src/HOL/SPARK/Manual/Example_Verification.thy
author huffman
Sun Apr 01 16:09:58 2012 +0200 (2012-04-01)
changeset 47255 30a1692557b0
parent 45044 2fae15f8984d
child 56798 939e88e79724
permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
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(*<*)
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theory Example_Verification
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imports "../Examples/Gcd/Greatest_Common_Divisor" Simple_Greatest_Common_Divisor
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begin
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(*>*)
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chapter {* Verifying an Example Program *}
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text {*
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\label{sec:example-verification}
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\begin{figure}
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\lstinputlisting{Gcd.ads}
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\lstinputlisting{Gcd.adb}
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\caption{\SPARK{} program for computing the greatest common divisor}
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\label{fig:gcd-prog}
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\end{figure}
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\begin{figure}
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\input{Greatest_Common_Divisor}
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\caption{Correctness proof for the greatest common divisor program}
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\label{fig:gcd-proof}
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\end{figure}
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We will now explain the usage of the \SPARK{} verification environment by proving
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the correctness of an example program. As an example, we use a program for computing
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the \emph{greatest common divisor} of two natural numbers shown in \figref{fig:gcd-prog},
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which has been taken from the book about \SPARK{} by Barnes \cite[\S 11.6]{Barnes}.
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*}
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section {* Importing \SPARK{} VCs into Isabelle *}
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text {*
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In order to specify that the \SPARK{} procedure \texttt{G\_C\_D} behaves like its
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mathematical counterpart, Barnes introduces a \emph{proof function} \texttt{Gcd}
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in the package specification. Invoking the \SPARK{} Examiner and Simplifier on
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this program yields a file \texttt{g\_c\_d.siv} containing the simplified VCs,
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as well as files \texttt{g\_c\_d.fdl} and \texttt{g\_c\_d.rls}, containing FDL
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declarations and rules, respectively. The files generated by \SPARK{} are assumed to reside in the
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subdirectory \texttt{greatest\_common\_divisor}. For \texttt{G\_C\_D} the
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Examiner generates ten VCs, eight of which are proved automatically by
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the Simplifier. We now show how to prove the remaining two VCs
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interactively using HOL-\SPARK{}. For this purpose, we create a \emph{theory}
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\texttt{Greatest\_Common\_Divisor}, which is shown in \figref{fig:gcd-proof}.
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A theory file always starts with the keyword \isa{\isacommand{theory}} followed
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by the name of the theory, which must be the same as the file name. The theory
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name is followed by the keyword \isa{\isacommand{imports}} and a list of theories
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imported by the current theory. All theories using the HOL-\SPARK{} verification
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environment must import the theory \texttt{SPARK}. In addition, we also include
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the \texttt{GCD} theory. The list of imported theories is followed by the
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\isa{\isacommand{begin}} keyword. In order to interactively process the theory
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shown in \figref{fig:gcd-proof}, we start Isabelle with the command
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\begin{verbatim}
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  isabelle emacs -l HOL-SPARK Greatest_Common_Divisor.thy
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\end{verbatim}
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The option ``\texttt{-l HOL-SPARK}'' instructs Isabelle to load the right
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object logic image containing the verification environment. Each proof function
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occurring in the specification of a \SPARK{} program must be linked with a
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corresponding Isabelle function. This is accomplished by the command
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\isa{\isacommand{spark\_proof\_functions}}, which expects a list of equations
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of the form \emph{name}\texttt{\ =\ }\emph{term}, where \emph{name} is the
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name of the proof function and \emph{term} is the corresponding Isabelle term.
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In the case of \texttt{gcd}, both the \SPARK{} proof function and its Isabelle
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counterpart happen to have the same name. Isabelle checks that the type of the
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term linked with a proof function agrees with the type of the function declared
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in the \texttt{*.fdl} file.
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It is worth noting that the
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\isa{\isacommand{spark\_proof\_functions}} command can be invoked both outside,
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i.e.\ before \isa{\isacommand{spark\_open}}, and inside the environment, i.e.\ after
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\isa{\isacommand{spark\_open}}, but before any \isa{\isacommand{spark\_vc}} command. The
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former variant is useful when having to declare proof functions that are shared by several
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procedures, whereas the latter has the advantage that the type of the proof function
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can be checked immediately, since the VCs, and hence also the declarations of proof
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functions in the \texttt{*.fdl} file have already been loaded.
