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\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

; In an unpublished paper, Textbook Examples of Recursion, Donald E.
; Knuth of Stanford University gives the following generalization of
; McCarthy's 91 function:

; Let a be a real, let b and d be positive reals, and let c be a
; positive integer.

; Define K( x ) for integer inputs x by

;   K( x ) <== if  x > a  then  x - b
;                         else  K( ... K( x+d ) ... ).

; Here the else-clause in this definition has c applications of the
; function K.

; When a = 100, b = 10, c = 2, and d = 11, the definition specializes
; to McCarthy's original 91 function:

;   K( x ) <== if  x > 100  then  x - 10
;                           else  K( K( x+11 ) ).

; Knuth calls the first definition of K given above, the generalized 
; 91 recursion scheme with parameters ( a,b,c,d ).  

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

; The purpose of this file of Boyer-Moore-Kaufmann events is to 
; provide mechanical verification of the following theorem given by 
; Knuth in his paper.

; Theorem.  The generalized 91 recursion with parameters ( a,b,c,d )
;           defines a total function on the integers if and only if
;           (c-1)b < d.  In such a case the values of K( x ) also
;           satisfy the much simpler recurrence

;           K( x )  =  if  x > a  then  x - b
;                                 else  K( x+d-(c-1)b ).

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

; The first problem to solve is: How can Knuth's problem be stated so
; the theorem prover can work on it?

; Define two mutually recursive ( partial ) functions:

;   K( a,b,c,d,x ) <== if  x > a  then  x - b      
;                                 else  IterateK( a,b,c,d,c,x+d ). 
                                                        
;   IterateK( a,b,c,d,e,x ) <== if  e <= 1                           
;                                   then  K( a,b,c,d,x )             
;                                   else  K( a,b,c,d,
;                                            IterateK( a,b,c,d,e-1,x))

; Knuth's parameters a, b, c, and d are included in the formal
; parameters of both K and IterateK because the theorem prover does 
; not allow functions with definitions which contain "global" 
; variables.

; Intuitively, IterateK iterates K e times, that is, K is applied e
; times.  

; When the specified number, e, of times K is be iterated is is not
; positive, K is iterated one time.  That is, when e<1 the result in
; IterateK is the same as if e were 1.  

; Thus K( a,b,c,d,x ) = IterateK( a,b,c,d,1,x ).  

; Since the theorem prover does not deal with reals, the parameters
; a,b,c, and d are assumed to be integers.  

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

; To avoid the complications of dealing with mutually recursive 
; partial functions, a suggestion of M. Kaufmann is followed:

; In the definition of IterateK, replace occurrences of K with the 
; body of K.

; Define a recursive partial function by                             

;  IterK( a,b,c,d,e,x ) <== if  1 < e  
;                               then  IterK( a,b,c,d,1, 
;                                            IterK( a,b,c,d,e-1,x ))
;                               else if  a < x 
;                                        then  x - b 
;                                        else  IterK( a,b,c,d,c,x+d).
                                                                     
; Then the partial function K can be defined by                      

;   K( a,b,c,d,x ) <== IterK( a,b,c,d,1,x ).                         
  
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

; K( a,b,c,d,x ) is said to exist just in case the tuple of integers 
; ( a,b,c,d,x ) is in the domain of those input values where the 
; partial function K halts and produces output values.  

; Knuth's theorem follows from parts 1, 4, 6, 8, and 9 of the 
; following Main Theorem. 

;  1. If  x > a, then  K( a,b,c,d,x )  exists.

;  2. If  x <= a, d <= 0, and c <= 1, then  K( a,b,c,d,x )  does not
;     exist.

;  3. If  x <= a, d <= 0, and c > 1, then  K( a,b,c,d,x )  does not 
;     exist.

;  4. If  x <= a, d > 0, and c <= 1, then  K( a,b,c,d,x )  exists.

;  5. If  x <= a, d > 0, c > 1, and b <= 0, then  K( a,b,c,d,x )  
;     exists.

;  6. If  x <= a, d > 0, c > 1, and b > 0, then  K( a,b,c,d,x )
;     exists if and only if  K( a,b,c,d,x+d-(c-1)b )  exists and
;     K( a,b,c,d,x ) = K( a,b,c,d,x+d-(c-1)b ).

;  7. If  d > 0, c > 1, b > 0, and (c-1)b < d, then  
;     K( a,b,c,d,x )  exists if and only if  K( a,b,1,d-(c-1)b,x )
;     exists and  K( a,b,c,d,x ) = K( a,b,1,d-(c-1)b,x ).

;  8. If  d > 0, c > 1, b > 0, and (c-1)b < d, then 
;     K( a,b,c,d,x ) exists.

;  9. If  x <= a, d > 0, c > 1, b > 0, and (c-1)b >= d, then 
;     K( a,b,c,d,x ) does not exist

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Use the library of integer facts.

\end{verbatim}
\noindent{\sc Event}:   Start with the library {\tt{"integers"}}.

\addvspace{18pt}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Define the partial function IterK using EVAL$,
; and define the partial function K using IterK.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{iterk} \\  
\=\+{\rm{iterk}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{e\/}}, {\it{x\/}})\index{iterk} \\ 
$=$$\;\;\;\;${\rm{eval\$}}\,(\=\+{\bf{t}}, \\ 
{\tt{'}}{\tt{(}}\=\+{\tt{if}} \\ 
{\tt{(}}{\tt{ilessp}}{\tt{ }}{\tt{'}}{\tt{1}}{\tt{ }}{\tt{e}}{\tt{)}} \\ 
{\tt{(}}\=\+{\tt{iterk}}{\tt{ }}{\tt{a}}{\tt{ }}{\tt{b}}{\tt{ }}{\tt{c}}{\tt{ }}{\tt{d}} \\ 
{\tt{'}}{\tt{1}} \\ 
{\tt{(}}{\tt{iterk}}{\tt{ }}{\tt{a}}{\tt{ }}{\tt{b}}{\tt{ }}{\tt{c}}{\tt{ }}{\tt{d}}{\tt{ }}{\tt{(}}{\tt{idifference}}{\tt{ }}{\tt{e}}{\tt{ }}{\tt{'}}{\tt{1}}{\tt{)}}{\tt{ }}{\tt{x}}{\tt{)}}{\tt{)}}\- \\ 
{\tt{(}}\=\+{\tt{if}} \\ 
{\tt{(}}{\tt{ilessp}}{\tt{ }}{\tt{a}}{\tt{ }}{\tt{x}}{\tt{)}} \\ 
{\tt{(}}{\tt{idifference}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{b}}{\tt{)}} \\ 
{\tt{(}}{\tt{iterk}}{\tt{ }}{\tt{a}}{\tt{ }}{\tt{b}}{\tt{ }}{\tt{c}}{\tt{ }}{\tt{d}}{\tt{ }}{\tt{c}}{\tt{ }}{\tt{(}}{\tt{iplus}}{\tt{ }}{\tt{x}}{\tt{ }}{\tt{d}}{\tt{)}}{\tt{)}}{\tt{)}}\-{\tt{)}}\-, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\tt{'}}{\tt{a}}, {\it{a\/}}), \\ 
{\rm{cons}}\,({\tt{'}}{\tt{b}}, {\it{b\/}}), \\ 
{\rm{cons}}\,({\tt{'}}{\tt{c}}, {\it{c\/}}), \\ 
{\rm{cons}}\,({\tt{'}}{\tt{d}}, {\it{d\/}}), \\ 
{\rm{cons}}\,({\tt{'}}{\tt{e}}, {\it{e\/}}), \\ 
{\rm{cons}}\,({\tt{'}}{\tt{x}}, {\it{x\/}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{k}$\;\;$
{\rm{k}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{x\/}})\index{k} $=$ {\rm{iterk}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\tt{1}}, {\it{x\/}})\index{iterk}
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Recursively distribute V&C-APPLY$ throughout the body of the 
; defintion of ITERK. Break the definition into several cases.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists \\ \index{iterk-exists-iff-body-exists}
\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{if}}, \\ 
{\rm{list}}\,(\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{ilessp}}, {\rm{list}}\,({\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, {\rm{cons}}\,({\it{e\/}}, {\tt{0}}))), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\it{e\/}}, \\ 
{\tt{0}})\-, \\ 
{\tt{'}}{\tt{(}}\=\+{\tt{(}}\=\+{\tt{1}} \\ 
{\tt{.\ }}{\tt{0}}{\tt{)}}\-{\tt{)}}\-)\-)\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{if}}, \\ 
{\rm{list}}\,(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{ilessp}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{a\/}}, {\tt{0}}), {\rm{cons}}\,({\it{x\/}}, {\tt{0}})))\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{x\/}}, {\tt{0}}), {\rm{cons}}\,({\it{b\/}}, {\tt{0}})))\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iplus}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,(\=\+{\it{x\/}}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\it{d\/}}, \\ 
{\tt{0}})\-)\-)\-)\-)\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value \\ \index{iterk-value=body-value}
\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{if}}, \\ 
{\rm{list}}\,(\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{ilessp}}, {\rm{list}}\,({\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, {\rm{cons}}\,({\it{e\/}}, {\tt{0}}))), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\it{e\/}}, \\ 
{\tt{0}})\-, \\ 
{\tt{'}}{\tt{(}}\=\+{\tt{(}}\=\+{\tt{1}} \\ 
{\tt{.\ }}{\tt{0}}{\tt{)}}\-{\tt{)}}\-)\-)\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{if}}, \\ 
{\rm{list}}\,(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{ilessp}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{x\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}))\-)\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iplus}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,(\=\+{\it{x\/}}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\it{d\/}}, \\ 
{\tt{0}})\-)\-)\-)\-)\-)\-)\-)\-)\-)\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost-is-a-numberp \\ \index{cost-is-a-numberp}
{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}})) $\in\;${\bf{N}}${}$
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost$>$0-if-fn-exists \\ \index{cost$>$0-if-fn-exists}
{\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}}) $\rightarrow$ ({\tt{0}} $<$ {\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}})))
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-cost$>$body-cost \\ \index{iterk-cost$>$body-cost}
\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{if}}, \\ 
{\rm{list}}\,(\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{ilessp}}, {\rm{list}}\,({\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, {\rm{cons}}\,({\it{e\/}}, {\tt{0}}))), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\it{e\/}}, \\ 
{\tt{0}})\-, \\ 
{\tt{'}}{\tt{(}}\=\+{\tt{(}}\=\+{\tt{1}} \\ 
{\tt{.\ }}{\tt{0}}{\tt{)}}\-{\tt{)}}\-)\-)\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{if}}, \\ 
{\rm{list}}\,(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{ilessp}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{x\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}))\-)\-, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iplus}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,(\=\+{\it{x\/}}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\it{d\/}}, \\ 
{\tt{0}})\-)\-)\-)\-)\-)\-)\-)\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  v\&c-apply\$-if \\ \index{v\&c-apply\$-if}
\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{if}}, {\it{args\/}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{car}}\,({\it{args\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{args\/}})$\;\;${\bf then }{\rm{fix-cost}}\,({\rm{cadr}}\,({\it{args\/}}), $1\;+$ {\rm{cdar}}\,({\it{args\/}})) \\ 
{\bf else }{\rm{fix-cost}}\,({\rm{caddr}}\,({\it{args\/}}), $1\;+$ {\rm{cdar}}\,({\it{args\/}}))$\;${\bf  endif}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists-when-e$>$1 \\ \index{iterk-exists-iff-body-exists-whe!n-e$>$1}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,(\=\+{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}\=\+{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}{\tt{)}}\-)\-)\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value-when-e$>$1 \\ \index{iterk-value=body-value-when-e$>$!1}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,(\=\+{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}\=\+{\tt{(}}\=\+{\tt{1}} \\ 
{\tt{.\ }}{\tt{0}}{\tt{)}}\-{\tt{)}}\-)\-)\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-cost$>$body-cost-when-e$>$1 \\ \index{iterk-cost$>$body-cost-when-e$>$!1}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,(\=\+{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}{\tt{)}})\-)\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  eq-args-give-eq-existence \\ \index{eq-args-give-eq-existence}
\=\+(\=\+({\it{fn\/}} $\not=$ {\tt{'}}{\tt{quote}}) \\ 
$\wedge$$\;\;\;\;$({\it{fn\/}} $\not=$ {\tt{'}}{\tt{if}}) \\ 
$\wedge$$\;\;\;\;$({\rm{strip-cars}}\,({\it{args1\/}}) $=$ {\rm{strip-cars}}\,({\it{args2\/}})) \\ 
$\wedge$$\;\;\;\;$(({\bf{f}} $\in$ {\it{args1\/}}) $=$ ({\bf{f}} $\in$ {\it{args2\/}})))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args1\/}}) $\leftrightarrow$ {\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args2\/}}))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  eq-args-give-eq-values \\ \index{eq-args-give-eq-values}
\=\+(\=\+({\it{fn\/}} $\not=$ {\tt{'}}{\tt{quote}}) \\ 
$\wedge$$\;\;\;\;$({\it{fn\/}} $\not=$ {\tt{'}}{\tt{if}}) \\ 
$\wedge$$\;\;\;\;$({\rm{strip-cars}}\,({\it{args1\/}}) $=$ {\rm{strip-cars}}\,({\it{args2\/}})) \\ 
$\wedge$$\;\;\;\;$(({\bf{f}} $\in$ {\it{args1\/}}) $=$ ({\bf{f}} $\in$ {\it{args2\/}})))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{car}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args1\/}})) $=$ {\rm{car}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args2\/}})))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  eq-args-cost-depends-on-cost-of-args \\ \index{eq-args-cost-depends-on-cost-of!-args}
\=\+(\=\+({\it{fn\/}} $\not=$ {\tt{'}}{\tt{if}}) \\ 
$\wedge$$\;\;\;\;$({\rm{sum-cdrs}}\,({\it{args1\/}}) $<$ {\rm{sum-cdrs}}\,({\it{args2\/}})) \\ 
$\wedge$$\;\;\;\;$({\rm{strip-cars}}\,({\it{args1\/}}) $=$ {\rm{strip-cars}}\,({\it{args2\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args2\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args1\/}})) $<$ {\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args2\/}})))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-v\&c-apply\$-idifference-exists \\ \index{iterk-v\&c-apply\$-idifference-e!xists}
\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{idifference}}, {\rm{cons}}\,({\rm{cons}}\,({\it{e\/}}, {\tt{0}}), {\tt{'}}{\tt{(}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}{\tt{)}})), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-v\&c-apply\$-idifference-value \\ \index{iterk-v\&c-apply\$-idifference-v!alue}
\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,({\rm{cons}}\,({\it{e\/}}, {\tt{0}}), {\tt{'}}{\tt{(}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}{\tt{)}}))\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-v\&c-apply\$-idifference-cost \\ \index{iterk-v\&c-apply\$-idifference-!cost}
\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{idifference}}, \\ 
{\rm{cons}}\,({\rm{cons}}\,({\it{e\/}}, {\tt{0}}), {\tt{'}}{\tt{(}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}{\tt{)}}))\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists-when-e$>$1-version-2 \\ \index{iterk-exists-iff-body-exists-whe!n-e$>$1-version-2}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value-when-e$>$1-version-2 \\ \index{iterk-value=body-value-when-e$>$!1-version-2}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  args-exist-when-fn-exists \\ \index{args-exist-when-fn-exists}
(({\it{fn\/}} $\not=$ {\tt{'}}{\tt{if}}) $\wedge$ {\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}})) $\rightarrow$ ({\bf{f}} $\not\in$ {\it{args\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-cost$>$body-cost-when-e$>$1-version-2 \\ \index{iterk-cost$>$body-cost-when-e$>$!1-version-2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists-when-e$>$1-version-3-case-1 \\ \index{iterk-exists-iff-body-exists-whe!n-e$>$1-version-3-case-1}
\=\+({\it{vc-x\/}} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists-when-e$>$1-version-3-case-2 \\ \index{iterk-exists-iff-body-exists-whe!n-e$>$1-version-3-case-2}
\=\+(({\it{vc-x\/}} $=$ {\bf{f}}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists-when-e$>$1-version-3 \\ \index{iterk-exists-iff-body-exists-whe!n-e$>$1-version-3}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value-when-e$>$1-version-3-case-1 \\ \index{iterk-value=body-value-when-e$>$!1-version-3-case-1}
\=\+({\it{vc-x\/}} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value-when-e$>$1-version-3-case-2 \\ \index{iterk-value=body-value-when-e$>$!1-version-3-case-2}
\=\+(({\it{vc-x\/}} $=$ {\bf{f}}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value-when-e$>$1-version-3 \\ \index{iterk-value=body-value-when-e$>$!1-version-3}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\tt{1}})\index{idifference}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-when-e$<$=1\&a$<$x \\ \index{iterk-exists-when-e$<$=1\&a$<$x}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value-when-e$<$=1\&a$<$x \\ \index{iterk-value-when-e$<$=1\&a$<$x}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists-when-e$<$=1\&a$>$=x \\ \index{iterk-exists-iff-body-exists-whe!n-e$<$=1\&a$>$=x}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iplus}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{x\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}})))\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value-when-e$<$=1\&a$>$=x \\ \index{iterk-value=body-value-when-e$<$!=1\&a$>$=x}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iplus}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{x\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}})))\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-cost$>$body-cost-when-e$<$=1\&a$>$=x \\ \index{iterk-cost$>$body-cost-when-e$<$!=1\&a$>$=x}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{iplus}}, {\rm{list}}\,({\rm{cons}}\,({\it{x\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}}))))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable iterk-exists-iff-body-exists.

