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\begin{verbatim}#|

 Copyright (C) 1994 by Yuan Yu.  All Rights Reserved.

 This script is hereby placed in the public domain, and therefore unlimited
 editing and redistribution is permitted.

 NO WARRANTY

 Yuan Yu PROVIDES ABSOLUTELY NO WARRANTY.  THE EVENT SCRIPT IS PROVIDED "AS IS"
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 IN NO EVENT WILL Yuan Yu BE LIABLE TO YOU FOR ANY DAMAGES, ANY LOST PROFITS,
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|#

\end{verbatim}
\noindent{\sc Event}:   Start with the library {\tt{"mc20-2"}} using the compiled version.

\addvspace{18pt}
\begin{verbatim};          Proof of the Correctness of the STRLEN Function
#|
This is part of our effort to verify the Berkeley string library.  The 
Berkeley string library is widely used as part of the Berkeley Unix OS.

This is the source code of strlen function in the Berkeley string library.

/* find the length of str[] */
size_t
strlen(str)
	const char *str;
{
	register const char *s;

	for (s = str; *s; ++s);
	return(s - str);
}

The MC68020 assembly code of the C function strlen on SUN-3 is given as 
follows.  This binary is generated by "gcc -O".

0x25b0 <strlen>:        linkw fp,#0
0x25b4 <strlen+4>:      movel fp@(8),d0
0x25b8 <strlen+8>:      moveal d0,a0
0x25ba <strlen+10>:     tstb a0@
0x25bc <strlen+12>:     beq 0x25c4 <strlen+20>
0x25be <strlen+14>:     addqw #1,a0
0x25c0 <strlen+16>:     tstb a0@
0x25c2 <strlen+18>:     bne 0x25be <strlen+14>
0x25c4 <strlen+20>:     subl a0,d0
0x25c6 <strlen+22>:     negl d0
0x25c8 <strlen+24>:     unlk fp
0x25ca <strlen+26>:     rts

The machine code of the above program is:

<strlen>:    0x4e56  0x0000  0x202e  0x0008  0x2040  0x4a10  0x6706  0x5248
<strlen+16>: 0x4a10  0x66fa  0x9088  0x4480  0x4e5e  0x4e75

'(78      86      0       0       32      46      0       8
  32      64      74      16      103     6       82      72
  74      16      102     250     144     136     68      128
  78      94      78      117)
|#

; in the logic, the above program is defined by (strlen-code).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strlen-code} \\  
\=\+{\sc{strlen-code}}\index{strlen-code} \\ 
$=$$\;\;\;\;${\tt{'}}{\tt{(}}\=\+{\tt{78}}{\tt{ }}{\tt{86}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{64}}{\tt{ }}{\tt{74}}{\tt{ }}{\tt{16}}{\tt{ }}{\tt{103}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{82}}{\tt{ }}{\tt{72}}{\tt{ }}{\tt{74}}{\tt{ }}{\tt{16}} \\ 
{\tt{102}}{\tt{ }}{\tt{250}}{\tt{ }}{\tt{144}}{\tt{ }}{\tt{136}}{\tt{ }}{\tt{68}}{\tt{ }}{\tt{128}}{\tt{ }}{\tt{78}}{\tt{ }}{\tt{94}}{\tt{ }}{\tt{78}}{\tt{ }}{\tt{117}}{\tt{)}}\-\-
\end{tabbing}

\begin{tabbing}{\sc Conservative Axiom}:  strlen-load \\ \index{strlen-load}
\=\+{\rm{strlen-loadp}}\,({\it{s\/}})\index{strlen-loadp} \\ 
$=$$\;\;\;\;$(\=\+{\rm{evenp}}\,({\sc{strlen-addr}}\index{strlen-addr})\index{evenp} \\ 
$\wedge$$\;\;\;\;$({\sc{strlen-addr}}\index{strlen-addr} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\sc{strlen-addr}}\index{strlen-addr}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{rom-addrp}}\,({\sc{strlen-addr}}\index{strlen-addr}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{28}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\sc{strlen-addr}}\index{strlen-addr}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{strlen-code}}\index{strlen-code})\index{mcode-addrp})\-\-
\end{tabbing}\noindent Simultaneously, we introduce the new function symbols {\it strlen-loadp\/} and {\it strlen-addr\/}.

