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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"
 WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT
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 SCRIPT IS WITH YOU.  SHOULD THE SCRIPT PROVE DEFECTIVE, YOU ASSUME THE COST OF
 ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

 IN NO EVENT WILL Yuan Yu BE LIABLE TO YOU FOR ANY DAMAGES, ANY LOST PROFITS,
 LOST MONIES, OR OTHER SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT
 OF THE USE OR INABILITY TO USE THIS SCRIPT (INCLUDING BUT NOT LIMITED TO LOSS
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 PARTIES), EVEN IF YOU HAVE ADVISED US OF THE POSSIBILITY OF SUCH DAMAGES, OR
 FOR ANY CLAIM BY ANY OTHER PARTY.

|#

\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 MEMCMP 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 memcmp function in the Berkeley string library.

int
memcmp(s1, s2, n)
	const void *s1, *s2;
	size_t n;
{
	if (n != 0) {
		register const unsigned char *p1 = s1, *p2 = s2;

		do {
			if (*p1++ != *p2++)
				return (*--p1 - *--p2);
		} while (--n != 0);
	}
	return (0);
}

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

0x2810 <memcmp>:        linkw fp,#0
0x2814 <memcmp+4>:      movel d2,sp@-
0x2816 <memcmp+6>:      movel fp@(16),d0
0x281a <memcmp+10>:     beq 0x2842 <memcmp+50>
0x281c <memcmp+12>:     moveal fp@(8),a1
0x2820 <memcmp+16>:     moveal fp@(12),a0
0x2824 <memcmp+20>:     andil #255,d1
0x282a <memcmp+26>:     andil #255,d2
0x2830 <memcmp+32>:     cmpmb a0@+,a1@+
0x2832 <memcmp+34>:     beq 0x283e <memcmp+46>
0x2834 <memcmp+36>:     moveb a1@-,d1
0x2836 <memcmp+38>:     moveb a0@-,d2
0x2838 <memcmp+40>:     movel d1,d0
0x283a <memcmp+42>:     subl d2,d0
0x283c <memcmp+44>:     bra 0x2844 <memcmp+52>
0x283e <memcmp+46>:     subl #1,d0
0x2840 <memcmp+48>:     bne 0x2830 <memcmp+32>
0x2842 <memcmp+50>:     clrl d0
0x2844 <memcmp+52>:     movel fp@(-4),d2
0x2848 <memcmp+56>:     unlk fp
0x284a <memcmp+58>:     rts

The machine code of the above program is:

<memcmp>:      0x4e56  0x0000  0x2f02  0x202e  0x0010  0x6726  0x226e  0x0008
<memcmp+16>:   0x206e  0x000c  0x0281  0x0000  0x00ff  0x0282  0x0000  0x00ff
<memcmp+32>:   0xb308  0x670a  0x1221  0x1420  0x2001  0x9082  0x6006  0x5380
<memcmp+48>:   0x66ee  0x4280  0x242e  0xfffc  0x4e5e  0x4e75

'(78      86      0       0       47      2       32      46
  0       16      103     38      34      110     0       8
  32      110     0       12      2       129     0       0
  0       255     2       130     0       0       0       255
  179     8       103     10      18      33      20      32
  32      1       144     130     96      6       83      128
  102     238     66      128     36      46      255     252
  78      94      78      117)
|#

; in the logic, the above program is defined by (memcmp-code).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memcmp-code} \\  
\=\+{\sc{memcmp-code}}\index{memcmp-code} \\ 
$=$$\;\;\;\;${\tt{'}}{\tt{(}}\=\+{\tt{78}}{\tt{ }}{\tt{86}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{47}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{16}}{\tt{ }}{\tt{103}}{\tt{ }}{\tt{38}}{\tt{ }}{\tt{34}}{\tt{ }}{\tt{110}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{110}}{\tt{ }}{\tt{0}} \\ 
{\tt{12}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{129}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{255}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{130}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{255}}{\tt{ }}{\tt{179}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{103}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{18}} \\ 
{\tt{33}}{\tt{ }}{\tt{20}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{144}}{\tt{ }}{\tt{130}}{\tt{ }}{\tt{96}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{83}}{\tt{ }}{\tt{128}}{\tt{ }}{\tt{102}}{\tt{ }}{\tt{238}}{\tt{ }}{\tt{66}}{\tt{ }}{\tt{128}}{\tt{ }}{\tt{36}} \\ 
{\tt{46}}{\tt{ }}{\tt{255}}{\tt{ }}{\tt{252}}{\tt{ }}{\tt{78}}{\tt{ }}{\tt{94}}{\tt{ }}{\tt{78}}{\tt{ }}{\tt{117}}{\tt{)}}\-\-
\end{tabbing}