\addvspace{18pt}

\noindent{\sc Event}:   Disable iterk-value=body-value.

\addvspace{18pt}

\noindent{\sc Event}:   Disable iterk-cost$>$body-cost.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  iterk-exists-iff-body-exists-when-e$<$=1\&a$>$=x-version-2 \\ \index{iterk-exists-iff-body-exists-whe!n-e$<$=1\&a$>$=x-version-2}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\it{d\/}})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value=body-value-when-e$<$=1\&a$>$=x-version-2 \\ \index{iterk-value=body-value-when-e$<$!=1\&a$>$=x-version-2}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\it{d\/}})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-cost$>$body-cost-when-e$<$=1\&a$>$=x-version-2 \\ \index{iterk-cost$>$body-cost-when-e$<$!=1\&a$>$=x-version-2}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\it{d\/}})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable iterk-exists-iff-body-exists-when-e$>$1.

\addvspace{18pt}

\noindent{\sc Event}:   Disable iterk-value=body-value-when-e$>$1.

\addvspace{18pt}

\noindent{\sc Event}:   Disable iterk-cost$>$body-cost-when-e$>$1.

\addvspace{18pt}


\noindent{\sc Event}:   Disable iterk-exists-iff-body-exists-when-e$<$=1\&a$>$=x.

\addvspace{18pt}

\noindent{\sc Event}:   Disable iterk-value=body-value-when-e$<$=1\&a$>$=x.

\addvspace{18pt}

\noindent{\sc Event}:   Disable iterk-cost$>$body-cost-when-e$<$=1\&a$>$=x.

\addvspace{18pt}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events are versions of Part 1 of the
; Main Theorem given above in the introduction.

;  1. If  x > a, then  K( a,b,c,d,x )  exists.

;     Proof.  By the definitions of IterK and K.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-exists-when-x$>$a \\ \index{k-exists-when-x$>$a}
\=\+{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{a\/}}, {\tt{0}}), {\rm{cons}}\,({\it{b\/}}, {\tt{0}}), {\rm{cons}}\,({\it{c\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}}), {\rm{cons}}\,({\it{x\/}}, {\tt{0}})))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-halts-when-x$>$a \\ \index{k-halts-when-x$>$a}
\=\+({\it{vc-a\/}} $\wedge$ {\it{vc-b\/}} $\wedge$ {\it{vc-c\/}} $\wedge$ {\it{vc-d\/}} $\wedge$ {\it{vc-x\/}} $\wedge$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 1. The cost of computing IterK is unbounded 
;          when x <= a, d <= 0, and c <= 1:

;  Assume  x <= a, d <= 0, c <= 1, i > 0, 
;   and 
;  IterK( a,b,c,d,1,x ) exists.

;  Then IterK( a,b,c,d,c,x+id ) exists 

;   and

;  cost[ IterK( a,b,c,d,c,x+id ) ] + i-1 
;   <
;  cost[ IterK( a,b,c,d,1,x ) ].

; Proof. By induction on i.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  count-sub1-x$<$count-x-when-x$>$0\&x$<$$>$1 \\ \index{count-sub1-x$<$count-x-when-x$>$!0\&x$<$$>$1}
({\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\wedge$ ({\it{x\/}} $\not=$ {\tt{1}})) $\rightarrow$ ({\rm{count}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{x\/}})\index{iplus}) $<$ {\rm{count}}\,({\it{x\/}}))
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{induct-hint-positive-int} \\  
\=\+{\rm{induct-hint-positive-int}}\,({\it{x\/}})\index{induct-hint-positive-int} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{x\/}} $=$ {\tt{1}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{induct-hint-positive-int}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{x\/}})\index{iplus})\index{induct-hint-positive-int}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  sub1-x$>$0-when-x$>$0\&x$<$$>$1 \\ \index{sub1-x$>$0-when-x$>$0\&x$<$$>$1}
({\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\wedge$ ({\it{x\/}} $\not=$ {\tt{1}})) $\rightarrow$ {\rm{ilessp}}\,({\tt{0}}, {\rm{iplus}}\,({\tt{-1}}, {\it{x\/}})\index{iplus})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x+\_i-1\_d$<$=a-when-x$<$=a\&d$<$=0\&i$>$0 \\ \index{x+\_i-1\_d$<$=a-when-x$<$=a\&d$<$!=0\&i$>$0}
\=\+(($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_x+\_i-1\_d\_implies\_iterk\_x+id-when-x$<$=a\&d$<$=0\&c$<$=1\&i$>$0 \\ \index{iterk\_x+\_i-1\_d\_implies\_ite!rk\_x+id-when-x$<$=a\&d$<$=0\&c$<$=1\&i$>$0}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e=1\&x-implies-iterk-e=c\&x+id-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{iterk-e=1\&x-implies-iterk-e=c\&!x+id-when-x$<$=a\&d$<$=0\&c$<$=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\tt{1}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{i\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  w-1+y$<$z\&x$<$y-implies-w+x$<$z \\ \index{w-1+y$<$z\&x$<$y-implies-w+x$<$!z}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{w\/}})\index{ilessp} $\wedge$ ({\it{x\/}} $<$ {\it{y\/}}) $\wedge$ (({\rm{iplus}}\,({\tt{-1}}, {\it{w\/}})\index{iplus} $+$ {\it{y\/}}) $<$ {\it{z\/}})) \\ 
$\rightarrow$$\;\;\;\;$((({\it{w\/}} $+$ {\it{x\/}}) $<$ {\it{z\/}}) $=$ {\bf{t}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost-induction-step-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{cost-induction-step-when-x$<$=a!\&d$<$=0\&c$<$=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$+$$\;\;\;\;${\rm{iplus}}\,({\tt{-1}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{iplus})\- \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$+$$\;\;\;\;${\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\- \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost-base-step-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{cost-base-step-when-x$<$=a\&d$<$!=0\&c$<$=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\tt{1}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\tt{1}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e=1\&x-cost$>$i-1+cost-iterk-e=c\&x+id-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{iterk-e=1\&x-cost$>$i-1+cost-ite!rk-e=c\&x+id-when-x$<$=a\&d$<$=0\&c$<$=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{i\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$+$$\;\;\;\;${\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\- \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  y$<$=z-when-x+y-1$<$z\&x$>$=0\&y$>$0 \\ \index{y$<$=z-when-x+y-1$<$z\&x$>$=0\&!y$>$0}
\=\+(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{y\/}})\index{ilessp} $\wedge$ (({\it{x\/}} $+$ {\rm{iplus}}\,({\tt{-1}}, {\it{y\/}})\index{iplus}) $<$ {\it{z\/}})) \\ 
$\rightarrow$$\;\;\;\;$(({\it{z\/}} $\not<$ {\it{y\/}}) $=$ {\bf{t}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-cost-is-unbounded-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{iterk-cost-is-unbounded-when-x$<$!=a\&d$<$=0\&c$<$=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$\not<$$\;\;\;\;${\it{i\/}})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ilessp-0-add1-x \\ \index{ilessp-0-add1-x}
{\rm{ilessp}}\,({\tt{0}}, $1\;+$ {\it{x\/}})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-does-not-exist-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{iterk-does-not-exist-when-x$<$=!a\&d$<$=0\&c$<$=1}
\=\+(($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events are versions of Part 2 of the
; Main Theorem given above in the introduction.

