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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 MEMCHR 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 memchr function in the Berkeley string library.

void *
memchr(s, c, n)
	const void *s;
	register unsigned char c;
	register size_t n;
{
	if (n != 0) {
		register const unsigned char *p = s;

		do {
			if (*p++ == c)
				return ((void *)(p - 1));
		} while (--n != 0);
	}
	return (NULL);
}

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

0x27e8 <memchr>:        linkw fp,#0
0x27ec <memchr+4>:      moveb fp@(15),d1
0x27f0 <memchr+8>:      movel fp@(16),d0
0x27f4 <memchr+12>:     beq 0x2808 <memchr+32>
0x27f6 <memchr+14>:     moveal fp@(8),a0
0x27fa <memchr+18>:     cmpb a0@+,d1
0x27fc <memchr+20>:     bne 0x2804 <memchr+28>
0x27fe <memchr+22>:     movel a0,d0
0x2800 <memchr+24>:     subl #1,d0
0x2802 <memchr+26>:     bra 0x280a <memchr+34>
0x2804 <memchr+28>:     subl #1,d0
0x2806 <memchr+30>:     bne 0x27fa <memchr+18>
0x2808 <memchr+32>:     clrl d0
0x280a <memchr+34>:     unlk fp
0x280c <memchr+36>:     rts

The machine code of the above program is:

<memchr>:        0x4e56  0x0000  0x122e  0x000f  0x202e  0x0010  0x6712  0x206e
<memchr+16>:     0x0008  0xb218  0x6606  0x2008  0x5380  0x6006  0x5380  0x66f2
<memchr+32>:     0x4280  0x4e5e  0x4e75

'(78      86      0       0       18      46      0       15
  32      46      0       16      103     18      32      110
  0       8       178     24      102     6       32      8
  83      128     96      6       83      128     102     242
  66      128     78      94      78      117)
|#

; in the logic, the above program is defined by (memchr-code).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memchr-code} \\  
\=\+{\sc{memchr-code}}\index{memchr-code} \\ 
$=$$\;\;\;\;${\tt{'}}{\tt{(}}\=\+{\tt{78}}{\tt{ }}{\tt{86}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{18}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{15}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{16}}{\tt{ }}{\tt{103}}{\tt{ }}{\tt{18}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{110}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{8}} \\ 
{\tt{178}}{\tt{ }}{\tt{24}}{\tt{ }}{\tt{102}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{83}}{\tt{ }}{\tt{128}}{\tt{ }}{\tt{96}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{83}}{\tt{ }}{\tt{128}}{\tt{ }}{\tt{102}}{\tt{ }}{\tt{242}}{\tt{ }}{\tt{66}}{\tt{ }}{\tt{128}} \\ 
{\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{memchr-t1} \\  
\=\+{\rm{memchr-t1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\it{ch\/}}$\;\;${\bf then }{\tt{7}} \\ 
{\bf elseif }({\it{n\/}} $-\;1$) $=$ {\tt{0}}$\;\;${\bf then }{\tt{7}} \\ 
{\bf else }{\rm{splus}}\,({\tt{4}}, {\rm{memchr-t1}}\,($1\;+$ {\it{i\/}}, {\it{n\/}} $-\;1$, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t1})\index{splus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{memchr-t} \\  
\=\+{\rm{memchr-t}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $=$ {\tt{0}}$\;\;${\bf then }{\tt{7}} \\ 
{\bf else }{\rm{splus}}\,({\tt{5}}, {\rm{memchr-t1}}\,({\tt{0}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t1})\index{splus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; an induction hint.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memchr-induct} \\  
\=\+{\rm{memchr-induct}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-induct} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\it{ch\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }({\it{n\/}} $-\;1$) $=$ {\tt{0}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{memchr-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{lst\/}}, {\it{ch\/}})\index{memchr-induct}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; the preconditions of the initial state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memchr-statep} \\  
\=\+{\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-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$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{38}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{memchr-code}}\index{memchr-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{20}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\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}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{20}}, {\it{str\/}}, {\it{n\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;$({\it{str\/}} $=$ {\rm{read-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{4}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{read-mem}) \\ 
$\wedge$$\;\;\;\;$({\it{ch\/}} $=$ {\rm{uread-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{11}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{1}})\index{uread-mem}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\rm{uread-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{12}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{uread-mem}) \\ 
$\wedge$$\;\;\;\;$({\rm{nat-to-uint}}\,({\it{str\/}})\index{nat-to-uint} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\rm{nat-to-uint}}\,({\it{str\/}})\index{nat-to-uint} $+$ {\it{n\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; an intermediate state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memchr-s0p} \\  
\=\+{\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-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$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{18}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{38}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{18}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{memchr-code}}\index{memchr-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{20}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\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}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{20}}, {\it{str\/}}, {\it{n\_\/}})\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{ch\/}} $=$ {\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{8}}, {\tt{1}}, {\it{s\/}})\index{read-dn})\index{nat-to-uint}) \\ 
$\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$$\;\;\;\;$({\it{i*\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{n\_\/}} $\in\;${\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 intial state s to exit: s --> sn.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memchr-s-sn \\ \index{memchr-s-sn}
\=\+({\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} $\wedge$ ({\it{n\/}} $=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-pc} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{read-dn} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\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{7}})\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}:  memchr-s-sn-rfile \\ \index{memchr-s-sn-rfile}
\=\+({\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} $\wedge$ ({\it{n\/}} $=$ {\tt{0}}) $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memchr-s-sn-mem \\ \index{memchr-s-sn-mem}
\=\+(\=\+{\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{20}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\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}:  memchr-s-s0 \\ \index{memchr-s-s0}
\=\+({\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{memchr-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn}, {\tt{0}}, {\tt{0}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\/}})\index{memchr-s0p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memchr-s-s0-else \\ \index{memchr-s-s0-else}
\=\+({\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn})\index{linked-a6} $=$ {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub})\-)\-\-
\end{tabbing}