\begin{verbatim}; the computation time of the program.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memcmp-t1} \\  
\=\+{\rm{memcmp-t1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-t1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth} \\ 
{\bf then }\=\+{\bf if }({\it{n\/}} $-\;1$) $=$ {\tt{0}}$\;\;${\bf then }{\tt{8}} \\ 
{\bf else }{\rm{splus}}\,({\tt{4}}, {\rm{memcmp-t1}}\,($1\;+$ {\it{i\/}}, {\it{n\/}} $-\;1$, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-t1})\index{splus}$\;${\bf  endif}\- \\ 
{\bf else }{\tt{10}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{memcmp-t} \\  
\=\+{\rm{memcmp-t}}\,({\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-t} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $=$ {\tt{0}}$\;\;${\bf then }{\tt{8}} \\ 
{\bf else }{\rm{splus}}\,({\tt{8}}, {\rm{memcmp-t1}}\,({\tt{0}}, {\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-t1})\index{splus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; an induction hint.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memcmp-induct} \\  
\=\+{\rm{memcmp-induct}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-induct} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth} \\ 
{\bf then }\=\+{\bf if }({\it{n\/}} $-\;1$) $=$ {\tt{0}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{memcmp-induct}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i\/}}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\it{lst1\/}}, \\ 
{\it{lst2\/}})\-\index{memcmp-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{memcmp-statep} \\  
\=\+{\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-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$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{60}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{memcmp-code}}\index{memcmp-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{24}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}}, {\it{lst1\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}}, {\it{lst2\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}}, {\it{str1\/}}, {\it{n\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}}, {\it{str2\/}}, {\it{n\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;$({\it{str1\/}} $=$ {\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$$\;\;\;\;$({\it{str2\/}} $=$ {\rm{read-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{8}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{read-mem}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\rm{uread-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{12}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{uread-mem}))\-\-
\end{tabbing}

\begin{verbatim}; an intermediate state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memcmp-s0p} \\  
\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-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$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{32}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{60}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{32}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{memcmp-code}}\index{memcmp-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{24}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\_\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\_\/}}, {\it{lst1\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\_\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\_\/}}, {\it{lst2\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\tt{24}}, {\it{str1\/}}, {\it{n\_\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\tt{24}}, {\it{str2\/}}, {\it{n\_\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{equal*}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{1}}, {\it{s\/}})\index{read-an}, {\rm{add}}\,({\tt{32}}, {\it{str1\/}}, {\it{i*\/}})\index{add})\index{equal*} \\ 
$\wedge$$\;\;\;\;${\rm{equal*}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{0}}, {\it{s\/}})\index{read-an}, {\rm{add}}\,({\tt{32}}, {\it{str2\/}}, {\it{i*\/}})\index{add})\index{equal*} \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\rm{read-rn}}\,({\tt{32}}, {\tt{1}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn}, {\tt{8}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\rm{read-rn}}\,({\tt{32}}, {\tt{2}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn}, {\tt{8}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{s\/}})\index{read-dn})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $=$ {\rm{nat-to-uint}}\,({\it{i*\/}})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$(({\it{i\/}} $+$ {\it{n\/}}) $\leq$ {\it{n\_\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{i*\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{i*\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n\_\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; from the initial state s to exit: s --> sn, when n = 0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memcmp-s-sn \\ \index{memcmp-s-sn}
\=\+({\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-statep} $\wedge$ ({\it{n\/}} $=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-pc} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{iread-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{iread-dn} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\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{8}})\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}:  memcmp-s-sn-rfile \\ \index{memcmp-s-sn-rfile}
\=\+(\=\+{\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{oplen\/}} $\leq$ {\tt{32}}) \\ 
$\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{8}})\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}:  memcmp-s-sn-mem \\ \index{memcmp-s-sn-mem}
\=\+(\=\+{\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\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}:  memcmp-s-s0 \\ \index{memcmp-s-s0}
\=\+({\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{memcmp-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn}, {\tt{0}}, {\tt{0}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\/}})\index{memcmp-s0p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memcmp-s-s0-else \\ \index{memcmp-s-s0-else}
\=\+({\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{linked-a6} $=$ {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub})\- \\ 
$\wedge$$\;\;\;\;$({\rm{rn-saved}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{rn-saved} $=$ {\rm{read-dn}}\,({\tt{32}}, {\tt{2}}, {\it{s\/}})\index{read-dn}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memcmp-s-s0-rfile \\ \index{memcmp-s-s0-rfile}
\=\+({\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}}) $\wedge$ {\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memcmp-s-s0-mem \\ \index{memcmp-s-s0-mem}
\=\+(\=\+{\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\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: s0 --> sn.
; base case 1:  s0 --> sn, when lst1[i] =\= lst2[i].
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memcmp-s0-sn-base1 \\ \index{memcmp-s0-sn-base1}
\=\+(\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $\not=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{10}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{10}})\index{stepn})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{iread-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{10}})\index{stepn})\index{iread-dn} \\ 
$=$$\;\;\;\;${\rm{idifference}}\,({\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth}, {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth})\index{idifference})\- \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{10}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{10}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{10}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}
		   
