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\begin{verbatim}#|
 Copyright (C) 1994 by Natarajan Shankar.  All Rights Reserved.

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

 NO WARRANTY

 Natarajan Shankar PROVIDES ABSOLUTELY NO WARRANTY.  THE EVENT SCRIPT IS
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 ARISING OUT OF THE USE OR INABILITY TO USE THIS SCRIPT (INCLUDING BUT NOT
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|#

;   The Mechanical Proof of Goedel's Incompleteness Theorem
;                            by
;                        N. Shankar




;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;  I:  Defining the Proof-checker for Z2         ;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;;;;;;;BOOT-STRAP loads basic library of axioms about lists, litatoms, and numbers.
;;;;;;;
\end{verbatim}
\noindent{\sc Event}:   Start with the initial {\bf thm} theory.

\addvspace{18pt}

\noindent{\sc Event}:   For efficiency, compile those definitions not yet compiled.

\addvspace{18pt}
\begin{verbatim};;;;;;;Turn off the use of the compiled lisp definitions for the non-BOOT-STRAP
;;;;;;;functions.
\end{verbatim}
\noindent{\sc Event}:   Enable defined functions; name this event `turn-off-lisp-defns'.\index{turn-off-lisp-defns}

\addvspace{18pt}
\begin{verbatim};;;;;;;Recognizers for function, variable, and predicate symbols.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{function}$\;\;$
{\rm{function}}\,({\it{fn\/}})\index{function} $=$ ({\rm{car}}\,({\it{fn\/}}) $=$ {\tt{2}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{variable}$\;\;$
{\rm{variable}}\,({\it{x\/}})\index{variable} $=$ ({\it{x\/}} $\in\;${\bf{N}}${}$)
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{predicate}$\;\;$
{\rm{predicate}}\,({\it{p\/}})\index{predicate} $=$ ({\rm{car}}\,({\it{p\/}}) $=$ {\tt{3}})
\end{tabbing}
\begin{verbatim};;;;;;;Degree (arity) and index (subscript) of a function/predicate symbol.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{degree}$\;\;$
{\rm{degree}}\,({\it{fn\/}})\index{degree} $=$ {\rm{fix}}\,({\rm{cddr}}\,({\it{fn\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{index}$\;\;$
{\rm{index}}\,({\it{fn\/}})\index{index} $=$ {\rm{fix}}\,({\rm{cadr}}\,({\it{fn\/}}))
\end{tabbing}
\begin{verbatim};;;;;;;(V X) generates the Xth variable
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{v}$\;\;$
{\rm{v}}\,({\it{x\/}})\index{v} $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numberp-fix \\ \index{numberp-fix}
{\rm{fix}}\,({\it{x\/}}) $\in\;${\bf{N}}${}$
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  variable-v \\ \index{variable-v}
{\rm{variable}}\,({\rm{v}}\,({\it{x\/}})\index{v})\index{variable}
\end{tabbing}
\begin{verbatim};;;;;;;FN generates function symbol with index X and arity Y.  
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{fn}$\;\;$
{\rm{fn}}\,({\it{x\/}}, {\it{y\/}})\index{fn} $=$ {\rm{cons}}\,({\tt{2}}, {\rm{cons}}\,({\rm{fix}}\,({\it{x\/}}), {\rm{fix}}\,({\it{y\/}})))
\end{tabbing}
\begin{verbatim};;;;;;;P generates the predicate symbol with index X and arity Y.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{p}$\;\;$
{\rm{p}}\,({\it{x\/}}, {\it{y\/}})\index{p} $=$ {\rm{cons}}\,({\tt{3}}, {\rm{cons}}\,({\rm{fix}}\,({\it{x\/}}), {\rm{fix}}\,({\it{y\/}})))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  function-fn \\ \index{function-fn}
{\rm{function}}\,({\rm{fn}}\,({\it{x\/}}, {\it{y\/}})\index{fn})\index{function}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  predicate-p \\ \index{predicate-p}
{\rm{predicate}}\,({\rm{p}}\,({\it{x\/}}, {\it{y\/}})\index{p})\index{predicate}
\end{tabbing}
\begin{verbatim};;;;;;;F-NOT is the negation shell.
\end{verbatim}
\noindent{\sc Event}:   Add the shell {\it{f-not\/}}\index{f-not}, with recognizer function symbol {\it{f-notp\/}} and 1 accessor: {\it{arg\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}
\begin{verbatim};;;;;;;F-OR is the disjunction shell.
\end{verbatim}
\noindent{\sc Event}:   Add the shell {\it{f-or\/}}\index{f-or}, with recognizer function symbol {\it{f-orp\/}} and 2 accessors: {\it{arg1\/}}, with type restriction (none-of) and default value zero; {\it{arg2\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}
\begin{verbatim};;;;;;;FORSOME is the existential quantification shell.
\end{verbatim}
\noindent{\sc Event}:   Add the shell {\it{forsome\/}}\index{forsome}, with recognizer function symbol {\it{forsomep\/}} and 2 accessors: {\it{bind\/}}, with type restriction (one-of numberp) and default value zero; {\it{body\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}
\begin{verbatim};;;;;;;EQL is the equality predicate for Z2 expressions.  Strict equality EQUAL cannot
;;;;;;;be used since expressions will be Goedel-encoded, and it is possible
;;;;;;;two different expressions to have the same Goedel-encoding.  EQL essentially 
;;;;;;;checks that its two arguments have the same encoding.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{fix1} \\  
\=\+{\rm{fix1}}\,({\it{x\/}})\index{fix1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\in\;${\bf{N}}${}$$\;\;${\bf then }{\it{x\/}} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  bind-count \\ \index{bind-count}
{\rm{forsomep}}\,({\it{x\/}})\index{forsomep} $\rightarrow$ ({\rm{count}}\,({\rm{bind}}\,({\it{x\/}})\index{bind}) $<$ {\rm{count}}\,({\it{x\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{eql} \\  
\=\+{\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{y\/}})$\;\;${\bf then }{\rm{eql}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{eql} $\wedge$ {\rm{eql}}\,({\rm{cdr}}\,({\it{x\/}}), {\rm{cdr}}\,({\it{y\/}}))\index{eql} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{y\/}})$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{x\/}})\index{f-notp} \\ 
{\bf then }\=\+{\bf if }{\rm{f-notp}}\,({\it{y\/}})\index{f-notp}$\;\;${\bf then }{\rm{eql}}\,({\rm{arg}}\,({\it{x\/}})\index{arg}, {\rm{arg}}\,({\it{y\/}})\index{arg})\index{eql} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{y\/}})\index{f-notp}$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{x\/}})\index{f-orp} \\ 
{\bf then }\=\+{\bf if }{\rm{f-orp}}\,({\it{y\/}})\index{f-orp}$\;\;${\bf then }{\rm{eql}}\,({\rm{arg1}}\,({\it{x\/}})\index{arg1}, {\rm{arg1}}\,({\it{y\/}})\index{arg1})\index{eql} $\wedge$ {\rm{eql}}\,({\rm{arg2}}\,({\it{x\/}})\index{arg2}, {\rm{arg2}}\,({\it{y\/}})\index{arg2})\index{eql} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{y\/}})\index{f-orp}$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{x\/}})\index{forsomep} \\ 
{\bf then }\=\+{\bf if }{\rm{forsomep}}\,({\it{y\/}})\index{forsomep} \\ 
{\bf then }({\rm{bind}}\,({\it{x\/}})\index{bind} $=$ {\rm{bind}}\,({\it{y\/}})\index{bind}) $\wedge$ {\rm{eql}}\,({\rm{body}}\,({\it{x\/}}), {\rm{body}}\,({\it{y\/}}))\index{eql} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{y\/}})\index{forsomep}$\;\;${\bf then }{\bf{f}} \\ 
{\bf else }{\rm{fix1}}\,({\it{x\/}})\index{fix1} $=$ {\rm{fix1}}\,({\it{y\/}})\index{fix1}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eql-comm \\ \index{eql-comm}
{\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql} $=$ {\rm{eql}}\,({\it{y\/}}, {\it{x\/}})\index{eql}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-eql \\ \index{equal-eql}
({\it{x\/}} $=$ {\it{y\/}}) $\rightarrow$ {\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numberp-eql \\ \index{numberp-eql}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\vee$ ({\it{y\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ ({\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql} $=$ ({\it{x\/}} $=$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  listp-eql \\ \index{listp-eql}
{\rm{eql}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}), {\rm{cons}}\,({\it{u\/}}, {\it{v\/}}))\index{eql} $=$ ({\rm{eql}}\,({\it{x\/}}, {\it{u\/}})\index{eql} $\wedge$ {\rm{eql}}\,({\it{y\/}}, {\it{v\/}})\index{eql})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  f-notp-eql \\ \index{f-notp-eql}
{\rm{eql}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\rm{f-not}}\,({\it{y\/}})\index{f-not})\index{eql} $=$ {\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  f-orp-eql \\ \index{f-orp-eql}
{\rm{eql}}\,({\rm{f-or}}\,({\it{x\/}}, {\it{y\/}})\index{f-or}, {\rm{f-or}}\,({\it{u\/}}, {\it{v\/}})\index{f-or})\index{eql} $=$ ({\rm{eql}}\,({\it{x\/}}, {\it{u\/}})\index{eql} $\wedge$ {\rm{eql}}\,({\it{y\/}}, {\it{v\/}})\index{eql})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  forsomep-eql \\ \index{forsomep-eql}
{\rm{eql}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{y\/}})\index{forsome}, {\rm{forsome}}\,({\it{u\/}}, {\it{v\/}})\index{forsome})\index{eql} $=$ (({\rm{fix}}\,({\it{x\/}}) $=$ {\rm{fix}}\,({\it{u\/}})) $\wedge$ {\rm{eql}}\,({\it{y\/}}, {\it{v\/}})\index{eql})
\end{tabbing}
\begin{verbatim};;;;;;;MEMB checks if X is a member of Y, using EQL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memb} \\  
\=\+{\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{eql}}\,({\it{x\/}}, {\rm{car}}\,({\it{y\/}}))\index{eql}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{memb}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{y\/}}))\index{memb}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  degree-fn \\ \index{degree-fn}
{\rm{degree}}\,({\rm{fn}}\,({\it{x\/}}, {\it{y\/}})\index{fn})\index{degree} $=$ {\rm{fix}}\,({\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  degree-p \\ \index{degree-p}
{\rm{degree}}\,({\rm{p}}\,({\it{x\/}}, {\it{y\/}})\index{p})\index{degree} $=$ {\rm{fix}}\,({\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  index-fn \\ \index{index-fn}
{\rm{index}}\,({\rm{fn}}\,({\it{x\/}}, {\it{y\/}})\index{fn})\index{index} $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  index-p \\ \index{index-p}
{\rm{index}}\,({\rm{p}}\,({\it{x\/}}, {\it{y\/}})\index{p})\index{index} $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-function \\ \index{not-function}
(({\it{x\/}} $\simeq\;${{\bf{nil}}}${}$) $\vee$ {\rm{predicate}}\,({\it{x\/}})\index{predicate}) $\rightarrow$ ($\neg$ {\rm{function}}\,({\it{x\/}})\index{function})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-predicate \\ \index{not-predicate}
(({\it{x\/}} $\simeq\;${{\bf{nil}}}${}$) $\vee$ {\rm{function}}\,({\it{x\/}})\index{function}) $\rightarrow$ ($\neg$ {\rm{predicate}}\,({\it{x\/}})\index{predicate})
\end{tabbing}

\noindent{\sc Event}:   Enable index; name this event `g0616'.\index{g0616}

\addvspace{18pt}

\noindent{\sc Event}:   Enable degree; name this event `g0617'.\index{g0617}

\addvspace{18pt}

\noindent{\sc Event}:   Enable v; name this event `g0618'.\index{g0618}

\addvspace{18pt}

\noindent{\sc Event}:   Enable fn; name this event `g0619'.\index{g0619}

\addvspace{18pt}

\noindent{\sc Event}:   Enable p; name this event `g0620'.\index{g0620}

\addvspace{18pt}

\noindent{\sc Event}:   Enable function; name this event `g0621'.\index{g0621}

\addvspace{18pt}

\noindent{\sc Event}:   Enable predicate; name this event `g0622'.\index{g0622}

\addvspace{18pt}
\begin{verbatim};;;;;;;X is a function or a predicate.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{func-pred}$\;\;$
{\rm{func-pred}}\,({\it{x\/}})\index{func-pred} $=$ ({\rm{function}}\,({\it{x\/}})\index{function} $\vee$ {\rm{predicate}}\,({\it{x\/}})\index{predicate})
\end{tabbing}
\begin{verbatim};;;;;;;APPEND concatenates two lists.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{append} \\  
\=\+{\rm{append}}\,({\it{x\/}}, {\it{y\/}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{cons}}\,({\rm{car}}\,({\it{x\/}}), {\rm{append}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{y\/}})) \\ 
{\bf else }{\it{y\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;DEL deletes all occurrences of X from list Y, using EQL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{del} \\  
\=\+{\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{eql}}\,({\it{x\/}}, {\rm{car}}\,({\it{y\/}}))\index{eql}$\;\;${\bf then }{\rm{del}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{y\/}}))\index{del} \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{y\/}}), {\rm{del}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{y\/}}))\index{del})$\;${\bf  endif}\- \\ 
{\bf else }{\it{y\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-memb-del \\ \index{not-memb-del}
$\neg$ {\rm{memb}}\,({\it{x\/}}, {\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del})\index{memb}
\end{tabbing}
\begin{verbatim};;;;;;;COLLECT-FREE collects a list of all the free variables in the
;;;;;;;expression/list of expressions EXP, depending on (ZEROP FLG).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{collect-free} \\  
\=\+{\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{variable}}\,({\it{exp\/}})\index{variable}$\;\;${\bf then }{\rm{cons}}\,({\it{exp\/}}, {\bf{nil}}) \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{collect-free}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\tt{0}})\index{collect-free} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }{\rm{append}}\,({\rm{collect-free}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\tt{0}})\index{collect-free}, {\rm{collect-free}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\tt{0}})\index{collect-free}) \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep} \\ 
{\bf then }{\rm{del}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\rm{collect-free}}\,({\rm{body}}\,({\it{exp\/}}), {\tt{0}})\index{collect-free})\index{del} \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }{\rm{collect-free}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}})\index{collect-free} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{append}}\,({\rm{collect-free}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{0}})\index{collect-free}, {\rm{collect-free}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}})\index{collect-free}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{sentence}$\;\;$
{\rm{sentence}}\,({\it{exp\/}})\index{sentence} $=$ ({\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}})
\end{tabbing}
\begin{verbatim};;;;;;;COVERING collects a list of all bound variables that govern free occurrences
;;;;;;;of VAR in EXP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{covering} \\  
\=\+{\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{covering}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{var\/}}, {\tt{0}})\index{covering} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }{\rm{append}}\,({\rm{covering}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{var\/}}, {\tt{0}})\index{covering}) \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep} \\ 
{\bf then }\=\+{\bf if }{\rm{memb}}\,({\it{var\/}}, {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})\index{memb} \\ 
{\bf then }{\rm{cons}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\rm{covering}}\,({\rm{body}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{0}})\index{covering}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }{\rm{covering}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{1}})\index{covering} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{append}}\,({\rm{covering}}\,({\rm{car}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{1}})\index{covering}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;X and Y have no common elements.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{nil-intersect} \\  
\=\+{\rm{nil-intersect}}\,({\it{x\/}}, {\it{y\/}})\index{nil-intersect} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }($\neg$ {\rm{memb}}\,({\rm{car}}\,({\it{x\/}}), {\it{y\/}})\index{memb}) $\wedge$ {\rm{nil-intersect}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{y\/}})\index{nil-intersect} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;TERM is FREE-FOR VAR in EXP, if no free occurrence of VAR in EXP is governed
;;;;;;;by a variable which occurs free in TERM.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{free-for} \\  
\=\+{\rm{free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{free-for} \\ 
$=$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering}, {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect}\-
\end{tabbing}
\begin{verbatim};;;;;;;(F-EQUAL X Y) is the Z2 expression for (x=y).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{f-equal}$\;\;$
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal} $=$ {\rm{list}}\,({\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  or-memb-append \\ \index{or-memb-append}
{\rm{memb}}\,({\it{x\/}}, {\rm{append}}\,({\it{y\/}}, {\it{z\/}}))\index{memb} $=$ ({\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb} $\vee$ {\rm{memb}}\,({\it{x\/}}, {\it{z\/}})\index{memb})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  memb-del \\ \index{memb-del}
($\neg$ {\rm{eql}}\,({\it{x\/}}, {\it{z\/}})\index{eql}) $\rightarrow$ ({\rm{memb}}\,({\it{x\/}}, {\rm{del}}\,({\it{z\/}}, {\it{y\/}})\index{del})\index{memb} $=$ {\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb})
\end{tabbing}
\begin{verbatim};;;;;;;VAR-LIST checks that LIST is a list of N variables.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{var-list} \\  
\=\+{\rm{var-list}}\,({\it{list\/}}, {\it{n\/}})\index{var-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $\simeq {\tt{0}}$) $\vee$ ({\it{list\/}} $\simeq\;${{\bf{nil}}}${}$)$\;\;${\bf then }({\it{n\/}} $\simeq {\tt{0}}$) $\wedge$ ({\it{list\/}} $\simeq\;${{\bf{nil}}}${}$) \\ 
{\bf else }{\rm{variable}}\,({\rm{car}}\,({\it{list\/}}))\index{variable} $\wedge$ {\rm{var-list}}\,({\rm{cdr}}\,({\it{list\/}}), {\it{n\/}} $-\;1$)\index{var-list}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;SYMB checks that X is either the equality symbol, or appears in list SYMBOLS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{symb} \\  
{\rm{symb}}\,({\it{x\/}}, {\it{symbols\/}})\index{symb} $=$ (({\it{x\/}} $=$ {\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}) $\vee$ {\rm{memb}}\,({\it{x\/}}, {\it{symbols\/}})\index{memb})
\end{tabbing}
\begin{verbatim};;;;;;;TERMP checks that EXP is either a variable or an n-ary function symbol (in SYMBOLS)
;;;;;;;applied to n smaller terms.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{termp} \\  
\=\+{\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\rm{function}}\,({\rm{car}}\,({\it{exp\/}}))\index{function} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\rm{car}}\,({\it{exp\/}}), {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{length} $=$ {\rm{degree}}\,({\rm{car}}\,({\it{exp\/}}))\index{degree})\- \\ 
{\bf else }{\rm{variable}}\,({\it{exp\/}})\index{variable}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{exp\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{termp}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;ATOMP checks that EXP is an n-ary predicate symbol applied to n terms.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{atomp} \\  
\=\+{\rm{atomp}}\,({\it{exp\/}}, {\it{symbols\/}})\index{atomp} \\ 
$=$$\;\;\;\;$(\=\+{\rm{predicate}}\,({\rm{car}}\,({\it{exp\/}}))\index{predicate} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\rm{car}}\,({\it{exp\/}}), {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{length} $=$ {\rm{degree}}\,({\rm{car}}\,({\it{exp\/}}))\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  atomp-reduc \\ \index{atomp-reduc}
\=\+{\rm{atomp}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}), {\it{symbols\/}})\index{atomp} \\ 
$=$$\;\;\;\;$(\=\+{\rm{predicate}}\,({\it{x\/}})\index{predicate} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{x\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{y\/}})\index{length} $=$ {\rm{degree}}\,({\it{x\/}})\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  atomp-f-equal \\ \index{atomp-f-equal}
\=\+{\rm{atomp}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{symbols\/}})\index{atomp} \\ 
$=$$\;\;\;\;$({\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{verbatim};;;;;;;SUBST substitutes TERM for all free occurrences of VAR in EXP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{subst} \\  
\=\+{\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{variable}}\,({\it{exp\/}})\index{variable} \\ 
{\bf then }\=\+{\bf if }{\it{exp\/}} $=$ {\it{var\/}}$\;\;${\bf then }{\it{term\/}} \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{f-not}}\,({\rm{subst}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-not} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }{\rm{f-or}}\,(\=\+{\rm{subst}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, \\ 
{\rm{subst}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\-\index{f-or} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep} \\ 
{\bf then }\=\+{\bf if }{\rm{eql}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\it{var\/}})\index{eql}$\;\;${\bf then }{\it{exp\/}} \\ 
{\bf else }{\rm{forsome}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\rm{subst}}\,({\rm{body}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{forsome}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{car}}\,({\it{exp\/}}), {\rm{subst}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{1}})\index{subst}) \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{subst}}\,({\rm{car}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{1}})\index{subst}) \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;SUB checks if every element of X occurs in Y, using EQL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{sub} \\  
\=\+{\rm{sub}}\,({\it{x\/}}, {\it{y\/}})\index{sub} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{memb}}\,({\rm{car}}\,({\it{x\/}}), {\it{y\/}})\index{memb} $\wedge$ {\rm{sub}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{y\/}})\index{sub} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;SUBSET checks if every element of X occurs in Y, using EQUAL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{subset} \\  
\=\+{\rm{subset}}\,({\it{x\/}}, {\it{y\/}})\index{subset} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }({\rm{car}}\,({\it{x\/}}) $\in$ {\it{y\/}}) $\wedge$ {\rm{subset}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{y\/}})\index{subset} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  sub-subset \\ \index{sub-subset}
{\rm{subset}}\,({\it{x\/}}, {\it{y\/}})\index{subset} $\rightarrow$ {\rm{sub}}\,({\it{x\/}}, {\it{y\/}})\index{sub}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-f-equal \\ \index{subst-f-equal}
\=\+{\rm{subst}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{f-equal}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-equal}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-equal; name this event `g0623'.\index{g0623}

