
|
wenzelm@30055
|
456 |
concepts. The following grammar describes the core language
|
wenzelm@30055
|
457 |
(category @{text "proof"}), which is embedded into theory
|
wenzelm@30055
|
458 |
specification elements such as @{command theorem}; see also
|
wenzelm@30055
|
459 |
\secref{sec:framework-stmt} for the separate category @{text
|
wenzelm@30055
|
460 |
"statement"}.
|
wenzelm@30055
|
461 |
|
wenzelm@30055
|
462 |
\medskip
|
wenzelm@30055
|
463 |
\begin{tabular}{rcl}
|
wenzelm@30055
|
464 |
@{text "theory\<dash>stmt"} & = & @{command "theorem"}~@{text "statement proof |"}~~@{command "definition"}~@{text "\<dots> | \<dots>"} \\[1ex]
|
wenzelm@30055
|
465 |
|
wenzelm@30055
|
466 |
@{text "proof"} & = & @{text "prfx\<^sup>*"}~@{command "proof"}~@{text "method\<^sup>? stmt\<^sup>*"}~@{command "qed"}~@{text "method\<^sup>?"} \\[1ex]
|
wenzelm@30055
|
467 |
|
wenzelm@30055
|
468 |
@{text prfx} & = & @{command "using"}~@{text "facts"} \\
|
wenzelm@30055
|
469 |
& @{text "|"} & @{command "unfolding"}~@{text "facts"} \\
|
wenzelm@30055
|
470 |
|
wenzelm@30055
|
471 |
@{text stmt} & = & @{command "{"}~@{text "stmt\<^sup>*"}~@{command "}"} \\
|
wenzelm@30055
|
472 |
& @{text "|"} & @{command "next"} \\
|
wenzelm@30055
|
473 |
& @{text "|"} & @{command "note"}~@{text "name = facts"} \\
|
wenzelm@30055
|
474 |
& @{text "|"} & @{command "let"}~@{text "term = term"} \\
|
wenzelm@30055
|
475 |
& @{text "|"} & @{command "fix"}~@{text "var\<^sup>+"} \\
|
wenzelm@30063
|
476 |
& @{text "|"} & @{command assume}~@{text "\<guillemotleft>inference\<guillemotright> name: props"} \\
|
wenzelm@30055
|
477 |
& @{text "|"} & @{command "then"}@{text "\<^sup>?"}~@{text goal} \\
|
wenzelm@30055
|
478 |
@{text goal} & = & @{command "have"}~@{text "name: props proof"} \\
|
wenzelm@30055
|
479 |
& @{text "|"} & @{command "show"}~@{text "name: props proof"} \\
|
wenzelm@30055
|
480 |
\end{tabular}
|
wenzelm@30055
|
481 |
|
wenzelm@30055
|
482 |
\medskip Simultaneous propositions or facts may be separated by the
|
wenzelm@30055
|
483 |
@{keyword "and"} keyword.
|
wenzelm@30055
|
484 |
|
wenzelm@30055
|
485 |
\medskip The syntax for terms and propositions is inherited from
|
wenzelm@30055
|
486 |
Pure (and the object-logic). A @{text "pattern"} is a @{text
|
wenzelm@30055
|
487 |
"term"} with schematic variables, to be bound by higher-order
|
wenzelm@30055
|
488 |
matching.
|
wenzelm@30055
|
489 |
|
wenzelm@30061
|
490 |
\medskip Facts may be referenced by name or proposition. For
|
wenzelm@30063
|
491 |
example, the result of ``@{command have}~@{text "a: A \<langle>proof\<rangle>"}''
|
wenzelm@30061
|
492 |
becomes available both as @{text "a"} and
|
wenzelm@30061
|
493 |
\isacharbackquoteopen@{text "A"}\isacharbackquoteclose. Moreover,
|
wenzelm@30061
|
494 |
fact expressions may involve attributes that modify either the
|
wenzelm@30061
|
495 |
theorem or the background context. For example, the expression
|
wenzelm@30061
|
496 |
``@{text "a [OF b]"}'' refers to the composition of two facts
|
wenzelm@30061
|
497 |
according to the @{inference resolution} inference of
|
wenzelm@30061
|
498 |
\secref{sec:framework-resolution}, while ``@{text "a [intro]"}''
|
wenzelm@30061
|
499 |
declares a fact as introduction rule in the context.
|
wenzelm@30055
|
500 |
|
wenzelm@30061
|
501 |
The special fact called ``@{fact this}'' always refers to the last
|
wenzelm@30063
|
502 |
result, as produced by @{command note}, @{command assume}, @{command
|
wenzelm@30063
|
503 |
have}, or @{command show}. Since @{command note} occurs
|
wenzelm@30063
|
504 |
frequently together with @{command then} we provide some
|
wenzelm@30063
|
505 |
abbreviations:
|
wenzelm@30055
|
506 |
|
wenzelm@30063
|
507 |
\medskip
|
wenzelm@30063
|
508 |
\begin{tabular}{rcl}
|
wenzelm@30063
|
509 |
@{command from}~@{text a} & @{text "\<equiv>"} & @{command note}~@{text a}~@{command then} \\
|
wenzelm@30063
|
510 |
@{command with}~@{text a} & @{text "\<equiv>"} & @{command from}~@{text "a \<AND> this"} \\
|
wenzelm@30063
|
511 |
\end{tabular}
|
wenzelm@30063
|
512 |
\medskip
|
wenzelm@30063
|
513 |
|
wenzelm@30063
|
514 |
The @{text "method"} category is essentially a parameter and may be
|
wenzelm@30063
|
515 |
populated later. Methods use the facts indicated by @{command
|
wenzelm@30063
|
516 |
"then"} or @{command using}, and then operate on the goal state.