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

\begin{tabbing}{\sc Theorem}:  memchr-s-s0-mem \\ \index{memchr-s-s0-mem}
\=\+(\=\+{\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{20}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\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 lst[i] = ch.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memchr-s0-sn-base1 \\ \index{memchr-s0-sn-base1}
\=\+({\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} $\wedge$ ({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\it{ch\/}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{read-dn} $=$ {\rm{add}}\,({\tt{32}}, {\it{str\/}}, {\it{i*\/}})\index{add}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memchr-s0-sn-rfile-base1 \\ \index{memchr-s0-sn-rfile-base1}
\=\+(\=\+{\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{verbatim}; base case 2: s0 --> sn, when lst[i] =\= ch, n-1 = 0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memchr-s0-sn-base2 \\ \index{memchr-s0-sn-base2}
\=\+(\=\+{\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $=$ {\tt{0}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{read-dn} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memchr-s0-sn-rfile-base2 \\ \index{memchr-s0-sn-rfile-base2}
\=\+(\=\+{\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $=$ {\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{7}})\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}:  memchr-s0-s0 \\ \index{memchr-s0-s0}
\=\+(\=\+{\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $\not=$ {\tt{0}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{memchr-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{4}})\index{stepn}, {\rm{add}}\,({\tt{32}}, {\it{i*\/}}, {\tt{1}})\index{add}, $1\;+$ {\it{i\/}}, {\it{str\/}}, {\it{n\/}} $-\;1$, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\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{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}:  memchr-s0-s0-rfile \\ \index{memchr-s0-s0-rfile}
\=\+(\=\+{\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i\/}}, {\it{lst\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $\not=$ {\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{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}:  memchr-s0-sn \\ \index{memchr-s0-sn}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memchr-t1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t1})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-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$$\;\;\;\;$(\=\+{\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{read-dn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{memchr1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr1} \\ 
{\bf then }{\rm{add}}\,({\tt{32}}, {\it{str\/}}, {\rm{memchr*}}\,({\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr*})\index{add} \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-)\- \\ 
$\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}:  memchr-s0-sn-rfile \\ \index{memchr-s0-sn-rfile}
\=\+({\rm{memchr-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memchr-s0p} $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{memchr-t1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t1})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\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 MEMCHR program.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memchr-correctness \\ \index{memchr-correctness}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memchr-t}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-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{20}})\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$$\;\;\;\;$(\=\+{\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{read-dn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{memchr}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr} \\ 
{\bf then }{\rm{add}}\,({\tt{32}}, {\it{str\/}}, {\rm{memchr*}}\,({\tt{0}}, {\tt{0}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr*})\index{add} \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-)\-)\-\-$\;${\bf  endlet}\-
\end{tabbing}


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

\addvspace{18pt}

\begin{verbatim}; memchr* --> memchr.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memchr*-memchr1 \\ \index{memchr*-memchr1}
\=\+(\=\+{\rm{memchr1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr1} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $=$ {\rm{nat-to-uint}}\,({\it{i*\/}})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{i*\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{i\/}} $+$ {\it{n\/}}, {\tt{32}})\index{uint-rangep})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nat-to-uint}}\,({\rm{memchr*}}\,({\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr*})\index{nat-to-uint} $=$ {\rm{memchr1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr1})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memchr-non-zerop-la \\ \index{memchr-non-zerop-la}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memchr-t}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+(\=\+{\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{str\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;$({\rm{nat-to-uint}}\,({\it{str\/}})\index{nat-to-uint} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\rm{nat-to-uint}}\,({\it{str\/}})\index{nat-to-uint} $+$ {\it{n\/}}, {\tt{32}})\index{uint-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{memchr}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{read-dn})\index{nat-to-uint} $\not=$ {\tt{0}})\-$\;${\bf  endlet}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memchr-non-zerop \\ \index{memchr-non-zerop}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memchr-t}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+({\rm{memchr-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr-statep} $\wedge$ {\rm{memchr}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memchr}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{read-dn})\index{nat-to-uint} $\not=$ {\tt{0}})\-$\;${\bf  endlet}\-
\end{tabbing}


\noindent{\sc Event}:   Disable memchr*.

\addvspace{18pt}

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