;  2. If  x <= a, d <= 0, and c <= 1, then  
;     K( a,b,c,d,x )  does not exist.

;     Proof. By Lemma 1 and the definition of K.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-does-not-exist-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{k-does-not-exist-when-x$<$=a\&d!$<$=0\&c$<$=1}
\=\+(($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-does-not-halt-when-x$<$=a\&d$<$=0\&c$<$=1 \\ \index{k-does-not-halt-when-x$<$=a\&d$<$!=0\&c$<$=1}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})))\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 2. When the number of iterates is positive, 
;          the cost of computing IterK is an order 
;          homomorhism of the number of iterates:

;  Assume that  0 < e1 < e2  
;   and 
;  IterK( a,b,c,d,e2,x ) exists.

;  Then  IterK( a,b,c,d,e1,x ) also exists 

;   and

;  cost[ IterK( a,b,c,d,e1,x ) ] 
;   <   
;  cost[ IterK( a,b,c,d,e2,x ) ].

; Proof.  Hold e1 constant and induct on e2.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  count-sub1-y$<$count-y-when-0$<$x$<$y \\ \index{count-sub1-y$<$count-y-when-0$<$!x$<$y}
({\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\it{x\/}}, {\it{y\/}})\index{ilessp}) $\rightarrow$ ({\rm{count}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{y\/}})\index{iplus}) $<$ {\rm{count}}\,({\it{y\/}}))
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{induct-hint-cost-hom} \\  
\=\+{\rm{induct-hint-cost-hom}}\,({\it{x\/}}, {\it{y\/}})\index{induct-hint-cost-hom} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }$\neg$ {\rm{ilessp}}\,({\it{x\/}}, {\it{y\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{x\/}} $=$ {\rm{iplus}}\,({\tt{-1}}, {\it{y\/}})\index{iplus}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{induct-hint-cost-hom}}\,({\it{x\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{y\/}})\index{iplus})\index{induct-hint-cost-hom}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  y$>$1-when-y$>$x\&x$>$0 \\ \index{y$>$1-when-y$>$x\&x$>$0}
({\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\it{x\/}}, {\it{y\/}})\index{ilessp}) $\rightarrow$ {\rm{ilessp}}\,({\tt{1}}, {\it{y\/}})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-at-e1-exists-when-iterk-at-e2-exists\&0$<$e1$<$e2-basic-step \\ \index{iterk-at-e1-exists-when-iterk-at!-e2-exists\&0$<$e1$<$e2-basic-step}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{e1\/}}, {\it{e2\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x$<$y-1-when-0$<$x\&x$<$y\&x$<$$>$y-1 \\ \index{x$<$y-1-when-0$<$x\&x$<$y\&x$<$!$>$y-1}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\it{x\/}}, {\it{y\/}})\index{ilessp} $\wedge$ ({\it{x\/}} $\not=$ {\rm{iplus}}\,({\tt{-1}}, {\it{y\/}})\index{iplus})) \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\it{x\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{y\/}})\index{iplus})\index{ilessp}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-at-e1-exists-when-iterk-at-e2-exists\&0$<$e1$<$e2 \\ \index{iterk-at-e1-exists-when-iterk-at!-e2-exists\&0$<$e1$<$e2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{e1\/}}, {\it{e2\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e1\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost$>$cost-of-args \\ \index{cost$>$cost-of-args}
\=\+(({\it{fn\/}} $\not=$ {\tt{'}}{\tt{if}}) $\wedge$ ({\bf{f}} $\not\in$ {\it{args\/}}) $\wedge$ {\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{sum-cdrs}}\,({\it{args\/}}) $<$ {\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}})))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  sum-cdrs$>$=member \\ \index{sum-cdrs$>$=member}
({\it{x\/}} $\in$ {\it{l\/}}) $\rightarrow$ ({\rm{sum-cdrs}}\,({\it{l\/}}) $\not<$ {\rm{cdr}}\,({\it{x\/}}))
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost$>$member \\ \index{cost$>$member}
\=\+({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}}) $\wedge$ ({\it{fn\/}} $\not=$ {\tt{'}}{\tt{if}}) $\wedge$ ({\it{x\/}} $\in$ {\it{args\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{cdr}}\,({\it{x\/}}) $<$ {\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}})))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost-iterk-at-e1$<$cost-iterk-at-e2-when-0$<$e1$<$e2-basic-step \\ \index{cost-iterk-at-e1$<$cost-iterk-at!-e2-when-0$<$e1$<$e2-basic-step}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{e1\/}}, {\it{e2\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable y$>$1-when-y$>$x\&x$>$0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  cost-iterk-at-e1$<$cost-iterk-at-e2-when-0$<$e1$<$e2 \\ \index{cost-iterk-at-e1$<$cost-iterk-at!-e2-when-0$<$e1$<$e2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{e1\/}}, {\it{e2\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e1\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 3. The cost of computing IterK is unbounded 
;          when x <= a, d <= 0, and c > 1:

;  Assume  x <= a, d <= 0, c > 1, i > 0, 
;   and 
;  IterK( a,b,c,d,1,x ) exists.

;  Then IterK( a,b,c,d,1,x+id ) exists 

;   and

;  cost[ IterK( a,b,c,d,1,x+id ) ] + i 
;   <
;  cost[ IterK( a,b,c,d,1,x ) ].

; Proof. By induction on i.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_x+\_i-1\_d\_implies\_iterk\_x+id-when-x$<$=a\&d$<$=0\&c$>$1\&i$>$0\&e=1 \\ \index{iterk\_x+\_i-1\_d\_implies\_ite!rk\_x+id-when-x$<$=a\&d$<$=0\&c$>$1\&i$>$0\&e=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-x-implies-iterk-x+id-when-x$<$=a\&d$<$=0\&c$>$1\&e=1 \\ \index{iterk-x-implies-iterk-x+id-when!-x$<$=a\&d$<$=0\&c$>$1\&e=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{i\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-x+d-cost$<$cost-iterk-x-when-a$>$=x\&1$<$c \\ \index{iterk-x+d-cost$<$cost-iterk-x-w!hen-a$>$=x\&1$<$c}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\it{d\/}})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cost-induction-step-when-x$<$=a\&d$<$=0\&c$>$1 \\ \index{cost-induction-step-when-x$<$=a!\&d$<$=0\&c$>$1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$+$$\;\;\;\;${\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\- \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$+$$\;\;\;\;${\it{i\/}})\- \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-x-cost$>$i+cost-iterk-x+id-when-x$<$=a\&d$<$=0\&c$>$1\&e=1 \\ \index{iterk-x-cost$>$i+cost-iterk-x+i!d-when-x$<$=a\&d$<$=0\&c$>$1\&e=1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{i\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$+$$\;\;\;\;${\it{i\/}})\- \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-cost-is-unbounded-when-x$<$=a\&d$<$=0\&c$>$1 \\ \index{iterk-cost-is-unbounded-when-x$<$!=a\&d$<$=0\&c$>$1}
\=\+(\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $<$ {\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-does-not-exist-when-x$<$=a\&d$<$=0\&c$>$1 \\ \index{iterk-does-not-exist-when-x$<$=!a\&d$<$=0\&c$>$1}
\=\+(($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events are versions of Part 3 of the
; Main Theorem given above in the introduction.

;  3. If  x <= a, d <= 0, and c > 1, then  
;     K( a,b,c,d,x )  does not exist.

;     Proof. By Lemma 3 and the definition of K.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-does-not-exist-when-x$<$=a\&d$<$=0\&c$>$1 \\ \index{k-does-not-exist-when-x$<$=a\&d!$<$=0\&c$>$1}
\=\+(($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-does-not-halt-when-x$<$=a\&d$<$=0\&c$>$1 \\ \index{k-does-not-halt-when-x$<$=a\&d$<$!=0\&c$>$1}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})))\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 4. IterK exists when d > 0, 
;          c <= 1, and e = c:

;  Assume  d > 0, and c <= 1. 

;  Then IterK( a,b,c,d,c,x ) exists.

; Proof. Hold the parameter a fixed and 
;        induct on the value given by

;        if  x > a  then  0
;                   else  1 + a - x.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{k-measure} \\  
\=\+{\rm{k-measure}}\,({\it{a\/}}, {\it{x\/}})\index{k-measure} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }{\rm{iplus}}\,({\tt{1}}, {\rm{iplus}}\,({\it{a\/}}, {\rm{ineg}}\,({\it{x\/}})\index{ineg})\index{iplus})\index{iplus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-measure\_x+d$<$k-measure\_x-when-0$<$d\&a$>$=x \\ \index{k-measure\_x+d$<$k-measure\_x-w!hen-0$<$d\&a$>$=x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{k-measure}}\,({\it{a\/}}, {\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus})\index{k-measure} $<$ {\rm{k-measure}}\,({\it{a\/}}, {\it{x\/}})\index{k-measure})\-
\end{tabbing}


\noindent{\sc Event}:   Disable k-measure.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{induct-hint-k-measure} \\  
\=\+{\rm{induct-hint-k-measure}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{x\/}})\index{induct-hint-k-measure} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{induct-hint-k-measure}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus})\index{induct-hint-k-measure}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-when-d$>$0\&c$<$=1\&e=c \\ \index{iterk-exists-when-d$>$0\&c$<$=1!\&e=c}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-when-x$<$=a\&d$>$0\&c$<$=1\&e=1 \\ \index{iterk-exists-when-x$<$=a\&d$>$0!\&c$<$=1\&e=1}
\=\+(($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events are versions of Part 4 of the
; Main Theorem given above in the introduction.

;  4. If  x <= a, d > 0, and c <= 1, then 
;      K( a,b,c,d,x )  exists.

;     Proof. By Lemma 4 and the definition of K.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-exists-when-x$<$=a\&d$>$0\&c$<$=1 \\ \index{k-exists-when-x$<$=a\&d$>$0\&c$<$!=1}
\=\+(($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{a\/}}, {\tt{0}}), {\rm{cons}}\,({\it{b\/}}, {\tt{0}}), {\rm{cons}}\,({\it{c\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}}), {\rm{cons}}\,({\it{x\/}}, {\tt{0}})))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-halts-when-x$<$=a\&d$>$0\&c$<$=1 \\ \index{k-halts-when-x$<$=a\&d$>$0\&c$<$!=1}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 5. IterK exists and has value > a
;          when x > a, b <= 0, and e > 0: 

;  Assume  x > a, b <= 0, and e > 0.

;  Then IterK( a,b,c,d,e,x ) exists 

;   and

;  IterK( a,b,c,d,e,x ) > a.

; Proof.  By induction on e.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk-exists-when-e$<$=1\&a$<$x-version-2 \\ \index{iterk-exists-when-e$<$=1\&a$<$x!-version-2}
\=\+({\it{vc-x\/}} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-value-when-e$<$=1\&a$<$x-version-2 \\ \index{iterk-value-when-e$<$=1\&a$<$x-ve!rsion-2}
\=\+({\it{vc-x\/}} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\it{vc-x\/}})\-)\-) \\ 
$=$$\;\;\;\;${\rm{idifference}}\,({\rm{car}}\,({\it{vc-x\/}}), {\it{b\/}})\index{idifference})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a+b$<$x-when-a$<$x\&b$<$=0 \\ \index{a+b$<$x-when-a$<$x\&b$<$=0}
({\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp})) $\rightarrow$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{b\/}})\index{iplus}, {\it{x\/}})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a$<$-b+x-when-a$<$x\&b$<$=0 \\ \index{a$<$-b+x-when-a$<$x\&b$<$=0}
({\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp})) $\rightarrow$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{iplus}}\,({\rm{ineg}}\,({\it{b\/}})\index{ineg}, {\it{x\/}})\index{iplus})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x$>$1-when-x$>$0\&x$<$$>$1 \\ \index{x$>$1-when-x$>$0\&x$<$$>$1}
({\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\wedge$ ({\it{x\/}} $\not=$ {\tt{1}})) $\rightarrow$ {\rm{ilessp}}\,({\tt{1}}, {\it{x\/}})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists\&iterk$>$a-when-x$>$a\&b$<$=0\&e$>$0 \\ \index{iterk-exists\&iterk$>$a-when-x$>$!a\&b$<$=0\&e$>$0}
\=\+({\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,(\=\+{\it{a\/}}, \\ 
{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\index{ilessp})\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 6. The number of iterates in the composition
;          of IterK with itself add up:

;  Let  e1 > 0  and  e2 > 0.  Then

;  IterK( a,b,c,d,e1,IterK( a,b,c,d,e2,x ) ) exists
;   iff  
;  IterK( a,b,c,d,e1+e2,x ) exists