|
wenzelm@30063
|
517 |
Some basic methods are predefined: ``@{method "-"}'' leaves the goal
|
wenzelm@30063
|
518 |
unchanged, ``@{method this}'' applies the facts as rules to the
|
wenzelm@30063
|
519 |
goal, ``@{method "rule"}'' applies the facts to another rule and the
|
wenzelm@30063
|
520 |
result to the goal (both ``@{method this}'' and ``@{method rule}''
|
wenzelm@30063
|
521 |
refer to @{inference resolution} of
|
wenzelm@30055
|
522 |
\secref{sec:framework-resolution}). The secondary arguments to
|
wenzelm@30055
|
523 |
``@{method rule}'' may be specified explicitly as in ``@{text "(rule
|
wenzelm@30055
|
524 |
a)"}'', or picked from the context. In the latter case, the system
|
wenzelm@30055
|
525 |
first tries rules declared as @{attribute (Pure) elim} or
|
wenzelm@30055
|
526 |
@{attribute (Pure) dest}, followed by those declared as @{attribute
|
wenzelm@30055
|
527 |
(Pure) intro}.
|
wenzelm@30055
|
528 |
|
wenzelm@30063
|
529 |
The default method for @{command proof} is ``@{method rule}''
|
wenzelm@30063
|
530 |
(arguments picked from the context), for @{command qed} it is
|
wenzelm@30055
|
531 |
``@{method "-"}''. Further abbreviations for terminal proof steps
|
wenzelm@30055
|
532 |
are ``@{command "by"}~@{text "method\<^sub>1 method\<^sub>2"}'' for
|
wenzelm@30063
|
533 |
``@{command proof}~@{text "method\<^sub>1"}~@{command qed}~@{text
|
wenzelm@30063
|
534 |
"method\<^sub>2"}'', and ``@{command ".."}'' for ``@{command
|
wenzelm@30063
|
535 |
"by"}~@{method rule}, and ``@{command "."}'' for ``@{command
|
wenzelm@30063
|
536 |
"by"}~@{method this}''. The @{command unfolding} element operates
|
wenzelm@30063
|
537 |
directly on the current facts and goal by applying equalities.
|
wenzelm@30055
|
538 |
|
wenzelm@30063
|
539 |
\medskip Block structure can be indicated explicitly by ``@{command
|
wenzelm@30063
|
540 |
"{"}~@{text "\<dots>"}~@{command "}"}'', although the body of a sub-proof
|
wenzelm@30063
|
541 |
already involves implicit nesting. In any case, @{command next}
|
wenzelm@30063
|
542 |
jumps into the next section of a block, i.e.\ it acts like closing
|
wenzelm@30063
|
543 |
an implicit block scope and opening another one; there is no direct
|
wenzelm@30063
|
544 |
correspondence to subgoals here.
|
wenzelm@30055
|
545 |
|
wenzelm@30063
|
546 |
The remaining elements @{command fix} and @{command assume} build up
|
wenzelm@30063
|
547 |
a local context (see \secref{sec:framework-context}), while
|
wenzelm@30063
|
548 |
@{command show} refines a pending sub-goal by the rule resulting
|
wenzelm@30055
|
549 |
from a nested sub-proof (see \secref{sec:framework-subproof}).
|
wenzelm@30055
|
550 |
Further derived concepts will support calculational reasoning (see
|
wenzelm@30055
|
551 |
\secref{sec:framework-calc}).
|
wenzelm@30055
|
552 |
*}
|
wenzelm@30055
|
553 |
|
wenzelm@30055
|
554 |
|
wenzelm@30055
|
555 |
subsection {* Context elements \label{sec:framework-context} *}
|
wenzelm@30055
|
556 |
|
wenzelm@30055
|
557 |
text {*
|
wenzelm@30055
|
558 |
In judgments @{text "\<Gamma> \<turnstile> \<phi>"} of the primitive framework, @{text "\<Gamma>"}
|
wenzelm@30055
|
559 |
essentially acts like a proof context. Isar elaborates this idea
|
wenzelm@30061
|
560 |
towards a higher-level notion, with additional information for
|
wenzelm@30055
|
561 |
type-inference, term abbreviations, local facts, hypotheses etc.
|
wenzelm@30055
|
562 |
|
wenzelm@30063
|
563 |
The element @{command fix}~@{text "x :: \<alpha>"} declares a local
|
wenzelm@30055
|
564 |
parameter, i.e.\ an arbitrary-but-fixed entity of a given type; in
|
wenzelm@30055
|
565 |
results exported from the context, @{text "x"} may become anything.
|
wenzelm@30063
|
566 |
The @{command assume}~@{text "\<guillemotleft>inference\<guillemotright>"} element provides a
|
wenzelm@30063
|
567 |
general interface to hypotheses: ``@{command assume}~@{text
|
wenzelm@30063
|
568 |
"\<guillemotleft>inference\<guillemotright> A"}'' produces @{text "A \<turnstile> A"} locally, while the
|
wenzelm@30063
|
569 |
included inference tells how to discharge @{text A} from results
|
wenzelm@30063
|
570 |
@{text "A \<turnstile> B"} later on. There is no user-syntax for @{text
|
wenzelm@30063
|
571 |
"\<guillemotleft>inference\<guillemotright>"}, i.e.\ it may only occur internally when derived
|
wenzelm@30063
|
572 |
commands are defined in ML.