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\begin{figure}
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\begin{flushleft}
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\tt
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Context: \\
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\ \\
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\begin{tabular}{ll}
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fixes & @{text "m ::"}\ "@{text int}" \\
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and   & @{text "n ::"}\ "@{text int}" \\
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and   & @{text "c ::"}\ "@{text int}" \\
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and   & @{text "d ::"}\ "@{text int}" \\
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assumes & @{text "g_c_d_rules1:"}\ "@{text "0 \<le> integer__size"}" \\
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and     & @{text "g_c_d_rules6:"}\ "@{text "0 \<le> natural__size"}" \\
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\multicolumn{2}{l}{notes definition} \\
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\multicolumn{2}{l}{\hspace{2ex}@{text "defns ="}\ `@{text "integer__first = - 2147483648"}`} \\
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\multicolumn{2}{l}{\hspace{4ex}`@{text "integer__last = 2147483647"}`} \\
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\multicolumn{2}{l}{\hspace{4ex}\dots}
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\end{tabular}\ \\[1.5ex]
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\ \\
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Definitions: \\
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\ \\
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\begin{tabular}{ll}
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@{text "g_c_d_rules2:"} & @{text "integer__first = - 2147483648"} \\
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@{text "g_c_d_rules3:"} & @{text "integer__last = 2147483647"} \\
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\dots
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\end{tabular}\ \\[1.5ex]
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\ \\
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Verification conditions: \\
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\ \\
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path(s) from assertion of line 10 to assertion of line 10 \\
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\ \\
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@{text procedure_g_c_d_4}\ (unproved) \\
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\ \ \begin{tabular}{ll}
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assumes & @{text "H1:"}\ "@{text "0 \<le> c"}" \\
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and     & @{text "H2:"}\ "@{text "0 < d"}" \\
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and     & @{text "H3:"}\ "@{text "gcd c d = gcd m n"}" \\
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\dots \\
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shows & "@{text "0 < c - c sdiv d * d"}" \\
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and   & "@{text "gcd d (c - c sdiv d * d) = gcd m n"}
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\end{tabular}\ \\[1.5ex]
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\ \\
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path(s) from assertion of line 10 to finish \\
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\ \\
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@{text procedure_g_c_d_11}\ (unproved) \\
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\ \ \begin{tabular}{ll}
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assumes & @{text "H1:"}\ "@{text "0 \<le> c"}" \\
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and     & @{text "H2:"}\ "@{text "0 < d"}" \\
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and     & @{text "H3:"}\ "@{text "gcd c d = gcd m n"}" \\
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\dots \\
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shows & "@{text "d = gcd m n"}"
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\end{tabular}
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\end{flushleft}
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\caption{Output of \isa{\isacommand{spark\_status}} for \texttt{g\_c\_d.siv}}
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\label{fig:gcd-status}
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\end{figure}
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We now instruct Isabelle to open
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a new verification environment and load a set of VCs. This is done using the
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command \isa{\isacommand{spark\_open}}, which must be given the name of a
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\texttt{*.siv} or \texttt{*.vcg} file as an argument. Behind the scenes, Isabelle
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parses this file and the corresponding \texttt{*.fdl} and \texttt{*.rls} files,
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and converts the VCs to Isabelle terms. Using the command \isa{\isacommand{spark\_status}},
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the user can display the current VCs together with their status (proved, unproved).
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The variants \isa{\isacommand{spark\_status}\ (proved)}
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and \isa{\isacommand{spark\_status}\ (unproved)} show only proved and unproved
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VCs, respectively. For \texttt{g\_c\_d.siv}, the output of
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\isa{\isacommand{spark\_status}} is shown in \figref{fig:gcd-status}.
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To minimize the number of assumptions, and hence the size of the VCs,
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FDL rules of the form ``\dots\ \texttt{may\_be\_replaced\_by}\ \dots'' are
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turned into native Isabelle definitions, whereas other rules are modelled
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as assumptions.
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*}
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section {* Proving the VCs *}
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text {*
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\label{sec:proving-vcs}
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The two open VCs are @{text procedure_g_c_d_4} and @{text procedure_g_c_d_11},
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both of which contain the @{text gcd} proof function that the \SPARK{} Simplifier
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does not know anything about. The proof of a particular VC can be started with
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the \isa{\isacommand{spark\_vc}} command, which is similar to the standard
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\isa{\isacommand{lemma}} and \isa{\isacommand{theorem}} commands, with the
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difference that it only takes a name of a VC but no formula as an argument.
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A VC can have several conclusions that can be referenced by the identifiers
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@{text "?C1"}, @{text "?C2"}, etc. If there is just one conclusion, it can
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also be referenced by @{text "?thesis"}. It is important to note that the
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\texttt{div} operator of FDL behaves differently from the @{text div} operator
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of Isabelle/HOL on negative numbers. The former always truncates towards zero,
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whereas the latter truncates towards minus infinity. This is why the FDL
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\texttt{div} operator is mapped to the @{text sdiv} operator in Isabelle/HOL,
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which is defined as
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@{thm [display] sdiv_def}
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For example, we have that
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@{lemma "-5 sdiv 4 = -1" by (simp add: sdiv_neg_pos)}, but
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@{lemma "(-5::int) div 4 = -2" by simp}.