;  and

;  IterK( a,b,c,d,e1,IterK( a,b,c,d,e2,x ) )
;   =   
;  IterK( a,b,c,d,e1+e2,x )

; Proof.  By induction on e1.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iplus\_1\_-1\_x=x\_when\_x$>$0 \\ \index{iplus\_1\_-1\_x=x\_when\_x$>$0}
{\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\rightarrow$ ({\rm{iplus}}\,({\tt{1}}, {\rm{iplus}}\,({\tt{-1}}, {\it{x\/}})\index{iplus})\index{iplus} $=$ {\it{x\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x+y$>$1-when-x$>$0\&x$<$$>$1\&0$<$y \\ \index{x+y$>$1-when-x$>$0\&x$<$$>$1\&0!$<$y}
({\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\wedge$ ({\it{x\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{y\/}})\index{ilessp}) $\rightarrow$ {\rm{ilessp}}\,({\tt{1}}, {\rm{iplus}}\,({\it{x\/}}, {\it{y\/}})\index{iplus})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nbr-of-iterates-sum-exists-step-1 \\ \index{nbr-of-iterates-sum-exists-step!-1}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} $\wedge$ ({\it{e1\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e1\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nbr-of-iterates-sum-values-step-1 \\ \index{nbr-of-iterates-sum-values-step!-1}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} $\wedge$ ({\it{e1\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e1\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\it{e2\/}}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nbr-of-iterates-sum-exists-step-2 \\ \index{nbr-of-iterates-sum-exists-step!-2}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} $\wedge$ ({\it{e1\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{e2\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e1\/}}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\it{e2\/}})\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nbr-of-iterates-sum-values-step-2 \\ \index{nbr-of-iterates-sum-values-step!-2}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} $\wedge$ ({\it{e1\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{e2\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e1\/}}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, \\ 
{\it{e2\/}})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nbr-of-iterates-sum-exists-step-3 \\ \index{nbr-of-iterates-sum-exists-step!-3}
\=\+(\=\+(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\tt{-1}}, {\rm{iplus}}\,({\it{e1\/}}, {\it{e2\/}})\index{iplus})\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nbr-of-iterates-sum-values-step-3 \\ \index{nbr-of-iterates-sum-values-step!-3}
\=\+(\=\+(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e1\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,({\it{e1\/}}, {\it{e2\/}})\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nbr-of-iterates-sum-exists\&values \\ \index{nbr-of-iterates-sum-exists\&val!ues}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{e1\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{e2\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e1\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e1\/}}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e1\/}}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e2\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e1\/}}, {\it{e2\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 7. The number of iterates of K accumulate 
;          when 1 < c, 0 < d, and x is "small."

;  Assume 1 < c, 0 < d, 0 < n, and a+d >= x+nd.

;  Then

;  IterK( a,b,c,d,1,x ) exists
;   iff  
;  IterK( a,b,c,d,1+n(c-1),x+nd ) exists

;  and

;  IterK( a,b,c,d,1,x ) 
;   =   
;  IterK( a,b,c,d,1+n(c-1),x+nd ).

; Proof.  By induction on n.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  n\_c-1\_+1=c-when-n=1\&c$>$1 \\ \index{n\_c-1\_+1=c-when-n=1\&c$>$1}
(({\it{n\/}} $=$ {\tt{1}}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) $\rightarrow$ ({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus} $=$ {\it{c\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nd=d-when-n=1\&d$>$0 \\ \index{nd=d-when-n=1\&d$>$0}
(({\it{n\/}} $=$ {\tt{1}}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) $\rightarrow$ ({\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes} $=$ {\it{d\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a$>$=x-when-a+d$>$=x+d \\ \index{a$>$=x-when-a+d$>$=x+d}
($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{d\/}})\index{iplus})\index{ilessp}) $\rightarrow$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-exists-base-step \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-exist!s-base-step}
\=\+(\=\+({\it{n\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-value-base-step \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-value!-base-step}
\=\+(\=\+({\it{n\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  n-1\_c-1$>$0-when-1$<$c\&0$<$n\&n$<$$>$1 \\ \index{n-1\_c-1$>$0-when-1$<$c\&0$<$n\&!n$<$$>$1}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} $\wedge$ ({\it{n\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{ilessp}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-exists-step-1 \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-exist!s-step-1}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} $\wedge$ ({\it{n\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,(\=\+{\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, \\ 
{\it{d\/}})\-\index{itimes})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-value-step-1 \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-value!-step-1}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} $\wedge$ ({\it{n\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\it{n\/}})\-\index{iplus}, \\ 
{\it{d\/}})\-\index{itimes})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-exists-step-2 \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-exist!s-step-2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-value-step-2 \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-value!-step-2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\it{d\/}})\index{itimes})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-exists-step-3 \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-exist!s-step-3}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} $\wedge$ ({\it{n\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-value-step-3 \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-value!-step-3}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} $\wedge$ ({\it{n\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a+d$>$=x+\_n-1\_d-when-a+d$>$=x+nd\&0$<$d\&0$<$n \\ \index{a+d$>$=x+\_n-1\_d-when-a+d$>$=x!+nd\&0$<$d\&0$<$n}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{n\/}})\index{iplus}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-exists-when-1$<$c\&0$<$d\&0$<$n\&a+d$>$=x+nd \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-exist!s-when-1$<$c\&0$<$d\&0$<$n\&a+d$>$=x+nd}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e=1+\_n\_c-1\&x+nd-value-when-1$<$c\&0$<$d\&0$<$n\&a+d$>$=x+nd \\ \index{iterk\_e=1+\_n\_c-1\&x+nd-value!-when-1$<$c\&0$<$d\&0$<$n\&a+d$>$=x+nd}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Define the function N( a,d,x ) recursively, 
;  so that whenever d > 0, N( a,d,x ) is the smallest 
;  nonnegative integer i such that x + id > a.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{n} \\  
\=\+{\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }{\rm{iplus}}\,({\tt{1}}, {\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\rm{iplus}}\,({\it{x\/}}, {\it{d\/}})\index{iplus})\index{n})\index{iplus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  n$>$=0 \\ \index{n$>$=0}
$\neg$ {\rm{ilessp}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\tt{0}})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  n$>$0-when-d$>$o\&x$<$=a \\ \index{n$>$0-when-d$>$o\&x$<$=a}
({\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) $\rightarrow$ {\rm{ilessp}}\,({\tt{0}}, {\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a$<$x+nd \\ \index{a$<$x+nd}
{\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\rightarrow$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a+d$>$=x+nd-when-d$>$0\&a$>$=x \\ \index{a+d$>$=x+nd-when-d$>$0\&a$>$=x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e=1\&x-iff-iterk\_e=1+\_n\_c-1\&x+nd-when-1$<$c\&0$<$d\&a$>$=x \\ \index{iterk-e=1\&x-iff-iterk\_e=1+\_n!\_c-1\&x+nd-when-1$<$c\&0$<$d\&a$>$=x}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-when-e=1+n\_c-1\&x=x+nd\&b$<$=0 \\ \index{iterk-exists-when-e=1+n\_c-1\&x!=x+nd\&b$<$=0}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-exists-when-x$<$=a\&d$>$0\&c$<$1\&b$<$=0 \\ \index{iterk-exists-when-x$<$=a\&d$>$0!\&c$<$1\&b$<$=0}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events are versions of Part 5 of the
; Main Theorem given above in the introduction.

;  5. If  x <= a, d > 0, c > 1, and b <= 0, then  
;     K( a,b,c,d,x ) exists.

;     Proof. By Lemma 7, the definition of the
;            function N, Part 1 of the Main Theorem,
;            and the definition of K. 

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-exists-when-x$<$=a\&d$>$0\&c$<$1\&b$<$=0 \\ \index{k-exists-when-x$<$=a\&d$>$0\&c$<$!1\&b$<$=0}
\=\+(($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{a\/}}, {\tt{0}}), {\rm{cons}}\,({\it{b\/}}, {\tt{0}}), {\rm{cons}}\,({\it{c\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}}), {\rm{cons}}\,({\it{x\/}}, {\tt{0}})))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-halts-when-x$<$=a\&d$>$0\&c$<$1\&b$<$=0 \\ \index{k-halts-when-x$<$=a\&d$>$0\&c$<$!1\&b$<$=0}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 8. Generalize Lemma 7 to the case when e > 1.

;  Assume 1 < c, 0 < d, 1 < e, 0 < n, and a+d >= x+nd.

;  Then

;  IterK( a,b,c,d,e,x ) exists
;   iff  
;  IterK( a,b,c,d,e+n(c-1),x+nd ) exists

;  and

;  IterK( a,b,c,d,e,x ) 
;   =   
;  IterK( a,b,c,d,e+n(c-1),x+nd ).

; Proof.  By Lemma 6 (with e2 = 1) and Lemma 7.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  x+-x+y=fix-int-y \\ \index{x+-x+y=fix-int-y}
{\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\rightarrow$ ({\rm{iplus}}\,({\it{x\/}}, {\rm{iplus}}\,($-$ {\it{x\/}}, {\it{y\/}})\index{iplus})\index{iplus} $=$ {\rm{fix-int}}\,({\it{y\/}})\index{fix-int})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  fix-int-x=x-when-x$>$1 \\ \index{fix-int-x=x-when-x$>$1}
{\rm{ilessp}}\,({\tt{1}}, {\it{x\/}})\index{ilessp} $\rightarrow$ ({\rm{fix-int}}\,({\it{x\/}})\index{fix-int} $=$ {\it{x\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-x-iff-iterk-e-1-iterk-1-x \\ \index{iterk-e-x-iff-iterk-e-1-iterk-1!-x}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-x=iterk-e-1-iterk-1-x \\ \index{iterk-e-x=iterk-e-1-iterk-1-x}
\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e+\_n\_c-1\&x+nd-exists-when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-1 \\ \index{iterk\_e+\_n\_c-1\&x+nd-exists-!when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-1}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{1}}, \\ 
{\rm{itimes}}\,(\=\+{\it{n\/}}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\-\index{itimes})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e+\_n\_c-1\&x+nd-value-when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-1 \\ \index{iterk\_e+\_n\_c-1\&x+nd-value-w!hen-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-1}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{1}}, \\ 
{\rm{itimes}}\,(\=\+{\it{n\/}}, \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\it{c\/}})\-\index{iplus})\-\index{itimes})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e+\_n\_c-1\&x+nd-exists-when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-2 \\ \index{iterk\_e+\_n\_c-1\&x+nd-exists-!when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-2}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\tt{1}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e\/}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e+\_n\_c-1\&x+nd-value-when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-2 \\ \index{iterk\_e+\_n\_c-1\&x+nd-value-w!hen-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd-step-2}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{1}}, \\ 
{\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e\/}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e+\_n\_c-1\&x+nd-exists-when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd \\ \index{iterk\_e+\_n\_c-1\&x+nd-exists-!when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e\/}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e+\_n\_c-1\&x+nd-value-when-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd \\ \index{iterk\_e+\_n\_c-1\&x+nd-value-w!hen-1$<$c\&0$<$d\&1$<$e\&0$<$n\&a+d$>$=x+nd}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e\/}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e\&x-iff-iterk\_e+\_n\_c-1\&x+nd-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1 \\ \index{iterk-e\&x-iff-iterk\_e+\_n\_c-!1\&x+nd-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{e\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e\&x=iterk\_e+\_n\_c-1\&x+nd-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1 \\ \index{iterk-e\&x=iterk\_e+\_n\_c-1\&x!+nd-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 9. The number of iterates of K can be reduced 
;          by one if the value of x is reduced by b
;          when a < x.

;  Assume  1 < e, and  a < x.

;  Then

;  IterK( a,b,c,d,e,x ) exists
;   iff  
;  IterK( a,b,c,d,e-1,x-b ) exists

;  and

;  IterK( a,b,c,d,e,x ) 
;   =   
;  IterK( a,b,c,d,e-1,x-b ).