|
wenzelm@30063
|
573 |
|
wenzelm@30063
|
574 |
At the user-level, the default inference for @{command assume} is
|
wenzelm@30063
|
575 |
@{inference discharge} as given below. The additional variants
|
wenzelm@30063
|
576 |
@{command presume} and @{command def} are defined as follows:
|
wenzelm@30063
|
577 |
|
wenzelm@30063
|
578 |
\medskip
|
wenzelm@30063
|
579 |
\begin{tabular}{rcl}
|
wenzelm@30063
|
580 |
@{command presume}~@{text A} & @{text "\<equiv>"} & @{command assume}~@{text "\<guillemotleft>weak\<dash>discharge\<guillemotright> A"} \\
|
wenzelm@30063
|
581 |
@{command def}~@{text "x \<equiv> a"} & @{text "\<equiv>"} & @{command fix}~@{text x}~@{command assume}~@{text "\<guillemotleft>expansion\<guillemotright> x \<equiv> a"} \\
|
wenzelm@30063
|
582 |
\end{tabular}
|
wenzelm@30063
|
583 |
\medskip
|
wenzelm@30055
|
584 |
|
wenzelm@30055
|
585 |
\[
|
wenzelm@30063
|
586 |
\infer[(@{inference_def discharge})]{@{text "\<strut>\<Gamma> - A \<turnstile> #A \<Longrightarrow> B"}}{@{text "\<strut>\<Gamma> \<turnstile> B"}}
|
wenzelm@30063
|
587 |
\]
|
wenzelm@30063
|
588 |
\[
|
wenzelm@30063
|
589 |
\infer[(@{inference_def "weak\<dash>discharge"})]{@{text "\<strut>\<Gamma> - A \<turnstile> A \<Longrightarrow> B"}}{@{text "\<strut>\<Gamma> \<turnstile> B"}}
|
wenzelm@30063
|
590 |
\]
|
wenzelm@30063
|
591 |
\[
|
wenzelm@30055
|
592 |
\infer[(@{inference_def expansion})]{@{text "\<strut>\<Gamma> - (x \<equiv> a) \<turnstile> B a"}}{@{text "\<strut>\<Gamma> \<turnstile> B x"}}
|
wenzelm@30055
|
593 |
\]
|
wenzelm@30055
|
594 |
|
wenzelm@30063
|
595 |
\medskip Note that @{inference discharge} and @{inference
|
wenzelm@30063
|
596 |
"weak\<dash>discharge"} differ in the marker for @{prop A}, which is
|
wenzelm@30063
|
597 |
relevant when the result of a @{command fix}-@{command
|
wenzelm@30063
|
598 |
assume}-@{command show} outline is composed with a pending goal,
|
wenzelm@30063
|
599 |
cf.\ \secref{sec:framework-subproof}.
|
wenzelm@30055
|
600 |
|
wenzelm@30063
|
601 |
The most interesting derived context element in Isar is @{command
|
wenzelm@30063
|
602 |
obtain} \cite[\S5.3]{Wenzel-PhD}, which supports generalized
|
wenzelm@30063
|
603 |
elimination steps in a purely forward manner. The @{command obtain}
|
wenzelm@30065
|
604 |
command takes a specification of parameters @{text "\<^vec>x"} and
|
wenzelm@30063
|
605 |
assumptions @{text "\<^vec>A"} to be added to the context, together
|
wenzelm@30063
|
606 |
with a proof of a case rule stating that this extension is
|
wenzelm@30063
|
607 |
conservative (i.e.\ may be removed from closed results later on):
|
wenzelm@30055
|
608 |
|
wenzelm@30055
|
609 |
\medskip
|
wenzelm@30055
|
610 |
\begin{tabular}{l}
|
wenzelm@30055
|
611 |
@{text "\<langle>facts\<rangle>"}~~@{command obtain}~@{text "\<^vec>x \<WHERE> \<^vec>A \<^vec>x \<langle>proof\<rangle> \<equiv>"} \\[0.5ex]
|
wenzelm@30055
|
612 |
\quad @{command have}~@{text "case: \<And>thesis. (\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<rangle>"} \\
|
wenzelm@30055
|
613 |
\quad @{command proof}~@{method "-"} \\
|
wenzelm@30055
|
614 |
\qquad @{command fix}~@{text thesis} \\
|
wenzelm@30055
|
615 |
\qquad @{command assume}~@{text "[intro]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis"} \\
|
wenzelm@30059
|
616 |
\qquad @{command show}~@{text thesis}~@{command using}~@{text "\<langle>facts\<rangle> \<langle>proof\<rangle>"} \\
|
wenzelm@30055
|
617 |
\quad @{command qed} \\
|
wenzelm@30063
|
618 |
\quad @{command fix}~@{text "\<^vec>x"}~@{command assume}~@{text "\<guillemotleft>elimination case\<guillemotright> \<^vec>A \<^vec>x"} \\
|
wenzelm@30055
|
619 |
\end{tabular}
|
wenzelm@30055
|
620 |
\medskip
|
wenzelm@30055
|
621 |
|
wenzelm@30055
|
622 |
\[
|
wenzelm@30055
|
623 |
\infer[(@{inference elimination})]{@{text "\<Gamma> \<turnstile> B"}}{
|
wenzelm@30055
|
624 |
\begin{tabular}{rl}
|
wenzelm@30055
|
625 |
@{text "case:"} &
|
wenzelm@30055
|
626 |
@{text "\<Gamma> \<turnstile> \<And>thesis. (\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis) \<Longrightarrow> thesis"} \\[0.2ex]
|
wenzelm@30055
|
627 |
@{text "result:"} &
|
wenzelm@30055
|
628 |
@{text "\<Gamma> \<union> \<^vec>A \<^vec>y \<turnstile> B"} \\[0.2ex]
|
wenzelm@30055
|
629 |
\end{tabular}}
|
wenzelm@30055
|
630 |
\]
|
wenzelm@30055
|
631 |
|
wenzelm@30055
|
632 |
\noindent Here the name ``@{text thesis}'' is a specific convention
|
wenzelm@30055
|
633 |
for an arbitrary-but-fixed proposition; in the primitive natural
|
wenzelm@30055
|
634 |
deduction rules shown before we have occasionally used @{text C}.