\begin{tabbing}{\sc Theorem}:  stepn-strlen-loadp \\ \index{stepn-strlen-loadp}
{\rm{strlen-loadp}}\,({\rm{stepn}}\,({\it{s\/}}, {\it{n\/}})\index{stepn})\index{strlen-loadp} $=$ {\rm{strlen-loadp}}\,({\it{s\/}})\index{strlen-loadp}
\end{tabbing}

\begin{verbatim}; the computation time of the program.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strlen1-t} \\  
\=\+{\rm{strlen1-t}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen1-t} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $<$ {\it{n\/}} \\ 
{\bf then }\=\+{\bf if }{\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\sc{null}}\index{null}$\;\;${\bf then }{\tt{6}} \\ 
{\bf else }{\rm{splus}}\,({\tt{3}}, {\rm{strlen1-t}}\,($1\;+$ {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen1-t})\index{splus}$\;${\bf  endif}\- \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{strlen-t} \\  
\=\+{\rm{strlen-t}}\,({\it{n\/}}, {\it{lst\/}})\index{strlen-t} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $=$ {\tt{0}}$\;\;${\bf then }{\tt{9}} \\ 
{\bf else }{\rm{splus}}\,({\tt{6}}, {\rm{strlen1-t}}\,({\tt{1}}, {\it{n\/}}, {\it{lst\/}})\index{strlen1-t})\index{splus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; an induction hint.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strlen-induct} \\  
\=\+{\rm{strlen-induct}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-induct} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $<$ {\it{n\/}} \\ 
{\bf then }\=\+{\bf if }{\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\sc{null}}\index{null}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{strlen-induct}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn}, {\rm{add}}\,({\tt{32}}, {\it{i*\/}}, {\tt{1}})\index{add}, $1\;+$ {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-induct}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; the preconditions of the initial state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strlen-statep} \\  
\=\+{\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} \\ 
$=$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{s\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;${\rm{evenp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{evenp} \\ 
$\wedge$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{28}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{strlen-code}}\index{strlen-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{12}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}}, {\it{lst\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{str\/}}, {\it{n\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{12}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;$({\it{str\/}} $=$ {\rm{read-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{4}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{read-mem}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\tt{0}}, {\it{n\/}}, {\it{lst\/}})\index{slen} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; an intermediate state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strlen-s0p} \\  
\=\+{\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} \\ 
$=$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{s\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;${\rm{evenp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{evenp} \\ 
$\wedge$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{16}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{28}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{16}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{strlen-code}}\index{strlen-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{12}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}}, {\it{lst\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{str\/}}, {\it{n\/}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{12}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{equal*}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{0}}, {\it{s\/}})\index{read-an}, {\rm{add}}\,({\tt{32}}, {\it{str\/}}, {\it{i*\/}})\index{add})\index{equal*} \\ 
$\wedge$$\;\;\;\;$({\it{str\/}} $=$ {\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{s\/}})\index{read-dn}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $=$ {\rm{nat-to-uint}}\,({\it{i*\/}})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{slen} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{i*\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{i*\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; from the intial state s to exit: s --> sn.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strlen-s-sn \\ \index{strlen-s-sn}
\=\+({\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} $\wedge$ ({\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-pc} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{read-dn})\index{nat-to-uint} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{7}}, {\it{s\/}})\index{read-an}, {\tt{4}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s-sn-rfile \\ \index{strlen-s-sn-rfile}
\=\+({\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} $\wedge$ ({\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $=$ {\tt{0}}) $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s-sn-mem \\ \index{strlen-s-sn-mem}
\=\+(\=\+{\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{12}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} $=$ {\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-
\end{tabbing}