\addvspace{18pt}

\noindent{\sc Event}:   Enable atomp; name this event `g0253'.\index{g0253}

\addvspace{18pt}
\begin{verbatim};;;;;;;FORMULA checks if EXP is a well-formed formula.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{formula} \\  
\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{formula}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{symbols\/}})\index{formula} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }{\rm{formula}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{symbols\/}})\index{formula} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep}$\;\;${\bf then }{\rm{formula}}\,({\rm{body}}\,({\it{exp\/}}), {\it{symbols\/}})\index{formula} \\ 
{\bf else }{\rm{atomp}}\,({\it{exp\/}}, {\it{symbols\/}})\index{atomp}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-f-equal \\ \index{formula-f-equal}
\=\+{\rm{formula}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$({\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{verbatim};;;;;;;(F-AND X Y) denotes (X ^ Y).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{f-and}$\;\;$
{\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and} $=$ {\rm{f-not}}\,({\rm{f-or}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\rm{f-not}}\,({\it{y\/}})\index{f-not})\index{f-or})\index{f-not}
\end{tabbing}
\begin{verbatim};;;;;;;(F-IMPLIES X Y) denotes (X => Y).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{f-implies}$\;\;$
{\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies} $=$ {\rm{f-or}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{y\/}})\index{f-or}
\end{tabbing}
\begin{verbatim};;;;;;;(FORALL X EXP) denotes (Ax exp).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{forall}$\;\;$
$\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall} $=$ {\rm{f-not}}\,({\rm{forsome}}\,({\it{var\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not})\index{forsome})\index{f-not}
\end{tabbing}
\begin{verbatim};;;;;;;PAIREQUALS returns a formula of the form
;;;;;;;((x1=y1) -> (...-> ((xn=yn) -> EXP))).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pairequals} \\  
\=\+{\rm{pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}})\index{pairequals} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars1\/}}) $\wedge$ {\rm{listp}}\,({\it{vars2\/}}) \\ 
{\bf then }{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{car}}\,({\it{vars1\/}}), {\rm{car}}\,({\it{vars2\/}}))\index{f-equal}, \\ 
{\rm{pairequals}}\,({\rm{cdr}}\,({\it{vars1\/}}), {\rm{cdr}}\,({\it{vars2\/}}), {\it{exp\/}})\index{pairequals})\-\index{f-implies} \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{second}$\;\;$
{\rm{second}}\,({\it{x\/}})\index{second} $=$ {\rm{cadr}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{third}$\;\;$
{\rm{third}}\,({\it{x\/}})\index{third} $=$ {\rm{caddr}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{fourth}$\;\;$
{\rm{fourth}}\,({\it{x\/}})\index{fourth} $=$ {\rm{cadddr}}\,({\it{x\/}})
\end{tabbing}
\begin{verbatim};;;;;;;FOL-AXIOM-PROOF checks if CONC is a logical axiom (in a theory with
;;;;;;;SYMBOLS), using hints from HINTS.  The axioms are: [1] propositional
;;;;;;;axioms of the form, "((not A) or A)";  [2] substitution axioms of
;;;;;;;the form, "[t/x]A -> (forsome x A)";  [3] Identity axioms of the form,
;;;;;;;"(x=x)";  [4] Equality axioms for functions, of the form, 
;;;;;;;"((x1=y1) -> (...-> ((xn=yn) -> ((fx1...xn)=(fy1...yn)))))";
;;;;;;;[5] Equality axioms for predicates, of the form, 
;;;;;;;"((x1=y1) -> (...-> ((xn=yn) -> ((px1...xn) -> (py1...yn)))))".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{fol-axiom-proof} \\  
\=\+{\rm{fol-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{fol-axiom-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{1}} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{f-or}}\,({\rm{f-not}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{f-not}, {\rm{second}}\,({\it{hints\/}})\index{second})\index{f-or})\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{2}} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{subst}}\,(\=\+{\rm{second}}\,({\it{hints\/}})\index{second}, \\ 
{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\tt{0}})\-\index{subst}, \\ 
{\rm{forsome}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{second}}\,({\it{hints\/}})\index{second})\index{forsome})\-\index{f-implies})\-\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{3}} \\ 
{\bf then }\=\+{\rm{termp}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{f-equal}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{second}}\,({\it{hints\/}})\index{second})\index{f-equal})\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{4}} \\ 
{\bf then }\=\+{\rm{termp}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{function}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{function} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{length} $=$ {\rm{degree}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{degree}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{length} $=$ {\rm{degree}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{pairequals}}\,(\=\+{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{cons}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third}), \\ 
{\rm{cons}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}))\-\index{f-equal})\-\index{pairequals})\-\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{5}} \\ 
{\bf then }\=\+{\rm{termp}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{predicate}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{predicate} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{length} $=$ {\rm{degree}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{degree}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{length} $=$ {\rm{degree}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{pairequals}}\,(\=\+{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{cons}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third}), \\ 
{\rm{cons}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}))\-\index{f-implies})\-\index{pairequals})\-\index{eql}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;FOL-RULES checks if CONC follows from the SUB-GOALS by an application
;;;;;;;of one of the logical rules of inference, using hints from HINTS.
;;;;;;;The rules of inference are:  [1] Expansion: derive "(A or B)" from "B";
;;;;;;;[2] Contraction: derive "A" from "(A or A)";  
;;;;;;;[3] Associativity: derive "((A or B) or C)" from "(A or (B or C))";  
;;;;;;;[4] Cut: derive "(B or C)" from "(A or B)" and "((not B) or C)";
;;;;;;;[5] E-introduction: derive "((forsome x A) -> B)" from "(A -> B)",
;;;;;;;provided "x" does not occur free in "B".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{fol-rules} \\  
\=\+{\rm{fol-rules}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{fol-rules} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{1}} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{car}}\,({\it{sub-goals\/}}), {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{f-or}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{car}}\,({\it{sub-goals\/}}))\index{f-or})\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{2}} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{sub-goals\/}}, {\rm{list}}\,({\rm{f-or}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{second}}\,({\it{hints\/}})\index{second})\index{f-or}))\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{second}}\,({\it{hints\/}})\index{second})\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{3}} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{list}}\,({\rm{f-or}}\,(\=\+{\rm{second}}\,({\it{hints\/}})\index{second}, \\ 
{\rm{f-or}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{f-or})\-\index{f-or}))\-\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{f-or}}\,({\rm{f-or}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{f-or}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{f-or})\-\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{4}} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{list}}\,(\=\+{\rm{f-or}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{f-or}, \\ 
{\rm{f-or}}\,({\rm{f-not}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{f-not}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{f-or})\-)\-\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{f-or}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{f-or})\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{5}} \\ 
{\bf then }\=\+{\rm{variable}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{memb}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{collect-free}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{0}})\index{collect-free})\index{memb}) \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{sub-goals\/}}, {\rm{list}}\,({\rm{f-implies}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{f-implies}))\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{forsome}, \\ 
{\rm{fourth}}\,({\it{hints\/}})\index{fourth})\-\index{f-implies})\-\index{eql}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;Some notation for: [1] Set-theoretic membership: "(x in y)";
;;;;;;;[2] Null Set: "(phi)";  [3] Pairing: "{x, y}";  
;;;;;;;[4] Union: "(x U y)";  [5] Integerhood:  "(I x)" defined as
;;;;;;;"x is transitive and linearly ordered";
;;;;;;;[6] Successor: "(S x)" defined as "(x U {x})", where "{x}" is "{x, x}";
;;;;;;;[7] If-and-only-if: "(A <-> B)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{isin}$\;\;$
{\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin} $=$ {\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{phi}$\;\;$
{\sc{phi}}\index{phi} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-pair}$\;\;$
{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-union}$\;\;$
{\rm{z-union}}\,({\it{x\/}}, {\it{y\/}})\index{z-union} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-int}$\;\;$
{\rm{z-int}}\,({\it{x\/}})\index{z-int} $=$ {\rm{list}}\,({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-succ}$\;\;$
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{f-iff}$\;\;$
{\rm{f-iff}}\,({\it{x\/}}, {\it{y\/}})\index{f-iff} $=$ {\rm{f-and}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\rm{f-implies}}\,({\it{y\/}}, {\it{x\/}})\index{f-implies})\index{f-and}
\end{tabbing}
\begin{verbatim};;;;;;;Z2-AXIOMS checks if CONC is a set-theoretic axiom from Z2.
;;;;;;;These axioms are: 
;;;;;;;[1] Extensionality: "(x=y) <-> (forall z ((z in x) <-> (z in y)))".
;;;;;;;[2] Null set: "(not (x in (phi)))".
;;;;;;;[3] Pairing: "(z in {x, y}) <-> ((z=x) v (z=y))".
;;;;;;;[4] Union: "(z in (x U y)) <-> ((z in x) v (z in y))".
;;;;;;;[5] Induction:  "A[(phi)] ^ (forall x (((I x) ^ A[x]) -> A[(S x)])
;;;;;;;                    -> (forall x ((I x) -> A[x]))"
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z2-axioms} \\  
\=\+{\rm{z2-axioms}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{z2-axioms} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{1}} \\ 
{\bf then }\=\+{\rm{variable}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{memb}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{collect-free}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\tt{0}})\index{collect-free})\index{memb}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{memb}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{collect-free}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{0}})\index{collect-free})\index{memb}) \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{f-equal}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{f-equal}, \\ 
$\forall\;${\it{fourth\/}}, {\it{hints\/}}$\;${\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\rm{second}}\,({\it{hints\/}})\index{second})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\rm{third}}\,({\it{hints\/}})\index{third})\-\index{isin})\-\index{f-iff}\index{forall})\-\index{f-iff})\-\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{2}} \\ 
{\bf then }\=\+{\rm{termp}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{f-not}}\,({\rm{isin}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\sc{phi}}\index{phi})\index{isin})\index{f-not})\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{3}} \\ 
{\bf then }\=\+{\rm{termp}}\,({\rm{list}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\rm{z-pair}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{z-pair})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{second}}\,({\it{hints\/}})\index{second})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{f-equal})\-\index{f-or})\-\index{f-iff})\-\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{4}} \\ 
{\bf then }\=\+{\rm{termp}}\,({\rm{list}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\rm{z-union}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{z-union})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{isin}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{second}}\,({\it{hints\/}})\index{second})\index{isin}, \\ 
{\rm{isin}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{isin})\-\index{f-or})\-\index{f-iff})\-\index{eql}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{hints\/}}) $=$ {\tt{5}} \\ 
{\bf then }\=\+{\rm{variable}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{subst}}\,(\=\+{\rm{second}}\,({\it{hints\/}})\index{second}, \\ 
{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\sc{phi}}\index{phi}, \\ 
{\tt{0}})\-\index{subst}, \\ 
$\forall\;${\it{third\/}}, {\it{hints\/}}$\;${\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{z-int}, \\ 
{\rm{second}}\,({\it{hints\/}})\index{second})\-\index{f-and}, \\ 
{\rm{subst}}\,(\=\+{\rm{second}}\,({\it{hints\/}})\index{second}, \\ 
{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\rm{z-succ}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{z-succ}, \\ 
{\tt{0}})\-\index{subst})\-\index{f-implies}\index{forall})\-\index{f-and}, \\ 
$\forall\;${\it{third\/}}, {\it{hints\/}}$\;${\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{z-int}, \\ 
{\rm{second}}\,({\it{hints\/}})\index{second})\-\index{f-implies}\index{forall})\-\index{f-implies})\-\index{eql}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;X is a list with no duplicates.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{set} \\  
\=\+{\rm{set}}\,({\it{x\/}})\index{set} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }($\neg$ {\rm{memb}}\,({\rm{car}}\,({\it{x\/}}), {\rm{cdr}}\,({\it{x\/}}))\index{memb}) $\wedge$ {\rm{set}}\,({\rm{cdr}}\,({\it{x\/}}))\index{set} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;Returns the element paired to X in the alist Y.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{assoc} \\  
\=\+{\rm{assoc}}\,({\it{x\/}}, {\it{y\/}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{eql}}\,({\rm{caar}}\,({\it{y\/}}), {\it{x\/}})\index{eql}$\;\;${\bf then }{\rm{cdar}}\,({\it{y\/}}) \\ 
{\bf else }{\rm{assoc}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{y\/}}))$\;${\bf  endif}\- \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;Returns the list of symbols following the last occurrence of SYM in SYMBOLS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{rest-of} \\  
\=\+{\rm{rest-of}}\,({\it{sym\/}}, {\it{symbols\/}})\index{rest-of} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{symbols\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{eql}}\,({\rm{car}}\,({\it{symbols\/}}), {\it{sym\/}})\index{eql} $\wedge$ ($\neg$ {\rm{memb}}\,({\it{sym\/}}, {\rm{cdr}}\,({\it{symbols\/}}))\index{memb}) \\ 
{\bf then }{\rm{cdr}}\,({\it{symbols\/}}) \\ 
{\bf else }{\rm{rest-of}}\,({\it{sym\/}}, {\rm{cdr}}\,({\it{symbols\/}}))\index{rest-of}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;SYMB-DEFN-PROOF checks if CONC is admissible as the definition of a new
;;;;;;;function or predicate symbol.  The definition of a predicate symbol
;;;;;;;must have the form, "(px1...xn) <-> A", where the "xi" are distinct
;;;;;;;variables, and "A" is a formula w.r.t. the symbols that follow "p" in SYMBOLS
;;;;;;;which contains no free variables other than the "xi".
;;;;;;;The definition of a function symbol must have the form, "[(fx1...xn)/x]A",
;;;;;;;where the "x" and the "xi" are distinct, "A" is a formula w.r.t. the
;;;;;;;symbols that follow "f" in SYMBOLS, and "(forsome-unique x A)" is
;;;;;;;provable in a theory which only contains the symbols that follow "f" in SYMBOLS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{symb-defn-proof} \\  
\=\+{\rm{symb-defn-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{defn\/}}, {\it{symbols\/}})\index{symb-defn-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{predicate}}\,({\rm{caar}}\,({\it{hints\/}}))\index{predicate} \\ 
{\bf then }\=\+{\rm{set}}\,({\rm{cdar}}\,({\it{hints\/}}))\index{set} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\rm{cdar}}\,({\it{hints\/}}), {\rm{degree}}\,({\rm{caar}}\,({\it{hints\/}}))\index{degree})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{f-iff}}\,({\rm{car}}\,({\it{hints\/}}), {\it{defn\/}})\index{f-iff})\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{sub}}\,({\rm{collect-free}}\,({\it{defn\/}}, {\tt{0}})\index{collect-free}, {\rm{cdar}}\,({\it{hints\/}}))\index{sub} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{defn\/}}, {\rm{rest-of}}\,({\rm{caar}}\,({\it{hints\/}}), {\it{symbols\/}})\index{rest-of})\index{formula}\- \\ 
{\bf elseif }{\rm{function}}\,({\rm{caar}}\,({\it{hints\/}}))\index{function} \\ 
{\bf then }\=\+{\rm{set}}\,({\rm{cons}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{cons}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{cdar}}\,({\it{hints\/}}))))\index{set} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\rm{cdar}}\,({\it{hints\/}}), {\rm{degree}}\,({\rm{caar}}\,({\it{hints\/}}))\index{degree})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{rest-of}}\,({\rm{caar}}\,({\it{hints\/}}), {\it{symbols\/}})\index{rest-of})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{sub}}\,(\=\+{\rm{collect-free}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{0}})\index{collect-free}, \\ 
{\rm{cons}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{cdar}}\,({\it{hints\/}})))\-\index{sub} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{car}}\,({\it{hints\/}}), {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\rm{subst}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{car}}\,({\it{hints\/}}), {\tt{0}})\index{subst})\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc\/}}, {\it{defn\/}})\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{list}}\,(\=\+{\rm{forsome}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third})\index{forsome}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\rm{subst}}\,(\=\+{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\rm{second}}\,({\it{hints\/}})\index{second}, \\ 
{\rm{fourth}}\,({\it{hints\/}})\index{fourth}, \\ 
{\tt{0}})\-\index{subst})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{f-equal})\-\index{f-implies})\-)\-\index{eql}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;A formal proof is represented as 4-tuple: <step hints conclusion sub-proofs>,
;;;;;;;where the conclusion is derived from the conclusions of the sub-proofs
;;;;;;;by means of a proof-step given by the combination of `step' and `hints'.
;;;;;;;CONC returns the conclusion of a proof/list of proofs.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{conc} \\  
\=\+{\rm{conc}}\,({\it{pf\/}}, {\it{flg\/}})\index{conc} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{third}}\,({\it{pf\/}})\index{third} \\ 
{\bf elseif }{\rm{listp}}\,({\it{pf\/}})$\;\;${\bf then }{\rm{cons}}\,({\rm{conc}}\,({\rm{car}}\,({\it{pf\/}}), {\tt{0}})\index{conc}, {\rm{conc}}\,({\rm{cdr}}\,({\it{pf\/}}), {\tt{1}})\index{conc}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;PRF checks if PF is well-formed proof in a theory whose function/predicate
;;;;;;;symbols appear in SYMBOLS, where the definitions of the defined symbols
;;;;;;;appear in DEFNS, and GIVEN is a list of assumptions.  The argument GIVEN is
;;;;;;;there merely to facilitate the proof of essential incompleteness, and is never
;;;;;;;actually used to demonstrate the existence of a proof.  One can obviously 
;;;;;;;introduce an inconsistency through GIVEN.
;;;;;;;PRF recurses down PF and checks if the conclusion is either [1] a logical axiom;
;;;;;;;[2] follows from the conclusion of the sub-proofs (which are themselves
;;;;;;;well-formed) by means of one of the rules of inference; [3] a set-theoretic
;;;;;;;axiom; [4] a function/predicate definition; [5] an assumption from GIVEN.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{prf} \\  
\=\+{\rm{prf}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{prf} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\it{pf\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }$\neg$ {\rm{formula}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\it{symbols\/}})\index{formula}$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{1}} \\ 
{\bf then }{\rm{fol-axiom-proof}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\rm{second}}\,({\it{pf\/}})\index{second}, {\it{symbols\/}})\index{fol-axiom-proof} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{2}} \\ 
{\bf then }\=\+{\rm{fol-rules}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\rm{second}}\,({\it{pf\/}})\index{second}, {\rm{conc}}\,({\rm{fourth}}\,({\it{pf\/}})\index{fourth}, {\tt{1}})\index{conc}, {\it{symbols\/}})\index{fol-rules} \\ 
$\wedge$$\;\;\;\;${\rm{prf}}\,({\rm{fourth}}\,({\it{pf\/}})\index{fourth}, {\it{given\/}}, {\it{defns\/}}, {\tt{1}}, {\it{symbols\/}})\index{prf}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{3}} \\ 
{\bf then }{\rm{z2-axioms}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\rm{second}}\,({\it{pf\/}})\index{second}, {\it{symbols\/}})\index{z2-axioms} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{4}} \\ 
{\bf then }\=\+{\rm{symb-defn-proof}}\,(\=\+{\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, \\ 
{\rm{second}}\,({\it{pf\/}})\index{second}, \\ 
{\rm{conc}}\,({\rm{fourth}}\,({\it{pf\/}})\index{fourth}, {\tt{1}})\index{conc}, \\ 
{\rm{assoc}}\,({\rm{caar}}\,({\rm{second}}\,({\it{pf\/}})\index{second}), {\it{defns\/}}), \\ 
{\it{symbols\/}})\-\index{symb-defn-proof} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\rm{function}}\,({\rm{caar}}\,({\rm{cadr}}\,({\it{pf\/}})))\index{function} \\ 
{\bf then }{\rm{prf}}\,(\=\+{\rm{cadddr}}\,({\it{pf\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\tt{1}}, \\ 
{\rm{rest-of}}\,({\rm{caar}}\,({\rm{cadr}}\,({\it{pf\/}})), {\it{symbols\/}})\index{rest-of})\-\index{prf} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\- \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{5}}$\;\;${\bf then }{\rm{memb}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\it{given\/}})\index{memb} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{pf\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{prf}}\,({\rm{car}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf} \\ 
$\wedge$$\;\;\;\;${\rm{prf}}\,({\rm{cdr}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{1}}, {\it{symbols\/}})\index{prf}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;PROVES checks that PF is a well-formed proof whose conclusion is EXP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{proves} \\  
\=\+{\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$=$$\;\;\;\;$(\=\+{\rm{eql}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\it{exp\/}})\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{prf}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;What follows is a series of rewrite rules which convert the proof-checker above
;;;;;;;into a set of rewrite rules.  Functions like PROP-AXIOM construct axioms
;;;;;;;whose proofs are constructed by `proof-constructors' like PROP-AXIOM-PROOF.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{prop-axiom}$\;\;$
{\rm{prop-axiom}}\,({\it{exp\/}})\index{prop-axiom} $=$ {\rm{f-or}}\,({\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{exp\/}})\index{f-or}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{subst-axiom} \\  
\=\+{\rm{subst-axiom}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}})\index{subst-axiom} \\ 
$=$$\;\;\;\;${\rm{f-implies}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{forsome}}\,({\it{var\/}}, {\it{exp\/}})\index{forsome})\index{f-implies}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ident-axiom}$\;\;$
{\rm{ident-axiom}}\,({\it{var\/}})\index{ident-axiom} $=$ {\rm{f-equal}}\,({\it{var\/}}, {\it{var\/}})\index{f-equal}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prop-axiom-proof} \\  
{\rm{prop-axiom-proof}}\,({\it{exp\/}})\index{prop-axiom-proof} $=$ {\rm{list}}\,({\tt{1}}, {\rm{list}}\,({\tt{1}}, {\it{exp\/}}), {\rm{prop-axiom}}\,({\it{exp\/}})\index{prop-axiom})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{subst-axiom-proof} \\  
\=\+{\rm{subst-axiom-proof}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}})\index{subst-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{1}}, {\rm{list}}\,({\tt{2}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}), {\rm{subst-axiom}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}})\index{subst-axiom})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ident-axiom-proof} \\  
{\rm{ident-axiom-proof}}\,({\it{var\/}})\index{ident-axiom-proof} $=$ {\rm{list}}\,({\tt{1}}, {\rm{list}}\,({\tt{3}}, {\it{var\/}}), {\rm{f-equal}}\,({\it{var\/}}, {\it{var\/}})\index{f-equal})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{equal-axiom1} \\  
\=\+{\rm{equal-axiom1}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{fn\/}})\index{equal-axiom1} \\ 
$=$$\;\;\;\;${\rm{pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\rm{f-equal}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{vars1\/}}), {\rm{cons}}\,({\it{fn\/}}, {\it{vars2\/}}))\index{f-equal})\index{pairequals}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{equal-axiom1-proof} \\  
\=\+{\rm{equal-axiom1-proof}}\,({\it{fn\/}}, {\it{vars1\/}}, {\it{vars2\/}})\index{equal-axiom1-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{1}}, {\rm{list}}\,({\tt{4}}, {\it{fn\/}}, {\it{vars1\/}}, {\it{vars2\/}}), {\rm{equal-axiom1}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{fn\/}})\index{equal-axiom1})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{equal-axiom2} \\  
\=\+{\rm{equal-axiom2}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{pr\/}})\index{equal-axiom2} \\ 
$=$$\;\;\;\;${\rm{pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\rm{f-implies}}\,({\rm{cons}}\,({\it{pr\/}}, {\it{vars1\/}}), {\rm{cons}}\,({\it{pr\/}}, {\it{vars2\/}}))\index{f-implies})\index{pairequals}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{equal-axiom2-proof} \\  
\=\+{\rm{equal-axiom2-proof}}\,({\it{pr\/}}, {\it{vars1\/}}, {\it{vars2\/}})\index{equal-axiom2-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{1}}, {\rm{list}}\,({\tt{5}}, {\it{pr\/}}, {\it{vars1\/}}, {\it{vars2\/}}), {\rm{equal-axiom2}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{pr\/}})\index{equal-axiom2})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-f-not \\ \index{formula-f-not}
{\rm{formula}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{symbols\/}})\index{formula} $=$ {\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-f-or \\ \index{formula-f-or}
{\rm{formula}}\,({\rm{f-or}}\,({\it{x\/}}, {\it{y\/}})\index{f-or}, {\it{symbols\/}})\index{formula} $=$ ({\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-forsome \\ \index{formula-forsome}
{\rm{formula}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{y\/}})\index{forsome}, {\it{symbols\/}})\index{formula} $=$ {\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula}
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below establishes that PROP-AXIOM-PROOF constructs a well-formed proof
;;;;;;;of a propositional axiom.  Once this is proved, the definition of PROP-AXIOM-PROOF
;;;;;;;is irrelevant and is disabled.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  prop-axiom-proves \\ \index{prop-axiom-proves}
\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{prop-axiom-proof}}\,({\it{exp\/}})\index{prop-axiom-proof}, {\rm{f-or}}\,({\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{exp\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-length \\ \index{subst-length}
({\it{flg\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ ({\rm{length}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst})\index{length} $=$ {\rm{length}}\,({\it{exp\/}})\index{length})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-termp \\ \index{subst-termp}
\=\+({\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{flg\/}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}

\noindent{\sc Event}:   Disable atomp; name this event `g0606'.\index{g0606}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subst-formula \\ \index{subst-formula}
\=\+({\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}

\noindent{\sc Event}:   Enable atomp; name this event `g0607'.\index{g0607}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subst-axiom-proves \\ \index{subst-axiom-proves}
\=\+(\=\+({\it{lhs\/}} $=$ {\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}) \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{free-for})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{subst-axiom-proof}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}})\index{subst-axiom-proof}, \\ 
{\rm{f-implies}}\,({\it{lhs\/}}, {\rm{forsome}}\,({\it{var\/}}, {\it{exp\/}})\index{forsome})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-pairequals \\ \index{formula-pairequals}
\=\+({\rm{termp}}\,({\it{vars1\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{vars2\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}})\index{pairequals}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-axiom1-proves \\ \index{equal-axiom1-proves}
\=\+(\=\+{\rm{function}}\,({\it{fn\/}})\index{function} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{fn\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{vars1\/}})\index{length} $=$ {\rm{degree}}\,({\it{fn\/}})\index{degree}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{vars2\/}})\index{length} $=$ {\rm{degree}}\,({\it{fn\/}})\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{vars1\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{vars2\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{equal-axiom1}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{fn\/}})\index{equal-axiom1}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{equal-axiom1-proof}}\,({\it{fn\/}}, {\it{vars1\/}}, {\it{vars2\/}})\index{equal-axiom1-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-axiom2-proves \\ \index{equal-axiom2-proves}
\=\+(\=\+{\rm{predicate}}\,({\it{pr\/}})\index{predicate} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{pr\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{vars1\/}})\index{length} $=$ {\rm{degree}}\,({\it{pr\/}})\index{degree}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{vars2\/}})\index{length} $=$ {\rm{degree}}\,({\it{pr\/}})\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{vars1\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{vars2\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{equal-axiom2}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{pr\/}})\index{equal-axiom2}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{equal-axiom2-proof}}\,({\it{pr\/}}, {\it{vars1\/}}, {\it{vars2\/}})\index{equal-axiom2-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ident-axiom-proves \\ \index{ident-axiom-proves}
\=\+{\rm{termp}}\,({\it{var\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{ident-axiom-proof}}\,({\it{var\/}})\index{ident-axiom-proof}, {\rm{f-equal}}\,({\it{var\/}}, {\it{var\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{assum-proof}$\;\;$
{\rm{assum-proof}}\,({\it{exp\/}})\index{assum-proof} $=$ {\rm{list}}\,({\tt{5}}, {\bf{nil}}, {\it{exp\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  assum-proof-proves \\ \index{assum-proof-proves}
\=\+(({\it{exp\/}} $\in$ {\it{given\/}}) $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{assum-proof}}\,({\it{exp\/}})\index{assum-proof}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  memb-member \\ \index{memb-member}
({\it{x\/}} $\in$ {\it{y\/}}) $\rightarrow$ {\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{fun-defn-proof} \\  
\=\+{\rm{fun-defn-proof}}\,({\it{fn\/}}, {\it{vars\/}}, {\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{fun-defn-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,(\=\+{\tt{4}}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{vars\/}}), {\it{x\/}}, {\it{exp\/}}, {\it{y\/}}), \\ 
{\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\rm{cons}}\,({\it{fn\/}}, {\it{vars\/}}), {\tt{0}})\index{subst}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  fun-defn-proof-proves \\ \index{fun-defn-proof-proves}
\=\+(\=\+{\rm{var-list}}\,({\it{vars\/}}, {\rm{degree}}\,({\it{fn\/}})\index{degree})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{y\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{set}}\,({\rm{cons}}\,({\it{y\/}}, {\rm{cons}}\,({\it{x\/}}, {\it{vars\/}})))\index{set} \\ 
$\wedge$$\;\;\;\;${\rm{function}}\,({\it{fn\/}})\index{function} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{fn\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{a-prime\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free}, {\rm{cons}}\,({\it{x\/}}, {\it{vars\/}}))\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\it{exp\/}}, {\it{x\/}}, {\rm{cons}}\,({\it{fn\/}}, {\it{vars\/}}), {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{assoc}}\,({\it{fn\/}}, {\it{defns\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\rm{forsome}}\,({\it{x\/}}, {\it{exp\/}})\index{forsome}, {\it{given\/}}, {\it{defns\/}}, {\rm{rest-of}}\,({\it{fn\/}}, {\it{symbols\/}})\index{rest-of})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{exp\/}}, {\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst})\index{f-and}, {\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\rm{rest-of}}\,({\it{fn\/}}, {\it{symbols\/}})\index{rest-of})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\rm{cons}}\,({\it{fn\/}}, {\it{vars\/}}), {\tt{0}})\index{subst}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{fun-defn-proof}}\,({\it{fn\/}}, {\it{vars\/}}, {\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{fun-defn-proof}, \\ 
{\it{a-prime\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable fun-defn-proof; name this event `g0591'.\index{g0591}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{pred-defn-proof} \\  
\=\+{\rm{pred-defn-proof}}\,({\it{pr\/}}, {\it{vars\/}}, {\it{defn\/}})\index{pred-defn-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{4}}, {\rm{list}}\,({\rm{cons}}\,({\it{pr\/}}, {\it{vars\/}})), {\rm{f-iff}}\,({\rm{cons}}\,({\it{pr\/}}, {\it{vars\/}}), {\it{defn\/}})\index{f-iff})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-list-termp \\ \index{var-list-termp}
{\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} $\rightarrow$ {\rm{termp}}\,({\it{vars\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-list-length \\ \index{var-list-length}
({\rm{length}}\,({\it{vars\/}})\index{length} $\not=$ {\rm{fix}}\,({\it{n\/}})) $\rightarrow$ ($\neg$ {\rm{var-list}}\,({\it{vars\/}}, {\it{n\/}})\index{var-list})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  memb-rest-of \\ \index{memb-rest-of}
($\neg$ {\rm{memb}}\,({\it{x\/}}, {\it{z\/}})\index{memb}) $\rightarrow$ ($\neg$ {\rm{memb}}\,({\it{x\/}}, {\rm{rest-of}}\,({\it{y\/}}, {\it{z\/}})\index{rest-of})\index{memb})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-rest-of \\ \index{termp-rest-of}
($\neg$ {\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp}) $\rightarrow$ ($\neg$ {\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\rm{rest-of}}\,({\it{symb\/}}, {\it{symbols\/}})\index{rest-of})\index{termp})
\end{tabbing}

\noindent{\sc Event}:   Disable atomp; name this event `g0594'.\index{g0594}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-rest-of \\ \index{formula-rest-of}
($\neg$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}) $\rightarrow$ ($\neg$ {\rm{formula}}\,({\it{exp\/}}, {\rm{rest-of}}\,({\it{symb\/}}, {\it{symbols\/}})\index{rest-of})\index{formula})
\end{tabbing}