|
wenzelm@30063
|
635 |
The whole statement of ``@{command obtain}~@{text x}~@{keyword
|
wenzelm@30055
|
636 |
"where"}~@{text "A x"}'' may be read as a claim that @{text "A x"}
|
wenzelm@30055
|
637 |
may be assumed for some arbitrary-but-fixed @{text "x"}. Also note
|
wenzelm@30063
|
638 |
that ``@{command obtain}~@{text "A \<AND> B"}'' without parameters
|
wenzelm@30063
|
639 |
is similar to ``@{command have}~@{text "A \<AND> B"}'', but the
|
wenzelm@30063
|
640 |
latter involves multiple sub-goals.
|
wenzelm@30055
|
641 |
|
wenzelm@30055
|
642 |
\medskip The subsequent Isar proof texts explain all context
|
wenzelm@30055
|
643 |
elements introduced above using the formal proof language itself.
|
wenzelm@30055
|
644 |
After finishing a local proof within a block, we indicate the
|
wenzelm@30065
|
645 |
exported result via @{command note}.
|
wenzelm@30055
|
646 |
*}
|
wenzelm@30055
|
647 |
|
wenzelm@30055
|
648 |
(*<*)
|
wenzelm@30055
|
649 |
theorem True
|
wenzelm@30055
|
650 |
proof
|
wenzelm@30055
|
651 |
(*>*)
|
wenzelm@30061
|
652 |
txt_raw {* \begin{minipage}[t]{0.4\textwidth} *}
|
wenzelm@30055
|
653 |
{
|
wenzelm@30055
|
654 |
fix x
|
wenzelm@30061
|
655 |
have "B x" sorry %noproof
|
wenzelm@30055
|
656 |
}
|
wenzelm@30055
|
657 |
note `\<And>x. B x`
|
wenzelm@30061
|
658 |
txt_raw {* \end{minipage}\quad\begin{minipage}[t]{0.4\textwidth} *}(*<*)next(*>*)
|
wenzelm@30061
|
659 |
{
|
wenzelm@30061
|
660 |
assume A
|
wenzelm@30061
|
661 |
have B sorry %noproof
|
wenzelm@30061
|
662 |
}
|
wenzelm@30061
|
663 |
note `A \<Longrightarrow> B`
|
wenzelm@30061
|
664 |
txt_raw {* \end{minipage}\\[3ex]\begin{minipage}[t]{0.4\textwidth} *}(*<*)next(*>*)
|
wenzelm@30055
|
665 |
{
|
wenzelm@30055
|
666 |
def x \<equiv> a
|
wenzelm@30061
|
667 |
have "B x" sorry %noproof
|
wenzelm@30055
|
668 |
}
|
wenzelm@30055
|
669 |
note `B a`
|
wenzelm@30061
|
670 |
txt_raw {* \end{minipage}\quad\begin{minipage}[t]{0.4\textwidth} *}(*<*)next(*>*)
|
wenzelm@30055
|
671 |
{
|
wenzelm@30061
|
672 |
obtain x where "A x" sorry %noproof
|
wenzelm@30059
|
673 |
have B sorry %noproof
|
wenzelm@30055
|
674 |
}
|
wenzelm@30055
|
675 |
note `B`
|
wenzelm@30055
|
676 |
txt_raw {* \end{minipage} *}
|
wenzelm@30055
|
677 |
(*<*)
|
wenzelm@30055
|
678 |
qed
|
wenzelm@30055
|
679 |
(*>*)
|
wenzelm@30055
|
680 |
|
wenzelm@30065
|
681 |
text {*
|
wenzelm@30065
|
682 |
\bigskip\noindent This illustrates the meaning of Isar context
|
wenzelm@30065
|
683 |
elements without goals getting in between.
|
wenzelm@30065
|
684 |
*}
|
wenzelm@30055
|
685 |
|
wenzelm@30055
|
686 |
subsection {* Structured statements \label{sec:framework-stmt} *}
|
wenzelm@30055
|
687 |
|
wenzelm@30055
|
688 |
text {*
|
wenzelm@30055
|
689 |
The category @{text "statement"} of top-level theorem specifications
|
wenzelm@30055
|
690 |
is defined as follows:
|
wenzelm@30055
|
691 |
|
wenzelm@30055
|
692 |
\medskip
|
wenzelm@30055
|
693 |
\begin{tabular}{rcl}
|
wenzelm@30055
|
694 |
@{text "statement"} & @{text "\<equiv>"} & @{text "name: props \<AND> \<dots>"} \\
|
wenzelm@30055
|
695 |
& @{text "|"} & @{text "context\<^sup>* conclusion"} \\[0.5ex]
|
wenzelm@30055
|
696 |
|
wenzelm@30055
|
697 |
@{text "context"} & @{text "\<equiv>"} & @{text "\<FIXES> vars \<AND> \<dots>"} \\
|
wenzelm@30055
|
698 |
& @{text "|"} & @{text "\<ASSUMES> name: props \<AND> \<dots>"} \\
|
wenzelm@30055
|
699 |
|
wenzelm@30055
|
700 |
@{text "conclusion"} & @{text "\<equiv>"} & @{text "\<SHOWS> name: props \<AND> \<dots>"} \\
|
wenzelm@30061
|
701 |
& @{text "|"} & @{text "\<OBTAINS> vars \<AND> \<dots> \<WHERE> name: props \<AND> \<dots>"} \\
|
wenzelm@30061
|
702 |
& & \quad @{text "\<BBAR> \<dots>"} \\
|
wenzelm@30055
|
703 |
\end{tabular}
|
wenzelm@30055
|
704 |
|
wenzelm@30055
|
705 |
\medskip\noindent A simple @{text "statement"} consists of named
|
wenzelm@30055
|
706 |
propositions. The full form admits local context elements followed
|
wenzelm@30055
|
707 |
by the actual conclusions, such as ``@{keyword "fixes"}~@{text
|
wenzelm@30055
|
708 |
x}~@{keyword "assumes"}~@{text "A x"}~@{keyword "shows"}~@{text "B
|
wenzelm@30055
|
709 |
x"}''. The final result emerges as a Pure rule after discharging
|
wenzelm@30055
|
710 |
the context: @{prop "\<And>x. A x \<Longrightarrow> B x"}.