\begin{tabbing}{\sc Theorem}:  memcmp-s0-sn-rfile-base1 \\ \index{memcmp-s0-sn-rfile-base1}
\=\+(\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $\not=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth}) \\ 
$\wedge$$\;\;\;\;$({\it{oplen\/}} $\leq$ {\tt{32}}) \\ 
$\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{10}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{head}}\,({\rm{rn-saved}}\,({\it{s\/}})\index{rn-saved}, {\it{oplen\/}})\index{head}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{verbatim}; base case 2: s0 --> sn, when lst[i] = lst2[i], lst[i] =\= 0, and n-1 = 0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memcmp-s0-sn-base2 \\ \index{memcmp-s0-sn-base2}
\=\+(\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $=$ {\tt{0}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{iread-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{iread-dn} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}			  

\begin{tabbing}{\sc Theorem}:  memcmp-s0-sn-rfile-base2 \\ \index{memcmp-s0-sn-rfile-base2}
\=\+(\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{oplen\/}} $\leq$ {\tt{32}}) \\ 
$\wedge$$\;\;\;\;${\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{head}}\,({\rm{rn-saved}}\,({\it{s\/}})\index{rn-saved}, {\it{oplen\/}})\index{head}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{verbatim}; induction case: s0 --> s0, lst[i] = lst2[i], lst[i] =\= 0 and n-1 =\= 0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memcmp-s0-s0 \\ \index{memcmp-s0-s0}
\=\+(\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $\not=$ {\tt{0}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{memcmp-s0p}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i\/}}, \\ 
{\it{str1\/}}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\it{lst1\/}}, \\ 
{\it{str2\/}}, \\ 
{\it{lst2\/}}, \\ 
{\it{n\_\/}})\-\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\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{4}})\index{stepn})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{rn-saved}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\index{stepn})\index{rn-saved} $=$ {\rm{rn-saved}}\,({\it{s\/}})\index{rn-saved}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}			  

\begin{tabbing}{\sc Theorem}:  memcmp-s0-s0-rfile \\ \index{memcmp-s0-s0-rfile}
\=\+(\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst1\/}})\index{get-nth} $=$ {\rm{get-nth}}\,({\it{i\/}}, {\it{lst2\/}})\index{get-nth}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\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}:  memcmp-s0-sn \\ \index{memcmp-s0-sn}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memcmp-t1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-t1})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-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$$\;\;\;\;$({\rm{iread-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{iread-dn} $=$ {\rm{memcmp1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp1}) \\ 
$\wedge$$\;\;\;\;$({\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}

\begin{tabbing}{\sc Theorem}:  memcmp-s0-sn-rfile \\ \index{memcmp-s0-sn-rfile}
\=\+(\=\+{\rm{memcmp-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}}, {\it{n\_\/}})\index{memcmp-s0p} \\ 
$\wedge$$\;\;\;\;${\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p} \\ 
$\wedge$$\;\;\;\;$({\it{oplen\/}} $\leq$ {\tt{32}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{memcmp-t1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-t1})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{head}}\,({\rm{rn-saved}}\,({\it{s\/}})\index{rn-saved}, {\it{oplen\/}})\index{head}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{verbatim}; the correctness of memcmp.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memcmp-correctness \\ \index{memcmp-correctness}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memcmp-t}}\,({\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{memcmp-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{lst2\/}})\index{memcmp-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-an}}\,({\tt{32}}, {\tt{7}}, {\it{s\/}})\index{read-an}, {\tt{4}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+({\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p} $\wedge$ ({\it{oplen\/}} $\leq$ {\tt{32}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\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{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}})\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{iread-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{iread-dn} $=$ {\rm{memcmp}}\,({\it{n\/}}, {\it{lst1\/}}, {\it{lst2\/}})\index{memcmp}))\-\-$\;${\bf  endlet}\-
\end{tabbing}


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

\addvspace{18pt}

\begin{verbatim}; some properties of memcmp.
; see file cstring.events.\end{verbatim}
%\end{list}
\printindex
\end{document}