\noindent{\sc Event}:   Enable atomp; name this event `g0595'.\index{g0595}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pred-defn-proof-proves \\ \index{pred-defn-proof-proves}
\=\+(\=\+{\rm{var-list}}\,({\it{vars\/}}, {\rm{degree}}\,({\it{pr\/}})\index{degree})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{set}}\,({\it{vars\/}})\index{set} \\ 
$\wedge$$\;\;\;\;${\rm{predicate}}\,({\it{pr\/}})\index{predicate} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{pr\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;$({\it{defn\/}} $=$ {\rm{assoc}}\,({\it{pr\/}}, {\it{defns\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{defn\/}}, {\rm{rest-of}}\,({\it{pr\/}}, {\it{symbols\/}})\index{rest-of})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{collect-free}}\,({\it{defn\/}}, {\tt{0}})\index{collect-free}, {\it{vars\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-iff}}\,({\rm{cons}}\,({\it{pr\/}}, {\it{vars\/}}), {\it{defn\/}})\index{f-iff}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{pred-defn-proof}}\,({\it{pr\/}}, {\it{vars\/}}, {\it{defn\/}})\index{pred-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable pred-defn-proof; name this event `g0592'.\index{g0592}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{expan-proof} \\  
{\rm{expan-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{expan-proof} $=$ {\rm{list}}\,({\tt{2}}, {\rm{list}}\,({\tt{1}}, {\it{a\/}}), {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\rm{list}}\,({\it{pf\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{contrac-proof}$\;\;$
{\rm{contrac-proof}}\,({\it{a\/}}, {\it{pf\/}})\index{contrac-proof} $=$ {\rm{list}}\,({\tt{2}}, {\rm{list}}\,({\tt{2}}, {\it{a\/}}), {\it{a\/}}, {\rm{list}}\,({\it{pf\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{assoc-proof} \\  
{\rm{assoc-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{assoc-proof} $=$ {\rm{list}}\,({\tt{2}}, {\rm{list}}\,({\tt{3}}, {\it{a\/}}, {\it{b\/}}, {\it{c\/}}), {\rm{f-or}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{c\/}})\index{f-or}, {\rm{list}}\,({\it{pf\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{cut-proof} \\  
{\rm{cut-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{cut-proof} $=$ {\rm{list}}\,({\tt{2}}, {\rm{list}}\,({\tt{4}}, {\it{a\/}}, {\it{b\/}}, {\it{c\/}}), {\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or}, {\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{forsome-intro-proof} \\  
\=\+{\rm{forsome-intro-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{forsome-intro-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{2}}, {\rm{list}}\,({\tt{5}}, {\it{var\/}}, {\it{a\/}}, {\it{b\/}}), {\rm{f-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{a\/}})\index{forsome}, {\it{b\/}})\index{f-implies}, {\rm{list}}\,({\it{pf\/}}))\-
\end{tabbing}

\noindent{\sc Event}:   Enable fol-axiom-proof; name this event `g0624'.\index{g0624}

\addvspace{18pt}

\noindent{\sc Event}:   Enable fun-defn-proof; name this event `g0625'.\index{g0625}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  expan-proof-proves \\ \index{expan-proof-proves}
\=\+({\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{proves}}\,({\it{pf\/}}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{expan-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{expan-proof}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  assoc-proof-proves \\ \index{assoc-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{a\/}}, {\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{assoc-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{assoc-proof}, {\rm{f-or}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{c\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  contrac-proof-proves \\ \index{contrac-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{a\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{contrac-proof}}\,({\it{a\/}}, {\it{pf\/}})\index{contrac-proof}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  cut-proof-proves \\ \index{cut-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-or}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\it{c\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or}, {\it{symbols\/}})\index{formula})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{cut-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{cut-proof}, {\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  forsome-intro-proves \\ \index{forsome-intro-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{memb}}\,({\it{var\/}}, {\rm{collect-free}}\,({\it{b\/}}, {\tt{0}})\index{collect-free})\index{memb}) \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{f-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{a\/}})\index{forsome}, {\it{b\/}})\index{f-implies}, {\it{symbols\/}})\index{formula})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-intro-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{forsome-intro-proof}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{a\/}})\index{forsome}, {\it{b\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable prop-axiom-proof; name this event `g2752'.\index{g2752}

\addvspace{18pt}

\noindent{\sc Event}:   Enable subst-axiom-proof; name this event `g2753'.\index{g2753}

\addvspace{18pt}

\noindent{\sc Event}:   Enable equal-axiom1-proof; name this event `g2754'.\index{g2754}

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\noindent{\sc Event}:   Enable equal-axiom2-proof; name this event `g2755'.\index{g2755}

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\noindent{\sc Event}:   Enable ident-axiom-proof; name this event `g2756'.\index{g2756}

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\noindent{\sc Event}:   Enable assum-proof; name this event `g2757'.\index{g2757}

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\noindent{\sc Event}:   Enable fun-defn-proof; name this event `g2758'.\index{g2758}

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\noindent{\sc Event}:   Enable expan-proof; name this event `g2759'.\index{g2759}

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\noindent{\sc Event}:   Enable contrac-proof; name this event `g2760'.\index{g2760}

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\noindent{\sc Event}:   Enable assoc-proof; name this event `g2761'.\index{g2761}

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\noindent{\sc Event}:   Enable cut-proof; name this event `g2762'.\index{g2762}

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\noindent{\sc Event}:   Enable forsome-intro-proof; name this event `g2763'.\index{g2763}

\addvspace{18pt}

\noindent{\sc Event}:   Enable fol-rules; name this event `g0626'.\index{g0626}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  nlistp-proves \\ \index{nlistp-proves}
({\it{pf\/}} $\simeq\;${{\bf{nil}}}${}$) $\rightarrow$ ($\neg$ {\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ext-axiom} \\  
\=\+{\rm{ext-axiom}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{ext-axiom} \\ 
$=$$\;\;\;\;${\rm{f-iff}}\,({\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, $\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-iff}\index{forall})\index{f-iff}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ext-axiom-proof} \\  
\=\+{\rm{ext-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{ext-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{3}}, {\rm{list}}\,({\tt{1}}, {\it{terma\/}}, {\it{termb\/}}, {\it{x\/}}), {\rm{ext-axiom}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{ext-axiom})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ext-axiom-proof-proves \\ \index{ext-axiom-proof-proves}
\=\+(\=\+($\neg$ {\rm{memb}}\,({\it{x\/}}, {\rm{collect-free}}\,({\it{terma\/}}, {\tt{0}})\index{collect-free})\index{memb}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{memb}}\,({\it{x\/}}, {\rm{collect-free}}\,({\it{termb\/}}, {\tt{0}})\index{collect-free})\index{memb}) \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{memb}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\it{symbols\/}})\index{memb} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terma\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{termb\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{ext-axiom-proof}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-iff}\index{forall})\-\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-axiom-proof; name this event `g0374'.\index{g0374}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{null-axiom}$\;\;$
{\rm{null-axiom}}\,({\it{term\/}})\index{null-axiom} $=$ {\rm{f-not}}\,({\rm{isin}}\,({\it{term\/}}, {\sc{phi}}\index{phi})\index{isin})\index{f-not}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{null-axiom-proof} \\  
{\rm{null-axiom-proof}}\,({\it{term\/}})\index{null-axiom-proof} $=$ {\rm{list}}\,({\tt{3}}, {\rm{list}}\,({\tt{2}}, {\it{term\/}}), {\rm{null-axiom}}\,({\it{term\/}})\index{null-axiom})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  null-axiom-proof-proves \\ \index{null-axiom-proof-proves}
\=\+({\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}), {\it{symbols\/}})\index{subset} $\wedge$ {\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{null-axiom-proof}}\,({\it{term\/}})\index{null-axiom-proof}, \\ 
{\rm{f-not}}\,({\rm{isin}}\,({\it{term\/}}, {\sc{phi}}\index{phi})\index{isin})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable null-axiom-proof; name this event `g0375'.\index{g0375}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{pair-axiom} \\  
\=\+{\rm{pair-axiom}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{pair-axiom} \\ 
$=$$\;\;\;\;${\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{terma\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{termb\/}})\index{f-equal})\index{f-or})\-\index{f-iff}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pair-axiom-proof} \\  
\=\+{\rm{pair-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{pair-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{3}}, {\rm{list}}\,({\tt{3}}, {\it{terma\/}}, {\it{termb\/}}, {\it{x\/}}), {\rm{pair-axiom}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{pair-axiom})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pair-axiom-proof-proves \\ \index{pair-axiom-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terma\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{termb\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{pair-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{pair-axiom-proof}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{terma\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{termb\/}})\index{f-equal})\index{f-or})\-\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable pair-axiom-proof; name this event `g0376'.\index{g0376}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{union-axiom} \\  
\=\+{\rm{union-axiom}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{union-axiom} \\ 
$=$$\;\;\;\;${\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{z-union}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-union})\index{isin}, {\rm{f-or}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-or})\index{f-iff}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{union-axiom-proof} \\  
\=\+{\rm{union-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{union-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{3}}, {\rm{list}}\,({\tt{4}}, {\it{terma\/}}, {\it{termb\/}}, {\it{x\/}}), {\rm{union-axiom}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{union-axiom})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  union-axiom-proof-proves \\ \index{union-axiom-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terma\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{termb\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{memb}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\it{symbols\/}})\index{memb} \\ 
$\wedge$$\;\;\;\;${\rm{memb}}\,({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\it{symbols\/}})\index{memb})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{union-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{union-axiom-proof}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-union}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-union})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-or})\-\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable union-axiom-proof; name this event `g0377'.\index{g0377}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ind-axiom} \\  
\=\+{\rm{ind-axiom}}\,({\it{form\/}}, {\it{x\/}})\index{ind-axiom} \\ 
$=$$\;\;\;\;${\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{subst}}\,({\it{form\/}}, {\it{x\/}}, {\sc{phi}}\index{phi}, {\tt{0}})\index{subst}, \\ 
$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-and}, \\ 
{\rm{subst}}\,({\it{form\/}}, {\it{x\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\tt{0}})\index{subst})\-\index{f-implies}\index{forall})\-\index{f-and}, \\ 
$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-implies}\index{forall})\-\index{f-implies}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ind-axiom-proof} \\  
{\rm{ind-axiom-proof}}\,({\it{form\/}}, {\it{x\/}})\index{ind-axiom-proof} $=$ {\rm{list}}\,({\tt{3}}, {\rm{list}}\,({\tt{5}}, {\it{form\/}}, {\it{x\/}}), {\rm{ind-axiom}}\,({\it{form\/}}, {\it{x\/}})\index{ind-axiom})
\end{tabbing}

\noindent{\sc Event}:   Disable predicate; name this event `g0627'.\index{g0627}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  ind-axiom-proof-proves \\ \index{ind-axiom-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{form\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{ind-axiom}}\,({\it{form\/}}, {\it{x\/}})\index{ind-axiom}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{ind-axiom-proof}}\,({\it{form\/}}, {\it{x\/}})\index{ind-axiom-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable predicate; name this event `g0629'.\index{g0629}

\addvspace{18pt}

\noindent{\sc Event}:   Enable ind-axiom-proof; name this event `g0378'.\index{g0378}

\addvspace{18pt}

\noindent{\sc Event}:   Enable prf; name this event `g0379'.\index{g0379}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  proves-is-formula \\ \index{proves-is-formula}
{\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} $\rightarrow$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  proves-is-formula-again \\ \index{proves-is-formula-again}
($\neg$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}) $\rightarrow$ ($\neg$ {\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})
\end{tabbing}

\noindent{\sc Event}:   Enable proves; name this event `g0380'.\index{g0380}

\addvspace{18pt}
\begin{verbatim};;;;;;;The commutativity of disjunction is the first example of a `derived inference rule'.
;;;;;;;COMMUT-PROOF constructs a proof of "(B or A)" given a proof of "(A or B)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{commut-proof} \\  
{\rm{commut-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{commut-proof} $=$ {\rm{cut-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{a\/}}, {\it{pf\/}}, {\rm{prop-axiom-proof}}\,({\it{a\/}})\index{prop-axiom-proof})\index{cut-proof}
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below establishes that this is the required proof.  The definition
;;;;;;;of COMMUT-PROOF then becomes irrelevant and is disabled.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  commut-proof-proves \\ \index{commut-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{commut-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{commut-proof}, {\rm{f-or}}\,({\it{b\/}}, {\it{a\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable commut-proof; name this event `g2766'.\index{g2766}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{detach-proof} \\  
\=\+{\rm{detach-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{detach-proof} \\ 
$=$$\;\;\;\;${\rm{contrac-proof}}\,(\=\+{\it{b\/}}, \\ 
{\rm{cut-proof}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{commut-proof}}\,({\it{b\/}}, {\it{a\/}}, {\rm{expan-proof}}\,({\it{b\/}}, {\it{a\/}}, {\it{pf1\/}})\index{expan-proof})\index{commut-proof}, \\ 
{\it{pf2\/}})\-\index{cut-proof})\-\index{contrac-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Enable proves-is-formula; name this event `g0503'.\index{g0503}

\addvspace{18pt}
\begin{verbatim};;;;;;;Modus Ponens is proved as a derived inference rule below.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  detach-proof-proves \\ \index{detach-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{detach-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{detach-proof}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable detach-proof; name this event `g2767'.\index{g2767}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-f-implies \\ \index{formula-f-implies}
\=\+{\rm{formula}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$({\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-f-implies \\ \index{covering-f-implies}
\=\+{\rm{covering}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\it{var\/}}, {\tt{0}})\index{covering} \\ 
$=$$\;\;\;\;${\rm{append}}\,({\rm{covering}}\,({\it{x\/}}, {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\it{y\/}}, {\it{var\/}}, {\tt{0}})\index{covering})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-append \\ \index{member-append}
({\it{x\/}} $\in$ {\rm{append}}\,({\it{y\/}}, {\it{z\/}})) $=$ (({\it{x\/}} $\in$ {\it{y\/}}) $\vee$ ({\it{x\/}} $\in$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-f-implies \\ \index{collect-free-f-implies}
\=\+{\rm{collect-free}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\tt{0}})\index{collect-free} \\ 
$=$$\;\;\;\;${\rm{append}}\,({\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}, {\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free})\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-implies; name this event `g0344'.\index{g0344}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-implies} \\  
\=\+{\rm{list-implies}}\,({\it{list\/}}, {\it{conc\/}})\index{list-implies} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{list\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{conc\/}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{list\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{f-implies}}\,({\rm{car}}\,({\it{list\/}}), {\it{conc\/}})\index{f-implies} \\ 
{\bf else }{\rm{f-implies}}\,({\rm{car}}\,({\it{list\/}}), {\rm{list-implies}}\,({\rm{cdr}}\,({\it{list\/}}), {\it{conc\/}})\index{list-implies})\index{f-implies}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-detach-proof} \\  
\=\+{\rm{list-detach-proof}}\,({\it{alist\/}}, {\it{b\/}}, {\it{pflist\/}}, {\it{pf2\/}})\index{list-detach-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{alist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{pf2\/}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{alist\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{detach-proof}}\,({\rm{car}}\,({\it{alist\/}}), {\it{b\/}}, {\rm{car}}\,({\it{pflist\/}}), {\it{pf2\/}})\index{detach-proof} \\ 
{\bf else }{\rm{list-detach-proof}}\,(\=\+{\rm{cdr}}\,({\it{alist\/}}), \\ 
{\it{b\/}}, \\ 
{\rm{cdr}}\,({\it{pflist\/}}), \\ 
{\rm{detach-proof}}\,(\=\+{\rm{car}}\,({\it{alist\/}}), \\ 
{\rm{list-implies}}\,({\rm{cdr}}\,({\it{alist\/}}), {\it{b\/}})\index{list-implies}, \\ 
{\rm{car}}\,({\it{pflist\/}}), \\ 
{\it{pf2\/}})\-\index{detach-proof})\-\index{list-detach-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-list-implies \\ \index{formula-list-implies}
\=\+({\rm{formula}}\,({\rm{list-implies}}\,({\it{alist\/}}, {\it{b\/}})\index{list-implies}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{listp}}\,({\it{alist\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{list-implies}}\,({\rm{cdr}}\,({\it{alist\/}}), {\it{b\/}})\index{list-implies}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{verbatim};;;;;;;PROVES-LIST checks that PFLIST is a list of proofs of the elements of
;;;;;;;the list EXPLIST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{proves-list} \\  
\=\+{\rm{proves-list}}\,({\it{pflist\/}}, {\it{explist\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{explist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{pflist\/}} $=$ {\bf{nil}} \\ 
{\bf else }\=\+{\rm{proves}}\,({\rm{car}}\,({\it{pflist\/}}), {\rm{car}}\,({\it{explist\/}}), {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,(\=\+{\rm{cdr}}\,({\it{pflist\/}}), \\ 
{\rm{cdr}}\,({\it{explist\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-list}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  detach-rule-corr \\ \index{detach-rule-corr}
\=\+(\=\+{\rm{proves-list}}\,({\it{pflist\/}}, {\it{alist\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{list-implies}}\,({\it{alist\/}}, {\it{b\/}})\index{list-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{list-detach-proof}}\,({\it{alist\/}}, {\it{b\/}}, {\it{pflist\/}}, {\it{pf2\/}})\index{list-detach-proof}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-detach-proof; name this event `g0220'.\index{g0220}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{rt-expan-proof} \\  
{\rm{rt-expan-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{rt-expan-proof} $=$ {\rm{commut-proof}}\,({\it{b\/}}, {\it{a\/}}, {\rm{expan-proof}}\,({\it{b\/}}, {\it{a\/}}, {\it{pf\/}})\index{expan-proof})\index{commut-proof}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rt-expan-proof-proves \\ \index{rt-expan-proof-proves}
\=\+({\rm{proves}}\,({\it{pf\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{rt-expan-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{rt-expan-proof}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable rt-expan-proof; name this event `g0227'.\index{g0227}

\addvspace{18pt}
\begin{verbatim};;;;;;;MAKE-DISJUNCT returns a disjunction of a given list of formulas.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{make-disjunct} \\  
\=\+{\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{flist\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{car}}\,({\it{flist\/}}) \\ 
{\bf else }{\rm{f-or}}\,({\rm{car}}\,({\it{flist\/}}), {\rm{make-disjunct}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{make-disjunct})\index{f-or}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{m1-proof} \\  
\=\+{\rm{m1-proof}}\,({\it{exp\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m1-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{flist\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{pf\/}} \\ 
{\bf elseif }{\it{exp\/}} $=$ {\rm{car}}\,({\it{flist\/}}) \\ 
{\bf then }{\rm{rt-expan-proof}}\,({\rm{car}}\,({\it{flist\/}}), {\rm{make-disjunct}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{make-disjunct}, {\it{pf\/}})\index{rt-expan-proof} \\ 
{\bf else }{\rm{expan-proof}}\,(\=\+{\rm{car}}\,({\it{flist\/}}), \\ 
{\rm{make-disjunct}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{make-disjunct}, \\ 
{\rm{m1-proof}}\,({\it{exp\/}}, {\rm{cdr}}\,({\it{flist\/}}), {\it{pf\/}})\index{m1-proof})\-\index{expan-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The rule below asserts that if "A" is provable, so is any disjunction 
;;;;;;;which contains "A" as a disjunct.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m1-proof-proves \\ \index{m1-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{exp\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{m1-proof}}\,({\it{exp\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m1-proof}, \\ 
{\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable m1-proof; name this event `g0228'.\index{g0228}

\addvspace{18pt}
\begin{verbatim};;;;;;;Note that the lemma in the above form contains MAKE-DISJUNCT in the
;;;;;;;conclusion which might prevent a match with a disjunction of the
;;;;;;;form "(A or B)".  This is remedied by moving it to the hypothesis in
;;;;;;;the rephrased form below.  The lemma had to first be proved in the
;;;;;;;above form to get the induction right.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m1-proof-proves1 \\ \index{m1-proof-proves1}
\=\+(\=\+{\rm{formula}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{exp\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{m1-proof}}\,({\it{exp\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m1-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable m1-proof-proves; name this event `g0504'.\index{g0504}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{rt-assoc-proof} \\  
\=\+{\rm{rt-assoc-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{rt-assoc-proof} \\ 
$=$$\;\;\;\;${\rm{commut-proof}}\,(\=\+{\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or}, \\ 
{\it{a\/}}, \\ 
{\rm{assoc-proof}}\,(\=\+{\it{b\/}}, \\ 
{\it{c\/}}, \\ 
{\it{a\/}}, \\ 
{\rm{commut-proof}}\,(\=\+{\rm{f-or}}\,({\it{c\/}}, {\it{a\/}})\index{f-or}, \\ 
{\it{b\/}}, \\ 
{\rm{assoc-proof}}\,(\=\+{\it{c\/}}, \\ 
{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{commut-proof}}\,(\=\+{\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or}, \\ 
{\it{c\/}}, \\ 
{\it{pf\/}})\-\index{commut-proof})\-\index{assoc-proof})\-\index{commut-proof})\-\index{assoc-proof})\-\index{commut-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rt-assoc-proof-proves \\ \index{rt-assoc-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{c\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{rt-assoc-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{rt-assoc-proof}, {\rm{f-or}}\,({\it{a\/}}, {\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable rt-assoc-proof; name this event `g0231'.\index{g0231}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{insert-proof} \\  
\=\+{\rm{insert-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{insert-proof} \\ 
$=$$\;\;\;\;${\rm{commut-proof}}\,(\=\+{\rm{f-or}}\,({\it{a\/}}, {\it{c\/}})\index{f-or}, \\ 
{\it{b\/}}, \\ 
{\rm{assoc-proof}}\,(\=\+{\it{a\/}}, \\ 
{\it{c\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{expan-proof}}\,({\it{a\/}}, {\rm{f-or}}\,({\it{c\/}}, {\it{b\/}})\index{f-or}, {\rm{commut-proof}}\,({\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{commut-proof})\index{expan-proof})\-\index{assoc-proof})\-\index{commut-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  insert-proof-proves \\ \index{insert-proof-proves}
\=\+({\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{insert-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{insert-proof}, {\rm{f-or}}\,({\it{b\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{c\/}})\index{f-or})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable insert-proof; name this event `g0232'.\index{g0232}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{m2-proof-step} \\  
\=\+{\rm{m2-proof-step}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m2-proof-step} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{flist\/}}) $\simeq\;${{\bf{nil}}}${}$ \\ 
{\bf then }\=\+{\bf if }{\it{exp2\/}} $=$ {\rm{car}}\,({\it{flist\/}})$\;\;${\bf then }{\it{pf\/}} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{exp2\/}} $=$ {\rm{car}}\,({\it{flist\/}}) \\ 
{\bf then }{\rm{rt-assoc-proof}}\,(\=\+{\it{exp1\/}}, \\ 
{\it{exp2\/}}, \\ 
{\rm{make-disjunct}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{make-disjunct}, \\ 
{\rm{rt-expan-proof}}\,(\=\+{\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}, \\ 
{\rm{make-disjunct}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{make-disjunct}, \\ 
{\it{pf\/}})\-\index{rt-expan-proof})\-\index{rt-assoc-proof} \\ 
{\bf else }{\rm{insert-proof}}\,(\=\+{\rm{car}}\,({\it{flist\/}}), \\ 
{\it{exp1\/}}, \\ 
{\rm{make-disjunct}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{make-disjunct}, \\ 
{\rm{m2-proof-step}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\rm{cdr}}\,({\it{flist\/}}), {\it{pf\/}})\index{m2-proof-step})\-\index{insert-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  m2-proof-step-proves \\ \index{m2-proof-step-proves}
\=\+(\=\+{\rm{formula}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{exp2\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{m2-proof-step}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m2-proof-step}, \\ 
{\rm{f-or}}\,({\it{exp1\/}}, {\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct})\index{f-or}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  m2-proof-step-proves1 \\ \index{m2-proof-step-proves1}
\=\+(\=\+{\rm{formula}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{exp2\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-or}}\,({\it{exp1\/}}, {\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct})\index{f-or}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{m2-proof-step}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m2-proof-step}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable m2-proof-step; name this event `g0233'.\index{g0233}

\addvspace{18pt}

\noindent{\sc Event}:   Enable m2-proof-step-proves; name this event `g0234'.\index{g0234}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{m2-proof} \\  
\=\+{\rm{m2-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m2-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\it{exp1\/}} $=$ {\it{exp2\/}}$\;\;${\bf then }{\rm{m1-proof}}\,({\it{exp1\/}}, {\it{flist\/}}, {\rm{contrac-proof}}\,({\it{exp1\/}}, {\it{pf\/}})\index{contrac-proof})\index{m1-proof} \\ 
{\bf elseif }{\it{exp1\/}} $=$ {\rm{car}}\,({\it{flist\/}}) \\ 
{\bf then }{\rm{m2-proof-step}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\rm{cdr}}\,({\it{flist\/}}), {\it{pf\/}})\index{m2-proof-step} \\ 
{\bf elseif }{\it{exp2\/}} $=$ {\rm{car}}\,({\it{flist\/}}) \\ 
{\bf then }{\rm{m2-proof-step}}\,({\it{exp2\/}}, {\it{exp1\/}}, {\rm{cdr}}\,({\it{flist\/}}), {\rm{commut-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{pf\/}})\index{commut-proof})\index{m2-proof-step} \\ 
{\bf else }{\rm{expan-proof}}\,(\=\+{\rm{car}}\,({\it{flist\/}}), \\ 
{\rm{make-disjunct}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{make-disjunct}, \\ 
{\rm{m2-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\rm{cdr}}\,({\it{flist\/}}), {\it{pf\/}})\index{m2-proof})\-\index{expan-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The rule below asserts that if "(A or B)" is provable, so is any
;;;;;;;disjunction which contains "A" and "B" as disjuncts.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m2-proof-proves \\ \index{m2-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{exp1\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{exp2\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{m2-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m2-proof}, \\ 
{\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  m2-proof-proves1 \\ \index{m2-proof-proves1}
\=\+(\=\+{\rm{formula}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{exp1\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{exp2\/}} $\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{m2-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist\/}}, {\it{pf\/}})\index{m2-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable m2-proof; name this event `g0235'.\index{g0235}