|
wenzelm@30055
|
711 |
|
wenzelm@30055
|
712 |
The @{keyword "obtains"} variant is another abbreviation defined
|
wenzelm@30055
|
713 |
below; unlike @{command obtain} (cf.\
|
wenzelm@30055
|
714 |
\secref{sec:framework-context}) there may be several ``cases''
|
wenzelm@30055
|
715 |
separated by ``@{text "\<BBAR>"}'', each consisting of several
|
wenzelm@30055
|
716 |
parameters (@{text "vars"}) and several premises (@{text "props"}).
|
wenzelm@30055
|
717 |
This specifies multi-branch elimination rules.
|
wenzelm@30055
|
718 |
|
wenzelm@30055
|
719 |
\medskip
|
wenzelm@30055
|
720 |
\begin{tabular}{l}
|
wenzelm@30055
|
721 |
@{text "\<OBTAINS> \<^vec>x \<WHERE> \<^vec>A \<^vec>x \<BBAR> \<dots> \<equiv>"} \\[0.5ex]
|
wenzelm@30055
|
722 |
\quad @{text "\<FIXES> thesis"} \\
|
wenzelm@30055
|
723 |
\quad @{text "\<ASSUMES> [intro]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis \<AND> \<dots>"} \\
|
wenzelm@30055
|
724 |
\quad @{text "\<SHOWS> thesis"} \\
|
wenzelm@30055
|
725 |
\end{tabular}
|
wenzelm@30055
|
726 |
\medskip
|
wenzelm@30055
|
727 |
|
wenzelm@30055
|
728 |
Presenting structured statements in such an ``open'' format usually
|
wenzelm@30055
|
729 |
simplifies the subsequent proof, because the outer structure of the
|
wenzelm@30055
|
730 |
problem is already laid out directly. E.g.\ consider the following
|
wenzelm@30055
|
731 |
canonical patterns for @{text "\<SHOWS>"} and @{text "\<OBTAINS>"},
|
wenzelm@30055
|
732 |
respectively:
|
wenzelm@30055
|
733 |
*}
|
wenzelm@30055
|
734 |
|
wenzelm@30055
|
735 |
text_raw {*\begin{minipage}{0.5\textwidth}*}
|
wenzelm@30055
|
736 |
|
wenzelm@30055
|
737 |
theorem
|
wenzelm@30055
|
738 |
fixes x and y
|
wenzelm@30055
|
739 |
assumes "A x" and "B y"
|
wenzelm@30055
|
740 |
shows "C x y"
|
wenzelm@30055
|
741 |
proof -
|
wenzelm@30055
|
742 |
from `A x` and `B y`
|
wenzelm@30059
|
743 |
show "C x y" sorry %noproof
|
wenzelm@30055
|
744 |
qed
|
wenzelm@30055
|
745 |
|
wenzelm@30055
|
746 |
text_raw {*\end{minipage}\begin{minipage}{0.5\textwidth}*}
|
wenzelm@30055
|
747 |
|
wenzelm@30055
|
748 |
theorem
|
wenzelm@30055
|
749 |
obtains x and y
|
wenzelm@30055
|
750 |
where "A x" and "B y"
|
wenzelm@30055
|
751 |
proof -
|
wenzelm@30059
|
752 |
have "A a" and "B b" sorry %noproof
|
wenzelm@30055
|
753 |
then show thesis ..
|
wenzelm@30055
|
754 |
qed
|
wenzelm@30055
|
755 |
|
wenzelm@30055
|
756 |
text_raw {*\end{minipage}*}
|
wenzelm@30055
|
757 |
|
wenzelm@30055
|
758 |
text {*
|
wenzelm@30055
|
759 |
\medskip\noindent Here local facts \isacharbackquoteopen@{text "A
|
wenzelm@30055
|
760 |
x"}\isacharbackquoteclose\ and \isacharbackquoteopen@{text "B
|
wenzelm@30055
|
761 |
y"}\isacharbackquoteclose\ are referenced immediately; there is no
|
wenzelm@30055
|
762 |
need to decompose the logical rule structure again. In the second
|
wenzelm@30055
|
763 |
proof the final ``@{command then}~@{command show}~@{text
|
wenzelm@30055
|
764 |
thesis}~@{command ".."}'' involves the local rule case @{text "\<And>x
|
wenzelm@30055
|
765 |
y. A x \<Longrightarrow> B y \<Longrightarrow> thesis"} for the particular instance of terms @{text
|
wenzelm@30055
|
766 |
"a"} and @{text "b"} produced in the body.
|
wenzelm@30055
|
767 |
*}
|
wenzelm@30055
|
768 |
|
wenzelm@30055
|
769 |
|
wenzelm@30055
|
770 |
subsection {* Structured proof refinement \label{sec:framework-subproof} *}
|
wenzelm@30055
|
771 |
|
wenzelm@30055
|
772 |
text {*
|
wenzelm@30055
|
773 |
By breaking up the grammar for the Isar proof language, we may
|
wenzelm@30055
|
774 |
understand a proof text as a linear sequence of individual proof
|
wenzelm@30055
|
775 |
commands. These are interpreted as transitions of the Isar virtual
|
wenzelm@30055
|
776 |
machine (Isar/VM), which operates on a block-structured
|
wenzelm@30055
|
777 |
configuration in single steps. This allows users to write proof
|
wenzelm@30055
|
778 |
texts in an incremental manner, and inspect intermediate
|
wenzelm@30055
|
779 |
configurations for debugging.