; Proof.  By Lemma 6 (with e2 = 1), and the definition 
;         of IterK for values of x larger than a.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_e\_x-iff-iterk\_e-1\_x-b-when-1$<$e\&x$>$a-step-1 \\ \index{iterk\_e\_x-iff-iterk\_e-1\_x-b!-when-1$<$e\&x$>$a-step-1}
\=\+({\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e\_x=iterk\_e-1\_x-b-when-1$<$e\&x$>$a-step-1 \\ \index{iterk\_e\_x=iterk\_e-1\_x-b-whe!n-1$<$e\&x$>$a-step-1}
\=\+({\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e\_x-iff-iterk\_e-1\_x-b-when-1$<$e\&x$>$a \\ \index{iterk\_e\_x-iff-iterk\_e-1\_x-b!-when-1$<$e\&x$>$a}
\=\+({\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e\_x=iterk\_e-1\_x-b-when-1$<$e\&x$>$a \\ \index{iterk\_e\_x=iterk\_e-1\_x-b-whe!n-1$<$e\&x$>$a}
\=\+({\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 10. A key fact noted by Knuth in his proof.

;  Assume 1 < c, 0 < d, 1 < e, and  x <= a.

;  Then

;  IterK( a,b,c,d,e,x ) exists
;   iff  
;  IterK( a,b,c,d,e-1+N( a,d,x )(c-1),x+N( a,d,x )d-b ) 
;  exists

;  and

;  IterK( a,b,c,d,e,x ) 
;   =   
;  IterK( a,b,c,d,e-1+N( a,d,x )(c-1),x+N( a,d,x )d-b ).

; Proof.  By Lemma 8, the definition of the function N,
;         and Lemma 9.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  e+n\_c-1\_$>$1-when-1$<$c\&1$<$e\&0$<$n \\ \index{e+n\_c-1\_$>$1-when-1$<$c\&1$<$e!\&0$<$n}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{iplus}}\,({\it{e\/}}, {\rm{itimes}}\,({\it{n\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus})\index{ilessp}\-
\end{tabbing}


\begin{tabbing}{\sc Theorem}:  iterk-e\&x-iff-iterk\_-1+e+\_n\_c-1\&x+nd-b-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1 \\ \index{iterk-e\&x-iff-iterk\_-1+e+\_n\_!c-1\&x+nd-b-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e\&x=iterk\_-1+e+\_n\_c-1\&x+nd-b-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1 \\ \index{iterk-e\&x=iterk\_-1+e+\_n\_c-1!\&x+nd-b-when-1$<$c\&0$<$d\&a$>$=x\&e$>$1}
\=\+({\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,(\=\+{\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\-\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\-\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 11. Another key fact noted by Knuth.

;  Assume 0 < b, 1 < c, 0 < d, 1 < e, and  x <= a.

;  Then

;  IterK( a,b,c,d,e-1,x-b ) exists
;   iff  
;  IterK( a,b,c,d,e-1+N( a,d,x )(c-1),x-b+N( a,d,x )d ) 
;  exists

;  and

;  IterK( a,b,c,d,e-1,x-b ) 
;   =   
;  IterK( a,b,c,d,e-1+N( a,d,x )(c-1),x-b+N( a,d,x )d ).

; Proof.  By Lemma 7 (if e = 2), Lemma 8 (if e > 2), and 
;         the definition of the function N.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  a+d$>$=x-b+nd-when-b$>$0\&a+d$>$=x+nd\&d$>$0 \\ \index{a+d$>$=x-b+nd-when-b$>$0\&a+d$>$!=x+nd\&d$>$0}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{n\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{d\/}})\index{iplus}, {\rm{iplus}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\rm{itimes}}\,({\it{n\/}}, {\it{d\/}})\index{itimes})\index{iplus})\index{ilessp})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-1\&x-b-iff-iterk\_e-1+\_n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e$>$2 \\ \index{iterk-e-1\&x-b-iff-iterk\_e-1+\_!n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e$>$2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{2}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-1\&x-b=iterk\_e-1+\_n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&e$>$2 \\ \index{iterk-e-1\&x-b=iterk\_e-1+\_n\_!c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&e$>$2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{2}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,(\=\+{\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\-\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\-\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x+2-1=x+1 \\ \index{x+2-1=x+1}
{\rm{iplus}}\,({\tt{-1}}, {\rm{iplus}}\,({\tt{2}}, {\it{x\/}})\index{iplus})\index{iplus} $=$ {\rm{iplus}}\,({\tt{1}}, {\it{x\/}})\index{iplus}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-1\&x-b-iff-iterk\_e-1+\_n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e=2 \\ \index{iterk-e-1\&x-b-iff-iterk\_e-1+\_!n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e=2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{e\/}} $=$ {\tt{2}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-1\&x-b=iterk\_e-1+\_n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e=2 \\ \index{iterk-e-1\&x-b=iterk\_e-1+\_n\_!c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e=2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{e\/}} $=$ {\tt{2}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,(\=\+{\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\-\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\-\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-1\&x-b-iff-iterk\_e-1+\_n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e$>$1 \\ \index{iterk-e-1\&x-b-iff-iterk\_e-1+\_!n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e$>$1}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e-1\&x-b=iterk\_e-1+\_n\_c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e$>$1 \\ \index{iterk-e-1\&x-b=iterk\_e-1+\_n\_!c-1\&x-b+nd-when-1$<$c\&0$<$d\&a$>$=x\&b$>$0\&e$>$1}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{e\/}}, \\ 
{\rm{itimes}}\,(\=\+{\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\-\index{itimes})\-\index{iplus})\-\index{iplus}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n}, {\it{d\/}})\index{itimes})\-\index{iplus}, \\ 
{\it{b\/}})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 12. The number of iterates of K can be reduced
;           by one if the value of x is reduced by b
;           when a >= x and restrictions are placed
;           on the parameters a,b,c, and d.

;  Assume  0 < b, 1 < c, 0 < d, 1 < e, and  a >= x.

;  Then

;  IterK( a,b,c,d,e,x ) exists
;   iff  
;  IterK( a,b,c,d,e-1,x-b ) exists

;  and

;  IterK( a,b,c,d,e,x ) 
;   =   
;  IterK( a,b,c,d,e-1,x-b ).

; Proof.  By Lemma 10 and Lemma 11.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_e\_x-iff-iterk\_e-1\_x-b-when-0$<$b\&1$<$c\&0$<$d\&1$<$e\&a$>$=x \\ \index{iterk\_e\_x-iff-iterk\_e-1\_x-b!-when-0$<$b\&1$<$c\&0$<$d\&1$<$e\&a$>$=x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e\_x=iterk\_e-1\_x-b-when-0$<$b\&1$<$c\&0$<$d\&1$<$e\&a$>$=x \\ \index{iterk\_e\_x=iterk\_e-1\_x-b-whe!n-0$<$b\&1$<$c\&0$<$d\&1$<$e\&a$>$=x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 13. Combine Lemma 9 and Lemma 12 about reducing
;           the number of iterates of K by one.

;  Assume  0 < b, 1 < c, 0 < d, and 1 < e.

;  Then

;  IterK( a,b,c,d,e,x ) exists
;   iff  
;  IterK( a,b,c,d,e-1,x-b ) exists

;  and

;  IterK( a,b,c,d,e,x ) 
;   =   
;  IterK( a,b,c,d,e-1,x-b ).

; Proof.  By Lemma 9 (if x > a) and Lemma 12 (if x <= a).

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_e\_x-iff-iterk\_e-1\_x-b-when-0$<$b\&1$<$c\&0$<$d\&1$<$e \\ \index{iterk\_e\_x-iff-iterk\_e-1\_x-b!-when-0$<$b\&1$<$c\&0$<$d\&1$<$e}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e\_x=iterk\_e-1\_x-b-when-0$<$b\&1$<$c\&0$<$d\&1$<$e \\ \index{iterk\_e\_x=iterk\_e-1\_x-b-whe!n-0$<$b\&1$<$c\&0$<$d\&1$<$e}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{e\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 14. Generalize Lemma 13 by reducing the number 
;           of iterates of K by more than one.

;  Assume  0 < b, 1 < c, 0 < d, 1 < e, and 0 < j < e.

;  Then

;  IterK( a,b,c,d,e,x ) exists
;   iff  
;  IterK( a,b,c,d,e-j,x-jb ) exists

;  and

;  IterK( a,b,c,d,e,x ) 
;   =   
;  IterK( a,b,c,d,e-j,x-jb ).

; Proof.  By induction on j.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_e\_x-iff-iterk\_e-j\_x-jb-when-0$<$b\&1$<$c\&0$<$d\&1$<$e\&0$<$j$<$e \\ \index{iterk\_e\_x-iff-iterk\_e-j\_x-j!b-when-0$<$b\&1$<$c\&0$<$d\&1$<$e\&0$<$j$<$e}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{j\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{j\/}}, {\it{e\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\it{j\/}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{j\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_e\_x=iterk\_e-j\_x-jb-when-0$<$b\&1$<$c\&0$<$d\&1$<$e\&0$<$j$<$e \\ \index{iterk\_e\_x=iterk\_e-j\_x-jb-whe!n-0$<$b\&1$<$c\&0$<$d\&1$<$e\&0$<$j$<$e}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{j\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{j\/}}, {\it{e\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{e\/}}, {\it{j\/}})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{j\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 15. A special case of Lemma 14.

;  Assume  0 < b, 1 < c, 0 < d, and  a >= x.

;  Then

;  IterK( a,b,c,d,1,x ) exists
;   iff  
;  IterK( a,b,c,d,1,x+d-(c-1)b ) exists

;  and

;  IterK( a,b,c,d,1,x ) 
;   =   
;  IterK( a,b,c,d,1,x+d-(c-1)b ).

; Proof.  By Lemma 14 with e = c, j = c-1, and 
;         x replaced by  x + d.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_x-iff-iterk\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x \\ \index{iterk\_x-iff-iterk\_x+d\_c-1\_b!-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{idifference}}\,(\=\+{\it{d\/}}, \\ 
{\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_x=iterk\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x \\ \index{iterk\_x=iterk\_x+d\_c-1\_b-whe!n-0$<$b\&1$<$c\&0$<$d\&a$>$=x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{idifference}}\,(\=\+{\it{d\/}}, \\ 
{\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events give a version of Part 6  of 
; the Main Theorem given above in the introduction.

;  6. If  x <= a, d > 0, c > 1, and b > 0, then  
;     K( a,b,c,d,x ) exists if and only if  
;     K( a,b,c,d,x+d-(c-1)b )  exists and
;     K( a,b,c,d,x ) = K( a,b,c,d,x+d-(c-1)b ).

;     Proof. By Lemma 15 and the definition of K.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k\_x-iff-k\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x \\ \index{k\_x-iff-k\_x+d\_c-1\_b-when-0$<$!b\&1$<$c\&0$<$d\&a$>$=x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{a\/}}, {\tt{0}}), {\rm{cons}}\,({\it{b\/}}, {\tt{0}}), {\rm{cons}}\,({\it{c\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}}), {\rm{cons}}\,({\it{x\/}}, {\tt{0}})))\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{idifference}}\,(\=\+{\it{d\/}}, \\ 
{\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_x=k\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x \\ \index{k\_x=k\_x+d\_c-1\_b-when-0$<$b\&!1$<$c\&0$<$d\&a$>$=x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{idifference}}\,(\=\+{\it{d\/}}, \\ 
{\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_x-iff-k\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x-version-2 \\ \index{k\_x-iff-k\_x+d\_c-1\_b-when-0$<$!b\&1$<$c\&0$<$d\&a$>$=x-version-2}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})) \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\rm{car}}\,({\it{vc-a\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-b\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-c\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-d\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{car}}\,({\it{vc-x\/}}), \\ 
{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_x=k\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x-version-2 \\ \index{k\_x=k\_x+d\_c-1\_b-when-0$<$b\&!1$<$c\&0$<$d\&a$>$=x-version-2}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\rm{car}}\,({\it{vc-a\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-b\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-c\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-d\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{car}}\,({\it{vc-x\/}}), \\ 
{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events give a more general version 
; of Part 6  of the Main Theorem.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k\_x-iff-k\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x-version-3 \\ \index{k\_x-iff-k\_x+d\_c-1\_b-when-0$<$!b\&1$<$c\&0$<$d\&a$>$=x-version-3}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})) \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\it{vc-c\/}}, \\ 
{\it{vc-d\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{car}}\,({\it{vc-x\/}}), \\ 
{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\it{cost\/}})\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_x=k\_x+d\_c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&a$>$=x-version-3 \\ \index{k\_x=k\_x+d\_c-1\_b-when-0$<$b\&!1$<$c\&0$<$d\&a$>$=x-version-3}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\it{vc-c\/}}, \\ 
{\it{vc-d\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{car}}\,({\it{vc-x\/}}), \\ 
{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\it{cost\/}})\-)\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 16. The parameter c can be replaced by 1 if
;           the parameter d is modified in a suitable
;           way, when certain restrictions are placed 
;           on the parameters.

