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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 STRCSPN 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 strcspn function in the Berkeley string library.

size_t
strcspn(s1, s2)
	const char *s1;
	register const char *s2;
{
	register const char *p, *spanp;
	register char c, sc;

	/*
	 * Stop as soon as we find any character from s2.  Note that there
	 * must be a NUL in s2; it suffices to stop when we find that, too.
	 */
	for (p = s1;;) {
		c = *p++;
		spanp = s2;
		do {
			if ((sc = *spanp++) == c)
				return (p - 1 - s1);
		} while (sc != 0);
	}
	/* NOTREACHED */
}

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

0x2578 <strcspn>:       linkw fp,#0
0x257c <strcspn+4>:     moveml d2-d3,sp@-
0x2580 <strcspn+8>:     movel fp@(8),d0
0x2584 <strcspn+12>:    movel fp@(12),d3
0x2588 <strcspn+16>:    moveal d0,a1
0x258a <strcspn+18>:    movel d0,d2
0x258c <strcspn+20>:    addql #1,d2
0x258e <strcspn+22>:    moveb a1@+,d1
0x2590 <strcspn+24>:    moveal d3,a0
0x2592 <strcspn+26>:    moveb a0@+,d0
0x2594 <strcspn+28>:    cmpb d0,d1
0x2596 <strcspn+30>:    beq 0x259e <strcspn+38>
0x2598 <strcspn+32>:    tstb d0
0x259a <strcspn+34>:    bne 0x2592 <strcspn+26>
0x259c <strcspn+36>:    bra 0x258e <strcspn+22>
0x259e <strcspn+38>:    movel a1,d0
0x25a0 <strcspn+40>:    subl d2,d0
0x25a2 <strcspn+42>:    moveml fp@(-8),d2-d3
0x25a8 <strcspn+48>:    unlk fp
0x25aa <strcspn+50>:    rts

The machine code of the above program is:

<strcspn>:     0x4e56  0x0000  0x48e7  0x3000  0x202e  0x0008  0x262e  0x000c
<strcspn+16>:  0x2240  0x2400  0x5282  0x1219  0x2043  0x1018  0xb200  0x6706
<strcspn+32>:  0x4a00  0x66f6  0x60f0  0x2009  0x9082  0x4cee  0x000c  0xfff8
<strcspn+48>:  0x4e5e  0x4e75

'(78      86      0       0       72      231     48      0
  32      46      0       8       38      46      0       12
  34      64      36      0       82      130     18      25
  32      67      16      24      178     0       103     6
  74      0       102     246     96      240     32      9
  144     130     76      238     0       12      255     248
  78      94      78      117)
|#

; in the logic, the above program is defined by (strcspn-code).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strcspn-code} \\  
\=\+{\sc{strcspn-code}}\index{strcspn-code} \\ 
$=$$\;\;\;\;${\tt{'}}{\tt{(}}\=\+{\tt{78}}{\tt{ }}{\tt{86}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{72}}{\tt{ }}{\tt{231}}{\tt{ }}{\tt{48}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{38}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{12}}{\tt{ }}{\tt{34}}{\tt{ }}{\tt{64}}{\tt{ }}{\tt{36}} \\ 
{\tt{0}}{\tt{ }}{\tt{82}}{\tt{ }}{\tt{130}}{\tt{ }}{\tt{18}}{\tt{ }}{\tt{25}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{67}}{\tt{ }}{\tt{16}}{\tt{ }}{\tt{24}}{\tt{ }}{\tt{178}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{103}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{74}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{102}}{\tt{ }}{\tt{246}} \\ 
{\tt{96}}{\tt{ }}{\tt{240}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{144}}{\tt{ }}{\tt{130}}{\tt{ }}{\tt{76}}{\tt{ }}{\tt{238}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{12}}{\tt{ }}{\tt{255}}{\tt{ }}{\tt{248}}{\tt{ }}{\tt{78}}{\tt{ }}{\tt{94}}{\tt{ }}{\tt{78}}{\tt{ }}{\tt{117}}{\tt{)}}\-\-
\end{tabbing}