|
wenzelm@30055
|
780 |
|
wenzelm@30055
|
781 |
The basic idea is analogous to evaluating algebraic expressions on a
|
wenzelm@30055
|
782 |
stack machine: @{text "(a + b) \<cdot> c"} then corresponds to a sequence
|
wenzelm@30055
|
783 |
of single transitions for each symbol @{text "(, a, +, b, ), \<cdot>, c"}.
|
wenzelm@30055
|
784 |
In Isar the algebraic values are facts or goals, and the operations
|
wenzelm@30055
|
785 |
are inferences.
|
wenzelm@30055
|
786 |
|
wenzelm@30055
|
787 |
\medskip The Isar/VM state maintains a stack of nodes, each node
|
wenzelm@30055
|
788 |
contains the local proof context, the linguistic mode, and a pending
|
wenzelm@30055
|
789 |
goal (optional). The mode determines the type of transition that
|
wenzelm@30055
|
790 |
may be performed next, it essentially alternates between forward and
|
wenzelm@30064
|
791 |
backward reasoning, with an intermediate stage for chained facts
|
wenzelm@30064
|
792 |
(see \figref{fig:isar-vm}).
|
wenzelm@30064
|
793 |
|
wenzelm@30064
|
794 |
\begin{figure}[htb]
|
wenzelm@30064
|
795 |
\begin{center}
|
wenzelm@30064
|
796 |
\includegraphics[width=0.8\textwidth]{Thy/document/isar-vm}
|
wenzelm@30064
|
797 |
\end{center}
|
wenzelm@30064
|
798 |
\caption{Isar/VM modes}\label{fig:isar-vm}
|
wenzelm@30064
|
799 |
\end{figure}
|
wenzelm@30064
|
800 |
|
wenzelm@30064
|
801 |
For example, in @{text "state"} mode Isar acts like a mathematical
|
wenzelm@30064
|
802 |
scratch-pad, accepting declarations like @{command fix}, @{command
|
wenzelm@30064
|
803 |
assume}, and claims like @{command have}, @{command show}. A goal
|
wenzelm@30064
|
804 |
statement changes the mode to @{text "prove"}, which means that we
|
wenzelm@30064
|
805 |
may now refine the problem via @{command unfolding} or @{command
|
wenzelm@30064
|
806 |
proof}. Then we are again in @{text "state"} mode of a proof body,
|
wenzelm@30064
|
807 |
which may issue @{command show} statements to solve pending
|
wenzelm@30064
|
808 |
sub-goals. A concluding @{command qed} will return to the original
|
wenzelm@30064
|
809 |
@{text "state"} mode one level upwards. The subsequent Isar/VM
|
wenzelm@30064
|
810 |
trace indicates block structure, linguistic mode, goal state, and
|
wenzelm@30064
|
811 |
inferences:
|
wenzelm@30055
|
812 |
*}
|
wenzelm@30055
|
813 |
|
wenzelm@30061
|
814 |
text_raw {* \begingroup\footnotesize *}
|
wenzelm@36367
|
815 |
(*<*)example_proof
|
wenzelm@30055
|
816 |
(*>*)
|
wenzelm@30061
|
817 |
txt_raw {* \begin{minipage}[t]{0.18\textwidth} *}
|
wenzelm@30055
|
818 |
have "A \<longrightarrow> B"
|
wenzelm@30055
|
819 |
proof
|
wenzelm@30055
|
820 |
assume A
|
wenzelm@30055
|
821 |
show B
|
wenzelm@30059
|
822 |
sorry %noproof
|
wenzelm@30055
|
823 |
qed
|
wenzelm@30055
|
824 |
txt_raw {* \end{minipage}\quad
|
wenzelm@30061
|
825 |
\begin{minipage}[t]{0.06\textwidth}
|
wenzelm@30055
|
826 |
@{text "begin"} \\
|
wenzelm@30055
|
827 |
\\
|
wenzelm@30055
|
828 |
\\
|
wenzelm@30055
|
829 |
@{text "begin"} \\
|
wenzelm@30055
|
830 |
@{text "end"} \\
|
wenzelm@30055
|
831 |
@{text "end"} \\
|
wenzelm@30055
|
832 |
\end{minipage}
|
wenzelm@30055
|
833 |
\begin{minipage}[t]{0.08\textwidth}
|
wenzelm@30055
|
834 |
@{text "prove"} \\
|
wenzelm@30055
|
835 |
@{text "state"} \\
|
wenzelm@30055
|
836 |
@{text "state"} \\
|
wenzelm@30055
|
837 |
@{text "prove"} \\
|
wenzelm@30055
|
838 |
@{text "state"} \\
|
wenzelm@30055
|
839 |
@{text "state"} \\
|
wenzelm@30061
|
840 |
\end{minipage}\begin{minipage}[t]{0.35\textwidth}
|
wenzelm@30055
|
841 |
@{text "(A \<longrightarrow> B) \<Longrightarrow> #(A \<longrightarrow> B)"} \\
|
wenzelm@30055
|
842 |
@{text "(A \<Longrightarrow> B) \<Longrightarrow> #(A \<longrightarrow> B)"} \\
|
wenzelm@30055
|
843 |
\\
|
wenzelm@30055
|
844 |
\\
|
wenzelm@30055
|
845 |
@{text "#(A \<longrightarrow> B)"} \\
|
wenzelm@30055
|
846 |
@{text "A \<longrightarrow> B"} \\
|
wenzelm@30061
|
847 |
\end{minipage}\begin{minipage}[t]{0.4\textwidth}
|
wenzelm@30055
|
848 |
@{text "(init)"} \\
|
wenzelm@30061
|
849 |
@{text "(resolution impI)"} \\
|
wenzelm@30055