\begin{verbatim}; from the initial state s to s0: s --> s0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strlen-s-s0 \\ \index{strlen-s-s0}
\=\+({\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} $\wedge$ ({\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{strlen-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn}, {\it{str\/}}, {\tt{1}}, {\tt{1}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s-s0-else \\ \index{strlen-s-s0-else}
\=\+({\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} $\wedge$ ({\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{linked-a6} $=$ {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s-s0-rfile \\ \index{strlen-s-s0-rfile}
\=\+({\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} $\wedge$ ({\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $\not=$ {\tt{0}}) $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s-s0-mem \\ \index{strlen-s-s0-mem}
\=\+(\=\+{\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\tt{0}}, {\it{lst\/}})\index{get-nth} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{12}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} $=$ {\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-
\end{tabbing}

\begin{verbatim}; from s0 to exit (base case), from s0 to s0 (induction case).
; base case: s0 --> exit.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strlen-s0-sn-base \\ \index{strlen-s0-sn-base}
\=\+({\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} $\wedge$ ({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{read-dn})\index{nat-to-uint} $=$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s0-sn-rfile-base \\ \index{strlen-s0-sn-rfile-base}
\=\+({\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p} $\wedge$ ({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{verbatim}; induction case: s0 --> s0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strlen-s0-s0 \\ \index{strlen-s0-s0}
\=\+({\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} $\wedge$ ({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{strlen-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn}, {\it{str\/}}, {\rm{add}}\,({\tt{32}}, {\it{i*\/}}, {\tt{1}})\index{add}, $1\;+$ {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s0-s0-rfile \\ \index{strlen-s0-s0-rfile}
\=\+({\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p} $\wedge$ ({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{verbatim}; put together (s0 --> exit).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strlen-s0p-info \\ \index{strlen-s0p-info}
{\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} $\rightarrow$ (({\it{i\/}} $<$ {\it{n\/}}) $=$ {\bf{t}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-s0-sn \\ \index{strlen-s0-sn}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strlen1-t}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen1-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{sn\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\it{sn\/}})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{read-dn})\index{nat-to-uint} $=$ {\rm{strlen}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{sn\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem} $=$ {\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem}))\-\-$\;${\bf  endlet}\-
\end{tabbing}


\noindent{\sc Event}:   Disable strlen-s0p-info.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  strlen-s0-sn-rfile \\ \index{strlen-s0-sn-rfile}
\=\+({\rm{strlen-s0p}}\,({\it{s\/}}, {\it{str\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-s0p} $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{strlen1-t}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen1-t})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{verbatim}; the correctness of the STRLEN program.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strlen-statep-info \\ \index{strlen-statep-info}
{\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} $\rightarrow$ ({\it{n\/}} $\not\simeq {\tt{0}}$)
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strlen-correctness \\ \index{strlen-correctness}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strlen-t}}\,({\it{n\/}}, {\it{lst\/}})\index{strlen-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{strlen-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}})\index{strlen-statep} \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{sn\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\it{sn\/}})\index{mc-pc} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{4}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{12}})\index{disjoint} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{sn\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\- \\ 
$\wedge$$\;\;\;\;$({\rm{uread-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{uread-dn} $=$ {\rm{strlen}}\,({\tt{0}}, {\it{n\/}}, {\it{lst\/}})\index{strlen}))\-\-$\;${\bf  endlet}\-
\end{tabbing}


\noindent{\sc Event}:   Disable strlen-t.

\addvspace{18pt}

\begin{verbatim}; some properties of strlen.
; see the file cstring.events.

\end{verbatim}
\noindent{\sc Event}:   Make the library {\tt{"strlen"}} and compile it.

\addvspace{18pt}
%\end{list}
\printindex
\end{document}