\begin{verbatim}; the computatin time of the program.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strcspn-t0} \\  
\=\+{\rm{strcspn-t0}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t0} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i2\/}} $<$ {\it{n2\/}} \\ 
{\bf then }\=\+{\bf if }{\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\it{ch\/}}$\;\;${\bf then }{\tt{8}} \\ 
{\bf elseif }{\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\tt{0}}$\;\;${\bf then }{\tt{6}} \\ 
{\bf else }{\rm{splus}}\,({\tt{5}}, {\rm{strcspn-t0}}\,($1\;+$ {\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t0})\index{splus}$\;${\bf  endif}\- \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{strcspn-t1} \\  
{\rm{strcspn-t1}}\,({\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t1} $=$ {\rm{splus}}\,({\tt{2}}, {\rm{strcspn-t0}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t0})\index{splus}
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{strcspn-t2} \\  
\=\+{\rm{strcspn-t2}}\,({\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-t2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i1\/}} $<$ {\it{n1\/}} \\ 
{\bf then }\=\+{\bf if }{\rm{strchr}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strchr} \\ 
{\bf then }{\rm{strcspn-t1}}\,({\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strcspn-t1} \\ 
{\bf else }{\rm{splus}}\,(\=\+{\rm{strcspn-t1}}\,({\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strcspn-t1}, \\ 
{\rm{strcspn-t2}}\,($1\;+$ {\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-t2})\-\index{splus}$\;${\bf  endif}\- \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{strcspn-t} \\  
{\rm{strcspn-t}}\,({\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-t} $=$ {\rm{splus}}\,({\tt{7}}, {\rm{strcspn-t2}}\,({\tt{0}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-t2})\index{splus}
\end{tabbing}