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For non-negative dividend and divisor, @{text sdiv} is equivalent to @{text div},
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as witnessed by theorem @{text sdiv_pos_pos}:
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@{thm [display,mode=no_brackets] sdiv_pos_pos}
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In contrast, the behaviour of the FDL \texttt{mod} operator is equivalent to
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the one of Isabelle/HOL. Moreover, since FDL has no counterpart of the \SPARK{}
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operator \textbf{rem}, the \SPARK{} expression \texttt{c}\ \textbf{rem}\ \texttt{d}
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just becomes @{text "c - c sdiv d * d"} in Isabelle. The first conclusion of
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@{text procedure_g_c_d_4} requires us to prove that the remainder of @{text c}
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and @{text d} is greater than @{text 0}. To do this, we use the theorem
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@{text zmod_zdiv_equality'} describing the correspondence between @{text div}
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and @{text mod}
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@{thm [display] zmod_zdiv_equality'}
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together with the theorem @{text pos_mod_sign} saying that the result of the
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@{text mod} operator is non-negative when applied to a non-negative divisor:
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@{thm [display] pos_mod_sign}
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We will also need the aforementioned theorem @{text sdiv_pos_pos} in order for
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the standard Isabelle/HOL theorems about @{text div} to be applicable
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to the VC, which is formulated using @{text sdiv} rather that @{text div}.
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Note that the proof uses \texttt{`@{text "0 \<le> c"}`} and \texttt{`@{text "0 < d"}`}
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rather than @{text H1} and @{text H2} to refer to the hypotheses of the current
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VC. While the latter variant seems more compact, it is not particularly robust,
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since the numbering of hypotheses can easily change if the corresponding
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program is modified, making the proof script hard to adjust when there are many hypotheses.
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Moreover, proof scripts using abbreviations like @{text H1} and @{text H2}
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are hard to read without assistance from Isabelle.
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The second conclusion of @{text procedure_g_c_d_4} requires us to prove that
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the @{text gcd} of @{text d} and the remainder of @{text c} and @{text d}
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is equal to the @{text gcd} of the original input values @{text m} and @{text n},
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which is the actual \emph{invariant} of the procedure. This is a consequence
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of theorem @{text gcd_non_0_int}
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@{thm [display] gcd_non_0_int}
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Again, we also need theorems @{text zmod_zdiv_equality'} and @{text sdiv_pos_pos}
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to justify that \SPARK{}'s \textbf{rem} operator is equivalent to Isabelle's
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@{text mod} operator for non-negative operands.
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The VC @{text procedure_g_c_d_11} says that if the loop invariant holds before
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the last iteration of the loop, the postcondition of the procedure will hold
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after execution of the loop body. To prove this, we observe that the remainder
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of @{text c} and @{text d}, and hence @{text "c mod d"} is @{text 0} when exiting
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the loop. This implies that @{text "gcd c d = d"}, since @{text c} is divisible
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by @{text d}, so the conclusion follows using the assumption @{text "gcd c d = gcd m n"}.
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This concludes the proofs of the open VCs, and hence the \SPARK{} verification
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environment can be closed using the command \isa{\isacommand{spark\_end}}.
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This command checks that all VCs have been proved and issues an error message
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if there are remaining unproved VCs. Moreover, Isabelle checks that there is
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no open \SPARK{} verification environment when the final \isa{\isacommand{end}}
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command of a theory is encountered.
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*}
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section {* Optimizing the proof *}
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text {*
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\begin{figure}
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\lstinputlisting{Simple_Gcd.adb}
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\input{Simple_Greatest_Common_Divisor}
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\caption{Simplified greatest common divisor program and proof}
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\label{fig:simple-gcd-proof}
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\end{figure}
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When looking at the program from \figref{fig:gcd-prog} once again, several
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optimizations come to mind. First of all, like the input parameters of the
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procedure, the local variables \texttt{C}, \texttt{D}, and \texttt{R} can
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be declared as \texttt{Natural} rather than \texttt{Integer}. Since natural
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numbers are non-negative by construction, the values computed by the algorithm
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are trivially proved to be non-negative. Since we are working with non-negative
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numbers, we can also just use \SPARK{}'s \textbf{mod} operator instead of
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\textbf{rem}, which spares us an application of theorems @{text zmod_zdiv_equality'}
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and @{text sdiv_pos_pos}. Finally, as noted by Barnes \cite[\S 11.5]{Barnes},
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we can simplify matters by placing the \textbf{assert} statement between
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\textbf{while} and \textbf{loop} rather than directly after the \textbf{loop}.
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In the former case, the loop invariant has to be proved only once, whereas in
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the latter case, it has to be proved twice: since the \textbf{assert} occurs after
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the check of the exit condition, the invariant has to be proved for the path
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from the \textbf{assert} statement to the \textbf{assert} statement, and for
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the path from the \textbf{assert} statement to the postcondition. In the case
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of the \texttt{G\_C\_D} procedure, this might not seem particularly problematic,
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since the proof of the invariant is very simple, but it can unnecessarily
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complicate matters if the proof of the invariant is non-trivial. The simplified
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program for computing the greatest common divisor, together with its correctness
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proof, is shown in \figref{fig:simple-gcd-proof}. Since the package specification
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has not changed, we only show the body of the packages. The two VCs can now be
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proved by a single application of Isabelle's proof method @{text simp}.
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*}
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(*<*)
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end
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(*>*)