src/HOL/SPARK/Manual/Example_Verification.thy

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