\begin{verbatim}; two induction hints.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strcspn-induct0} \\  
\=\+{\rm{strcspn-induct0}}\,({\it{s\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-induct0} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i2\/}} $<$ {\it{n2\/}} \\ 
{\bf then }\=\+{\bf if }{\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\it{ch\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\tt{0}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{strcspn-induct0}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i2*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i2\/}}, \\ 
{\it{n2\/}}, \\ 
{\it{lst2\/}}, \\ 
{\it{ch\/}})\-\index{strcspn-induct0}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{strcspn-induct1} \\  
\=\+{\rm{strcspn-induct1}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-induct1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i1\/}} $<$ {\it{n1\/}} \\ 
{\bf then }\=\+{\bf if }{\rm{strchr}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strchr}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{strcspn-induct1}}\,(\=\+{\rm{stepn}}\,(\=\+{\it{s\/}}, \\ 
{\rm{strcspn-t1}}\,(\=\+{\it{n2\/}}, \\ 
{\it{lst2\/}}, \\ 
{\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\-\index{strcspn-t1})\-\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i1*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i1\/}}, \\ 
{\it{n1\/}}, \\ 
{\it{lst1\/}}, \\ 
{\it{n2\/}}, \\ 
{\it{lst2\/}})\-\index{strcspn-induct1}$\;${\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{strcspn-statep} \\  
\=\+{\rm{strcspn-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-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$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{52}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{strcspn-code}}\index{strcspn-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{12}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{24}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n1\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n1\/}}, {\it{lst1\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n2\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n2\/}}, {\it{lst2\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{12}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}}, {\it{str1\/}}, {\it{n1\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{12}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}}, {\it{str2\/}}, {\it{n2\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;$({\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$$\;\;\;\;$({\it{str2\/}} $=$ {\rm{read-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{8}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{read-mem}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\tt{0}}, {\it{n1\/}}, {\it{lst1\/}})\index{slen} $<$ {\it{n1\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}})\index{slen} $<$ {\it{n2\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n1\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{n2\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n1\/}}, {\tt{32}})\index{uint-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n2\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; an intermediate state s0.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strcspn-s0p} \\  
\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-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$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{22}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{52}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{22}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{strcspn-code}}\index{strcspn-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{24}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n1\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n1\/}}, {\it{lst1\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n2\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n2\/}}, {\it{lst2\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\tt{24}}, {\it{str1\/}}, {\it{n1\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\tt{24}}, {\it{str2\/}}, {\it{n2\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{equal*}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{1}}, {\it{s\/}})\index{read-an}, {\rm{add}}\,({\tt{32}}, {\it{str1\/}}, {\it{i1*\/}})\index{add})\index{equal*} \\ 
$\wedge$$\;\;\;\;$({\it{str2\/}} $=$ {\rm{read-dn}}\,({\tt{32}}, {\tt{3}}, {\it{s\/}})\index{read-dn}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{2}}, {\it{s\/}})\index{read-dn} $=$ {\rm{add}}\,({\tt{32}}, {\it{str1\/}}, {\tt{1}})\index{add}) \\ 
$\wedge$$\;\;\;\;$({\it{i1*\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{i1*\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;$({\it{i1\/}} $=$ {\rm{nat-to-uint}}\,({\it{i1*\/}})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}})\index{slen} $<$ {\it{n1\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}})\index{slen} $<$ {\it{n2\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n1\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{n2\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n1\/}}, {\tt{32}})\index{uint-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n2\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; an intermediate state s1.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{strcspn-s1-1p} \\  
\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$=$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{s\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;${\rm{evenp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{evenp} \\ 
$\wedge$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{26}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{52}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{26}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{strcspn-code}}\index{strcspn-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{24}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n2\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str2\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n2\/}}, {\it{lst2\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\tt{24}}, {\it{str2\/}}, {\it{n2\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{equal*}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{1}}, {\it{s\/}})\index{read-an}, {\rm{add}}\,({\tt{32}}, {\it{str1\/}}, {\it{i1*\/}})\index{add})\index{equal*} \\ 
$\wedge$$\;\;\;\;${\rm{equal*}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{0}}, {\it{s\/}})\index{read-an}, {\rm{add}}\,({\tt{32}}, {\it{str2\/}}, {\it{i2*\/}})\index{add})\index{equal*} \\ 
$\wedge$$\;\;\;\;$({\it{str2\/}} $=$ {\rm{read-dn}}\,({\tt{32}}, {\tt{3}}, {\it{s\/}})\index{read-dn}) \\ 
$\wedge$$\;\;\;\;$({\it{ch\/}} $=$ {\rm{uread-dn}}\,({\tt{8}}, {\tt{1}}, {\it{s\/}})\index{uread-dn}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{2}}, {\it{s\/}})\index{read-dn} $=$ {\rm{add}}\,({\tt{32}}, {\it{str1\/}}, {\tt{1}})\index{add}) \\ 
$\wedge$$\;\;\;\;$({\it{i1*\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{i1*\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;$({\it{i1\/}} $=$ {\rm{nat-to-uint}}\,({\it{i1*\/}})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$({\it{i2*\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{i2*\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;$({\it{i2\/}} $=$ {\rm{nat-to-uint}}\,({\it{i2*\/}})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}})\index{slen} $<$ {\it{n2\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{slen} $<$ {\it{n2\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n2\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n2\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{strcspn-s1p} \\  
\=\+{\rm{strcspn-s1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1p} \\ 
$=$$\;\;\;\;$(\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n1\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str1\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n1\/}}, {\it{lst1\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\tt{24}}, {\it{str1\/}}, {\it{n1\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;$({\rm{slen}}\,({\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}})\index{slen} $<$ {\it{n1\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n1\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n1\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; from the initial state s to s0: s --> s0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s-s0 \\ \index{strcspn-s-s0}
\=\+{\rm{strcspn-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-statep} \\ 
$\rightarrow$$\;\;\;\;${\rm{strcspn-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn}, {\tt{0}}, {\tt{0}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s-s0-else \\ \index{strcspn-s-s0-else}
\=\+{\rm{strcspn-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-statep} \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{linked-a6} $=$ {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{movem-saved}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved} \\ 
$=$$\;\;\;\;${\rm{readm-rn}}\,({\tt{32}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{readm-rn})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s-s0-rfile \\ \index{strcspn-s-s0-rfile}
\=\+({\rm{strcspn-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-statep} $\wedge$ {\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p}) \\ 
$\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}:  strcspn-s-s0-mem \\ \index{strcspn-s-s0-mem}
\=\+(\=\+{\rm{strcspn-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-statep} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{12}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{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}; loop 0.
; from s0 to s1: s0 --> s1.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s0-s1-1 \\ \index{strcspn-s0-s1-1}
\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\rightarrow$$\;\;\;\;${\rm{strcspn-s1-1p}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{2}})\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i1*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i1\/}}, \\ 
{\it{str1\/}}, \\ 
{\it{n1\/}}, \\ 
{\it{lst1\/}}, \\ 
{\tt{0}}, \\ 
{\tt{0}}, \\ 
{\it{str2\/}}, \\ 
{\it{n2\/}}, \\ 
{\it{lst2\/}}, \\ 
{\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\-\index{strcspn-s1-1p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strchr-la \\ \index{strchr-la}
(({\rm{slen}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{slen} $<$ {\it{n2\/}}) $\wedge$ ($\neg$ {\rm{strchr}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strchr})) $\rightarrow$ ({\it{ch\/}} $\not=$ {\tt{0}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s0-s1 \\ \index{strcspn-s0-s1}
\=\+(\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{strchr}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strchr}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{strcspn-s1p}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{2}})\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i1*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i1\/}}, \\ 
{\it{str1\/}}, \\ 
{\it{n1\/}}, \\ 
{\it{lst1\/}}, \\ 
{\tt{0}}, \\ 
{\tt{0}}, \\ 
{\it{str2\/}}, \\ 
{\it{n2\/}}, \\ 
{\it{lst2\/}}, \\ 
{\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\-\index{strcspn-s1p}\-
\end{tabbing}
	      