\addvspace{18pt}

\noindent{\sc Event}:   Enable m2-proof-proves; name this event `g0236'.\index{g0236}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{m3-proof} \\  
\=\+{\rm{m3-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{m3-proof} \\ 
$=$$\;\;\;\;${\rm{contrac-proof}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\rm{contrac-proof}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct})\-\index{f-or}, \\ 
{\rm{m2-proof}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct})\-\index{f-or}, \\ 
{\it{exp1\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct})\-\index{f-or}, \\ 
{\rm{cons}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{flist2\/}})\-)\-, \\ 
{\rm{assoc-proof}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{exp1\/}}, \\ 
{\rm{commut-proof}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{exp1\/}})\-\index{f-or}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\rm{m2-proof}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{exp1\/}})\-\index{f-or}, \\ 
{\it{exp2\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{exp1\/}})\-\index{f-or}, \\ 
{\it{flist2\/}})\-, \\ 
{\rm{assoc-proof}}\,(\=\+{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{exp1\/}}, \\ 
{\it{exp2\/}}, \\ 
{\rm{commut-proof}}\,(\=\+{\rm{f-or}}\,(\=\+{\it{exp1\/}}, \\ 
{\it{exp2\/}})\-\index{f-or}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{pf\/}})\-\index{commut-proof})\-\index{assoc-proof})\-\index{m2-proof})\-\index{commut-proof})\-\index{assoc-proof})\-\index{m2-proof})\-\index{contrac-proof})\-\index{contrac-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-list} \\  
\=\+{\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{flist\/}}) \\ 
{\bf then }{\rm{formula}}\,({\rm{car}}\,({\it{flist\/}}), {\it{symbols\/}})\index{formula} $\wedge$ {\rm{form-list}}\,({\rm{cdr}}\,({\it{flist\/}}), {\it{symbols\/}})\index{form-list} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subset-cons \\ \index{subset-cons}
{\rm{subset}}\,({\it{x\/}}, {\it{y\/}})\index{subset} $\rightarrow$ {\rm{subset}}\,({\it{x\/}}, {\rm{cons}}\,({\it{z\/}}, {\it{y\/}}))\index{subset}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formlist-formula-make-disj \\ \index{formlist-formula-make-disj}
\=\+({\rm{listp}}\,({\it{flist\/}}) $\wedge$ {\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  m3-proof-proves \\ \index{m3-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{make-disjunct}}\,({\rm{cons}}\,({\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}, {\it{flist2\/}}))\index{make-disjunct}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{exp1\/}} $\in$ {\it{flist2\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{exp2\/}} $\in$ {\it{flist2\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{m3-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{m3-proof}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable m3-proof; name this event `g0222'.\index{g0222}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  m3-proof-proves1 \\ \index{m3-proof-proves1}
\=\+(\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{make-disjunct}}\,({\rm{cons}}\,({\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}, {\it{flist2\/}}))\index{make-disjunct}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{exp1\/}} $\in$ {\it{flist2\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{exp2\/}} $\in$ {\it{flist2\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{m3-proof}}\,({\it{exp1\/}}, {\it{exp2\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{m3-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable m3-proof-proves; name this event `g0229'.\index{g0229}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{m-proof} \\  
\=\+{\rm{m-proof}}\,({\it{flist1\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{m-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flist1\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{cdr}}\,({\it{flist1\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{m1-proof}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}}, {\it{pf\/}})\index{m1-proof} \\ 
{\bf elseif }{\rm{cddr}}\,({\it{flist1\/}}) $\simeq\;${{\bf{nil}}}${}$ \\ 
{\bf then }{\rm{m2-proof}}\,({\rm{car}}\,({\it{flist1\/}}), {\rm{cadr}}\,({\it{flist1\/}}), {\it{flist2\/}}, {\it{pf\/}})\index{m2-proof} \\ 
{\bf else }{\rm{m3-proof}}\,(\=\+{\rm{car}}\,({\it{flist1\/}}), \\ 
{\rm{cadr}}\,({\it{flist1\/}}), \\ 
{\it{flist2\/}}, \\ 
{\rm{m-proof}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{f-or}}\,({\rm{car}}\,({\it{flist1\/}}), {\rm{cadr}}\,({\it{flist1\/}}))\index{f-or}, \\ 
{\rm{cddr}}\,({\it{flist1\/}}))\-, \\ 
{\rm{cons}}\,(\=\+{\rm{f-or}}\,({\rm{car}}\,({\it{flist1\/}}), {\rm{cadr}}\,({\it{flist1\/}}))\index{f-or}, \\ 
{\it{flist2\/}})\-, \\ 
{\rm{assoc-proof}}\,(\=\+{\rm{car}}\,({\it{flist1\/}}), \\ 
{\rm{cadr}}\,({\it{flist1\/}}), \\ 
{\rm{make-disjunct}}\,({\rm{cddr}}\,({\it{flist1\/}}))\index{make-disjunct}, \\ 
{\it{pf\/}})\-\index{assoc-proof})\-\index{m-proof})\-\index{m3-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The rule below is the important lemma on disjuncts which asserts that
;;;;;;;if a disjunction of formulas {"A1",...,"Am"} is provable, so is any
;;;;;;;disjunction of formulas {"B1",...,"Bn"} which contains all the "Ai".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-proof-proves \\ \index{m-proof-proves}
\=\+(\=\+{\rm{form-list}}\,({\it{flist2\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{flist1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{flist2\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\it{flist1\/}}, {\it{flist2\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{make-disjunct}}\,({\it{flist1\/}})\index{make-disjunct}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{m-proof}}\,({\it{flist1\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{m-proof}, \\ 
{\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{termp1} \\  
\=\+{\rm{termp1}}\,({\it{exp\/}}, {\it{flg\/}})\index{termp1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\rm{function}}\,({\rm{car}}\,({\it{exp\/}}))\index{function} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{length} $=$ {\rm{degree}}\,({\rm{car}}\,({\it{exp\/}}))\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{termp1}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}})\index{termp1}\- \\ 
{\bf else }{\rm{variable}}\,({\it{exp\/}})\index{variable}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{exp\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{termp1}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{0}})\index{termp1} $\wedge$ {\rm{termp1}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}})\index{termp1}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-termp1 \\ \index{termp-termp1}
{\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp} $\rightarrow$ {\rm{termp1}}\,({\it{exp\/}}, {\it{flg\/}})\index{termp1}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{atomp1} \\  
\=\+{\rm{atomp1}}\,({\it{exp\/}})\index{atomp1} \\ 
$=$$\;\;\;\;$(\=\+{\rm{predicate}}\,({\rm{car}}\,({\it{exp\/}}))\index{predicate} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{length} $=$ {\rm{degree}}\,({\rm{car}}\,({\it{exp\/}}))\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{termp1}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}})\index{termp1})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable atomp; name this event `g0345'.\index{g0345}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  atomp-atomp1 \\ \index{atomp-atomp1}
{\rm{atomp}}\,({\it{exp\/}}, {\it{symbols\/}})\index{atomp} $\rightarrow$ {\rm{atomp1}}\,({\it{exp\/}})\index{atomp1}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{dble-neg-type} \\  
{\rm{dble-neg-type}}\,({\it{exp\/}})\index{dble-neg-type} $=$ ({\rm{f-notp}}\,({\it{exp\/}})\index{f-notp} $\wedge$ {\rm{f-notp}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg})\index{f-notp})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{neg} \\  
{\rm{neg}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{neg} $=$ (({\it{exp1\/}} $=$ {\rm{f-not}}\,({\it{exp2\/}})\index{f-not}) $\vee$ ({\it{exp2\/}} $=$ {\rm{f-not}}\,({\it{exp1\/}})\index{f-not}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{neg-list} \\  
\=\+{\rm{neg-list}}\,({\it{exp\/}}, {\it{list\/}})\index{neg-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{arg}}\,({\it{exp\/}})\index{arg} $\in$ {\it{list\/}} \\ 
{\bf else }{\rm{f-not}}\,({\it{exp\/}})\index{f-not} $\in$ {\it{list\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{elem-form} \\  
\=\+{\rm{elem-form}}\,({\it{exp\/}})\index{elem-form} \\ 
$=$$\;\;\;\;$(\=\+{\rm{atomp1}}\,({\it{exp\/}})\index{atomp1} \\ 
$\vee$$\;\;\;\;$({\rm{forsomep}}\,({\it{exp\/}})\index{forsomep} $\wedge$ ({\it{exp\/}} $=$ {\rm{forsome}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\rm{body}}\,({\it{exp\/}}))\index{forsome})))\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{neg-elem-form} \\  
{\rm{neg-elem-form}}\,({\it{exp\/}})\index{neg-elem-form} $=$ ({\rm{f-notp}}\,({\it{exp\/}})\index{f-notp} $\wedge$ {\rm{elem-form}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg})\index{elem-form})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prop-atomp} \\  
{\rm{prop-atomp}}\,({\it{exp\/}})\index{prop-atomp} $=$ ({\rm{elem-form}}\,({\it{exp\/}})\index{elem-form} $\vee$ {\rm{neg-elem-form}}\,({\it{exp\/}})\index{neg-elem-form})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-count} \\  
\=\+{\rm{form-count}}\,({\it{exp\/}})\index{form-count} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }$1\;+$ {\rm{form-count}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg})\index{form-count} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }$1\;+$ ({\rm{form-count}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1})\index{form-count} $+$ {\rm{form-count}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2})\index{form-count}) \\ 
{\bf else }{\tt{1}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-count} \\  
\=\+{\rm{list-count}}\,({\it{list\/}})\index{list-count} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}})$\;\;${\bf then }{\rm{form-count}}\,({\rm{car}}\,({\it{list\/}}))\index{form-count} $+$ {\rm{list-count}}\,({\rm{cdr}}\,({\it{list\/}}))\index{list-count} \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{nor-type}$\;\;$
{\rm{nor-type}}\,({\it{exp\/}})\index{nor-type} $=$ ({\rm{f-notp}}\,({\it{exp\/}})\index{f-notp} $\wedge$ {\rm{f-orp}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg})\index{f-orp})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-list-count \\ \index{lessp-list-count}
{\rm{listp}}\,({\it{flist\/}}) $\rightarrow$ ({\rm{list-count}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{list-count} $<$ {\rm{list-count}}\,({\it{flist\/}})\index{list-count})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  f-orp-list-count \\ \index{f-orp-list-count}
\=\+({\rm{f-orp}}\,({\rm{car}}\,({\it{flist\/}}))\index{f-orp} $\wedge$ {\rm{listp}}\,({\it{flist\/}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{list-count}}\,({\rm{cons}}\,({\rm{arg1}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg1}, {\rm{cons}}\,({\rm{arg2}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg2}, {\rm{cdr}}\,({\it{flist\/}}))))\index{list-count} \\ 
$<$$\;\;\;\;${\rm{list-count}}\,({\it{flist\/}})\index{list-count})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nor-type-list-count1 \\ \index{nor-type-list-count1}
\=\+({\rm{listp}}\,({\it{flist\/}}) $\wedge$ {\rm{nor-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{nor-type}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{list-count}}\,({\rm{cons}}\,({\rm{f-not}}\,({\rm{arg1}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg1})\index{f-not}, {\rm{cdr}}\,({\it{flist\/}})))\index{list-count} \\ 
$<$$\;\;\;\;${\rm{list-count}}\,({\it{flist\/}})\index{list-count})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nor-type-list-count2 \\ \index{nor-type-list-count2}
\=\+({\rm{listp}}\,({\it{flist\/}}) $\wedge$ {\rm{nor-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{nor-type}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{list-count}}\,({\rm{cons}}\,({\rm{f-not}}\,({\rm{arg2}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg2})\index{f-not}, {\rm{cdr}}\,({\it{flist\/}})))\index{list-count} \\ 
$<$$\;\;\;\;${\rm{list-count}}\,({\it{flist\/}})\index{list-count})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  dble-neg-list-count \\ \index{dble-neg-list-count}
\=\+({\rm{listp}}\,({\it{flist\/}}) $\wedge$ {\rm{dble-neg-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{dble-neg-type}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{list-count}}\,({\rm{cons}}\,({\rm{arg}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg}, {\rm{cdr}}\,({\it{flist\/}})))\index{list-count} \\ 
$<$$\;\;\;\;${\rm{list-count}}\,({\it{flist\/}})\index{list-count})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-count; name this event `g0240'.\index{g0240}

\addvspace{18pt}
\begin{verbatim};;;;;;;TAUTOLOGYP1 is the tautology-checker.  It recurses through a list of
;;;;;;;formulas FLIST and builds up a list of literals AUXLIST.  If AUXLIST
;;;;;;;is ever to contain a literal and its negation, TAUTOLOGYP1 returns T.
;;;;;;;Otherwise, it returns F.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{tautologyp1} \\  
\=\+{\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{flist\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{f-orp}}\,({\rm{car}}\,({\it{flist\/}}))\index{f-orp} \\ 
{\bf then }{\rm{tautologyp1}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{arg1}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg1}, \\ 
{\rm{cons}}\,({\rm{arg2}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg2}, {\rm{cdr}}\,({\it{flist\/}})))\-, \\ 
{\it{auxlist\/}})\-\index{tautologyp1} \\ 
{\bf elseif }{\rm{nor-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{nor-type} \\ 
{\bf then }\=\+{\rm{tautologyp1}}\,(\=\+{\rm{cons}}\,({\rm{f-not}}\,({\rm{arg1}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg1})\index{f-not}, {\rm{cdr}}\,({\it{flist\/}})), \\ 
{\it{auxlist\/}})\-\index{tautologyp1} \\ 
$\wedge$$\;\;\;\;${\rm{tautologyp1}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{f-not}}\,({\rm{arg2}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg2})\index{f-not}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-, \\ 
{\it{auxlist\/}})\-\index{tautologyp1}\- \\ 
{\bf elseif }{\rm{dble-neg-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{dble-neg-type} \\ 
{\bf then }{\rm{tautologyp1}}\,({\rm{cons}}\,({\rm{arg}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg}, {\rm{cdr}}\,({\it{flist\/}})), {\it{auxlist\/}})\index{tautologyp1} \\ 
{\bf else }\=\+{\rm{neg-list}}\,({\rm{car}}\,({\it{flist\/}}), {\it{auxlist\/}})\index{neg-list} \\ 
$\vee$$\;\;\;\;${\rm{tautologyp1}}\,(\=\+{\rm{cdr}}\,({\it{flist\/}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{flist\/}}), {\it{auxlist\/}}))\-\index{tautologyp1}\-$\;${\bf  endif}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-list-append \\ \index{form-list-append}
\=\+({\rm{form-list}}\,({\it{x\/}}, {\it{symbols\/}})\index{form-list} $\wedge$ {\rm{form-list}}\,({\it{y\/}}, {\it{symbols\/}})\index{form-list}) \\ 
$\rightarrow$$\;\;\;\;${\rm{form-list}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}), {\it{symbols\/}})\index{form-list}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{neg-proof} \\  
\=\+{\rm{neg-proof}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{neg-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{exp1\/}} $=$ {\rm{f-not}}\,({\it{exp2\/}})\index{f-not}$\;\;${\bf then }{\rm{prop-axiom-proof}}\,({\it{exp2\/}})\index{prop-axiom-proof} \\ 
{\bf else }{\rm{commut-proof}}\,({\it{exp2\/}}, {\it{exp1\/}}, {\rm{prop-axiom-proof}}\,({\it{exp1\/}})\index{prop-axiom-proof})\index{commut-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  neg-proof-proves \\ \index{neg-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{exp1\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp2\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{neg}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{neg} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-or}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{f-or}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{neg-proof}}\,({\it{exp1\/}}, {\it{exp2\/}})\index{neg-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{neg-list-proof} \\  
\=\+{\rm{neg-list-proof}}\,({\it{exp\/}}, {\it{flist\/}})\index{neg-list-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{f-not}}\,({\it{exp\/}})\index{f-not} $\in$ {\it{flist\/}} \\ 
{\bf then }{\rm{m2-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{f-not}}\,({\it{exp\/}})\index{f-not}, \\ 
{\rm{cons}}\,({\it{exp\/}}, {\it{flist\/}}), \\ 
{\rm{neg-proof}}\,({\it{exp\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not})\index{neg-proof})\-\index{m2-proof} \\ 
{\bf else }{\rm{m2-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{arg}}\,({\it{exp\/}})\index{arg}, \\ 
{\rm{cons}}\,({\it{exp\/}}, {\it{flist\/}}), \\ 
{\rm{neg-proof}}\,({\it{exp\/}}, {\rm{arg}}\,({\it{exp\/}})\index{arg})\index{neg-proof})\-\index{m2-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  neg-list-proof-proves \\ \index{neg-list-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{neg-list}}\,({\it{exp\/}}, {\it{flist\/}})\index{neg-list} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{cons}}\,({\it{exp\/}}, {\it{flist\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{neg-list-proof}}\,({\it{exp\/}}, {\it{flist\/}})\index{neg-list-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subset-ident \\ \index{subset-ident}
{\rm{subset}}\,({\it{x\/}}, {\it{x\/}})\index{subset}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subset-car \\ \index{subset-car}
{\rm{subset}}\,({\it{x\/}}, {\rm{cons}}\,({\it{y\/}}, {\it{x\/}}))\index{subset}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subset-append \\ \index{subset-append}
{\rm{subset}}\,({\rm{cons}}\,({\it{exp\/}}, {\it{list2\/}}), {\rm{append}}\,({\rm{cons}}\,({\it{exp\/}}, {\it{list1\/}}), {\it{list2\/}}))\index{subset}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  m-proof-proves1 \\ \index{m-proof-proves1}
\=\+(\=\+{\rm{form-list}}\,({\it{flist2\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\it{flist1\/}}, {\it{flist2\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{make-disjunct}}\,({\it{flist1\/}})\index{make-disjunct}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\it{flist2\/}})\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{m-proof}}\,({\it{flist1\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{m-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable m-proof-proves; name this event `g0505'.\index{g0505}