;  Assume  0 < b, 1 < c, 0 < d, and  (c-1)b < d.

;  Then

;  IterK( a,b,c,d,1,x ) exists
;   iff  
;  IterK( a,b,1,d-(c-1)b,1,x ) exists

;  and

;  IterK( a,b,c,d,1,x ) 
;   =   
;  IterK( a,b,1,d-(c-1)b,1,x ).

; Proof.  By induction on the value given by

;         if  x > a  then  0
;                    else  1 + a - x.

;         Also use Lemma 15.         

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-measure\_x+d-\_c-1\_b$<$k-measure\_x-when-\_c-1\_b$<$d\&a$>$=x \\ \index{k-measure\_x+d-\_c-1\_b$<$k-mea!sure\_x-when-\_c-1\_b$<$d\&a$>$=x}
\=\+({\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{c\/}})\index{itimes}, {\rm{iplus}}\,({\it{b\/}}, {\it{d\/}})\index{iplus})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{k-measure}}\,({\it{a\/}}, {\rm{iplus}}\,({\it{b\/}}, {\rm{iplus}}\,({\it{d\/}}, {\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{c\/}})\index{itimes})\index{ineg})\index{iplus})\index{iplus})\index{iplus})\index{k-measure} \\ 
$<$$\;\;\;\;${\rm{k-measure}}\,({\it{a\/}}, {\it{x\/}})\index{k-measure})\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{induct-hint-1-k-measure} \\  
\=\+{\rm{induct-hint-1-k-measure}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{x\/}})\index{induct-hint-1-k-measure} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{induct-hint-1-k-measure}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\it{c\/}}, \\ 
{\it{d\/}}, \\ 
{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{idifference}}\,(\=\+{\it{d\/}}, \\ 
{\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\it{c\/}})\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus})\-\index{induct-hint-1-k-measure}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_c\&d-iff-iterk\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d \\ \index{iterk\_c\&d-iff-iterk\_1\&d-c-1!\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{d\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{idifference}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_c\&d=iterk\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d \\ \index{iterk\_c\&d=iterk\_1\&d-c-1\_b-!when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,({\it{d\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{idifference}, \\ 
{\tt{0}})\-, \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events give a version of Part 7  of 
; the Main Theorem given above in the introduction.

;  7. If  d > 0, c > 1, b > 0, and (c-1)b < d, then  
;     K( a,b,c,d,x )  exists if and only if  
;     K( a,b,1,d-(c-1)b,x ) exists and  
;     K( a,b,c,d,x ) = K( a,b,1,d-(c-1)b,x ).

;     Proof. By Lemma 16.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k\_c\&d-iff-k\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d \\ \index{k\_c\&d-iff-k\_1\&d-c-1\_b-when!-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{a\/}}, {\tt{0}}), {\rm{cons}}\,({\it{b\/}}, {\tt{0}}), {\rm{cons}}\,({\it{c\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}}), {\rm{cons}}\,({\it{x\/}}, {\tt{0}})))\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{d\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{idifference}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_c\&d=k\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d \\ \index{k\_c\&d=k\_1\&d-c-1\_b-when-0$<$!b\&1$<$c\&0$<$d\&b\_c-1\_$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,({\it{d\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{idifference}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_c\&d-iff-k\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-2 \\ \index{k\_c\&d-iff-k\_1\&d-c-1\_b-when!-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-2}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})) \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\rm{car}}\,({\it{vc-a\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-b\/}}), {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\-\index{itimes})\-\index{idifference}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-x\/}}), {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_c\&d=k\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-2 \\ \index{k\_c\&d=k\_1\&d-c-1\_b-when-0$<$!b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-2}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\rm{car}}\,({\it{vc-a\/}}), {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-b\/}}), {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference}, \\ 
{\tt{0}})\-, \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{vc-x\/}}), {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events give a more general version 
; of Part 7  of the Main Theorem.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k\_c\&d-iff-k\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-3 \\ \index{k\_c\&d-iff-k\_1\&d-c-1\_b-when!-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-3}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})) \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\rm{cons}}\,({\tt{1}}, {\it{cost1\/}}), \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\-\index{itimes})\-\index{idifference}, \\ 
{\it{cost2\/}})\-, \\ 
{\it{vc-x\/}})\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k\_c\&d=k\_1\&d-c-1\_b-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-3 \\ \index{k\_c\&d=k\_1\&d-c-1\_b-when-0$<$!b\&1$<$c\&0$<$d\&b\_c-1\_$<$d-version-3}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\rm{cons}}\,({\tt{1}}, {\it{cost1\/}}), \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference}, \\ 
{\it{cost2\/}})\-, \\ 
{\it{vc-x\/}})\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 17. Essentially the if-part of Knuth's 
;           theorem characterizing the parameters 
;           when K is total.

;  Assume  0 < b, 1 < c, 0 < d, and  (c-1)b < d.

;  Then IterK( a,b,c,d,1,x ) exists.

; Proof.  By Lemma 4 and Lemma 16.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk-exists-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d \\ \index{iterk-exists-when-0$<$b\&1$<$c\&!0$<$d\&b\_c-1\_$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events are versions of Part 8  of the
; Main Theorem given above in the introduction.

;  8. If  d > 0, c > 1, b > 0, and (c-1)b < d, then 
;     K( a,b,c,d,x ) exists.

;     Proof. By Lemma 17.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-exists-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d \\ \index{k-exists-when-0$<$b\&1$<$c\&0$<$!d\&b\_c-1\_$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{a\/}}, {\tt{0}}), {\rm{cons}}\,({\it{b\/}}, {\tt{0}}), {\rm{cons}}\,({\it{c\/}}, {\tt{0}}), {\rm{cons}}\,({\it{d\/}}, {\tt{0}}), {\rm{cons}}\,({\it{x\/}}, {\tt{0}})))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-halts-when-0$<$b\&1$<$c\&0$<$d\&b\_c-1\_$<$d \\ \index{k-halts-when-0$<$b\&1$<$c\&0$<$!d\&b\_c-1\_$<$d}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 18. The cost of computing K is lowered when d is
;           added to x.

;  Assume  1 < c, a >= x, and IterK( a,b,c,d,1,x ) exists.

;  Then IterK( a,b,c,d,1,x+d ) exists

;   and

;  cost[ IterK( a,b,c,d,1,x+d ) ]
;   <
;  cost[ IterK( a,b,c,d,1,x ) ]

; Proof. By the definition of IterK and Lemma 2.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk-x+d-exists-when-iterk-exists\&1$<$c\&a$>$=x \\ \index{iterk-x+d-exists-when-iterk-exi!sts\&1$<$c\&a$>$=x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-x+d-cost$<$cost-iterk-when-iterk-exists\&1$<$c\&a$>$=x \\ \index{iterk-x+d-cost$<$cost-iterk-whe!n-iterk-exists\&1$<$c\&a$>$=x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 19. The cost of computing K is lowered
;           when K is iterated c times and 
;           "enough" d's are added to x.

;  Assume  1 < c, a >= x, 0 < d, and  
;  IterK( a,b,c,d,1,x ) exists.

;  Then IterK( a,b,c,d,c,x+N( a,d,x )d ) exists

;   and

;  cost[ IterK( a,b,c,d,c,x+N( a,d,x )d ) ]
;   <
;  cost[ IterK( a,b,c,d,1,x ) ]

; Proof. Hold a fixed and induct on the
;        value given by

;        if  x > a  then  0
;                   else  1 + a - x.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  n=1-when-x+d$>$a\&a$>$=x\&d$>$0 \\ \index{n=1-when-x+d$>$a\&a$>$=x\&d$>$0}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n} $=$ {\tt{1}})\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{induct-hint-2-k-measure} \\  
\=\+{\rm{induct-hint-2-k-measure}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{induct-hint-2-k-measure} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{ilessp}}\,({\it{a\/}}, {\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus})\index{ilessp}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{induct-hint-2-k-measure}}\,({\it{a\/}}, {\it{d\/}}, {\rm{iplus}}\,({\it{d\/}}, {\it{x\/}})\index{iplus})\index{induct-hint-2-k-measure}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-x+nd-exists-when-iterk-exists\&1$<$c\&a$>$=x\&0$<$d \\ \index{iterk-x+nd-exists-when-iterk-exi!sts\&1$<$c\&a$>$=x\&0$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-x+nd-cost$<$cost-iterk-when-iterk-exists\&1$<$c\&a$>$=x\&0$<$d \\ \index{iterk-x+nd-cost$<$cost-iterk-whe!n-iterk-exists\&1$<$c\&a$>$=x\&0$<$d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n})\index{itimes})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 20. The number of iterates of K can be
;           reduced to 1 if "enough" b's are
;           sutracted from x.

;  Assume  0 < b, 1 < c, 0 < d, 1 < e, 0 < i < e,
;  and IterK( a,b,c,d,e,x ) exists.

;  Then IterK( a,b,c,d,1,x-ib ) exists.

; Proof. By Lemma 14 and Lemma 2.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  e-i=1-when-1+i$>$=e\&i$<$e \\ \index{e-i=1-when-1+i$>$=e\&i$<$e}
(($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\it{e\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\it{i\/}}, {\it{e\/}})\index{ilessp}) $\rightarrow$ ({\rm{iplus}}\,({\it{e\/}}, {\rm{ineg}}\,({\it{i\/}})\index{ineg})\index{iplus} $=$ {\tt{1}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib-exists-when-iterk\_e\_x-exists\&0$<$b\&1$<$c\&0$<$d\&1$<$e\&0$<$i$<$e \\ \index{iterk\_1\_x-ib-exists-when-iter!k\_e\_x-exists\&0$<$b\&1$<$c\&0$<$d\&1$<$e\&0$<$i$<$e}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{i\/}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 21. Another lemma reducing the number of
;           iterates of K and subtracting b's 
;           from x. 

;  Assume  0 < b, 1 < c, 0 < d, 0 < i, a+ib < x+b,
;  and IterK( a,b,c,d,i+1,x ) exists.

;  Then IterK( a,b,c,d,1,x-ib ) exists

;   and

;  cost[ IterK( a,b,c,d,1,x-ib ) ]
;   <
;  cost[ IterK( a,b,c,d,i+1,x ) ]

; Proof. By Lemma 20 (for the exists portion) 
;        and induction on i (for the cost portion).

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-base-case \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-base-case}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  fix-int-x=x-when-0$<$x \\ \index{fix-int-x=x-when-0$<$x}
{\rm{ilessp}}\,({\tt{0}}, {\it{x\/}})\index{ilessp} $\rightarrow$ ({\rm{fix-int}}\,({\it{x\/}})\index{fix-int} $=$ {\it{x\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-step-1 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-step-1}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_i\_x-iff-iterk\_1\_x-ib-when-0$<$b\&1$<$c\&0$<$d\&i$>$0\&i$<$$>$1 \\ \index{iterk\_i\_x-iff-iterk\_1\_x-ib-!when-0$<$b\&1$<$c\&0$<$d\&i$>$0\&i$<$$>$1}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} $\wedge$ ({\it{i\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_i\_x=iterk\_1\_x-ib-when-0$<$b\&1$<$c\&0$<$d\&i$>$0\&i$<$$>$1 \\ \index{iterk\_i\_x=iterk\_1\_x-ib-when!-0$<$b\&1$<$c\&0$<$d\&i$>$0\&i$<$$>$1}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} $\wedge$ ({\it{i\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{itimes})\index{ineg})\index{iplus}, \\ 
{\tt{0}})\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-exists-step-2 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-exists-step-2}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} $\wedge$ ({\it{i\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\it{i\/}})\-\index{iplus})\-\index{itimes})\index{ineg})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-step-2 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-step-2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\- \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\-\index{itimes})\index{ineg})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\it{i\/}})\-\index{iplus})\-\index{itimes})\index{ineg})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_i+1\_exists-iff-iterk\_1\_iterk\_i\_exists \\ \index{iterk\_i+1\_exists-iff-iterk\_1!\_iterk\_i\_exists}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} $\wedge$ ({\it{i\/}} $\not=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-step-1\&2 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-step-1\&2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\it{i\/}})\-\index{iplus})\-\index{itimes})\index{ineg})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-exists-step-3 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-exists-step-3}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\-\index{itimes})\index{ineg})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-step-3 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-step-3}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\- \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\-\index{itimes})\index{ineg})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,(\=\+{\it{b\/}}, \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\it{i\/}})\-\index{iplus})\-\index{itimes})\index{ineg})\-\index{iplus}, \\ 
{\tt{0}})\-)\-)\-)\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_i+1\_x-iff-iterk\_1\_x-ib-when-0$<$b\&1$<$c\&0$<$d\&i$>$0 \\ \index{iterk\_i+1\_x-iff-iterk\_1\_x-i!b-when-0$<$b\&1$<$c\&0$<$d\&i$>$0}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\leftrightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-step-1\&2\&3 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-step-1\&2\&3}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{ineg})\index{iplus}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a+bi$<$x+b-when-a+bi$<$b+b+x\&0$<$b \\ \index{a+bi$<$x+b-when-a+bi$<$b+b+x\&0!$<$b}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{b\/}}, {\rm{iplus}}\,({\it{b\/}}, {\it{x\/}})\index{iplus})\index{iplus})\index{ilessp}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-e=i-exists-when-iterk-e=i+1-exists \\ \index{iterk-e=i-exists-when-iterk-e=i!+1-exists}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{i\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x-induction-step \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x-induction-step}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{idifference}}\,(\=\+{\it{x\/}}, \\ 
{\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{itimes})\-\index{idifference}, \\ 
{\tt{0}})\-)\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_i+1\_x \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_i+1\_x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 22. A generalization of Lemma 21. 