\begin{tabbing}{\sc Theorem}:  strcspn-s0-s1-else \\ \index{strcspn-s0-s1-else}
\=\+{\bf let }\=\+{\it{s1\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\tt{2}})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s1\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\it{s1\/}})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\it{s1\/}})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{movem-saved}}\,({\it{s1\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved} $=$ {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s1\/}})\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}:  strcspn-s0-s1-rfile \\ \index{strcspn-s0-s1-rfile}
\=\+({\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} $\wedge$ {\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{2}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{verbatim}; loop 1.
; base case 1. from s1 to exit: s1 --> sn, when lst1[i] = lst2[j].
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s1-sn-base \\ \index{strcspn-s1-sn-base}
\=\+(\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\it{ch\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\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{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{8}})\index{stepn})\index{read-dn} $=$ {\rm{sub}}\,({\tt{32}}, {\tt{1}}, {\it{i1*\/}})\index{sub}) \\ 
$\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}:  strcspn-s1-sn-rfile-base \\ \index{strcspn-s1-sn-rfile-base}
\=\+(\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$({\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} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{get-vlst}}\,({\it{oplen\/}}, {\tt{0}}, {\it{rn\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}}, {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved})\index{get-vlst}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{verbatim}; base case 2.  s1 --> s0, when lst1[i] =\= lst2[j] and lst2[j] = 0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s1-s0-base \\ \index{strcspn-s1-s0-base}
\=\+(\=\+{\rm{strcspn-s1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\tt{0}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{strcspn-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{movem-saved}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{6}})\index{stepn}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved} $=$ {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved}) \\ 
$\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}:  strcspn-s1-s0-rfile-base \\ \index{strcspn-s1-s0-rfile-base}
\=\+(\=\+{\rm{strcspn-s1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\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{verbatim}; induction case. s1 --> s1, when lst1[i] =\= lst2[j] and lst2[j] =\= 0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s1-s1-1 \\ \index{strcspn-s1-s1-1}
\=\+(\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\tt{0}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{strcspn-s1-1p}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn}, \\ 
{\it{i1*\/}}, \\ 
{\it{i1\/}}, \\ 
{\it{str1\/}}, \\ 
{\it{n1\/}}, \\ 
{\it{lst1\/}}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i2*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i2\/}}, \\ 
{\it{str2\/}}, \\ 
{\it{n2\/}}, \\ 
{\it{lst2\/}}, \\ 
{\it{ch\/}})\-\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{movem-saved}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved} $=$ {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s1-s1 \\ \index{strcspn-s1-s1}
\=\+(\=\+{\rm{strcspn-s1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\tt{0}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{strcspn-s1p}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{5}})\index{stepn}, \\ 
{\it{i1*\/}}, \\ 
{\it{i1\/}}, \\ 
{\it{str1\/}}, \\ 
{\it{n1\/}}, \\ 
{\it{lst1\/}}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i2*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i2\/}}, \\ 
{\it{str2\/}}, \\ 
{\it{n2\/}}, \\ 
{\it{lst2\/}}, \\ 
{\it{ch\/}})\-\index{strcspn-s1p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s1-s1-rfile \\ \index{strcspn-s1-s1-rfile}
\=\+(\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\it{ch\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{get-nth}}\,({\it{i2\/}}, {\it{lst2\/}})\index{get-nth} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\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{verbatim}; put together.
; case 1. s1 --> exit, when (strchr i2 n2 lst2 ch).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s1-1p-info \\ \index{strcspn-s1-1p-info}
\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\rightarrow$$\;\;\;\;$(({\it{i2\/}} $<$ {\it{n2\/}}) $=$ {\bf{t}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s1-sn \\ \index{strcspn-s1-sn}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t0}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t0})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+(\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;${\rm{strchr}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strchr})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\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} $=$ {\rm{sub}}\,({\tt{32}}, {\tt{1}}, {\it{i1*\/}})\index{sub}) \\ 
$\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}:  strcspn-s1-sn-rfile \\ \index{strcspn-s1-sn-rfile}
\=\+(\=\+{\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p} \\ 
$\wedge$$\;\;\;\;${\rm{strchr}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strchr} \\ 
$\wedge$$\;\;\;\;${\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{strcspn-t0}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t0})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{get-vlst}}\,({\it{oplen\/}}, {\tt{0}}, {\it{rn\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}}, {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved})\index{get-vlst}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{verbatim}; case 2. s1 --> s0, when (not (strchr i2 n2 lst2 ch)).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s1p-s1-1p \\ \index{strcspn-s1p-s1-1p}
\=\+{\rm{strcspn-s1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1p} \\ 
$\rightarrow$$\;\;\;\;${\rm{strcspn-s1-1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1-1p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s1-s0 \\ \index{strcspn-s1-s0}
\=\+{\bf let }\=\+{\it{s0\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t0}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t0})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+(\=\+{\rm{strcspn-s1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1p} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{strchr}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strchr}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{strcspn-s0p}}\,({\it{s0\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s0\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\it{s0\/}})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\it{s0\/}})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{movem-saved}}\,({\it{s0\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved} $=$ {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s0\/}})\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 strcspn-s1-1p-info.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  strcspn-s1-s0-rfile \\ \index{strcspn-s1-s0-rfile}
\=\+(\=\+{\rm{strcspn-s1p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{i2*\/}}, {\it{i2\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-s1p} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{strchr}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strchr}) \\ 
$\wedge$$\;\;\;\;${\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t0}}\,({\it{i2\/}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t0})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{verbatim}; from s0 --> exit. s0 --> sn.
; base case:  s0 --> sn, when (strchr i2 n2 lst2 ch).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s0-sn-base \\ \index{strcspn-s0-sn-base}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t1}}\,({\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strcspn-t1})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+(\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;${\rm{strchr}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strchr})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\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} $=$ {\rm{head}}\,({\it{i1*\/}}, {\tt{32}})\index{head}) \\ 
$\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}:  strcspn-s0-sn-rfile-base \\ \index{strcspn-s0-sn-rfile-base}
\=\+{\bf let }\=\+{\it{ch\/}}{\bf $\;$ be$\;\;$}{\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth}\- \\ 
{\bf in} \\ 
\=\+(\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;${\rm{strchr}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strchr} \\ 
$\wedge$$\;\;\;\;$({\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\/}}, {\rm{strcspn-t1}}\,({\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t1})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{get-vlst}}\,(\=\+{\it{oplen\/}}, \\ 
{\tt{0}}, \\ 
{\it{rn\/}}, \\ 
{\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}}, \\ 
{\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved})\-\index{get-vlst}$\;${\bf  endif}\-)\-\-$\;${\bf  endlet}\-
\end{tabbing}