\addvspace{18pt}

\noindent{\sc Event}:   Enable neg-list-proof; name this event `g0256'.\index{g0256}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{prop-atom-proof1} \\  
\=\+{\rm{prop-atom-proof1}}\,({\it{flist1\/}}, {\it{flist2\/}})\index{prop-atom-proof1} \\ 
$=$$\;\;\;\;${\rm{m-proof}}\,(\=\+{\rm{cons}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}}), \\ 
{\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}), \\ 
{\rm{neg-list-proof}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})\index{neg-list-proof})\-\index{m-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prop-atom-proof1-proves \\ \index{prop-atom-proof1-proves}
\=\+(\=\+{\rm{form-list}}\,({\it{flist1\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{flist1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{flist2\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{neg-list}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})\index{neg-list} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{prop-atom-proof1}}\,({\it{flist1\/}}, {\it{flist2\/}})\index{prop-atom-proof1}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable prop-atom-proof1; name this event `g0259'.\index{g0259}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subset-append-car \\ \index{subset-append-car}
{\rm{subset}}\,({\rm{append}}\,({\it{list1\/}}, {\rm{cons}}\,({\it{exp\/}}, {\it{list2\/}})), {\rm{append}}\,({\rm{cons}}\,({\it{exp\/}}, {\it{list1\/}}), {\it{list2\/}}))\index{subset}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-list-append-car \\ \index{form-list-append-car}
\=\+(\=\+{\rm{form-list}}\,({\rm{cons}}\,({\it{exp\/}}, {\it{list1\/}}), {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;$(({\it{list2\/}} $=$ {\bf{nil}}) $\vee$ {\rm{form-list}}\,({\it{list2\/}}, {\it{symbols\/}})\index{form-list}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{form-list}}\,({\rm{append}}\,({\it{list1\/}}, {\rm{cons}}\,({\it{exp\/}}, {\it{list2\/}})), {\it{symbols\/}})\index{form-list}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prop-atom-proof2} \\  
\=\+{\rm{prop-atom-proof2}}\,({\it{flist1\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{prop-atom-proof2} \\ 
$=$$\;\;\;\;${\rm{m-proof}}\,(\=\+{\rm{append}}\,({\rm{cdr}}\,({\it{flist1\/}}), {\rm{cons}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})), \\ 
{\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}), \\ 
{\it{pf\/}})\-\index{m-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prop-atom-proof2-proves \\ \index{prop-atom-proof2-proves}
\=\+(\=\+{\rm{form-list}}\,({\it{flist1\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{flist1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{flist2\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{make-disjunct}}\,({\rm{append}}\,({\rm{cdr}}\,({\it{flist1\/}}), {\rm{cons}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})))\index{make-disjunct}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{prop-atom-proof2}}\,({\it{flist1\/}}, {\it{flist2\/}}, {\it{pf\/}})\index{prop-atom-proof2}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable prop-atom-proof2; name this event `g0258'.\index{g0258}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{cancel-proof} \\  
\=\+{\rm{cancel-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{cancel-proof} \\ 
$=$$\;\;\;\;${\rm{contrac-proof}}\,({\it{b\/}}, {\rm{cut-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{b\/}}, {\it{pf2\/}}, {\rm{rt-expan-proof}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\it{b\/}}, {\it{pf1\/}})\index{rt-expan-proof})\index{cut-proof})\index{contrac-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  cancel-proof-proves \\ \index{cancel-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{cancel-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{cancel-proof}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable cancel-proof; name this event `g0255'.\index{g0255}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{nlistp-nor-type-proof} \\  
\=\+{\rm{nlistp-nor-type-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{nlistp-nor-type-proof} \\ 
$=$$\;\;\;\;${\rm{cancel-proof}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}, \\ 
{\it{pf2\/}}, \\ 
{\rm{cancel-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{f-or}}\,({\it{b\/}}, {\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not})\index{f-or}, \\ 
{\it{pf1\/}}, \\ 
{\rm{m-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}, {\it{a\/}}, {\it{b\/}}), \\ 
{\rm{list}}\,({\it{a\/}}, {\it{b\/}}, {\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}), \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{prop-axiom-proof})\-\index{m-proof})\-\index{cancel-proof})\-\index{cancel-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{listp-nor-type-proof} \\  
\=\+{\rm{listp-nor-type-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{listp-nor-type-proof} \\ 
$=$$\;\;\;\;${\rm{m-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}, {\it{c\/}}, {\it{c\/}}), \\ 
{\rm{list}}\,({\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}, {\it{c\/}}), \\ 
{\rm{rt-assoc-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}, \\ 
{\it{c\/}}, \\ 
{\it{c\/}}, \\ 
{\rm{cut-proof}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-or}}\,({\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}, {\it{c\/}})\index{f-or}, \\ 
{\it{c\/}}, \\ 
{\rm{rt-assoc-proof}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{f-not}, \\ 
{\it{c\/}}, \\ 
{\rm{cut-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{f-or}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\index{f-not})\-\index{f-or}, \\ 
{\it{c\/}}, \\ 
{\rm{m-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\index{f-not}, \\ 
{\it{a\/}}, \\ 
{\it{b\/}})\-, \\ 
{\rm{list}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\index{f-not})\-, \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\index{prop-axiom-proof})\-\index{m-proof}, \\ 
{\it{pf1\/}})\-\index{cut-proof})\-\index{rt-assoc-proof}, \\ 
{\it{pf2\/}})\-\index{cut-proof})\-\index{rt-assoc-proof})\-\index{m-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{nor-proof} \\  
\=\+{\rm{nor-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{clist\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{nor-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{clist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{nlistp-nor-type-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{nlistp-nor-type-proof} \\ 
{\bf else }{\rm{listp-nor-type-proof}}\,({\it{a\/}}, {\it{b\/}}, {\rm{make-disjunct}}\,({\it{clist\/}})\index{make-disjunct}, {\it{pf1\/}}, {\it{pf2\/}})\index{listp-nor-type-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{nlistp-dble-neg-proof} \\  
\=\+{\rm{nlistp-dble-neg-proof}}\,({\it{a\/}}, {\it{pf\/}})\index{nlistp-dble-neg-proof} \\ 
$=$$\;\;\;\;${\rm{contrac-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\rm{cut-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\rm{rt-expan-proof}}\,({\it{a\/}}, {\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, {\it{pf\/}})\index{rt-expan-proof}, \\ 
{\rm{commut-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{prop-axiom-proof})\-\index{commut-proof})\-\index{cut-proof})\-\index{contrac-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{listp-dble-neg-proof} \\  
\=\+{\rm{listp-dble-neg-proof}}\,({\it{a\/}}, {\it{c\/}}, {\it{pf\/}})\index{listp-dble-neg-proof} \\ 
$=$$\;\;\;\;${\rm{commut-proof}}\,(\=\+{\it{c\/}}, \\ 
{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\rm{cut-proof}}\,(\=\+{\it{a\/}}, \\ 
{\it{c\/}}, \\ 
{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\it{pf\/}}, \\ 
{\rm{commut-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{prop-axiom-proof})\-\index{commut-proof})\-\index{cut-proof})\-\index{commut-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{dble-neg-proof} \\  
\=\+{\rm{dble-neg-proof}}\,({\it{a\/}}, {\it{clist\/}}, {\it{pf\/}})\index{dble-neg-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{clist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{nlistp-dble-neg-proof}}\,({\it{a\/}}, {\it{pf\/}})\index{nlistp-dble-neg-proof} \\ 
{\bf else }{\rm{listp-dble-neg-proof}}\,({\it{a\/}}, {\rm{make-disjunct}}\,({\it{clist\/}})\index{make-disjunct}, {\it{pf\/}})\index{listp-dble-neg-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{f-orp-proof} \\  
\=\+{\rm{f-orp-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{clist\/}}, {\it{pf\/}})\index{f-orp-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{clist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{pf\/}} \\ 
{\bf else }{\rm{assoc-proof}}\,({\it{a\/}}, {\it{b\/}}, {\rm{make-disjunct}}\,({\it{clist\/}})\index{make-disjunct}, {\it{pf\/}})\index{assoc-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;TAUT-PROOF1 is the proof-constructor that constructs the proof for a formula
;;;;;;;asserted to be a tautology by TAUTOLOGYP1.  TAUT-PROOF1 and TAUTOLOGYP1 recurse
;;;;;;;in the same way.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{taut-proof1} \\  
\=\+{\rm{taut-proof1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{taut-proof1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{flist\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{f-orp}}\,({\rm{car}}\,({\it{flist\/}}))\index{f-orp} \\ 
{\bf then }{\rm{f-orp-proof}}\,(\=\+{\rm{arg1}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg1}, \\ 
{\rm{arg2}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg2}, \\ 
{\rm{append}}\,({\rm{cdr}}\,({\it{flist\/}}), {\it{auxlist\/}}), \\ 
{\rm{taut-proof1}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{arg1}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg1}, \\ 
{\rm{cons}}\,(\=\+{\rm{arg2}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg2}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-)\-, \\ 
{\it{auxlist\/}})\-\index{taut-proof1})\-\index{f-orp-proof} \\ 
{\bf elseif }{\rm{nor-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{nor-type} \\ 
{\bf then }{\rm{nor-proof}}\,(\=\+{\rm{arg1}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg1}, \\ 
{\rm{arg2}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg2}, \\ 
{\rm{append}}\,({\rm{cdr}}\,({\it{flist\/}}), {\it{auxlist\/}}), \\ 
{\rm{taut-proof1}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{f-not}}\,({\rm{arg1}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg1})\index{f-not}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-, \\ 
{\it{auxlist\/}})\-\index{taut-proof1}, \\ 
{\rm{taut-proof1}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{f-not}}\,({\rm{arg2}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg2})\index{f-not}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-, \\ 
{\it{auxlist\/}})\-\index{taut-proof1})\-\index{nor-proof} \\ 
{\bf elseif }{\rm{dble-neg-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{dble-neg-type} \\ 
{\bf then }{\rm{dble-neg-proof}}\,(\=\+{\rm{arg}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg}, \\ 
{\rm{append}}\,({\rm{cdr}}\,({\it{flist\/}}), {\it{auxlist\/}}), \\ 
{\rm{taut-proof1}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{arg}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-, \\ 
{\it{auxlist\/}})\-\index{taut-proof1})\-\index{dble-neg-proof} \\ 
{\bf elseif }{\rm{neg-list}}\,({\rm{car}}\,({\it{flist\/}}), {\it{auxlist\/}})\index{neg-list} \\ 
{\bf then }{\rm{prop-atom-proof1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{prop-atom-proof1} \\ 
{\bf else }{\rm{prop-atom-proof2}}\,(\=\+{\it{flist\/}}, \\ 
{\it{auxlist\/}}, \\ 
{\rm{taut-proof1}}\,(\=\+{\rm{cdr}}\,({\it{flist\/}}), \\ 
{\rm{cons}}\,({\rm{car}}\,({\it{flist\/}}), {\it{auxlist\/}}))\-\index{taut-proof1})\-\index{prop-atom-proof2}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below is a form of the tautology theorem.  If (TAUTOLOGYP1 FLIST AUXLIST)
;;;;;;;is T, then (TAUT-PROOF1 FLIST AUXLIST) constructs a correct proof of
;;;;;;;the disjunction of the formulas in FLIST and AUXLIST.  We still have to
;;;;;;;show that the tautology-checker TAUTOLOGYP1 is correct w.r.t the truth-table
;;;;;;;definition of a tautology.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  taut-thm1 \\ \index{taut-thm1}
\=\+(\=\+{\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{auxlist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{taut-proof1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{taut-proof1}, \\ 
{\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist\/}}, {\it{auxlist\/}}))\index{make-disjunct}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable taut-proof1; name this event `g0275'.\index{g0275}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  taut-thm2 \\ \index{taut-thm2}
\=\+(\=\+{\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{auxlist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist\/}}, {\it{auxlist\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{taut-proof1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{taut-proof1}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;;EVAL evaluates the truth-value of the Boolean expression EXP,
;;;;;;;where the truth-value assigned to atoms are specified in ALIST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{eval} \\  
\=\+{\rm{eval}}\,({\it{exp\/}}, {\it{alist\/}})\index{eval} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }$\neg$ {\rm{eval}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{alist\/}})\index{eval} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }{\rm{eval}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{alist\/}})\index{eval} $\vee$ {\rm{eval}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{alist\/}})\index{eval} \\ 
{\bf else }{\it{exp\/}} $\in$ {\it{alist\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-eval \\ \index{member-eval}
(({\it{exp\/}} $\in$ {\it{flist\/}}) $\wedge$ {\rm{eval}}\,({\it{exp\/}}, {\it{alist\/}})\index{eval}) $\rightarrow$ {\rm{eval}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{alist\/}})\index{eval}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-make-disjunct \\ \index{eval-make-disjunct}
\=\+({\rm{listp}}\,({\it{list1\/}}) $\wedge$ {\rm{listp}}\,({\it{list2\/}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{eval}}\,({\rm{make-disjunct}}\,({\rm{append}}\,({\it{list1\/}}, {\it{list2\/}}))\index{make-disjunct}, {\it{alist\/}})\index{eval} \\ 
$=$$\;\;\;\;$(\=\+{\rm{eval}}\,({\rm{make-disjunct}}\,({\it{list1\/}})\index{make-disjunct}, {\it{alist\/}})\index{eval} \\ 
$\vee$$\;\;\;\;${\rm{eval}}\,({\rm{make-disjunct}}\,({\it{list2\/}})\index{make-disjunct}, {\it{alist\/}})\index{eval})\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  neg-list-eval \\ \index{neg-list-eval}
\=\+({\rm{listp}}\,({\it{flist1\/}}) $\wedge$ {\rm{neg-list}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})\index{neg-list}) \\ 
$\rightarrow$$\;\;\;\;${\rm{eval}}\,({\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}))\index{make-disjunct}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-prop-atomp \\ \index{eval-prop-atomp}
\=\+(\=\+{\rm{listp}}\,({\it{flist1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{eval}}\,(\=\+{\rm{make-disjunct}}\,({\rm{append}}\,({\rm{cdr}}\,({\it{flist1\/}}), {\rm{cons}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})))\index{make-disjunct}, \\ 
{\it{alist\/}})\-\index{eval})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{eval}}\,({\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}))\index{make-disjunct}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-prop-atomp1 \\ \index{eval-prop-atomp1}
\=\+(\=\+{\rm{listp}}\,({\it{flist1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{eval}}\,(\=\+{\rm{make-disjunct}}\,({\rm{append}}\,({\rm{cdr}}\,({\it{flist1\/}}), {\rm{cons}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})))\index{make-disjunct}, \\ 
{\it{alist\/}})\-\index{eval} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{eval}}\,({\it{concl\/}}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  neg-list-eval1 \\ \index{neg-list-eval1}
\=\+(\=\+{\rm{listp}}\,({\it{flist1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{neg-list}}\,({\rm{car}}\,({\it{flist1\/}}), {\it{flist2\/}})\index{neg-list} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{eval}}\,({\it{concl\/}}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below is one half of the correctness of the tautology-checker.
;;;;;;;It asserts that any formula checked to be a tautology, evaluates to
;;;;;;;true under any assignment of truth-values to the atoms.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  taut-eval \\ \index{taut-eval}
\=\+{\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1} \\ 
$\rightarrow$$\;\;\;\;${\rm{eval}}\,({\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist\/}}, {\it{auxlist\/}}))\index{make-disjunct}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{verbatim};;;;;;;FALSIFY attempts to return a falsifying assignment corresponding to a list of literals.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{falsify} \\  
\=\+{\rm{falsify}}\,({\it{list\/}})\index{falsify} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{list\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{f-notp}}\,({\rm{car}}\,({\it{list\/}}))\index{f-notp}$\;\;${\bf then }{\rm{cons}}\,({\rm{arg}}\,({\rm{car}}\,({\it{list\/}}))\index{arg}, {\rm{falsify}}\,({\rm{cdr}}\,({\it{list\/}}))\index{falsify}) \\ 
{\bf else }{\rm{falsify}}\,({\rm{cdr}}\,({\it{list\/}}))\index{falsify}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;FALSIFY-TAUT attempts to return an assignment which falsifies a non-tautology.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{falsify-taut} \\  
\=\+{\rm{falsify-taut}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{falsify-taut} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{flist\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{f-orp}}\,({\rm{car}}\,({\it{flist\/}}))\index{f-orp} \\ 
{\bf then }{\rm{falsify-taut}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{arg1}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg1}, \\ 
{\rm{cons}}\,({\rm{arg2}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg2}, {\rm{cdr}}\,({\it{flist\/}})))\-, \\ 
{\it{auxlist\/}})\-\index{falsify-taut} \\ 
{\bf elseif }{\rm{nor-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{nor-type} \\ 
{\bf then }\=\+{\bf if }{\rm{falsify-taut}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{f-not}}\,({\rm{arg2}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg2})\index{f-not}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-, \\ 
{\it{auxlist\/}})\-\index{falsify-taut} \\ 
{\bf then }{\rm{falsify-taut}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{f-not}}\,({\rm{arg2}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg2})\index{f-not}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-, \\ 
{\it{auxlist\/}})\-\index{falsify-taut} \\ 
{\bf else }{\rm{falsify-taut}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{f-not}}\,({\rm{arg1}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg1})\index{f-not}, \\ 
{\rm{cdr}}\,({\it{flist\/}}))\-, \\ 
{\it{auxlist\/}})\-\index{falsify-taut}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{dble-neg-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{dble-neg-type} \\ 
{\bf then }{\rm{falsify-taut}}\,({\rm{cons}}\,({\rm{arg}}\,({\rm{arg}}\,({\rm{car}}\,({\it{flist\/}}))\index{arg})\index{arg}, {\rm{cdr}}\,({\it{flist\/}})), {\it{auxlist\/}})\index{falsify-taut} \\ 
{\bf elseif }{\rm{neg-list}}\,({\rm{car}}\,({\it{flist\/}}), {\it{auxlist\/}})\index{neg-list}$\;\;${\bf then }{\bf{f}} \\ 
{\bf else }{\rm{falsify-taut}}\,({\rm{cdr}}\,({\it{flist\/}}), {\rm{cons}}\,({\rm{car}}\,({\it{flist\/}}), {\it{auxlist\/}}))\index{falsify-taut}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{falsify}}\,({\it{auxlist\/}})\index{falsify}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-falsify-taut \\ \index{not-falsify-taut}
{\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1} $=$ ($\neg$ {\rm{falsify-taut}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{falsify-taut})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-member \\ \index{eval-member}
\=\+(($\neg$ {\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}) $\wedge$ ($\neg$ {\rm{f-orp}}\,({\it{exp\/}})\index{f-orp}) $\wedge$ ({\it{exp\/}} $\in$ {\it{alist\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{eval}}\,({\it{exp\/}}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prop-atomp-list} \\  
\=\+{\rm{prop-atomp-list}}\,({\it{flist\/}})\index{prop-atomp-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{flist\/}}) \\ 
{\bf then }\=\+($\neg$ {\rm{f-orp}}\,({\rm{car}}\,({\it{flist\/}}))\index{f-orp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{nor-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{nor-type}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{dble-neg-type}}\,({\rm{car}}\,({\it{flist\/}}))\index{dble-neg-type}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{neg-list}}\,({\rm{car}}\,({\it{flist\/}}), {\rm{cdr}}\,({\it{flist\/}}))\index{neg-list}) \\ 
$\wedge$$\;\;\;\;${\rm{prop-atomp-list}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{prop-atomp-list}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-falsify \\ \index{member-falsify}
({\it{exp\/}} $\in$ {\rm{falsify}}\,({\it{list\/}})\index{falsify}) $=$ ({\rm{f-not}}\,({\it{exp\/}})\index{f-not} $\in$ {\it{list\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  alist-cons-eval2 \\ \index{alist-cons-eval2}
\=\+(\=\+({\it{x\/}} $\not\in$ {\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{flist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{prop-atomp-list}}\,({\it{flist\/}})\index{prop-atomp-list} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{eval}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{alist\/}})\index{eval}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{eval}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\rm{cons}}\,({\it{x\/}}, {\it{alist\/}}))\index{eval})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prop-atomp-list-eval \\ \index{prop-atomp-list-eval}
\=\+({\rm{listp}}\,({\it{flist\/}}) $\wedge$ {\rm{prop-atomp-list}}\,({\it{flist\/}})\index{prop-atomp-list}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{eval}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\rm{falsify}}\,({\it{flist\/}})\index{falsify})\index{eval})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-eval-prop-atomp \\ \index{not-eval-prop-atomp}
\=\+{\rm{eval}}\,({\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist1\/}}, {\rm{cons}}\,({\it{exp\/}}, {\it{flist2\/}})))\index{make-disjunct}, {\it{alist\/}})\index{eval} \\ 
$=$$\;\;\;\;${\rm{eval}}\,({\rm{make-disjunct}}\,({\rm{cons}}\,({\it{exp\/}}, {\rm{append}}\,({\it{flist1\/}}, {\it{flist2\/}})))\index{make-disjunct}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below is the second half of the correctness of the tautology-checker.
;;;;;;;It asserts that if AUXLIST does not contain both a literal and its negation,
;;;;;;;then any formula which is checked to not be a tautology is falsifiable.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  not-taut-false \\ \index{not-taut-false}
\=\+({\rm{prop-atomp-list}}\,({\it{auxlist\/}})\index{prop-atomp-list} $\wedge$ ($\neg$ {\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{eval}}\,(\=\+{\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist\/}}, {\it{auxlist\/}}))\index{make-disjunct}, \\ 
{\rm{falsify-taut}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{falsify-taut})\-\index{eval})\-
\end{tabbing}
\begin{verbatim};;;;;;;TAUTOLOGYP is TAUTOLOGYP1 with AUXLIST set to NIL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{tautologyp}$\;\;$
{\rm{tautologyp}}\,({\it{flist\/}})\index{tautologyp} $=$ {\rm{tautologyp1}}\,({\it{flist\/}}, {\bf{nil}})\index{tautologyp1}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{taut-proof}$\;\;$
{\rm{taut-proof}}\,({\it{flist\/}})\index{taut-proof} $=$ {\rm{taut-proof1}}\,({\it{flist\/}}, {\bf{nil}})\index{taut-proof1}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-list-append-nil \\ \index{form-list-append-nil}
{\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist\/}}, {\bf{nil}}))\index{make-disjunct} $=$ {\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  tautology-theorem \\ \index{tautology-theorem}
\=\+(\=\+{\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{tautologyp}}\,({\it{flist\/}})\index{tautologyp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{taut-proof}}\,({\it{flist\/}})\index{taut-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  taut-eval2 \\ \index{taut-eval2}
\=\+(\=\+{\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist\/}}, {\it{auxlist\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{eval}}\,({\it{concl\/}}, {\it{alist\/}})\index{eval}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  tautologies-are-true \\ \index{tautologies-are-true}
({\rm{listp}}\,({\it{flist\/}}) $\wedge$ {\rm{tautologyp}}\,({\it{flist\/}})\index{tautologyp}) $\rightarrow$ {\rm{eval}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\it{alist\/}})\index{eval}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-taut-falsify2 \\ \index{not-taut-falsify2}
\=\+(\=\+{\rm{prop-atomp-list}}\,({\it{auxlist\/}})\index{prop-atomp-list} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{tautologyp1}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{tautologyp1}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-disjunct}}\,({\rm{append}}\,({\it{flist\/}}, {\it{auxlist\/}}))\index{make-disjunct}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{eval}}\,({\it{concl\/}}, {\rm{falsify-taut}}\,({\it{flist\/}}, {\it{auxlist\/}})\index{falsify-taut})\index{eval})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  truths-are-tautologies \\ \index{truths-are-tautologies}
\=\+($\neg$ {\rm{tautologyp}}\,({\it{flist\/}})\index{tautologyp}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{eval}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\rm{falsify-taut}}\,({\it{flist\/}}, {\bf{nil}})\index{falsify-taut})\index{eval})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{lis-not} \\  
\=\+{\rm{lis-not}}\,({\it{flist\/}})\index{lis-not} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flist\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,({\rm{f-not}}\,({\rm{car}}\,({\it{flist\/}}))\index{f-not}, {\rm{lis-not}}\,({\rm{cdr}}\,({\it{flist\/}}))\index{lis-not})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{taut-conseq} \\  
{\rm{taut-conseq}}\,({\it{flist\/}}, {\it{exp\/}})\index{taut-conseq} $=$ {\rm{tautologyp}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})))\index{tautologyp}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{tautconseq-proof} \\  
\=\+{\rm{tautconseq-proof}}\,({\it{flist\/}}, {\it{exp\/}}, {\it{pflist\/}})\index{tautconseq-proof} \\ 
$=$$\;\;\;\;${\rm{list-detach-proof}}\,(\=\+{\it{flist\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{pflist\/}}, \\ 
{\rm{taut-proof}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})))\index{taut-proof})\-\index{list-detach-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Disable f-implies; name this event `g0873'.\index{g0873}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  eval-f-implies \\ \index{eval-f-implies}
{\rm{eval}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\it{alist\/}})\index{eval} $=$ ({\rm{eval}}\,({\it{x\/}}, {\it{alist\/}})\index{eval} $\rightarrow$ {\rm{eval}}\,({\it{y\/}}, {\it{alist\/}})\index{eval})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-implies-reduc \\ \index{list-implies-reduc}
\=\+{\rm{list-implies}}\,({\it{flist\/}}, {\it{exp\/}})\index{list-implies} \\ 
$=$$\;\;\;\;${\rm{make-disjunct}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})))\index{make-disjunct}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-implies; name this event `g0874'.\index{g0874}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  append-exp-form-list \\ \index{append-exp-form-list}
\=\+({\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list} $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{form-list}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})), {\it{symbols\/}})\index{form-list}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-tautologyp \\ \index{eval-tautologyp}
{\rm{eval}}\,({\rm{make-disjunct}}\,({\it{flist\/}})\index{make-disjunct}, {\rm{falsify-taut}}\,({\it{flist\/}}, {\bf{nil}})\index{falsify-taut})\index{eval} $\rightarrow$ {\rm{tautologyp}}\,({\it{flist\/}})\index{tautologyp}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  proves-list-form-list \\ \index{proves-list-form-list}
{\rm{proves-list}}\,({\it{pflist\/}}, {\it{flist\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} $\rightarrow$ {\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  taut-conseq-proves \\ \index{taut-conseq-proves}
\=\+(\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{taut-conseq}}\,({\it{flist\/}}, {\it{exp\/}})\index{taut-conseq} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\it{flist\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{tautconseq-proof}}\,({\it{flist\/}}, {\it{exp\/}}, {\it{pflist\/}})\index{tautconseq-proof}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable tautconseq-proof; name this event `g0276'.\index{g0276}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  eval-tautconseq \\ \index{eval-tautconseq}
\=\+(\=\+{\rm{form-list}}\,({\it{flist\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eval}}\,(\=\+{\rm{make-disjunct}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})))\index{make-disjunct}, \\ 
{\rm{falsify-taut}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})), {\bf{nil}})\index{falsify-taut})\-\index{eval})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{taut-conseq}}\,({\it{flist\/}}, {\it{exp\/}})\index{taut-conseq}\-
\end{tabbing}

\noindent{\sc Event}:   Enable taut-conseq; name this event `g0277'.\index{g0277}

\addvspace{18pt}

\noindent{\sc Event}:   Enable falsify-taut; name this event `g0282'.\index{g0282}

\addvspace{18pt}
\begin{verbatim};;;;;;;The lemma below is the frequently used form of the tautology theorem.
;;;;;;;It asserts that if EXP is a tautological consequence of a list of
;;;;;;;provable formulas FLIST, then EXP is provable.  "B" is the 
;;;;;;;tautological consequence of {"A1",...,"An"} if "(A1->(...->(An->B)))" is
;;;;;;;a tautology.  A formula is a tautology if it evaluates to true under its
;;;;;;;falsifying assignment.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  eval-tautconseq-proof-proves \\ \index{eval-tautconseq-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eval}}\,(\=\+{\rm{make-disjunct}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})))\index{make-disjunct}, \\ 
{\rm{falsify-taut}}\,({\rm{append}}\,({\rm{lis-not}}\,({\it{flist\/}})\index{lis-not}, {\rm{cons}}\,({\it{exp\/}}, {\bf{nil}})), {\bf{nil}})\index{falsify-taut})\-\index{eval} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\it{flist\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{tautconseq-proof}}\,({\it{flist\/}}, {\it{exp\/}}, {\it{pflist\/}})\index{tautconseq-proof}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable taut-conseq-proves; name this event `g0283'.\index{g0283}

\addvspace{18pt}
\begin{verbatim};;;;;;;The rules below are some properties of the various logical connectives which
;;;;;;;follow as straightforward consequences of the tautology theorem.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{f-iff-mp-proof1} \\  
\=\+{\rm{f-iff-mp-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{f-iff-mp-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{a\/}}), {\it{b\/}}, {\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The rule of detachment for `<->' (first form).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-iff-mp-proves1 \\ \index{f-iff-mp-proves1}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{f-iff-mp-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{f-iff-mp-proof1}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-iff-mp-proof1; name this event `g0284'.\index{g0284}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{f-iff-mp-proof2} \\  
\=\+{\rm{f-iff-mp-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{f-iff-mp-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{b\/}}), {\it{a\/}}, {\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The rule of detachment for `<->' (second form).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-iff-mp-proves2 \\ \index{f-iff-mp-proves2}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{f-iff-mp-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{f-iff-mp-proof2}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{trans-implies-proof} \\  
\=\+{\rm{trans-implies-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{trans-implies-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\rm{f-implies}}\,({\it{b\/}}, {\it{c\/}})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\it{c\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The transitivity of `->'.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  transitivity-of-implies \\ \index{transitivity-of-implies}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{c\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{trans-implies-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{trans-implies-proof}, \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\it{c\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable trans-implies-proof; name this event `g0285'.\index{g0285}

\addvspace{18pt}
\begin{verbatim};;;;;;;The transitivity of `<->'.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{trans-f-iff-proof} \\  
\=\+{\rm{trans-f-iff-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{trans-f-iff-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\rm{f-iff}}\,({\it{b\/}}, {\it{c\/}})\index{f-iff}), {\rm{f-iff}}\,({\it{a\/}}, {\it{c\/}})\index{f-iff}, {\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  transitivity-of-iff \\ \index{transitivity-of-iff}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-iff}}\,({\it{b\/}}, {\it{c\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{trans-f-iff-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{trans-f-iff-proof}, \\ 
{\rm{f-iff}}\,({\it{a\/}}, {\it{c\/}})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable trans-f-iff-proof; name this event `g0286'.\index{g0286}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{and-split-proof1} \\  
\=\+{\rm{and-split-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{and-split-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}), {\it{a\/}}, {\rm{list}}\,({\it{pf1\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "A" from "(A and B)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  and-split1 \\ \index{and-split1}
\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{and-split-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{and-split-proof1}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{and-split-proof2} \\  
\=\+{\rm{and-split-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{and-split-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}), {\it{b\/}}, {\rm{list}}\,({\it{pf1\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "B" from "(A and B)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  and-split2 \\ \index{and-split2}
\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{and-split-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{and-split-proof2}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{and-join-proof} \\  
\=\+{\rm{and-join-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{and-join-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\it{a\/}}, {\it{b\/}}), {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(A and B)" from "A" and "B".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  and-join \\ \index{and-join}
\=\+({\rm{proves}}\,({\it{pf1\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} $\wedge$ {\rm{proves}}\,({\it{pf2\/}}, {\it{b\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{and-join-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{and-join-proof}, {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable and-split-proof1; name this event `g0287'.\index{g0287}

\addvspace{18pt}

\noindent{\sc Event}:   Enable and-split-proof2; name this event `g0288'.\index{g0288}

\addvspace{18pt}

\noindent{\sc Event}:   Enable and-join-proof; name this event `g0289'.\index{g0289}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{iff-split-proof1} \\  
\=\+{\rm{iff-split-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{iff-split-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}), {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\rm{list}}\,({\it{pf1\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(A -> B)" from "(A <-> B)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  iff-split1 \\ \index{iff-split1}
\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{iff-split-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{iff-split-proof1}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable iff-split-proof1; name this event `g0290'.\index{g0290}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{iff-split-proof2} \\  
\=\+{\rm{iff-split-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{iff-split-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}), {\rm{f-implies}}\,({\it{b\/}}, {\it{a\/}})\index{f-implies}, {\rm{list}}\,({\it{pf1\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(B -> A)" from "(A <-> B)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  iff-split2 \\ \index{iff-split2}
\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{iff-split-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{iff-split-proof2}, {\rm{f-implies}}\,({\it{b\/}}, {\it{a\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable iff-split-proof2; name this event `g0291'.\index{g0291}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{iff-join-proof} \\  
\=\+{\rm{iff-join-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{iff-join-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\rm{f-implies}}\,({\it{b\/}}, {\it{a\/}})\index{f-implies}), \\ 
{\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(A <-> B)" from "(A -> B)" and "(B -> A)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  iff-join \\ \index{iff-join}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{a\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{iff-join-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{iff-join-proof}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable iff-join-proof; name this event `g0292'.\index{g0292}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{contra-pos-proof1} \\  
\=\+{\rm{contra-pos-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{contra-pos-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\rm{f-not}}\,({\it{b\/}})\index{f-not}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "((not B) -> (not A))" from "(A -> B)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  contra-pos1 \\ \index{contra-pos1}
\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{contra-pos-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{contra-pos-proof1}, \\ 
{\rm{f-implies}}\,({\rm{f-not}}\,({\it{b\/}})\index{f-not}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable contra-pos-proof1; name this event `g0293'.\index{g0293}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{contra-pos-proof2} \\  
\=\+{\rm{contra-pos-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{contra-pos-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\rm{f-not}}\,({\it{b\/}})\index{f-not}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(A -> B)" from "((not B) -> (not A))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  contra-pos2 \\ \index{contra-pos2}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\rm{f-not}}\,({\it{b\/}})\index{f-not}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{contra-pos-proof2}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{contra-pos-proof2}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable contra-pos-proof2; name this event `g0294'.\index{g0294}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{iff-commute-proof} \\  
\=\+{\rm{iff-commute-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{iff-commute-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}), {\rm{f-iff}}\,({\it{b\/}}, {\it{a\/}})\index{f-iff}, {\rm{list}}\,({\it{pf\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(B <-> A)" from "(A <-> B)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  iff-commute-proof-proves \\ \index{iff-commute-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{iff-commute-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{iff-commute-proof}, {\rm{f-iff}}\,({\it{b\/}}, {\it{a\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable iff-commute-proof; name this event `g0884'.\index{g0884}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-f-iff \\ \index{formula-f-iff}
{\rm{formula}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{symbols\/}})\index{formula} $=$ ({\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-f-iff \\ \index{collect-free-f-iff}
\=\+({\it{var\/}} $\in$ {\rm{collect-free}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\tt{0}})\index{collect-free}) \\ 
$=$$\;\;\;\;$(({\it{var\/}} $\in$ {\rm{collect-free}}\,({\it{a\/}}, {\tt{0}})\index{collect-free}) $\vee$ ({\it{var\/}} $\in$ {\rm{collect-free}}\,({\it{b\/}}, {\tt{0}})\index{collect-free}))\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-f-iff \\ \index{covering-f-iff}
\=\+{\rm{covering}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{var\/}}, {\tt{0}})\index{covering} \\ 
$=$$\;\;\;\;${\rm{append}}\,(\=\+{\rm{append}}\,({\rm{covering}}\,({\it{a\/}}, {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\it{b\/}}, {\it{var\/}}, {\tt{0}})\index{covering}), \\ 
{\rm{append}}\,({\rm{covering}}\,({\it{b\/}}, {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\it{a\/}}, {\it{var\/}}, {\tt{0}})\index{covering}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-f-iff \\ \index{eval-f-iff}
{\rm{eval}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{alist\/}})\index{eval} $=$ ({\rm{eval}}\,({\it{a\/}}, {\it{alist\/}})\index{eval} $=$ {\rm{eval}}\,({\it{b\/}}, {\it{alist\/}})\index{eval})
\end{tabbing}