;  Assume  0 < b, 1 < c, 0 < d, 0 < i, a+ib < x+b,
;  i < e, and IterK( a,b,c,d,e,x ) exists.

;  Then IterK( a,b,c,d,1,x-ib ) exists

;   and

;  cost[ IterK( a,b,c,d,1,x-ib ) ]
;   <
;  cost[ IterK( a,b,c,d,e,x ) ]

; Proof. Lemma 2 and Lemma 21.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_e\_x-case-1 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_e\_x-case-1}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_e\_x-case-2 \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_e\_x-case-2}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus} $=$ {\it{e\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  e=i+1-or-e$>$i+1-when-e$>$i$>$0 \\ \index{e=i+1-or-e$>$i+1-when-e$>$i$>$0}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\it{i\/}}, {\it{e\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(({\it{e\/}} $=$ {\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}) $\vee$ {\rm{ilessp}}\,({\rm{iplus}}\,({\tt{1}}, {\it{i\/}})\index{iplus}, {\it{e\/}})\index{ilessp})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-ib\_cost$<$cost-iterk\_e\_x \\ \index{iterk\_1\_x-ib\_cost$<$cost-ite!rk\_e\_x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{i\/}}, {\it{e\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{x\/}}, {\it{b\/}})\index{iplus})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\it{i\/}})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{e\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Define the function J( a,b,x ) recursively, 
;  so that whenever b > 0, J( a,b,x ) is the smallest 
;  nonnegative integer i such that x <= a + ib.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{how-far-above-a} \\  
\=\+{\rm{how-far-above-a}}\,({\it{a\/}}, {\it{x\/}})\index{how-far-above-a} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }{\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\it{a\/}})\index{ineg})\index{iplus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  numberp\_x-a\_when\_x$>$a \\ \index{numberp\_x-a\_when\_x$>$a}
{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\rightarrow$ ({\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\it{a\/}})\index{ineg})\index{iplus} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x-a-b$<$x-a-when-0$<$b\&a$<$x\&a+b$<$x \\ \index{x-a-b$<$x-a-when-0$<$b\&a$<$x\&!a+b$<$x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\it{b\/}})\index{iplus}, {\it{x\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{iplus}}\,({\it{x\/}}, {\rm{iplus}}\,({\rm{ineg}}\,({\it{a\/}})\index{ineg}, {\rm{ineg}}\,({\it{b\/}})\index{ineg})\index{iplus})\index{iplus} $<$ {\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\it{a\/}})\index{ineg})\index{iplus})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a$<$$>$x-when-a$<$x \\ \index{a$<$$>$x-when-a$<$x}
{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} $\rightarrow$ ({\rm{fix-int}}\,({\it{x\/}})\index{fix-int} $\not=$ {\rm{fix-int}}\,({\it{a\/}})\index{fix-int})
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{j} \\  
\=\+{\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j} \\ 
$=$$\;\;\;\;$\=\+{\bf if }$\neg$ {\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp}$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }$\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }{\rm{iplus}}\,({\tt{1}}, {\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\rm{iplus}}\,({\it{x\/}}, {\rm{ineg}}\,({\it{b\/}})\index{ineg})\index{iplus})\index{j})\index{iplus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j$>$=0 \\ \index{j$>$=0}
$\neg$ {\rm{ilessp}}\,({\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j}, {\tt{0}})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j$>$0-when-x$>$a\&b$>$0 \\ \index{j$>$0-when-x$>$a\&b$>$0}
({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) $\rightarrow$ {\rm{ilessp}}\,({\tt{0}}, {\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x$<$=a+jb-when-0$<$b \\ \index{x$<$=a+jb-when-0$<$b}
{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\rightarrow$ ($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j})\index{itimes})\index{iplus}, {\it{x\/}})\index{ilessp})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a+jb$<$x+b-when-0$<$b\&a$<$x \\ \index{a+jb$<$x+b-when-0$<$b\&a$<$x}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j})\index{itimes})\index{iplus}, {\rm{iplus}}\,({\it{b\/}}, {\it{x\/}})\index{iplus})\index{ilessp}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x$<$z-when-x$<$y\&y$<$=z \\ \index{x$<$z-when-x$<$y\&y$<$=z}
({\rm{ilessp}}\,({\it{x\/}}, {\it{y\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{z\/}}, {\it{y\/}})\index{ilessp})) $\rightarrow$ {\rm{ilessp}}\,({\it{x\/}}, {\it{z\/}})\index{ilessp}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j$<$c-when-0$<$b\&a$<$x\&c$>$1\&a+\_c-1\_b$>$=x \\ \index{j$<$c-when-0$<$b\&a$<$x\&c$>$1\&!a+\_c-1\_b$>$=x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\it{x\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j}, {\it{c\/}})\index{ilessp}\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 23. A special case of Lemma 22.

;  Assume  0 < b, 1 < c, 0 < d, a < x, a+(c-1)b >= x,
;  and IterK( a,b,c,d,c,x ) exists.

;  Then IterK( a,b,c,d,1,x-J( a,b,x )b ) exists

;   and

;  cost[ IterK( a,b,c,d,1,x-J( a,b,x )b ) ]
;   <
;  cost[ IterK( a,b,c,d,c,x ) ]

; Proof. By Lemma 22 and the definition of J.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-jb-exists-when-iterk\_e=c\_x-exists\&0$<$b\&1$<$c\&0$<$d\&etc \\ \index{iterk\_1\_x-jb-exists-when-iter!k\_e=c\_x-exists\&0$<$b\&1$<$c\&0$<$d\&etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk\_1\_x-jb\_cost$<$cost-iterk\_e=c\_x \\ \index{iterk\_1\_x-jb\_cost$<$cost-ite!rk\_e=c\_x}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{idifference}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\it{x\/}})\index{j})\index{itimes})\index{idifference}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Define the function L( a,b,d,x ) to be the following
;  combination of  x, b, d, and the functions J and N:

;     x + N( a,d,x ) d - J( a,b,x+N( a,d,x )d ) b.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{l} \\  
\=\+{\rm{l}}\,({\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{l} \\ 
$=$$\;\;\;\;${\rm{iplus}}\,(\=\+{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n})\index{itimes})\index{iplus}, \\ 
{\rm{ineg}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{j}}\,({\it{a\/}}, {\it{b\/}}, {\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n})\index{itimes})\index{iplus})\index{j})\index{itimes})\index{ineg})\-\index{iplus}\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 24. Computing K with L( a,b,d,x ) is cheaper
;           than computing K with x.

;  Assume  0 < b, 1 < c, 0 < d, a >= x, (c-1)b >= d,
;  and IterK( a,b,c,d,1,x ) exists.

;  Then IterK( a,b,c,d,1,L( a,b,d,x ) ) exists

;   and

;  cost[ IterK( a,b,c,d,1,L( a,b,d,x ) ) ]
;   <
;  cost[ IterK( a,b,c,d,1,x ) ]

; Proof. By Lemma 19 and Lemma 23.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  x$>$=z-when-x$>$=y\&y$>$=z \\ \index{x$>$=z-when-x$>$=y\&y$>$=z}
(($\neg$ {\rm{ilessp}}\,({\it{x\/}}, {\it{y\/}})\index{ilessp}) $\wedge$ ($\neg$ {\rm{ilessp}}\,({\it{y\/}}, {\it{z\/}})\index{ilessp})) $\rightarrow$ ($\neg$ {\rm{ilessp}}\,({\it{x\/}}, {\it{z\/}})\index{ilessp})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  a+\_c-1\_b$>$=x+nd-when-0$<$d\&a$>$=x\&\_c-1\_b$>$=d \\ \index{a+\_c-1\_b$>$=x+nd-when-0$<$d\&!a$>$=x\&\_c-1\_b$>$=d}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,(\=\+{\rm{iplus}}\,({\it{a\/}}, {\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes})\index{iplus}, \\ 
{\rm{iplus}}\,({\it{x\/}}, {\rm{itimes}}\,({\it{d\/}}, {\rm{n}}\,({\it{a\/}}, {\it{d\/}}, {\it{x\/}})\index{n})\index{itimes})\index{iplus})\-\index{ilessp})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-l-exists-when-iterk-x-exists\&0$<$b\&1$<$c\&0$<$d\&etc \\ \index{iterk-l-exists-when-iterk-x-exi!sts\&0$<$b\&1$<$c\&0$<$d\&etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{l}}\,({\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{l}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-l-cost$<$cost-iterk-x\&0$<$b\&1$<$c\&0$<$d-etc \\ \index{iterk-l-cost$<$cost-iterk-x\&0$<$!b\&1$<$c\&0$<$d-etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{l}}\,({\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{l}, {\tt{0}}))\-)\-) \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Define the function IterL( i,a,b,d,x ) which iterates
;  the function L i times.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  count\_x-1$<$count\_x-when-1$<$x \\ \index{count\_x-1$<$count\_x-when-1$<$!x}
{\rm{ilessp}}\,({\tt{1}}, {\it{x\/}})\index{ilessp} $\rightarrow$ ({\rm{count}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{x\/}})\index{iplus}) $<$ {\rm{count}}\,({\it{x\/}}))
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{iterl} \\  
\=\+{\rm{iterl}}\,({\it{i\/}}, {\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{iterl} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{ilessp}}\,({\tt{1}}, {\it{i\/}})\index{ilessp}$\;\;${\bf then }{\rm{l}}\,({\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\rm{iterl}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, {\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{iterl})\index{l} \\ 
{\bf else }{\rm{l}}\,({\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{l}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l$<$=a \\ \index{l$<$=a}
{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\rightarrow$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{l}}\,({\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{l})\index{ilessp})
\end{tabbing}


\noindent{\sc Event}:   Disable l.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  iterl$<$=a \\ \index{iterl$<$=a}
{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\rightarrow$ ($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\rm{iterl}}\,({\it{i\/}}, {\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{iterl})\index{ilessp})
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Lemma 25. The cost of computing K is unbounded
;           when (c-1)b >= d

;  Assume  0 < b, 1 < c, 0 < d, a >= x, (c-1)b >= d,
;  i > 0, and IterK( a,b,c,d,1,x ) exists.