\begin{verbatim}; induction case: s0 --> s0, when (not (strchr i2 n2 lst2 ch)).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s0-s0 \\ \index{strcspn-s0-s0}
\=\+{\bf let }\=\+{\it{s0\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t1}}\,({\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strcspn-t1})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+(\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{strchr}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth})\index{strchr}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{strcspn-s0p}}\,(\=\+{\it{s0\/}}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i1*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i1\/}}, \\ 
{\it{str1\/}}, \\ 
{\it{n1\/}}, \\ 
{\it{lst1\/}}, \\ 
{\it{str2\/}}, \\ 
{\it{n2\/}}, \\ 
{\it{lst2\/}})\-\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s0\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\it{s0\/}})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\it{s0\/}})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{movem-saved}}\,({\it{s0\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved} $=$ {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s0\/}})\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}:  strcspn-s0-s0-rfile \\ \index{strcspn-s0-s0-rfile}
\=\+{\bf let }\=\+{\it{ch\/}}{\bf $\;$ be$\;\;$}{\rm{get-nth}}\,({\it{i1\/}}, {\it{lst1\/}})\index{get-nth}\- \\ 
{\bf in} \\ 
\=\+(\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{strchr}}\,({\tt{0}}, {\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strchr}) \\ 
$\wedge$$\;\;\;\;${\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t1}}\,({\it{n2\/}}, {\it{lst2\/}}, {\it{ch\/}})\index{strcspn-t1})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-$\;${\bf  endlet}\-
\end{tabbing}