\noindent{\sc Event}:   Enable f-iff; name this event `g0885'.\index{g0885}

\addvspace{18pt}
\begin{verbatim};;;;;;;Now we get into some rules about quantifiers.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{forall-intro-proof} \\  
\=\+{\rm{forall-intro-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{var\/}}, {\it{pf\/}})\index{forall-intro-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\rm{f-not}}\,({\it{b\/}})\index{f-not})\index{forsome}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\rm{f-not}}\,({\rm{forsome}}\,({\it{var\/}}, {\rm{f-not}}\,({\it{b\/}})\index{f-not})\index{forsome})\index{f-not})\index{f-implies}, \\ 
{\rm{list}}\,({\rm{forsome-intro-proof}}\,(\=\+{\it{var\/}}, \\ 
{\rm{f-not}}\,({\it{b\/}})\index{f-not}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{contra-pos-proof1}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{contra-pos-proof1})\-\index{forsome-intro-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  memb-member-numberp \\ \index{memb-member-numberp}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb} $=$ ({\it{x\/}} $\in$ {\it{y\/}}))
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(A -> (forall x B))" from "(A -> B)", provided "x" does not occur free in "A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forall-intro-proof-proves \\ \index{forall-intro-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\it{a\/}}, $\forall\;${\it{var\/}}$\;${\it{b\/}}\index{forall})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{forall-intro-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{var\/}}, {\it{pf\/}})\index{forall-intro-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forall-intro-proof; name this event `g0295'.\index{g0295}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{generalise-proof} \\  
\=\+{\rm{generalise-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{pf1\/}})\index{generalise-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\rm{f-not}}\,($\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall})\index{f-not}, $\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall})\index{f-implies}), \\ 
$\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall}, \\ 
{\rm{list}}\,({\rm{forall-intro-proof}}\,(\=\+{\rm{f-not}}\,($\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall})\index{f-not}, \\ 
{\it{a\/}}, \\ 
{\it{var\/}}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\it{a\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-not}}\,($\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall})\index{f-not}, \\ 
{\it{a\/}})\-\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}))\-\index{tautconseq-proof})\-\index{forall-intro-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-flg \\ \index{collect-free-flg}
({\it{flg\/}} $\not\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free} $=$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-flg \\ \index{covering-flg}
({\it{flg\/}} $\not\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering} $=$ {\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\tt{0}})\index{covering})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-flg \\ \index{subst-flg}
({\it{flg\/}} $\not\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst} $=$ {\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  variablep-collect-free \\ \index{variablep-collect-free}
($\neg$ {\rm{variable}}\,({\it{var\/}})\index{variable}) $\rightarrow$ ({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-del \\ \index{member-del}
{\it{x\/}} $\not\in$ {\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  forsome-collect-free-reduc \\ \index{forsome-collect-free-reduc}
({\it{var\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{exp\/}})\index{forsome}, {\tt{0}})\index{collect-free})
\end{tabbing}
\begin{verbatim};;;;;;;Derive "(forall x A)" from "A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  generalise-proof-proves \\ \index{generalise-proof-proves}
\=\+({\rm{proves}}\,({\it{pf1\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} $\wedge$ {\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ ({\it{concl\/}} $=$ $\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{generalise-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{pf1\/}})\index{generalise-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable generalise-proof; name this event `g0300'.\index{g0300}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{subst-rule-proof} \\  
\=\+{\rm{subst-rule-proof}}\,({\it{a\/}}, {\it{var\/}}, {\it{term\/}}, {\it{pf1\/}})\index{subst-rule-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-not}}\,({\rm{subst}}\,({\it{a\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-not}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome})\-\index{f-implies}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,({\it{var\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome})\index{f-not})\-, \\ 
{\rm{subst}}\,({\it{a\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, \\ 
{\rm{list}}\,(\=\+{\rm{subst-axiom-proof}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\it{var\/}}, {\it{term\/}})\index{subst-axiom-proof}, \\ 
{\rm{generalise-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{pf1\/}})\index{generalise-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Derive "[t/x]A" from "A", where "t" is free for "x" in "A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  subst-rule-proof-proves \\ \index{subst-rule-proof-proves}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\it{a\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{subst}}\,({\it{a\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{subst-rule-proof}}\,({\it{a\/}}, {\it{var\/}}, {\it{term\/}}, {\it{pf1\/}})\index{subst-rule-proof}, {\it{a-prime\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable subst-rule-proof; name this event `g0886'.\index{g0886}

\addvspace{18pt}
\begin{verbatim};;;;;;;"[x/x]A" is equal to "A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  subst-var-var \\ \index{subst-var-var}
{\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{var\/}}, {\it{flg\/}})\index{subst} $=$ {\it{exp\/}}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{e-intro-proof}$\;\;$
{\rm{e-intro-proof}}\,({\it{var\/}}, {\it{c\/}})\index{e-intro-proof} $=$ {\rm{subst-axiom-proof}}\,({\it{c\/}}, {\it{var\/}}, {\it{var\/}})\index{subst-axiom-proof}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  append-nil-intersect \\ \index{append-nil-intersect}
\=\+{\rm{nil-intersect}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}), {\it{z\/}})\index{nil-intersect} \\ 
$=$$\;\;\;\;$({\rm{nil-intersect}}\,({\it{x\/}}, {\it{z\/}})\index{nil-intersect} $\wedge$ {\rm{nil-intersect}}\,({\it{y\/}}, {\it{z\/}})\index{nil-intersect})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-covering-var \\ \index{var-covering-var}
{\it{var\/}} $\not\in$ {\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-cons \\ \index{nil-intersect-cons}
(($\neg$ {\rm{memb}}\,({\it{z\/}}, {\it{x\/}})\index{memb}) $\wedge$ {\rm{nil-intersect}}\,({\it{x\/}}, {\it{y\/}})\index{nil-intersect}) $\rightarrow$ {\rm{nil-intersect}}\,({\it{x\/}}, {\rm{cons}}\,({\it{z\/}}, {\it{y\/}}))\index{nil-intersect}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-nil \\ \index{nil-intersect-nil}
{\rm{nil-intersect}}\,({\it{x\/}}, {\bf{nil}})\index{nil-intersect}
\end{tabbing}
\begin{verbatim};;;;;;;"(C -> (forsome x C))" is a theorem.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  e-intro-proof-proves \\ \index{e-intro-proof-proves}
\=\+({\rm{formula}}\,({\it{c\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{variable}}\,({\it{var\/}})\index{variable}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{e-intro-proof}}\,({\it{var\/}}, {\it{c\/}})\index{e-intro-proof}, \\ 
{\rm{f-implies}}\,({\it{c\/}}, {\rm{forsome}}\,({\it{var\/}}, {\it{c\/}})\index{forsome})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable e-intro-proof; name this event `g0306'.\index{g0306}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{all-elim-proof} \\  
\=\+{\rm{all-elim-proof}}\,({\it{var\/}}, {\it{c\/}})\index{all-elim-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\rm{f-not}}\,({\it{c\/}})\index{f-not}, {\rm{forsome}}\,({\it{var\/}}, {\rm{f-not}}\,({\it{c\/}})\index{f-not})\index{forsome})\index{f-implies}), \\ 
{\rm{f-implies}}\,($\forall\;${\it{var\/}}$\;${\it{c\/}}\index{forall}, {\it{c\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\rm{e-intro-proof}}\,({\it{var\/}}, {\rm{f-not}}\,({\it{c\/}})\index{f-not})\index{e-intro-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;"((forall x C) -> C)" is a theorem.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  all-elim-proof-proves \\ \index{all-elim-proof-proves}
\=\+({\rm{formula}}\,({\it{c\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{variable}}\,({\it{var\/}})\index{variable}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{all-elim-proof}}\,({\it{var\/}}, {\it{c\/}})\index{all-elim-proof}, {\rm{f-implies}}\,($\forall\;${\it{var\/}}$\;${\it{c\/}}\index{forall}, {\it{c\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable all-elim-proof; name this event `g0307'.\index{g0307}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  member-del1 \\ \index{member-del1}
($\neg$ {\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql}) $\rightarrow$ (({\it{x\/}} $\in$ {\rm{del}}\,({\it{y\/}}, {\it{z\/}})\index{del}) $=$ ({\it{x\/}} $\in$ {\it{z\/}}))
\end{tabbing}
\begin{verbatim};;;;;;;"[t/x]A" is equal to "A", if "x" does not occur free in "A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  subst-ident \\ \index{subst-ident}
({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) $\rightarrow$ ({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst} $=$ {\it{exp\/}})
\end{tabbing}
\begin{verbatim};;;;;;;"y" is free in "[t/x]A" iff either "y" is free in "t" and "x" is free in "A"
;;;;;;;                                or "y" is not "x" and "y" is free in "A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  collect-free-subst \\ \index{collect-free-subst}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ {\rm{variable}}\,({\it{var1\/}})\index{variable} $\wedge$ ({\it{var\/}} $\not\in$ {\rm{covering}}\,({\it{exp\/}}, {\it{var1\/}}, {\it{flg\/}})\index{covering})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\it{var\/}} $\in$ {\rm{collect-free}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var1\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{flg\/}})\index{collect-free}) \\ 
$=$$\;\;\;\;$(\=\+(({\it{var\/}} $\not=$ {\it{var1\/}}) $\wedge$ ({\it{var\/}} $\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})) \\ 
$\vee$$\;\;\;\;$(\=\+({\it{var1\/}} $\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\in$ {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free}))\-)\-)\-\-
\end{tabbing}
\begin{verbatim};;;;;;;If "y" is a `new' variable, the bound variables governing free occurrences
;;;;;;;of "y" in "[y/x]A" are the ones that govern free occurrences of "x" in "A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  free-for-subst \\ \index{free-for-subst}
\=\+(\=\+{\rm{variable}}\,({\it{aux\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\rm{collect-free}}\,({\it{a\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\rm{covering}}\,({\it{a\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{covering}}\,({\rm{subst}}\,({\it{a\/}}, {\it{var\/}}, {\it{aux\/}}, {\it{flg\/}})\index{subst}, {\it{aux\/}}, {\it{flg\/}})\index{covering} $=$ {\rm{covering}}\,({\it{a\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-covering \\ \index{not-free-covering}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ ({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering} $=$ {\bf{nil}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-member-delete2 \\ \index{not-member-delete2}
({\it{x\/}} $\in$ {\rm{del}}\,({\it{y\/}}, {\it{z\/}})\index{del}) $=$ (($\neg$ {\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql}) $\wedge$ ({\it{x\/}} $\in$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-subst1 \\ \index{collect-free-subst1}
\=\+(\=\+{\rm{variable}}\,({\it{var1\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var1\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{flg\/}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-subst2 \\ \index{collect-free-subst2}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var1\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not=$ {\it{var1\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{var\/}} $\in$ {\rm{collect-free}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var1\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{flg\/}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-covering-unchanged \\ \index{subst-covering-unchanged}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var1\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not=$ {\it{var1\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{covering}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var1\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{var\/}}, {\it{flg\/}})\index{covering} $=$ {\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{var-set} \\  
\=\+{\rm{var-set}}\,({\it{list\/}}, {\it{n\/}})\index{var-set} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{list\/}} $=$ {\bf{nil}} \\ 
{\bf else }\=\+{\rm{variable}}\,({\rm{car}}\,({\it{list\/}}))\index{variable} \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{list\/}}) $\not\in$ {\rm{cdr}}\,({\it{list\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{cdr}}\,({\it{list\/}}), {\it{n\/}} $-\;1$)\index{var-set}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-append \\ \index{nil-intersect-append}
\=\+{\rm{nil-intersect}}\,({\it{x\/}}, {\rm{append}}\,({\it{y\/}}, {\it{z\/}}))\index{nil-intersect} \\ 
$=$$\;\;\;\;$({\rm{nil-intersect}}\,({\it{x\/}}, {\it{y\/}})\index{nil-intersect} $\wedge$ {\rm{nil-intersect}}\,({\it{x\/}}, {\it{z\/}})\index{nil-intersect})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-set-length \\ \index{var-set-length}
{\rm{var-set}}\,({\it{x\/}}, {\it{count\/}})\index{var-set} $\rightarrow$ {\rm{var-set}}\,({\it{x\/}}, {\rm{length}}\,({\it{x\/}})\index{length})\index{var-set}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-subst \\ \index{nil-intersect-subst}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{x\/}}, {\rm{length}}\,({\it{x\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{x\/}}, {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{x\/}}, {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{nil-intersect}}\,({\it{x\/}}, {\rm{collect-free}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{flg\/}})\index{collect-free})\index{nil-intersect}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-cons2 \\ \index{nil-intersect-cons2}
{\rm{nil-intersect}}\,({\it{x\/}}, {\rm{cons}}\,({\it{y\/}}, {\it{z\/}}))\index{nil-intersect} $=$ (($\neg$ {\rm{memb}}\,({\it{y\/}}, {\it{x\/}})\index{memb}) $\wedge$ {\rm{nil-intersect}}\,({\it{x\/}}, {\it{z\/}})\index{nil-intersect})
\end{tabbing}
\begin{verbatim};;;;;;;PAR-SUBST is the first definition of simultaneous substitution.
;;;;;;;It returns "[t1/x1, t2/x2,...,tn/xn]A".  It employs a list of
;;;;;;;new variables "yi" and evaluates "[t1/y1]...[tn/yn][yn/xn]...[y1/x1]A".
;;;;;;;This is in order to avoid substituting for an occurrence of an "xi"
;;;;;;;in some "tj".  This definition of simultaneous substitution while
;;;;;;;clumsy, turns out to be convenient for the proof of instantiation rule,
;;;;;;;which by necessity must be derived from the substitution rule above.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{par-subst} \\  
\=\+{\rm{par-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}})\index{par-subst} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) $\wedge$ {\rm{listp}}\,({\it{terms\/}}) $\wedge$ {\rm{listp}}\,({\it{auxvars\/}}) \\ 
{\bf then }{\rm{subst}}\,(\=\+{\rm{par-subst}}\,(\=\+{\rm{subst}}\,({\it{exp\/}}, {\rm{car}}\,({\it{vars\/}}), {\rm{car}}\,({\it{auxvars\/}}), {\tt{0}})\index{subst}, \\ 
{\rm{cdr}}\,({\it{vars\/}}), \\ 
{\rm{cdr}}\,({\it{terms\/}}), \\ 
{\rm{cdr}}\,({\it{auxvars\/}}))\-\index{par-subst}, \\ 
{\rm{car}}\,({\it{auxvars\/}}), \\ 
{\rm{car}}\,({\it{terms\/}}), \\ 
{\tt{0}})\-\index{subst} \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The unsubstituted free variables in EXP remain free after PAR-SUBST.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  par-subst-collect-free \\ \index{par-subst-collect-free}
\=\+(\=\+{\rm{variable}}\,({\it{aux\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{auxlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{varlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\it{varlist\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\it{auxlist\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{aux\/}} $\in$ {\rm{collect-free}}\,({\rm{par-subst}}\,({\it{exp\/}}, {\it{varlist\/}}, {\it{termlist\/}}, {\it{auxlist\/}})\index{par-subst}, {\tt{0}})\index{collect-free})\-
\end{tabbing}
\begin{verbatim};;;;;;;The COVERING of an unsubsubstituted free variables in EXP is unchanged
;;;;;;;after PAR-SUBST, provided it does not occur free in TERMLIST. 
\end{verbatim}\begin{tabbing}{\sc Theorem}:  par-subst-covering \\ \index{par-subst-covering}
\=\+(\=\+({\rm{length}}\,({\it{termlist\/}})\index{length} $=$ {\rm{length}}\,({\it{varlist\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\rm{collect-free}}\,({\it{termlist\/}}, {\tt{1}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\it{auxlist\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\it{varlist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{varlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{auxlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{aux\/}})\index{variable})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{covering}}\,({\rm{par-subst}}\,({\it{exp\/}}, {\it{varlist\/}}, {\it{termlist\/}}, {\it{auxlist\/}})\index{par-subst}, {\it{aux\/}}, {\tt{0}})\index{covering} \\ 
$=$$\;\;\;\;${\rm{covering}}\,({\it{exp\/}}, {\it{aux\/}}, {\tt{0}})\index{covering})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;FREE-FOR-LIST checks if each term in TERMLIST is free for the corresponding
;;;;;;;variable in VARLIST, in A.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{free-for-list} \\  
\=\+{\rm{free-for-list}}\,({\it{a\/}}, {\it{varlist\/}}, {\it{termlist\/}})\index{free-for-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{varlist\/}}) $\wedge$ {\rm{listp}}\,({\it{termlist\/}}) \\ 
{\bf then }\=\+{\rm{free-for}}\,({\it{a\/}}, {\rm{car}}\,({\it{varlist\/}}), {\rm{car}}\,({\it{termlist\/}}), {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\rm{cdr}}\,({\it{varlist\/}}), {\rm{cdr}}\,({\it{termlist\/}}))\index{free-for-list}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The variables covering AUX in (SUBST EXP VAR AUX 0) are the ones covering VAR
;;;;;;;in EXP, and that is unaffected by PAR-SUBST, when the hypotheses below hold.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  covering-par-subst \\ \index{covering-par-subst}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{aux\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\rm{collect-free}}\,({\it{termlist\/}}, {\tt{1}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\tt{0}})\index{covering}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\it{auxlist\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\it{varlist\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{varlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{auxlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{termlist\/}})\index{length} $=$ {\rm{length}}\,({\it{varlist\/}})\index{length}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{covering}}\,(\=\+{\rm{par-subst}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{aux\/}}, {\tt{0}})\index{subst}, {\it{varlist\/}}, {\it{termlist\/}}, {\it{auxlist\/}})\index{par-subst}, \\ 
{\it{aux\/}}, \\ 
{\tt{0}})\-\index{covering} \\ 
$=$$\;\;\;\;${\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\tt{0}})\index{covering})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;INST-CONDITIONS lays down the various conditions for PAR-SUBST to be a proper
;;;;;;;simultaneous substitution: (1) VARS and AUXVARS must be sets of variables,
;;;;;;;(2) TERMS must be a list of terms, (3) AUXVARS must be a list of new variables,
;;;;;;;(4) the AUXVARS must be free the respective VARS in A, and (5) the TERMS must
;;;;;;;be free for the respective VARS in A.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{inst-conditions} \\  
\=\+{\rm{inst-conditions}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}}, {\it{symbols\/}})\index{inst-conditions} \\ 
$=$$\;\;\;\;$(\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{auxvars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{terms\/}})\index{length} $=$ {\rm{length}}\,({\it{vars\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{auxvars\/}}, {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{auxvars\/}}, {\rm{collect-free}}\,({\it{a\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{auxvars\/}}, {\it{vars\/}})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{auxvars\/}})\index{free-for-list} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;FREE-FOR-LIST is unaffected by an irrelevant substitution into A.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  free-for-list-elim \\ \index{free-for-list-elim}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{aux\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\it{vars\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{aux\/}} $\not\in$ {\it{vars\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{free-for-list}}\,({\rm{subst}}\,({\it{a\/}}, {\it{var\/}}, {\it{aux\/}}, {\tt{0}})\index{subst}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list}\-
\end{tabbing}
\begin{verbatim};;;;;;;PAR-SUBST-PROOF constructs the proof of the instance of A, given the proof of A.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{par-subst-proof} \\  
\=\+{\rm{par-subst-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}}, {\it{pf1\/}})\index{par-subst-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) $\wedge$ {\rm{listp}}\,({\it{terms\/}}) $\wedge$ {\rm{listp}}\,({\it{auxvars\/}}) \\ 
{\bf then }{\rm{subst-rule-proof}}\,(\=\+{\rm{par-subst}}\,(\=\+{\rm{subst}}\,({\it{a\/}}, {\rm{car}}\,({\it{vars\/}}), {\rm{car}}\,({\it{auxvars\/}}), {\tt{0}})\index{subst}, \\ 
{\rm{cdr}}\,({\it{vars\/}}), \\ 
{\rm{cdr}}\,({\it{terms\/}}), \\ 
{\rm{cdr}}\,({\it{auxvars\/}}))\-\index{par-subst}, \\ 
{\rm{car}}\,({\it{auxvars\/}}), \\ 
{\rm{car}}\,({\it{terms\/}}), \\ 
{\rm{par-subst-proof}}\,(\=\+{\rm{subst}}\,(\=\+{\it{a\/}}, \\ 
{\rm{car}}\,({\it{vars\/}}), \\ 
{\rm{car}}\,({\it{auxvars\/}}), \\ 
{\tt{0}})\-\index{subst}, \\ 
{\rm{cdr}}\,({\it{vars\/}}), \\ 
{\rm{cdr}}\,({\it{terms\/}}), \\ 
{\rm{cdr}}\,({\it{auxvars\/}}), \\ 
{\rm{subst-rule-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{car}}\,({\it{vars\/}}), \\ 
{\rm{car}}\,({\it{auxvars\/}}), \\ 
{\it{pf1\/}})\-\index{subst-rule-proof})\-\index{par-subst-proof})\-\index{subst-rule-proof} \\ 
{\bf else }{\it{pf1\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below is the first form of the Instantiation Rule.
;;;;;;;It asserts that if A is provable, so is any instance of it as
;;;;;;;constructed by PAR-SUBST.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  par-subst-proof-proves \\ \index{par-subst-proof-proves}
\=\+(\=\+{\rm{inst-conditions}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}}, {\it{symbols\/}})\index{inst-conditions} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{par-subst-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}}, {\it{pf1\/}})\index{par-subst-proof}, \\ 
{\rm{par-subst}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}})\index{par-subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;;Since PAR-SUBST is not a very efficient way to carry out simultaneous
;;;;;;;substitution, we will prove it to be equivalent to a different
;;;;;;;form of simultaneous substitution.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{del-pairs} \\  
\=\+{\rm{del-pairs}}\,({\it{var\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) $\wedge$ {\rm{listp}}\,({\it{terms\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{eql}}\,({\rm{car}}\,({\it{vars\/}}), {\it{var\/}})\index{eql}$\;\;${\bf then }{\rm{del-pairs}}\,({\it{var\/}}, {\rm{cdr}}\,({\it{vars\/}}), {\rm{cdr}}\,({\it{terms\/}}))\index{del-pairs} \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{terms\/}}), {\rm{del-pairs}}\,({\it{var\/}}, {\rm{cdr}}\,({\it{vars\/}}), {\rm{cdr}}\,({\it{terms\/}}))\index{del-pairs})$\;${\bf  endif}\- \\ 
{\bf else }{\it{terms\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{getsub} \\  
\=\+{\rm{getsub}}\,({\it{var\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getsub} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) $\wedge$ {\rm{listp}}\,({\it{terms\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{eql}}\,({\rm{car}}\,({\it{vars\/}}), {\it{var\/}})\index{eql}$\;\;${\bf then }{\rm{car}}\,({\it{terms\/}}) \\ 
{\bf else }{\rm{getsub}}\,({\it{var\/}}, {\rm{cdr}}\,({\it{vars\/}}), {\rm{cdr}}\,({\it{terms\/}}))\index{getsub}$\;${\bf  endif}\- \\ 
{\bf else }{\it{var\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  del-pairs-getsub \\ \index{del-pairs-getsub}
\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\rightarrow$$\;\;\;\;$({\rm{getsub}}\,({\it{var\/}}, {\rm{del}}\,({\it{var\/}}, {\it{vars\/}})\index{del}, {\rm{del-pairs}}\,({\it{var\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs})\index{getsub} $=$ {\it{var\/}})\-
\end{tabbing}
\begin{verbatim};;;;;;;PARALLEL-SUBST below is the more efficient form of simultaneous substitution.
;;;;;;;It recurses down EXP and replaces any free occurrence of a variable in
;;;;;;;VARS by the corresponding term in TERMS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{parallel-subst} \\  
\=\+{\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{variable}}\,({\it{exp\/}})\index{variable}$\;\;${\bf then }{\rm{getsub}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getsub} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp} \\ 
{\bf then }{\rm{f-not}}\,({\rm{parallel-subst}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{f-not} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }{\rm{f-or}}\,(\=\+{\rm{parallel-subst}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\index{f-or} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep} \\ 
{\bf then }{\rm{forsome}}\,(\=\+{\rm{bind}}\,({\it{exp\/}})\index{bind}, \\ 
{\rm{parallel-subst}}\,(\=\+{\rm{body}}\,({\it{exp\/}}), \\ 
{\rm{del}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\it{vars\/}})\index{del}, \\ 
{\rm{del-pairs}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs}, \\ 
{\tt{0}})\-\index{parallel-subst})\-\index{forsome} \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{car}}\,({\it{exp\/}}), {\rm{parallel-subst}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{vars\/}}, {\it{terms\/}}, {\tt{1}})\index{parallel-subst}) \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{parallel-subst}}\,({\rm{car}}\,({\it{exp\/}}), {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{vars\/}}, {\it{terms\/}}, {\tt{1}})\index{parallel-subst})\- \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  getsub-not-memb \\ \index{getsub-not-memb}
($\neg$ {\rm{memb}}\,({\it{var\/}}, {\it{varlist\/}})\index{memb}) $\rightarrow$ ({\rm{getsub}}\,({\it{var\/}}, {\it{varlist\/}}, {\it{terms\/}})\index{getsub} $=$ {\it{var\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eql-trans \\ \index{eql-trans}
({\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql} $\wedge$ {\rm{eql}}\,({\it{y\/}}, {\it{z\/}})\index{eql}) $\rightarrow$ {\rm{eql}}\,({\it{x\/}}, {\it{z\/}})\index{eql}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  getsub-del-pairs \\ \index{getsub-del-pairs}
\=\+({\rm{getsub}}\,({\it{var\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getsub} $=$ {\it{var\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{getsub}}\,({\it{var\/}}, {\rm{del}}\,({\it{var1\/}}, {\it{vars\/}})\index{del}, {\rm{del-pairs}}\,({\it{var1\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs})\index{getsub} $=$ {\it{var\/}})\-
\end{tabbing}