|
850 |
\\
|
wenzelm@30055
|
851 |
\\
|
wenzelm@30055
|
852 |
@{text "(refinement #A \<Longrightarrow> B)"} \\
|
wenzelm@30055
|
853 |
@{text "(finish)"} \\
|
wenzelm@30055
|
854 |
\end{minipage} *}
|
wenzelm@30055
|
855 |
(*<*)
|
wenzelm@30055
|
856 |
qed
|
wenzelm@30055
|
857 |
(*>*)
|
wenzelm@30061
|
858 |
text_raw {* \endgroup *}
|
wenzelm@30055
|
859 |
|
wenzelm@30055
|
860 |
text {*
|
wenzelm@30061
|
861 |
\noindent Here the @{inference refinement} inference from
|
wenzelm@30055
|
862 |
\secref{sec:framework-resolution} mediates composition of Isar
|
wenzelm@30055
|
863 |
sub-proofs nicely. Observe that this principle incorporates some
|
wenzelm@30055
|
864 |
degree of freedom in proof composition. In particular, the proof
|
wenzelm@30055
|
865 |
body allows parameters and assumptions to be re-ordered, or commuted
|
wenzelm@30055
|
866 |
according to Hereditary Harrop Form. Moreover, context elements
|
wenzelm@30055
|
867 |
that are not used in a sub-proof may be omitted altogether. For
|
wenzelm@30055
|
868 |
example:
|
wenzelm@30055
|
869 |
*}
|
wenzelm@30055
|
870 |
|
wenzelm@30055
|
871 |
text_raw {*\begin{minipage}{0.5\textwidth}*}
|
wenzelm@30055
|
872 |
|
wenzelm@30055
|
873 |
(*<*)
|
wenzelm@36367
|
874 |
example_proof
|
wenzelm@30055
|
875 |
(*>*)
|
wenzelm@30055
|
876 |
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
|
wenzelm@30055
|
877 |
proof -
|
wenzelm@30055
|
878 |
fix x and y
|
wenzelm@30055
|
879 |
assume "A x" and "B y"
|
wenzelm@30059
|
880 |
show "C x y" sorry %noproof
|
wenzelm@30055
|
881 |
qed
|
wenzelm@30055
|
882 |
|
wenzelm@30055
|
883 |
txt_raw {*\end{minipage}\begin{minipage}{0.5\textwidth}*}
|
wenzelm@30055
|
884 |
|
wenzelm@30055
|
885 |
(*<*)
|
wenzelm@30055
|
886 |
next
|
wenzelm@30055
|
887 |
(*>*)
|
wenzelm@30055
|
888 |
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
|
wenzelm@30055
|
889 |
proof -
|
wenzelm@30055
|
890 |
fix x assume "A x"
|
wenzelm@30055
|
891 |
fix y assume "B y"
|
wenzelm@30059
|
892 |
show "C x y" sorry %noproof
|
wenzelm@30055
|
893 |
qed
|
wenzelm@30055
|
894 |
|
wenzelm@30061
|
895 |
txt_raw {*\end{minipage}\\[3ex]\begin{minipage}{0.5\textwidth}*}
|
wenzelm@30055
|
896 |
|
wenzelm@30055
|
897 |
(*<*)
|
wenzelm@30055
|
898 |
next
|
wenzelm@30055
|
899 |
(*>*)
|
wenzelm@30055
|
900 |
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
|
wenzelm@30055
|
901 |
proof -
|
wenzelm@30055
|
902 |
fix y assume "B y"
|
wenzelm@30055
|
903 |
fix x assume "A x"
|
wenzelm@30055
|
904 |
show "C x y" sorry
|
wenzelm@30055
|
905 |
qed
|
wenzelm@30055
|
906 |
|
wenzelm@30055
|
907 |
txt_raw {*\end{minipage}\begin{minipage}{0.5\textwidth}*}
|
wenzelm@30055
|
908 |
(*<*)
|
wenzelm@30055
|
909 |
next
|
wenzelm@30055
|
910 |
(*>*)
|
wenzelm@30055
|
911 |
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
|
wenzelm@30055
|
912 |
proof -
|
wenzelm@30055
|
913 |
fix y assume "B y"
|
wenzelm@30055
|
914 |
fix x
|
wenzelm@30055
|
915 |
show "C x y" sorry
|
wenzelm@30055
|
916 |
qed
|
wenzelm@30055
|
917 |
(*<*)
|
wenzelm@30055
|
918 |
qed
|
wenzelm@30055
|
919 |
(*>*)
|
wenzelm@30055
|
920 |
|
wenzelm@30055
|
921 |
text_raw {*\end{minipage}*}
|
wenzelm@30055
|
922 |
|
wenzelm@30055
|
923 |
text {*
|
wenzelm@30061
|
924 |
\medskip\noindent Such ``peephole optimizations'' of Isar texts are
|
wenzelm@30055
|
925 |
practically important to improve readability, by rearranging
|
wenzelm@30055
|
926 |
contexts elements according to the natural flow of reasoning in the
|
wenzelm@30055
|
927 |
body, while still observing the overall scoping rules.
|
wenzelm@30055
|
928 |
|
wenzelm@30055
|
929 |
\medskip This illustrates the basic idea of structured proof
|
wenzelm@30055
|
930 |
processing in Isar. The main mechanisms are based on natural
|
wenzelm@30055
|
931 |
deduction rule composition within the Pure framework. In
|
wenzelm@30055
|
932 |
particular, there are no direct operations on goal states within the
|
wenzelm@30055
|
933 |
proof body. Moreover, there is no hidden automated reasoning
|
wenzelm@30055
|
934 |
involved, just plain unification.