\begin{verbatim}; put together: s0 --> sn.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-s0p-info \\ \index{strcspn-s0p-info}
\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\rightarrow$$\;\;\;\;$((({\it{i1\/}} $<$ {\it{n1\/}}) $=$ {\bf{t}}) $\wedge$ ({\rm{nat-to-uint}}\,({\rm{head}}\,({\it{i1*\/}}, {\tt{32}})\index{head})\index{nat-to-uint} $=$ {\rm{fix}}\,({\it{i1\/}})))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  strcspn-s0-sn \\ \index{strcspn-s0-sn}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t2}}\,({\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-t2})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-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$$\;\;\;\;$({\rm{uread-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{uread-dn} $=$ {\rm{strcspn}}\,({\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn}) \\ 
$\wedge$$\;\;\;\;$({\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}:  strcspn-s0-sn-rfile \\ \index{strcspn-s0-sn-rfile}
\=\+(\=\+{\rm{strcspn-s0p}}\,({\it{s\/}}, {\it{i1*\/}}, {\it{i1\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-s0p} \\ 
$\wedge$$\;\;\;\;${\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p} \\ 
$\wedge$$\;\;\;\;$({\it{oplen\/}} $\leq$ {\tt{32}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t2}}\,({\it{i1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-t2})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d4-7a2-5p}}\,({\it{rn\/}})\index{d4-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{get-vlst}}\,({\it{oplen\/}}, {\tt{0}}, {\it{rn\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}}, {\rm{movem-saved}}\,({\it{s\/}}, {\tt{4}}, {\tt{8}}, {\tt{2}})\index{movem-saved})\index{get-vlst}$\;${\bf  endif}\-)\-\-
\end{tabbing}


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

\addvspace{18pt}

\begin{verbatim}; now, finally, the correctness of strcspn.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  strcspn-correctness \\ \index{strcspn-correctness}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{strcspn-t}}\,({\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{strcspn-statep}}\,({\it{s\/}}, {\it{str1\/}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{str2\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn-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{12}}, {\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{uread-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{uread-dn} $=$ {\rm{strcspn}}\,({\tt{0}}, {\it{n1\/}}, {\it{lst1\/}}, {\it{n2\/}}, {\it{lst2\/}})\index{strcspn}))\-\-$\;${\bf  endlet}\-
\end{tabbing}


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

\addvspace{18pt}

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