\noindent{\sc Event}:   Enable eql-trans; name this event `g0636'.\index{g0636}

\addvspace{18pt}
\begin{verbatim};;;;;;;The lemma below asserts that the effect of a single substitution 
;;;;;;;followed by a simultaneous substitution, can be got by extending the
;;;;;;;simultaneous substitution.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  newvar-unchanged-by-parallel-subst \\ \index{newvar-unchanged-by-parallel-su!bst}
\=\+(\=\+({\rm{getsub}}\,({\it{newvar\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getsub} $=$ {\it{newvar\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{newvar\/}})\index{variable})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{parallel-subst}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{newvar\/}}, {\it{flg\/}})\index{subst}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{parallel-subst}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{var\/}}, {\it{vars\/}}), {\rm{cons}}\,({\it{newvar\/}}, {\it{terms\/}}), {\it{flg\/}})\index{parallel-subst})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-in-terms-step \\ \index{not-free-in-terms-step}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{newvar\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not=$ {\it{var\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\rm{getsub}}\,({\it{var\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getsub}, {\tt{0}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-terms-del-pairs \\ \index{collect-terms-del-pairs}
\=\+({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free}) \\ 
$\rightarrow$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\rm{del-pairs}}\,({\it{var\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs}, {\tt{1}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-parallel-subst \\ \index{collect-free-parallel-subst}
\=\+(\=\+{\rm{variable}}\,({\it{newvar\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst}, {\it{flg\/}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-not-free-var \\ \index{parallel-subst-not-free-var}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ ({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{parallel-subst}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{var\/}}, {\it{vars\/}}), {\rm{cons}}\,({\it{term\/}}, {\it{terms\/}}), {\it{flg\/}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-parallel-subst \\ \index{subst-parallel-subst}
\=\+(\=\+{\rm{variable}}\,({\it{newvar\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{subst}}\,({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst}, {\it{newvar\/}}, {\it{newterm\/}}, {\it{flg\/}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;A PARALLEL-SUBST where VAR is replaced by NEWVAR, followed by a single substitution
;;;;;;;where NEWVAR is replaced by NEWTERM, is equivalent to the same PARALLEL-SUBST
;;;;;;;with VAR replaced by NEWTERM instead.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  parallel-subst-subst \\ \index{parallel-subst-subst}
\=\+(\=\+{\rm{variable}}\,({\it{newvar\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{newvar\/}} $\not\in$ {\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{subst}}\,(\=\+{\rm{parallel-subst}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{var\/}}, {\it{vars\/}}), {\rm{cons}}\,({\it{newvar\/}}, {\it{terms\/}}), {\it{flg\/}})\index{parallel-subst}, \\ 
{\it{newvar\/}}, \\ 
{\it{newterm\/}}, \\ 
{\it{flg\/}})\-\index{subst} \\ 
$=$$\;\;\;\;${\rm{parallel-subst}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{var\/}}, {\it{vars\/}}), {\rm{cons}}\,({\it{newterm\/}}, {\it{terms\/}}), {\it{flg\/}})\index{parallel-subst})\-\-
\end{tabbing}

\begin{verbatim};;;;;;;A null parallel-substitution leaved EXP unchanged.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  parallel-subst-nil \\ \index{parallel-subst-nil}
\=\+(({\it{vars\/}} $\simeq\;${{\bf{nil}}}${}$) $\vee$ ({\it{terms\/}} $\simeq\;${{\bf{nil}}}${}$)) \\ 
$\rightarrow$$\;\;\;\;$({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst} $=$ {\it{exp\/}})\-
\end{tabbing}
\begin{verbatim};;;;;;;Finally, the desired lemma that PAR-SUBST is equivalent to PARALLEL-SUBST,
;;;;;;;if the hypothesis conditions hold.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  parallel-subst-same-as-par-subst \\ \index{parallel-subst-same-as-par-subst}
\=\+(\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{auxvars\/}}, {\rm{length}}\,({\it{auxvars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{auxvars\/}})\index{length} $=$ {\rm{length}}\,({\it{vars\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{auxvars\/}}, {\it{vars\/}})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{auxvars\/}})\index{free-for-list} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{auxvars\/}}, {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{auxvars\/}}, {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{par-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}})\index{par-subst} \\ 
$=$$\;\;\;\;${\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;We still have to show that list of new variables AUXVARS can be constructed.
;;;;;;;That is done in the sequence of events (definitions and lemmas) below.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{make-newvars} \\  
\=\+{\rm{make-newvars}}\,({\it{index\/}}, {\it{number\/}})\index{make-newvars} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{number\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,($1\;+$ {\it{index\/}}, {\rm{make-newvars}}\,($1\;+$ {\it{index\/}}, {\it{number\/}} $-\;1$)\index{make-newvars})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{find-hi-index} \\  
\=\+{\rm{find-hi-index}}\,({\it{varlist\/}})\index{find-hi-index} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{varlist\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{varlist\/}}) $<$ {\rm{find-hi-index}}\,({\rm{cdr}}\,({\it{varlist\/}}))\index{find-hi-index} \\ 
{\bf then }{\rm{find-hi-index}}\,({\rm{cdr}}\,({\it{varlist\/}}))\index{find-hi-index} \\ 
{\bf else }{\rm{fix}}\,({\rm{car}}\,({\it{varlist\/}}))$\;${\bf  endif}\- \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{make-auxvars} \\  
\=\+{\rm{make-auxvars}}\,({\it{varlist\/}}, {\it{number\/}})\index{make-auxvars} \\ 
$=$$\;\;\;\;${\rm{make-newvars}}\,({\rm{find-hi-index}}\,({\it{varlist\/}})\index{find-hi-index}, {\it{number\/}})\index{make-newvars}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-member-varlist \\ \index{not-member-varlist}
({\rm{find-hi-index}}\,({\it{varlist\/}})\index{find-hi-index} $<$ {\it{var\/}}) $\rightarrow$ ({\it{var\/}} $\not\in$ {\it{varlist\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  newvars-make-newvars \\ \index{newvars-make-newvars}
\=\+({\rm{find-hi-index}}\,({\it{varlist\/}})\index{find-hi-index} $\leq$ {\it{index\/}}) \\ 
$\rightarrow$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{make-newvars}}\,({\it{index\/}}, {\it{number\/}})\index{make-newvars}, {\it{varlist\/}})\index{nil-intersect}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-set-make-newvars-step \\ \index{var-set-make-newvars-step}
({\it{var\/}} $\leq$ {\it{index\/}}) $\rightarrow$ ({\it{var\/}} $\not\in$ {\rm{make-newvars}}\,({\it{index\/}}, {\it{number\/}})\index{make-newvars})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  append-var-list \\ \index{append-var-list}
\=\+({\rm{var-list}}\,({\it{x\/}}, {\rm{length}}\,({\it{x\/}})\index{length})\index{var-list} $\wedge$ {\rm{var-list}}\,({\it{y\/}}, {\rm{length}}\,({\it{y\/}})\index{length})\index{var-list}) \\ 
$\rightarrow$$\;\;\;\;${\rm{var-list}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}), {\rm{length}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}))\index{length})\index{var-list}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{covering-list} \\  
\=\+{\rm{covering-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{flg\/}})\index{covering-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) \\ 
{\bf then }{\rm{append}}\,(\=\+{\rm{covering}}\,({\it{exp\/}}, {\rm{car}}\,({\it{vars\/}}), {\it{flg\/}})\index{covering}, \\ 
{\rm{covering-list}}\,({\it{exp\/}}, {\rm{cdr}}\,({\it{vars\/}}), {\it{flg\/}})\index{covering-list})\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  free-for-list-newvars-covering-list \\ \index{free-for-list-newvars-covering-!list}
\=\+(\=\+{\rm{var-set}}\,({\it{newvars\/}}, {\rm{length}}\,({\it{newvars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\it{newvars\/}}, {\rm{covering-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\tt{0}})\index{covering-list})\index{nil-intersect})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{free-for-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{newvars\/}})\index{free-for-list}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-list-var-set \\ \index{var-list-var-set}
{\rm{var-set}}\,({\it{vars\/}}, {\it{length\/}})\index{var-set} $\rightarrow$ {\rm{var-list}}\,({\it{vars\/}}, {\it{length\/}})\index{var-list}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  find-hi-index-append \\ \index{find-hi-index-append}
\=\+{\rm{find-hi-index}}\,({\rm{append}}\,({\it{list1\/}}, {\it{list2\/}}))\index{find-hi-index} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{find-hi-index}}\,({\it{list1\/}})\index{find-hi-index} $<$ {\rm{find-hi-index}}\,({\it{list2\/}})\index{find-hi-index} \\ 
{\bf then }{\rm{find-hi-index}}\,({\it{list2\/}})\index{find-hi-index} \\ 
{\bf else }{\rm{find-hi-index}}\,({\it{list1\/}})\index{find-hi-index}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{collect-vars} \\  
\=\+{\rm{collect-vars}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{collect-vars} \\ 
$=$$\;\;\;\;${\rm{append}}\,(\=\+{\it{vars\/}}, \\ 
{\rm{append}}\,(\=\+{\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free}, \\ 
{\rm{append}}\,({\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free}, {\rm{covering-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\tt{0}})\index{covering-list}))\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-newvars-vars \\ \index{nil-intersect-newvars-vars}
{\rm{nil-intersect}}\,(\=\+{\rm{make-newvars}}\,(\=\+{\rm{find-hi-index}}\,({\rm{collect-vars}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{collect-vars})\index{find-hi-index}, \\ 
{\it{length\/}})\-\index{make-newvars}, \\ 
{\it{vars\/}})\-\index{nil-intersect}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-newvars-terms \\ \index{nil-intersect-newvars-terms}
{\rm{nil-intersect}}\,(\=\+{\rm{make-newvars}}\,(\=\+{\rm{find-hi-index}}\,({\rm{collect-vars}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{collect-vars})\index{find-hi-index}, \\ 
{\it{length\/}})\-\index{make-newvars}, \\ 
{\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free})\-\index{nil-intersect}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-newvars-exp \\ \index{nil-intersect-newvars-exp}
{\rm{nil-intersect}}\,(\=\+{\rm{make-newvars}}\,(\=\+{\rm{find-hi-index}}\,({\rm{collect-vars}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{collect-vars})\index{find-hi-index}, \\ 
{\it{length\/}})\-\index{make-newvars}, \\ 
{\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})\-\index{nil-intersect}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-set-make-newvars-again \\ \index{var-set-make-newvars-again}
{\rm{var-set}}\,({\rm{make-newvars}}\,({\it{index\/}}, {\it{number\/}})\index{make-newvars}, {\it{number\/}})\index{var-set}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-make-newvars \\ \index{length-make-newvars}
{\rm{length}}\,({\rm{make-newvars}}\,({\it{x\/}}, {\it{n\/}})\index{make-newvars})\index{length} $=$ {\rm{fix}}\,({\it{n\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  free-for-list-newvars-exp \\ \index{free-for-list-newvars-exp}
{\rm{free-for-list}}\,(\=\+{\it{exp\/}}, \\ 
{\it{vars\/}}, \\ 
{\rm{make-newvars}}\,(\=\+{\rm{find-hi-index}}\,({\rm{collect-vars}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{collect-vars})\index{find-hi-index}, \\ 
{\it{length\/}})\-\index{make-newvars})\-\index{free-for-list}
\end{tabbing}

\noindent{\sc Event}:   Enable make-newvars; name this event `g0347'.\index{g0347}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{instance-proof} \\  
\=\+{\rm{instance-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{pf1\/}})\index{instance-proof} \\ 
$=$$\;\;\;\;${\rm{par-subst-proof}}\,(\=\+{\it{a\/}}, \\ 
{\it{vars\/}}, \\ 
{\it{terms\/}}, \\ 
{\rm{make-auxvars}}\,({\rm{collect-vars}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{collect-vars}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{make-auxvars}, \\ 
{\it{pf1\/}})\-\index{par-subst-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  par-subst-proof-proves1 \\ \index{par-subst-proof-proves1}
\=\+(\=\+({\it{concl\/}} $=$ {\rm{par-subst}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}})\index{par-subst}) \\ 
$\wedge$$\;\;\;\;${\rm{inst-conditions}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}}, {\it{symbols\/}})\index{inst-conditions} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{par-subst-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{auxvars\/}}, {\it{pf1\/}})\index{par-subst-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable par-subst-proof; name this event `g0346'.\index{g0346}

\addvspace{18pt}

\noindent{\sc Event}:   Enable find-hi-index; name this event `g0350'.\index{g0350}

\addvspace{18pt}

\noindent{\sc Event}:   Enable find-hi-index-append; name this event `g0351'.\index{g0351}

\addvspace{18pt}
\begin{verbatim};;;;;;;The Instantiation Rule can then be phrased as shown below, in terms
;;;;;;;of PARALLEL-SUBST instead of PAR-SUBST.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  instance-proof-proves \\ \index{instance-proof-proves}
\=\+(\=\+({\it{a-prime\/}} $=$ {\rm{parallel-subst}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}) \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{terms\/}})\index{length} $=$ {\rm{length}}\,({\it{vars\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\it{a\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{instance-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{pf1\/}})\index{instance-proof}, {\it{a-prime\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-exis-quantify} \\  
\=\+{\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) \\ 
{\bf then }{\rm{forsome}}\,({\rm{car}}\,({\it{vars\/}}), {\rm{list-exis-quantify}}\,({\rm{cdr}}\,({\it{vars\/}}), {\it{a\/}})\index{list-exis-quantify})\index{forsome} \\ 
{\bf else }{\it{a\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-member-del-pairs \\ \index{not-member-del-pairs}
($\neg$ {\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb}) $\rightarrow$ ({\rm{del-pairs}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{del-pairs} $=$ {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-member-delete-equals \\ \index{not-member-delete-equals}
($\neg$ {\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb}) $\rightarrow$ ({\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del} $=$ {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pairlist} \\  
\=\+{\rm{pairlist}}\,({\it{vars\/}}, {\it{terms\/}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) $\wedge$ {\rm{listp}}\,({\it{terms\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{cons}}\,({\rm{car}}\,({\it{vars\/}}), {\rm{car}}\,({\it{terms\/}})), {\rm{pairlist}}\,({\rm{cdr}}\,({\it{vars\/}}), {\rm{cdr}}\,({\it{terms\/}}))) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-list-exis-quant \\ \index{parallel-subst-list-exis-quant}
\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{parallel-subst}}\,({\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify})\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-e-intro-proof} \\  
\=\+{\rm{list-e-intro-proof}}\,({\it{varlist\/}}, {\it{a\/}})\index{list-e-intro-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{varlist\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{varlist\/}})) \\ 
{\bf then }{\rm{trans-implies-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{list-exis-quantify}}\,({\rm{cdr}}\,({\it{varlist\/}}), {\it{a\/}})\index{list-exis-quantify}, \\ 
{\rm{list-exis-quantify}}\,({\it{varlist\/}}, {\it{a\/}})\index{list-exis-quantify}, \\ 
{\rm{list-e-intro-proof}}\,({\rm{cdr}}\,({\it{varlist\/}}), {\it{a\/}})\index{list-e-intro-proof}, \\ 
{\rm{e-intro-proof}}\,(\=\+{\rm{car}}\,({\it{varlist\/}}), \\ 
{\rm{list-exis-quantify}}\,(\=\+{\rm{cdr}}\,({\it{varlist\/}}), \\ 
{\it{a\/}})\-\index{list-exis-quantify})\-\index{e-intro-proof})\-\index{trans-implies-proof} \\ 
{\bf else }{\rm{e-intro-proof}}\,({\rm{car}}\,({\it{varlist\/}}), {\it{a\/}})\index{e-intro-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{prop-axiom-proof}}\,({\it{a\/}})\index{prop-axiom-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-list-exis \\ \index{formula-list-exis}
\=\+({\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}

\noindent{\sc Event}:   Enable instance-proof; name this event `g0353'.\index{g0353}

\addvspace{18pt}

\noindent{\sc Event}:   Disable f-implies; name this event `g0354'.\index{g0354}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  a-implies-a \\ \index{a-implies-a}
\=\+{\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{prop-axiom-proof}}\,({\it{a\/}})\index{prop-axiom-proof}, {\rm{f-implies}}\,({\it{a\/}}, {\it{a\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-f-implies \\ \index{parallel-subst-f-implies}
\=\+{\rm{parallel-subst}}\,({\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{f-implies}}\,(\=\+{\rm{parallel-subst}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\it{b\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\index{f-implies}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-f-implies \\ \index{subst-f-implies}
\=\+{\rm{subst}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{f-implies}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-implies}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-implies; name this event `g0355'.\index{g0355}

\addvspace{18pt}
\begin{verbatim};;;;;;;The iterated forsome-introduction rule:
;;;;;;;   "A -> (forsome x1 (...(forsome xn A)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  list-e-intro-proof-proves1 \\ \index{list-e-intro-proof-proves1}
\=\+({\rm{var-set}}\,({\it{varlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{list-e-intro-proof}}\,({\it{varlist\/}}, {\it{a\/}})\index{list-e-intro-proof}, \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\rm{list-exis-quantify}}\,({\it{varlist\/}}, {\it{a\/}})\index{list-exis-quantify})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-e-intro-proof; name this event `g0348'.\index{g0348}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  list-e-intro-proof-proves \\ \index{list-e-intro-proof-proves}
\=\+(\=\+{\rm{var-set}}\,({\it{varlist\/}}, {\rm{length}}\,({\it{varlist\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{list-exis-quantify}}\,({\it{varlist\/}}, {\it{a\/}})\index{list-exis-quantify}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{list-e-intro-proof}}\,({\it{varlist\/}}, {\it{a\/}})\index{list-e-intro-proof}, \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\it{a-prime\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{subst-thm-a-proof} \\  
\=\+{\rm{subst-thm-a-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{subst-thm-a-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{f-implies}}\,({\it{a\/}}, {\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify})\index{f-implies}, \\ 
{\it{vars\/}}, \\ 
{\it{terms\/}}, \\ 
{\rm{list-e-intro-proof}}\,({\it{vars\/}}, {\it{a\/}})\index{list-e-intro-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-list-exis \\ \index{not-free-list-exis}
\=\+({\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} $\wedge$ ({\it{var\/}} $\in$ {\it{vars\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify}, {\tt{0}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-memb-del2 \\ \index{not-memb-del2}
($\neg$ {\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb}) $\rightarrow$ ($\neg$ {\rm{memb}}\,({\it{x\/}}, {\rm{del}}\,({\it{z\/}}, {\it{y\/}})\index{del})\index{memb})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-list-exis \\ \index{covering-list-exis}
\=\+({\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} $\wedge$ ({\it{var\/}} $\in$ {\it{vars\/}}) $\wedge$ ({\it{var\/}} $\in\;${\bf{N}}${}$)) \\ 
$\rightarrow$$\;\;\;\;$({\rm{covering}}\,({\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\bf{nil}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  free-for-list-step1 \\ \index{free-for-list-step1}
\=\+(\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{bvars\/}}, {\rm{length}}\,({\it{bvars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\it{vars\/}}, {\it{bvars\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{free-for-list}}\,({\rm{f-implies}}\,({\it{a\/}}, {\rm{list-exis-quantify}}\,({\it{bvars\/}}, {\it{a\/}})\index{list-exis-quantify})\index{f-implies}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list}\-
\end{tabbing}
\begin{verbatim};;;;;;;The first substitution theorem:
;;;;;;; "[t1/x1, t2/x2,...,tn/xn]A -> (forsome x1 (...(forsome xn A)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  subst-thm-a \\ \index{subst-thm-a}
\=\+(\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{terms\/}})\index{length} $=$ {\rm{length}}\,({\it{vars\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{parallel-subst}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-exis-quantify})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{subst-thm-a-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{subst-thm-a-proof}, {\it{a-prime\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable subst-thm-a-proof; name this event `g0356'.\index{g0356}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-all-quantify} \\  
\=\+{\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}})$\;\;${\bf then }$\forall\;${\it{car\/}}, {\it{vars\/}}$\;${\rm{list-all-quantify}}\,({\rm{cdr}}\,({\it{vars\/}}), {\it{a\/}})\index{list-all-quantify}\index{forall} \\ 
{\bf else }{\it{a\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-list-all-quant \\ \index{parallel-subst-list-all-quant}
\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{parallel-subst}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify})\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-all-elim-proof} \\  
\=\+{\rm{list-all-elim-proof}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-elim-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{cdr}}\,({\it{vars\/}}) $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\rm{all-elim-proof}}\,({\rm{car}}\,({\it{vars\/}}), {\it{a\/}})\index{all-elim-proof} \\ 
{\bf else }{\rm{trans-implies-proof}}\,(\=\+{\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, \\ 
{\rm{list-all-quantify}}\,({\rm{cdr}}\,({\it{vars\/}}), {\it{a\/}})\index{list-all-quantify}, \\ 
{\it{a\/}}, \\ 
{\rm{all-elim-proof}}\,(\=\+{\rm{car}}\,({\it{vars\/}}), \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{cdr}}\,({\it{vars\/}}), \\ 
{\it{a\/}})\-\index{list-all-quantify})\-\index{all-elim-proof}, \\ 
{\rm{list-all-elim-proof}}\,({\rm{cdr}}\,({\it{vars\/}}), {\it{a\/}})\index{list-all-elim-proof})\-\index{trans-implies-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{prop-axiom-proof}}\,({\it{a\/}})\index{prop-axiom-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-list-all-quant \\ \index{formula-list-all-quant}
\=\+({\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  all-elim-proof-proves-again \\ \index{all-elim-proof-proves-again}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ ({\it{a-prime\/}} $=$ {\rm{f-implies}}\,($\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall}, {\it{a\/}})\index{f-implies})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{all-elim-proof}}\,({\it{var\/}}, {\it{a\/}})\index{all-elim-proof}, {\it{a-prime\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-all-elim-proof-proves1 \\ \index{list-all-elim-proof-proves1}
\=\+({\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{list-all-elim-proof}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-elim-proof}, \\ 
{\rm{f-implies}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, {\it{a\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-all-elim-proof; name this event `g0357'.\index{g0357}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  not-free-list-all \\ \index{not-free-list-all}
\=\+({\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} $\wedge$ ({\it{var\/}} $\in$ {\it{vars\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, {\tt{0}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-list-all \\ \index{covering-list-all}
\=\+({\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} $\wedge$ ({\it{var\/}} $\in$ {\it{vars\/}}) $\wedge$ ({\it{var\/}} $\in\;${\bf{N}}${}$)) \\ 
$\rightarrow$$\;\;\;\;$({\rm{covering}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\bf{nil}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  free-for-list-step2 \\ \index{free-for-list-step2}
\=\+(\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{bvars\/}}, {\rm{length}}\,({\it{bvars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\it{vars\/}}, {\it{bvars\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{free-for-list}}\,({\rm{f-implies}}\,({\rm{list-all-quantify}}\,({\it{bvars\/}}, {\it{a\/}})\index{list-all-quantify}, {\it{a\/}})\index{f-implies}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{subst-thm-b-proof} \\  
\=\+{\rm{subst-thm-b-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{subst-thm-b-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{f-implies}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, {\it{a\/}})\index{f-implies}, \\ 
{\it{vars\/}}, \\ 
{\it{terms\/}}, \\ 
{\rm{list-all-elim-proof}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-elim-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-all-elim-proof-proves \\ \index{list-all-elim-proof-proves}
\=\+(\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{f-implies}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, {\it{a\/}})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{list-all-elim-proof}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-elim-proof}, {\it{a-prime\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;;The second substitution theorem:  
;;;;;;;    "(forall x1 (...(forall xn A))) -> [t1/x1, ..., tn/xn]A".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  subst-thm-b \\ \index{subst-thm-b}
\=\+(\=\+{\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{terms\/}})\index{length} $=$ {\rm{length}}\,({\it{vars\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{a\/}})\index{list-all-quantify}, \\ 
{\rm{parallel-subst}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{subst-thm-b-proof}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}})\index{subst-thm-b-proof}, {\it{a-prime\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable subst-thm-b-proof; name this event `g0358'.\index{g0358}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{distrib-rule-proof-a} \\  
\=\+{\rm{distrib-rule-proof-a}}\,({\it{var\/}}, {\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{distrib-rule-proof-a} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{var\/}}, \\ 
{\it{a\/}}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{b\/}})\index{forsome}, \\ 
{\rm{trans-implies-proof}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{b\/}})\index{forsome}, \\ 
{\it{pf1\/}}, \\ 
{\rm{e-intro-proof}}\,({\it{var\/}}, {\it{b\/}})\index{e-intro-proof})\-\index{trans-implies-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The first distribution rule:  Derive "(forsome x A) -> (forsome x B)" from "A -> B".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  distrib-rule-proof-a-proves \\ \index{distrib-rule-proof-a-proves}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ {\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{distrib-rule-proof-a}}\,({\it{var\/}}, {\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{distrib-rule-proof-a}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{var\/}}, {\it{b\/}})\index{forsome})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable distrib-rule-proof-a; name this event `g0360'.\index{g0360}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{distrib-rule-proof-b} \\  
\=\+{\rm{distrib-rule-proof-b}}\,({\it{var\/}}, {\it{a\/}}, {\it{b\/}}, {\it{pf1\/}})\index{distrib-rule-proof-b} \\ 
$=$$\;\;\;\;${\rm{forall-intro-proof}}\,(\=\+$\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall}, \\ 
{\it{b\/}}, \\ 
{\it{var\/}}, \\ 
{\rm{trans-implies-proof}}\,(\=\+$\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall}, \\ 
{\it{a\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{all-elim-proof}}\,({\it{var\/}}, {\it{a\/}})\index{all-elim-proof}, \\ 
{\it{pf1\/}})\-\index{trans-implies-proof})\-\index{forall-intro-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The second distribution rule:  Derive "(forall x A) -> (forall x B)" from "A -> B".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  distrib-rule-proof-b-proves \\ \index{distrib-rule-proof-b-proves}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{a-prime\/}} $=$ {\rm{f-implies}}\,($\forall\;${\it{var\/}}$\;${\it{a\/}}\index{forall}, $\forall\;${\it{var\/}}$\;${\it{b\/}}\index{forall})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{distrib-rule-proof-b}}\,({\it{var\/}}, {\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{distrib-rule-proof-b}, {\it{a-prime\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable distrib-rule-proof-b; name this event `g0361'.\index{g0361}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-forall \\ \index{formula-forall}
{\rm{formula}}\,($\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall}, {\it{symbols\/}})\index{formula} $=$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-forsome \\ \index{collect-free-forsome}
\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\rightarrow$$\;\;\;\;$({\rm{collect-free}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{exp\/}})\index{forsome}, {\tt{0}})\index{collect-free} $=$ {\rm{del}}\,({\it{var\/}}, {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})\index{del})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-forall \\ \index{collect-free-forall}
\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\rightarrow$$\;\;\;\;$({\rm{collect-free}}\,($\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall}, {\tt{0}})\index{collect-free} $=$ {\rm{del}}\,({\it{var\/}}, {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})\index{del})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-forall \\ \index{covering-forall}
\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{covering}}\,($\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall}, {\it{x\/}}, {\tt{0}})\index{covering} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{memb}}\,({\it{x\/}}, {\rm{collect-free}}\,($\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall}, {\tt{0}})\index{collect-free})\index{memb} \\ 
{\bf then }{\rm{cons}}\,({\it{var\/}}, {\rm{covering}}\,({\it{exp\/}}, {\it{x\/}}, {\tt{0}})\index{covering}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-forall \\ \index{parallel-subst-forall}
\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{parallel-subst}}\,($\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;$$\forall\;${\it{var\/}}$\;${\rm{parallel-subst}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{del}}\,({\it{var\/}}, {\it{vars\/}})\index{del}, \\ 
{\rm{del-pairs}}\,({\it{var\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs}, \\ 
{\tt{0}})\-\index{parallel-subst}\index{forall})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-forall \\ \index{subst-forall}
\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{subst}}\,($\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall}, {\it{x\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{var\/}} $=$ {\it{x\/}}$\;\;${\bf then }$\forall\;${\it{var\/}}$\;${\it{exp\/}}\index{forall} \\ 
{\bf else }$\forall\;${\it{var\/}}$\;${\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{term\/}}, {\tt{0}})\index{subst}\index{forall}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable forall; name this event `g0363'.\index{g0363}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-f-iff} \\  
\=\+{\rm{list-f-iff}}\,({\it{lhs\/}}, {\it{rhs\/}})\index{list-f-iff} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{lhs\/}}) $\wedge$ {\rm{listp}}\,({\it{rhs\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{f-iff}}\,({\rm{car}}\,({\it{lhs\/}}), {\rm{car}}\,({\it{rhs\/}}))\index{f-iff}, {\rm{list-f-iff}}\,({\rm{cdr}}\,({\it{lhs\/}}), {\rm{cdr}}\,({\it{rhs\/}}))\index{list-f-iff}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{f-iff-distrib-rule-a-proof} \\  
\=\+{\rm{f-iff-distrib-rule-a-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{a-prime\/}}, {\it{pf\/}})\index{f-iff-distrib-rule-a-proof} \\ 
$=$$\;\;\;\;${\rm{iff-join-proof}}\,(\=\+{\rm{forsome}}\,({\it{var\/}}, {\it{a\/}})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{a-prime\/}})\index{forsome}, \\ 
{\rm{distrib-rule-proof-a}}\,(\=\+{\it{var\/}}, \\ 
{\it{a\/}}, \\ 
{\it{a-prime\/}}, \\ 
{\rm{iff-split-proof1}}\,({\it{a\/}}, {\it{a-prime\/}}, {\it{pf\/}})\index{iff-split-proof1})\-\index{distrib-rule-proof-a}, \\ 
{\rm{distrib-rule-proof-a}}\,(\=\+{\it{var\/}}, \\ 
{\it{a-prime\/}}, \\ 
{\it{a\/}}, \\ 
{\rm{iff-split-proof2}}\,({\it{a\/}}, {\it{a-prime\/}}, {\it{pf\/}})\index{iff-split-proof2})\-\index{distrib-rule-proof-a})\-\index{iff-join-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  f-iff-distrib-rule-a-proof-proves \\ \index{f-iff-distrib-rule-a-proof-prove!s}
\=\+({\rm{proves}}\,({\it{pf\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{a-prime\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} $\wedge$ {\rm{variable}}\,({\it{var\/}})\index{variable}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{f-iff-distrib-rule-a-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{a-prime\/}}, {\it{pf\/}})\index{f-iff-distrib-rule-a-proof}, \\ 
{\rm{f-iff}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{var\/}}, {\it{a-prime\/}})\index{forsome})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{not-iff-proof} \\  
\=\+{\rm{not-iff-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{not-iff-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}), {\rm{f-iff}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\rm{f-not}}\,({\it{b\/}})\index{f-not})\index{f-iff}, {\rm{list}}\,({\it{pf\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-iff-proof-proves \\ \index{not-iff-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{not-iff-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{not-iff-proof}, \\ 
{\rm{f-iff}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\rm{f-not}}\,({\it{b\/}})\index{f-not})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable not-iff-proof; name this event `g0317'.\index{g0317}