;  Then IterK( a,b,c,d,1,IterL( i,a,b,d,x ) ) exists

;   and

;  cost[ IterK( a,b,c,d,1,IterL( i,a,b,d,x ) ) ] + i-1
;   <
;  cost[ IterK( a,b,c,d,1,x ) ]

; Proof. By induction on i using Lemma 24.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  iterk-iterl-exists-when-iterk-x-exists\&0$<$b\&1$<$c\&0$<$d\&etc \\ \index{iterk-iterl-exists-when-iterk-x!-exists\&0$<$b\&1$<$c\&0$<$d\&etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iterl}}\,({\it{i\/}}, {\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{iterl}, {\tt{0}}))\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  fix-int-cost=cost \\ \index{fix-int-cost=cost}
{\rm{fix-int}}\,({\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}})))\index{fix-int} $=$ {\rm{cdr}}\,({\rm{v\&c-apply\$}}\,({\it{fn\/}}, {\it{args\/}}))
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  x+i-1$<$z-when-x$<$y\&-1-1+i+y$<$z \\ \index{x+i-1$<$z-when-x$<$y\&-1-1+i+y$<$!z}
\=\+(({\it{x\/}} $<$ {\it{y\/}}) $\wedge$ ({\rm{iplus}}\,({\tt{-1}}, {\rm{iplus}}\,({\tt{-1}}, {\rm{iplus}}\,({\it{i\/}}, {\it{y\/}})\index{iplus})\index{iplus})\index{iplus} $<$ {\it{z\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{iplus}}\,({\tt{-1}}, {\rm{iplus}}\,({\it{i\/}}, {\it{x\/}})\index{iplus})\index{iplus} $<$ {\it{z\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-iterl-cost+i-1$<$cost-iterk-x\&0$<$b\&1$<$c\&0$<$d-etc-induction-step \\ \index{iterk-iterl-cost+i-1$<$cost-ite!rk-x\&0$<$b\&1$<$c\&0$<$d-etc-induction-step}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{i\/}}, \\ 
{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iterl}}\,(\=\+{\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, \\ 
{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\it{d\/}}, \\ 
{\it{x\/}})\-\index{iterl}, \\ 
{\tt{0}})\-)\-)\-))\-\index{iplus})\-\index{iplus})\-\index{iplus} \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\- \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\it{c\/}})\index{itimes}, {\rm{iplus}}\,({\it{b\/}}, {\it{d\/}})\index{iplus})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{i\/}}, \\ 
{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,(\=\+{\rm{l}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\it{d\/}}, \\ 
{\rm{iterl}}\,({\rm{iplus}}\,({\tt{-1}}, {\it{i\/}})\index{iplus}, {\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{iterl})\-\index{l}, \\ 
{\tt{0}})\-)\-)\-))\-\index{iplus})\-\index{iplus} \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-iterl-cost+i-1$<$cost-iterk-x\&0$<$b\&1$<$c\&0$<$d-etc \\ \index{iterk-iterl-cost+i-1$<$cost-ite!rk-x\&0$<$b\&1$<$c\&0$<$d-etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{iplus}}\,(\=\+{\it{i\/}}, \\ 
{\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\rm{iterl}}\,({\it{i\/}}, {\it{a\/}}, {\it{b\/}}, {\it{d\/}}, {\it{x\/}})\index{iterl}, {\tt{0}}))\-)\-))\-\index{iplus})\-\index{iplus} \\ 
$<$$\;\;\;\;${\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i$<$y-when-0$<$x\&-1+i+x$<$y \\ \index{i$<$y-when-0$<$x\&-1+i+x$<$y}
(({\tt{0}} $<$ {\it{x\/}}) $\wedge$ ({\rm{iplus}}\,({\tt{-1}}, {\rm{iplus}}\,({\it{i\/}}, {\it{x\/}})\index{iplus})\index{iplus} $<$ {\it{y\/}})) $\rightarrow$ ({\it{i\/}} $<$ {\it{y\/}})
\end{tabbing}


\noindent{\sc Event}:   Disable iterl.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  iterk-cost-is-unbounded-when-\_c-1\_b$>$=d\&x$<$=a\&etc \\ \index{iterk-cost-is-unbounded-when-\_!c-1\_b$>$=d\&x$<$=a\&etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{i\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $<$ {\rm{cdr}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-))\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  iterk-does-not-exist-when-\_c-1\_b$>$=d\&x$<$=a\&etc \\ \index{iterk-does-not-exist-when-\_c-1!\_b$>$=d\&x$<$=a\&etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{iterk}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; The next two events are versions of Part 9 of the
; Main Theorem given above in the introduction.

;  9. If  x <= a, d > 0, c > 1, b > 0, and 
;     (c-1)b >= d, then K( a,b,c,d,x ) does 
;     not exist.

;     Proof. By Lemma 25 and the definition of K.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-does-not-exist-when-\_c-1\_b$>$=d\&x$<$=a\&etc \\ \index{k-does-not-exist-when-\_c-1\_b$>$!=d\&x$<$=a\&etc}
\=\+(\=\+{\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-)\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-does-not-halt-when-\_c-1\_b$>$=d\&x$<$=a\&etc \\ \index{k-does-not-halt-when-\_c-1\_b$>$!=d\&x$<$=a\&etc}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})))\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Here a quantifier is used to define the concept that K
; is a total function of x with parameters a,b,c, and d.

; This is the only place in this file of events that
; an explicit quantifier is used.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{k-is-total} \\  
\=\+{\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total} \\ 
$\leftrightarrow$$\;\;\;\;$$\forall\;${\it{vc-x\/}}$\;$({\it{vc-x\/}} $\rightarrow$ {\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})))\index{forall}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-is-total-suff \\ \index{k-is-total-suff}
\=\+(\=\+{\rm{vc-x}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{vc-x} \\ 
$\rightarrow$$\;\;\;\;${\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\rm{vc-x}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{vc-x})))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-is-total-necc \\ \index{k-is-total-necc}
\=\+($\neg$ ({\it{vc-x\/}} $\rightarrow$ {\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}})))) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total})\-
\end{tabbing}


\noindent{\sc Event}:   Disable k.

\addvspace{18pt}

\noindent{\sc Event}:   Disable k-is-total.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-if-part-case-1 \\ \index{knuth-theorem-if-part-case-1}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;${\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-if-part-case-2 \\ \index{knuth-theorem-if-part-case-2}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp}) \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;${\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-if-part \\ \index{knuth-theorem-if-part}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;${\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-only-if-part-case-1 \\ \index{knuth-theorem-only-if-part-case!-1}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total} \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-only-if-part-case-2 \\ \index{knuth-theorem-only-if-part-case!-2}
\=\+({\rm{ilessp}}\,({\tt{0}}, {\it{b\/}})\index{ilessp} $\wedge$ ($\neg$ {\rm{ilessp}}\,({\tt{1}}, {\it{c\/}})\index{ilessp}) $\wedge$ {\rm{ilessp}}\,({\tt{0}}, {\it{d\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\it{b\/}}, {\rm{iplus}}\,({\tt{-1}}, {\it{c\/}})\index{iplus})\index{itimes}, {\it{d\/}})\index{ilessp}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-only-if-part \\ \index{knuth-theorem-only-if-part}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total} \\ 
$\rightarrow$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Here is a version of Knuth's theorem:

; Theorem.  The generalized 91 recursion with parameters ( a,b,c,d )
;           defines a total function on the integers if and only if
;           (c-1)b < d. 

\end{verbatim}\begin{tabbing}{\sc Theorem}:  knuth-theorem \\ \index{knuth-theorem}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{k-is-total}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}})\index{k-is-total} \\ 
$\leftrightarrow$$\;\;\;\;${\rm{ilessp}}\,({\rm{itimes}}\,({\rm{car}}\,({\it{vc-b\/}}), {\rm{iplus}}\,({\tt{-1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{iplus})\index{itimes}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-value-when-a$<$x \\ \index{k-value-when-a$<$x}
\=\+{\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{c\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{idifference}}\,({\it{x\/}}, {\it{b\/}})\index{idifference})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-value-when-a$<$x-version-2 \\ \index{k-value-when-a$<$x-version-2}
\=\+({\it{vc-a\/}} $\wedge$ {\it{vc-b\/}} $\wedge$ {\it{vc-c\/}} $\wedge$ {\it{vc-d\/}} $\wedge$ {\it{vc-x\/}} $\wedge$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;${\rm{idifference}}\,({\rm{car}}\,({\it{vc-x\/}}), {\rm{car}}\,({\it{vc-b\/}}))\index{idifference})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-part-2-case-1 \\ \index{knuth-theorem-part-2-case-1}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{1}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp} \\ 
{\bf then }{\rm{idifference}}\,({\rm{car}}\,({\it{vc-x\/}}), {\rm{car}}\,({\it{vc-b\/}}))\index{idifference} \\ 
{\bf else }{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\it{vc-c\/}}, \\ 
{\it{vc-d\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{car}}\,({\it{vc-x\/}}), \\ 
{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\it{cost\/}})\-)\-)\-)$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-value=body-value-when-a$>$=x\&c=1 \\ \index{k-value=body-value-when-a$>$=x\&!c=1}
\=\+($\neg$ {\rm{ilessp}}\,({\it{a\/}}, {\it{x\/}})\index{ilessp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{x\/}}, {\tt{0}}))\-)\-) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{a\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\it{b\/}}, {\tt{0}}), \\ 
{\tt{'}}{\tt{(}}{\tt{1}}{\tt{ .\ }}{\tt{0}}{\tt{)}}, \\ 
{\rm{cons}}\,({\it{d\/}}, {\tt{0}}), \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\it{x\/}}, {\it{d\/}})\index{iplus}, {\tt{0}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-value=body-value-when-a$>$=x\&c=1-version-2 \\ \index{k-value=body-value-when-a$>$=x\&!c=1-version-2}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{vc-c\/}}) $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;${\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\it{vc-c\/}}, \\ 
{\it{vc-d\/}}, \\ 
{\rm{cons}}\,({\rm{iplus}}\,({\rm{car}}\,({\it{vc-x\/}}), {\rm{car}}\,({\it{vc-d\/}}))\index{iplus}, {\it{cost\/}}))\-)\-))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  knuth-theorem-part-2-case-2 \\ \index{knuth-theorem-part-2-case-2}
\=\+({\it{vc-a\/}} $\wedge$ {\it{vc-b\/}} $\wedge$ {\it{vc-c\/}} $\wedge$ {\it{vc-d\/}} $\wedge$ {\it{vc-x\/}} $\wedge$ ({\rm{car}}\,({\it{vc-c\/}}) $=$ {\tt{1}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp} \\ 
{\bf then }{\rm{idifference}}\,({\rm{car}}\,({\it{vc-x\/}}), {\rm{car}}\,({\it{vc-b\/}}))\index{idifference} \\ 
{\bf else }{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\it{vc-c\/}}, \\ 
{\it{vc-d\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{car}}\,({\it{vc-x\/}}), \\ 
{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\it{cost\/}})\-)\-)\-)$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
; Here is a version of that part of Knuth's theorem 
; which gives the circumstances when K satisfies a
; simpler recurrence.

; Theorem.  In such a case the values of K( x ) also
;           satisfy the much simpler recurrence

;           K( x )  =  if  x > a  then  x - b
;                                 else  K( x+d-(c-1)b ).

\end{verbatim}\begin{tabbing}{\sc Theorem}:  knuth-theorem-part-2 \\ \index{knuth-theorem-part-2}
\=\+(\=\+{\it{vc-a\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-b\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-c\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-d\/}} \\ 
$\wedge$$\;\;\;\;${\it{vc-x\/}} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-b\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-c\/}}))\index{ilessp} \\ 
$\wedge$$\;\;\;\;${\rm{ilessp}}\,({\tt{0}}, {\rm{car}}\,({\it{vc-d\/}}))\index{ilessp})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{car}}\,({\rm{v\&c-apply\$}}\,({\tt{'}}{\tt{k}}, {\rm{list}}\,({\it{vc-a\/}}, {\it{vc-b\/}}, {\it{vc-c\/}}, {\it{vc-d\/}}, {\it{vc-x\/}}))) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{ilessp}}\,({\rm{car}}\,({\it{vc-a\/}}), {\rm{car}}\,({\it{vc-x\/}}))\index{ilessp} \\ 
{\bf then }{\rm{idifference}}\,({\rm{car}}\,({\it{vc-x\/}}), {\rm{car}}\,({\it{vc-b\/}}))\index{idifference} \\ 
{\bf else }{\rm{car}}\,({\rm{v\&c-apply\$}}\,(\=\+{\tt{'}}{\tt{k}}, \\ 
{\rm{list}}\,(\=\+{\it{vc-a\/}}, \\ 
{\it{vc-b\/}}, \\ 
{\it{vc-c\/}}, \\ 
{\it{vc-d\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{iplus}}\,(\=\+{\rm{car}}\,({\it{vc-x\/}}), \\ 
{\rm{idifference}}\,(\=\+{\rm{car}}\,({\it{vc-d\/}}), \\ 
{\rm{itimes}}\,(\=\+{\rm{car}}\,({\it{vc-b\/}}), \\ 
{\rm{iplus}}\,(\=\+{\tt{-1}}, \\ 
{\rm{car}}\,({\it{vc-c\/}}))\-\index{iplus})\-\index{itimes})\-\index{idifference})\-\index{iplus}, \\ 
{\it{cost\/}})\-)\-)\-)$\;${\bf  endif}\-)\-\-
\end{tabbing}

%\end{list}
\printindex
\end{document}