|
wenzelm@30055
|
935 |
*}
|
wenzelm@30055
|
936 |
|
wenzelm@30055
|
937 |
|
wenzelm@30055
|
938 |
subsection {* Calculational reasoning \label{sec:framework-calc} *}
|
wenzelm@30055
|
939 |
|
wenzelm@30055
|
940 |
text {*
|
wenzelm@30061
|
941 |
The existing Isar infrastructure is sufficiently flexible to support
|
wenzelm@30055
|
942 |
calculational reasoning (chains of transitivity steps) as derived
|
wenzelm@30055
|
943 |
concept. The generic proof elements introduced below depend on
|
wenzelm@30058
|
944 |
rules declared as @{attribute trans} in the context. It is left to
|
wenzelm@30055
|
945 |
the object-logic to provide a suitable rule collection for mixed
|
wenzelm@30058
|
946 |
relations of @{text "="}, @{text "<"}, @{text "\<le>"}, @{text "\<subset>"},
|
wenzelm@30058
|
947 |
@{text "\<subseteq>"} etc. Due to the flexibility of rule composition
|
wenzelm@30055
|
948 |
(\secref{sec:framework-resolution}), substitution of equals by
|
wenzelm@30055
|
949 |
equals is covered as well, even substitution of inequalities
|
wenzelm@30055
|
950 |
involving monotonicity conditions; see also \cite[\S6]{Wenzel-PhD}
|
wenzelm@30055
|
951 |
and \cite{Bauer-Wenzel:2001}.
|
wenzelm@30055
|
952 |
|
wenzelm@30055
|
953 |
The generic calculational mechanism is based on the observation that
|
wenzelm@30061
|
954 |
rules such as @{text "trans:"}~@{prop "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z"}
|
wenzelm@30061
|
955 |
proceed from the premises towards the conclusion in a deterministic
|
wenzelm@30061
|
956 |
fashion. Thus we may reason in forward mode, feeding intermediate
|
wenzelm@30061
|
957 |
results into rules selected from the context. The course of
|
wenzelm@30061
|
958 |
reasoning is organized by maintaining a secondary fact called
|
wenzelm@30061
|
959 |
``@{fact calculation}'', apart from the primary ``@{fact this}''
|
wenzelm@30061
|
960 |
already provided by the Isar primitives. In the definitions below,
|
wenzelm@30061
|
961 |
@{attribute OF} refers to @{inference resolution}
|
wenzelm@30061
|
962 |
(\secref{sec:framework-resolution}) with multiple rule arguments,
|
wenzelm@30061
|
963 |
and @{text "trans"} represents to a suitable rule from the context:
|
wenzelm@30055
|
964 |
|
wenzelm@30055
|
965 |
\begin{matharray}{rcl}
|
wenzelm@30055
|
966 |
@{command "also"}@{text "\<^sub>0"} & \equiv & @{command "note"}~@{text "calculation = this"} \\
|
wenzelm@30055
|
967 |
@{command "also"}@{text "\<^sub>n\<^sub>+\<^sub>1"} & \equiv & @{command "note"}~@{text "calculation = trans [OF calculation this]"} \\[0.5ex]
|
wenzelm@30055
|
968 |
@{command "finally"} & \equiv & @{command "also"}~@{command "from"}~@{text calculation} \\
|
wenzelm@30055
|
969 |
\end{matharray}
|
wenzelm@30055
|
970 |
|
wenzelm@30055
|
971 |
\noindent The start of a calculation is determined implicitly in the
|
wenzelm@30055
|
972 |
text: here @{command also} sets @{fact calculation} to the current
|
wenzelm@30055
|
973 |
result; any subsequent occurrence will update @{fact calculation} by
|
wenzelm@30055
|
974 |
combination with the next result and a transitivity rule. The
|
wenzelm@30055
|
975 |
calculational sequence is concluded via @{command finally}, where
|
wenzelm@30055
|
976 |
the final result is exposed for use in a concluding claim.
|
wenzelm@30055
|
977 |
|
wenzelm@30055
|
978 |
Here is a canonical proof pattern, using @{command have} to
|
wenzelm@30055
|
979 |
establish the intermediate results:
|
wenzelm@30055
|
980 |
*}
|
wenzelm@30055
|
981 |
|
wenzelm@30055
|
982 |
(*<*)
|
wenzelm@36367
|
983 |
example_proof
|
wenzelm@30055
|
984 |
(*>*)
|
wenzelm@30055
|
985 |
have "a = b" sorry
|
wenzelm@30055
|
986 |
also have "\<dots> = c" sorry
|
wenzelm@30055
|
987 |
also have "\<dots> = d" sorry
|
wenzelm@30055
|
988 |
finally have "a = d" .
|
wenzelm@30055
|
989 |
(*<*)
|
wenzelm@30055
|
990 |
qed
|
wenzelm@30055
|
991 |
(*>*)
|
wenzelm@30055
|
992 |
|
wenzelm@30055
|
993 |
text {*
|
wenzelm@30055
|
994 |
\noindent The term ``@{text "\<dots>"}'' above is a special abbreviation
|
wenzelm@30055
|
995 |
provided by the Isabelle/Isar syntax layer: it statically refers to
|
wenzelm@30055
|
996 |
the right-hand side argument of the previous statement given in the
|
wenzelm@30055
|
997 |
text. Thus it happens to coincide with relevant sub-expressions in
|
wenzelm@30055
|
998 |
the calculational chain, but the exact correspondence is dependent
|
wenzelm@30055
|
999 |
on the transitivity rules being involved.
|
wenzelm@30055
|
1000 |
|
wenzelm@30055
|
1001 |
\medskip Symmetry rules such as @{prop "x = y \<Longrightarrow> y = x"} are like
|
wenzelm@30055
|
1002 |
transitivities with only one premise. Isar maintains a separate
|
wenzelm@30055
|
1003 |
rule collection declared via the @{attribute sym} attribute, to be
|
wenzelm@30055
|
1004 |
used in fact expressions ``@{text "a [symmetric]"}'', or single-step
|
wenzelm@30055
|
1005 |
proofs ``@{command assume}~@{text "x = y"}~@{command then}~@{command
|
wenzelm@30055
|
1006 |
have}~@{text "y = x"}~@{command ".."}''.
|
wenzelm@30055
|
1007 |
*}
|
wenzelm@30055
|
1008 |
|
wenzelm@30068
|
1009 |
end |