\addvspace{18pt}

\noindent{\sc Event}:   Enable f-iff-distrib-rule-a-proof; name this event `g0318'.\index{g0318}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{or-iff-proof} \\  
\=\+{\rm{or-iff-proof}}\,({\it{a1\/}}, {\it{a2\/}}, {\it{b1\/}}, {\it{b2\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{or-iff-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-iff}}\,({\it{a1\/}}, {\it{b1\/}})\index{f-iff}, {\rm{f-iff}}\,({\it{a2\/}}, {\it{b2\/}})\index{f-iff}), \\ 
{\rm{f-iff}}\,({\rm{f-or}}\,({\it{a1\/}}, {\it{a2\/}})\index{f-or}, {\rm{f-or}}\,({\it{b1\/}}, {\it{b2\/}})\index{f-or})\index{f-iff}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  or-iff-proof-proves \\ \index{or-iff-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-iff}}\,({\it{a1\/}}, {\it{b1\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-iff}}\,({\it{a2\/}}, {\it{b2\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{or-iff-proof}}\,({\it{a1\/}}, {\it{a2\/}}, {\it{b1\/}}, {\it{b2\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{or-iff-proof}, \\ 
{\rm{f-iff}}\,({\rm{f-or}}\,({\it{a1\/}}, {\it{a2\/}})\index{f-or}, {\rm{f-or}}\,({\it{b1\/}}, {\it{b2\/}})\index{f-or})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable or-iff-proof; name this event `g0319'.\index{g0319}

\addvspace{18pt}
\begin{verbatim};;;;;;;EQN-FORM checks if A and A-PRIME are identical modulo the equivalences
;;;;;;;given in the list GIVEN.  It is used to prove the equivalence theorem.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{eqn-form} \\  
\=\+{\rm{eqn-form}}\,({\it{a-prime\/}}, {\it{a\/}}, {\it{given\/}})\index{eqn-form} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{a-prime\/}} $=$ {\it{a\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{f-iff}}\,({\it{a\/}}, {\it{a-prime\/}})\index{f-iff} $\in$ {\it{given\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{a\/}})\index{f-notp} $\wedge$ {\rm{f-notp}}\,({\it{a-prime\/}})\index{f-notp} \\ 
{\bf then }{\rm{eqn-form}}\,({\rm{arg}}\,({\it{a-prime\/}})\index{arg}, {\rm{arg}}\,({\it{a\/}})\index{arg}, {\it{given\/}})\index{eqn-form} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{a\/}})\index{f-orp} $\wedge$ {\rm{f-orp}}\,({\it{a-prime\/}})\index{f-orp} \\ 
{\bf then }\=\+{\rm{eqn-form}}\,({\rm{arg1}}\,({\it{a-prime\/}})\index{arg1}, {\rm{arg1}}\,({\it{a\/}})\index{arg1}, {\it{given\/}})\index{eqn-form} \\ 
$\wedge$$\;\;\;\;${\rm{eqn-form}}\,({\rm{arg2}}\,({\it{a-prime\/}})\index{arg2}, {\rm{arg2}}\,({\it{a\/}})\index{arg2}, {\it{given\/}})\index{eqn-form}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{a\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{a-prime\/}})\index{forsomep} \\ 
{\bf then }\=\+({\rm{bind}}\,({\it{a\/}})\index{bind} $=$ {\rm{bind}}\,({\it{a-prime\/}})\index{bind}) \\ 
$\wedge$$\;\;\;\;${\rm{eqn-form}}\,({\rm{body}}\,({\it{a-prime\/}}), {\rm{body}}\,({\it{a\/}}), {\it{given\/}})\index{eqn-form}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{getpf} \\  
\=\+{\rm{getpf}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{getpf} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) $\wedge$ {\rm{listp}}\,({\it{z\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $=$ {\rm{car}}\,({\it{y\/}})$\;\;${\bf then }{\rm{car}}\,({\it{z\/}}) \\ 
{\bf else }{\rm{getpf}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{y\/}}), {\rm{cdr}}\,({\it{z\/}}))\index{getpf}$\;${\bf  endif}\- \\ 
{\bf else }{\it{x\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  getpf-proves \\ \index{getpf-proves}
\=\+({\rm{proves-list}}\,({\it{pflist\/}}, {\it{list\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} $\wedge$ ({\it{concl\/}} $\in$ {\it{list\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{getpf}}\,({\it{concl\/}}, {\it{list\/}}, {\it{pflist\/}})\index{getpf}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Disable proves; name this event `g0596'.\index{g0596}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  eql-trans2 \\ \index{eql-trans2}
({\rm{eql}}\,({\it{x\/}}, {\it{z\/}})\index{eql} $\wedge$ {\rm{eql}}\,({\it{y\/}}, {\it{z\/}})\index{eql}) $\rightarrow$ {\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eql-trans3 \\ \index{eql-trans3}
({\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql} $\wedge$ {\rm{eql}}\,({\it{x\/}}, {\it{z\/}})\index{eql}) $\rightarrow$ {\rm{eql}}\,({\it{y\/}}, {\it{z\/}})\index{eql}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  proves-eql \\ \index{proves-eql}
\=\+(\=\+{\rm{proves}}\,({\it{pf\/}}, {\it{conc1\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{eql}}\,({\it{conc1\/}}, {\it{conc2\/}})\index{eql} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{conc2\/}}, {\it{symbols\/}})\index{formula})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\it{conc2\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable proves; name this event `g0598'.\index{g0598}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  getsub-proves \\ \index{getsub-proves}
\=\+(\=\+{\rm{proves-list}}\,({\it{pflist\/}}, {\it{list\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\wedge$$\;\;\;\;${\rm{memb}}\,({\it{concl\/}}, {\it{list\/}})\index{memb} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{concl\/}}, {\it{symbols\/}})\index{formula})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{getsub}}\,({\it{concl\/}}, {\it{list\/}}, {\it{pflist\/}})\index{getsub}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable proves-eql; name this event `g0600'.\index{g0600}

\addvspace{18pt}
\begin{verbatim};;;;;;;EQN-FORM-PROOF is the proof-constructor which constructs the proof of the
;;;;;;;equivalence of A and A-PRIME from the proofs of the equivalences in GIVEN.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{eqn-form-proof} \\  
\=\+{\rm{eqn-form-proof}}\,({\it{a-prime\/}}, {\it{a\/}}, {\it{given\/}}, {\it{pflist\/}})\index{eqn-form-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{a-prime\/}} $=$ {\it{a\/}}$\;\;${\bf then }{\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-iff}}\,({\it{a\/}}, {\it{a\/}})\index{f-iff}, {\bf{nil}})\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{f-iff}}\,({\it{a\/}}, {\it{a-prime\/}})\index{f-iff} $\in$ {\it{given\/}} \\ 
{\bf then }{\rm{getpf}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{a-prime\/}})\index{f-iff}, {\it{given\/}}, {\it{pflist\/}})\index{getpf} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{a\/}})\index{f-notp} $\wedge$ {\rm{f-notp}}\,({\it{a-prime\/}})\index{f-notp} \\ 
{\bf then }{\rm{not-iff-proof}}\,(\=\+{\rm{arg}}\,({\it{a\/}})\index{arg}, \\ 
{\rm{arg}}\,({\it{a-prime\/}})\index{arg}, \\ 
{\rm{eqn-form-proof}}\,({\rm{arg}}\,({\it{a-prime\/}})\index{arg}, {\rm{arg}}\,({\it{a\/}})\index{arg}, {\it{given\/}}, {\it{pflist\/}})\index{eqn-form-proof})\-\index{not-iff-proof} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{a\/}})\index{f-orp} $\wedge$ {\rm{f-orp}}\,({\it{a-prime\/}})\index{f-orp} \\ 
{\bf then }{\rm{or-iff-proof}}\,(\=\+{\rm{arg1}}\,({\it{a\/}})\index{arg1}, \\ 
{\rm{arg2}}\,({\it{a\/}})\index{arg2}, \\ 
{\rm{arg1}}\,({\it{a-prime\/}})\index{arg1}, \\ 
{\rm{arg2}}\,({\it{a-prime\/}})\index{arg2}, \\ 
{\rm{eqn-form-proof}}\,({\rm{arg1}}\,({\it{a-prime\/}})\index{arg1}, {\rm{arg1}}\,({\it{a\/}})\index{arg1}, {\it{given\/}}, {\it{pflist\/}})\index{eqn-form-proof}, \\ 
{\rm{eqn-form-proof}}\,({\rm{arg2}}\,({\it{a-prime\/}})\index{arg2}, {\rm{arg2}}\,({\it{a\/}})\index{arg2}, {\it{given\/}}, {\it{pflist\/}})\index{eqn-form-proof})\-\index{or-iff-proof} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{a\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{a-prime\/}})\index{forsomep} \\ 
{\bf then }{\rm{f-iff-distrib-rule-a-proof}}\,(\=\+{\rm{bind}}\,({\it{a\/}})\index{bind}, \\ 
{\rm{body}}\,({\it{a\/}}), \\ 
{\rm{body}}\,({\it{a-prime\/}}), \\ 
{\rm{eqn-form-proof}}\,(\=\+{\rm{body}}\,({\it{a-prime\/}}), \\ 
{\rm{body}}\,({\it{a\/}}), \\ 
{\it{given\/}}, \\ 
{\it{pflist\/}})\-\index{eqn-form-proof})\-\index{f-iff-distrib-rule-a-proof} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The equivalence theorem:  If A and A-PRIME are identical modulo the
;;;;;;;proven equivalences in GIVEN (whose proofs are in PFLIST), then
;;;;;;; "A <-> A-PRIME" is provable.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  eqn-form-proof-proves1 \\ \index{eqn-form-proof-proves1}
\=\+(\=\+{\rm{formula}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{a-prime\/}})\index{f-iff}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{eqn-form}}\,({\it{a-prime\/}}, {\it{a\/}}, {\it{list\/}})\index{eqn-form} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\it{list\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{eqn-form-proof}}\,({\it{a-prime\/}}, {\it{a\/}}, {\it{list\/}}, {\it{pflist\/}})\index{eqn-form-proof}, \\ 
{\rm{f-iff}}\,({\it{a\/}}, {\it{a-prime\/}})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable eqn-form-proof; name this event `g0338'.\index{g0338}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{elim-link-proof} \\  
\=\+{\rm{elim-link-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{elim-link-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{y\/}}, {\it{z\/}})\index{f-implies})\index{f-implies}, {\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\it{z\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  elim-link-proof-proves \\ \index{elim-link-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{y\/}}, {\it{z\/}})\index{f-implies})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{elim-link-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{elim-link-proof}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\it{z\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable elim-link-proof; name this event `g0499'.\index{g0499}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{elim-args-proof} \\  
\=\+{\rm{elim-args-proof}}\,({\it{hyp\/}}, {\it{args\/}}, {\it{exp\/}}, {\it{pflist\/}}, {\it{pf\/}})\index{elim-args-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{args\/}}) \\ 
{\bf then }{\rm{elim-args-proof}}\,(\=\+{\it{hyp\/}}, \\ 
{\rm{cdr}}\,({\it{args\/}}), \\ 
{\it{exp\/}}, \\ 
{\rm{cdr}}\,({\it{pflist\/}}), \\ 
{\rm{elim-link-proof}}\,(\=\+{\it{hyp\/}}, \\ 
{\rm{car}}\,({\it{args\/}}), \\ 
{\rm{list-implies}}\,({\rm{cdr}}\,({\it{args\/}}), {\it{exp\/}})\index{list-implies}, \\ 
{\it{pf\/}}, \\ 
{\rm{car}}\,({\it{pflist\/}}))\-\index{elim-link-proof})\-\index{elim-args-proof} \\ 
{\bf else }{\it{pf\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{dist-f-implies} \\  
\=\+{\rm{dist-f-implies}}\,({\it{hyp\/}}, {\it{list\/}})\index{dist-f-implies} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{f-implies}}\,({\it{hyp\/}}, {\rm{car}}\,({\it{list\/}}))\index{f-implies}, {\rm{dist-f-implies}}\,({\it{hyp\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{dist-f-implies}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-implies-reduc; name this event `g0506'.\index{g0506}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  elim-args-proof-proves \\ \index{elim-args-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\rm{dist-f-implies}}\,({\it{hyp\/}}, {\it{args\/}})\index{dist-f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{hyp\/}}, {\rm{list-implies}}\,({\it{args\/}}, {\it{exp\/}})\index{list-implies})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{elim-args-proof}}\,({\it{hyp\/}}, {\it{args\/}}, {\it{exp\/}}, {\it{pflist\/}}, {\it{pf\/}})\index{elim-args-proof}, \\ 
{\rm{f-implies}}\,({\it{hyp\/}}, {\it{exp\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable elim-args-proof; name this event `g0507'.\index{g0507}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{symmetry-proof1} \\  
\=\+{\rm{symmetry-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{symmetry-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\it{x\/}})\index{ident-axiom-proof}, \\ 
{\rm{equal-axiom2-proof}}\,({\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}, {\rm{list}}\,({\it{x\/}}, {\it{x\/}}), {\rm{list}}\,({\it{y\/}}, {\it{x\/}}))\index{equal-axiom2-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Disable f-equal; name this event `g0365'.\index{g0365}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  equal-equal-axiom2 \\ \index{equal-equal-axiom2}
\=\+{\rm{formula}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{u\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal})\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{symbols\/}})\-\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-axiom2-proof}}\,({\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}, {\rm{list}}\,({\it{x\/}}, {\it{u\/}}), {\rm{list}}\,({\it{y\/}}, {\it{v\/}}))\index{equal-axiom2-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{u\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal})\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-equal; name this event `g0368'.\index{g0368}

\addvspace{18pt}
\begin{verbatim};;;;;;;The symmetry theorem: "(x = y) -> (y = x)"
;;;;;;;The next few events lead up to the proof of the equality theorem.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  symmetry-theorem1 \\ \index{symmetry-theorem1}
\=\+{\rm{formula}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{symmetry-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{symmetry-proof1}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable symmetry-proof1; name this event `g0372'.\index{g0372}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-f-and} \\  
\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{list\/}}))$\;\;${\bf then }{\rm{f-and}}\,({\rm{car}}\,({\it{list\/}}), {\rm{list-f-and}}\,({\rm{cdr}}\,({\it{list\/}}))\index{list-f-and})\index{f-and} \\ 
{\bf else }{\rm{car}}\,({\it{list\/}})$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-list-f-and \\ \index{formula-list-f-and}
\=\+({\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} $\wedge$ {\rm{listp}}\,({\it{list\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-f-and-member-proof} \\  
\=\+{\rm{list-f-and-member-proof}}\,({\it{x\/}}, {\it{list\/}})\index{list-f-and-member-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{list\/}})) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $=$ {\rm{car}}\,({\it{list\/}}) \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\it{x\/}}, \\ 
{\rm{list-f-and}}\,({\rm{cdr}}\,({\it{list\/}}))\index{list-f-and})\-\index{f-and}, \\ 
{\it{x\/}})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{list-f-and}}\,({\rm{cdr}}\,({\it{list\/}}))\index{list-f-and}, \\ 
{\it{x\/}})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\it{x\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\rm{list-f-and-member-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{cdr}}\,({\it{list\/}}))\-\index{list-f-and-member-proof}))\-\index{tautconseq-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-implies}}\,({\it{x\/}}, {\it{x\/}})\index{f-implies}, {\bf{nil}})\index{tautconseq-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-f-and-member-proof-proves \\ \index{list-f-and-member-proof-proves}
\=\+({\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} $\wedge$ ({\it{x\/}} $\in$ {\it{list\/}}) $\wedge$ {\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{list-f-and-member-proof}}\,({\it{x\/}}, {\it{list\/}})\index{list-f-and-member-proof}, \\ 
{\rm{f-implies}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\it{x\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-f-and; name this event `g0508'.\index{g0508}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{add-hyp-proof} \\  
\=\+{\rm{add-hyp-proof}}\,({\it{hyp\/}}, {\it{exp\/}}, {\it{pf\/}})\index{add-hyp-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\it{exp\/}}), {\rm{f-implies}}\,({\it{hyp\/}}, {\it{exp\/}})\index{f-implies}, {\rm{list}}\,({\it{pf\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  add-hyp-proof-proves \\ \index{add-hyp-proof-proves}
\=\+({\rm{formula}}\,({\it{hyp\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{add-hyp-proof}}\,({\it{hyp\/}}, {\it{exp\/}}, {\it{pf\/}})\index{add-hyp-proof}, \\ 
{\rm{f-implies}}\,({\it{hyp\/}}, {\it{exp\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable add-hyp-proof; name this event `g0509'.\index{g0509}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-f-equal} \\  
\=\+{\rm{list-f-equal}}\,({\it{lista\/}}, {\it{listb\/}})\index{list-f-equal} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{lista\/}}) $\wedge$ {\rm{listp}}\,({\it{listb\/}}) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{f-equal}}\,({\rm{car}}\,({\it{lista\/}}), {\rm{car}}\,({\it{listb\/}}))\index{f-equal}, \\ 
{\rm{list-f-equal}}\,({\rm{cdr}}\,({\it{lista\/}}), {\rm{cdr}}\,({\it{listb\/}}))\index{list-f-equal})\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{flg-f-implies-equal} \\  
\=\+{\rm{flg-f-implies-equal}}\,({\it{hyp\/}}, {\it{terma\/}}, {\it{termb\/}}, {\it{flg\/}})\index{flg-f-implies-equal} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{f-implies}}\,({\it{hyp\/}}, {\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal})\index{f-implies} \\ 
{\bf else }{\rm{dist-f-implies}}\,({\it{hyp\/}}, {\rm{list-f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{list-f-equal})\index{dist-f-implies}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{flg-proves} \\  
\=\+{\rm{flg-proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{flg\/}})\index{flg-proves} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
{\bf else }{\rm{proves-list}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{term-list} \\  
\=\+{\rm{term-list}}\,({\it{terms\/}}, {\it{symbols\/}})\index{term-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{terms\/}}) \\ 
{\bf then }{\rm{termp}}\,({\rm{car}}\,({\it{terms\/}}), {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{term-list}}\,({\rm{cdr}}\,({\it{terms\/}}), {\it{symbols\/}})\index{term-list} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{flg-termp} \\  
\=\+{\rm{flg-termp}}\,({\it{term\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{flg-termp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
{\bf else }{\rm{term-list}}\,({\it{term\/}}, {\it{symbols\/}})\index{term-list}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-term-list \\ \index{termp-term-list}
{\rm{termp}}\,({\it{exp\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} $\rightarrow$ {\rm{term-list}}\,({\it{exp\/}}, {\it{symbols\/}})\index{term-list}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pairequals-reduc \\ \index{pairequals-reduc}
{\rm{pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}})\index{pairequals} $=$ {\rm{list-implies}}\,({\rm{list-f-equal}}\,({\it{vars1\/}}, {\it{vars2\/}})\index{list-f-equal}, {\it{exp\/}})\index{list-implies}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-list-implies1 \\ \index{formula-list-implies1}
\=\+{\rm{formula}}\,({\rm{list-implies}}\,({\it{list\/}}, {\it{exp\/}})\index{list-implies}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$({\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-list-f-equal \\ \index{form-list-f-equal}
\=\+({\rm{term-list}}\,({\it{terms1\/}}, {\it{symbols\/}})\index{term-list} $\wedge$ {\rm{term-list}}\,({\it{terms2\/}}, {\it{symbols\/}})\index{term-list}) \\ 
$\rightarrow$$\;\;\;\;${\rm{form-list}}\,({\rm{list-f-equal}}\,({\it{terms1\/}}, {\it{terms2\/}})\index{list-f-equal}, {\it{symbols\/}})\index{form-list}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{symmetry-proof} \\  
\=\+{\rm{symmetry-proof}}\,({\it{a\/}}, {\it{b\/}})\index{symmetry-proof} \\ 
$=$$\;\;\;\;${\rm{iff-join-proof}}\,(\=\+{\rm{f-equal}}\,({\it{a\/}}, {\it{b\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{b\/}}, {\it{a\/}})\index{f-equal}, \\ 
{\rm{symmetry-proof1}}\,({\it{a\/}}, {\it{b\/}})\index{symmetry-proof1}, \\ 
{\rm{symmetry-proof1}}\,({\it{b\/}}, {\it{a\/}})\index{symmetry-proof1})\-\index{iff-join-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  symmetry-theorem \\ \index{symmetry-theorem}
\=\+{\rm{formula}}\,({\rm{f-equal}}\,({\it{a\/}}, {\it{b\/}})\index{f-equal}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{symmetry-proof}}\,({\it{a\/}}, {\it{b\/}})\index{symmetry-proof}, \\ 
{\rm{f-iff}}\,({\rm{f-equal}}\,({\it{a\/}}, {\it{b\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{b\/}}, {\it{a\/}})\index{f-equal})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable symmetry-proof; name this event `g0373'.\index{g0373}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{commute-equal-proof} \\  
\=\+{\rm{commute-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pf\/}})\index{commute-equal-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\it{pf\/}}, {\rm{symmetry-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{symmetry-proof1})\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  commute-equal-proof-proves \\ \index{commute-equal-proof-proves}
\=\+({\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} $\wedge$ ({\it{concl\/}} $=$ {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{commute-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pf\/}})\index{commute-equal-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable commute-equal-proof; name this event `g0370'.\index{g0370}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-commute-equal-proof} \\  
\=\+{\rm{list-commute-equal-proof}}\,({\it{lista\/}}, {\it{listb\/}}, {\it{pflist\/}})\index{list-commute-equal-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{lista\/}}) $\wedge$ {\rm{listp}}\,({\it{listb\/}}) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{commute-equal-proof}}\,({\rm{car}}\,({\it{lista\/}}), {\rm{car}}\,({\it{listb\/}}), {\rm{car}}\,({\it{pflist\/}}))\index{commute-equal-proof}, \\ 
{\rm{list-commute-equal-proof}}\,({\rm{cdr}}\,({\it{lista\/}}), {\rm{cdr}}\,({\it{listb\/}}), {\rm{cdr}}\,({\it{pflist\/}}))\index{list-commute-equal-proof})\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-commute-equal-proof-proves \\ \index{list-commute-equal-proof-proves}
\=\+{\rm{proves-list}}\,({\it{pflist\/}}, {\rm{list-f-equal}}\,({\it{lista\/}}, {\it{listb\/}})\index{list-f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves-list}}\,(\=\+{\rm{list-commute-equal-proof}}\,({\it{lista\/}}, {\it{listb\/}}, {\it{pflist\/}})\index{list-commute-equal-proof}, \\ 
{\rm{list-f-equal}}\,({\it{listb\/}}, {\it{lista\/}})\index{list-f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-list}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-commute-equal-proof; name this event `g0371'.\index{g0371}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{forall-over-implies-proof} \\  
\=\+{\rm{forall-over-implies-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forall-over-implies-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}\index{forall}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\-\index{f-implies})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}\index{forall}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{forsome-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{a\/}}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}\index{forall}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\-\index{f-implies}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-implies}\index{forall}, \\ 
{\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-implies}\index{forall}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{all-elim-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-implies})\-\index{all-elim-proof}, \\ 
{\rm{e-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{e-intro-proof})\-)\-\index{tautconseq-proof})\-\index{forsome-intro-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  forall-over-implies-proof-proves \\ \index{forall-over-implies-proof-prove!s}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forall-over-implies-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forall-over-implies-proof}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}\index{forall}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forall-over-implies-proof; name this event `g0910'.\index{g0910}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{forall-over-iff-proof} \\  
\=\+{\rm{forall-over-iff-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forall-over-iff-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, \\ 
{\rm{f-implies}}\,($\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}\index{forall}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome}, \\ 
{\rm{f-implies}}\,($\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}\index{forall}, {\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome})\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}\index{forall}, \\ 
{\rm{f-iff}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-iff})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{forsome-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{a\/}}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}\index{forall}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\-\index{f-implies}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff}\index{forall}, \\ 
{\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff}\index{forall}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{all-elim-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff})\-\index{all-elim-proof}, \\ 
{\rm{e-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{e-intro-proof})\-)\-\index{tautconseq-proof})\-\index{forsome-intro-proof}, \\ 
{\rm{forsome-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}\index{forall}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome})\-\index{f-implies}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff}\index{forall}, \\ 
{\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{a\/}})\-\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff}\index{forall}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{a\/}})\-\index{forsome})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{all-elim-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-iff}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-iff})\-\index{all-elim-proof}, \\ 
{\rm{e-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{a\/}})\-\index{e-intro-proof})\-)\-\index{tautconseq-proof})\-\index{forsome-intro-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  forall-over-iff-proof-proves \\ \index{forall-over-iff-proof-proves}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forall-over-iff-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forall-over-iff-proof}, \\ 
{\rm{f-implies}}\,($\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}\index{forall}, {\rm{f-iff}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-iff})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forall-over-iff-proof; name this event `g0911'.\index{g0911}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{elim-free-terms} \\  
\=\+{\rm{elim-free-terms}}\,({\it{x\/}}, {\it{list\/}})\index{elim-free-terms} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $\in$ {\rm{collect-free}}\,({\rm{car}}\,({\it{list\/}}), {\tt{0}})\index{collect-free} \\ 
{\bf then }{\rm{elim-free-terms}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{elim-free-terms} \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{list\/}}), {\rm{elim-free-terms}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{elim-free-terms})$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subset-elim-free-terms \\ \index{subset-elim-free-terms}
{\rm{subset}}\,({\rm{elim-free-terms}}\,({\it{x\/}}, {\it{list\/}})\index{elim-free-terms}, {\it{list\/}})\index{subset}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-elim-free-terms \\ \index{not-free-elim-free-terms}
{\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\rm{elim-free-terms}}\,({\it{x\/}}, {\it{list\/}})\index{elim-free-terms}, {\tt{1}})\index{collect-free}
\end{tabbing}

\noindent{\sc Event}:   Disable list-f-and; name this event `g0912'.\index{g0912}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  collect-free-list-f-and \\ \index{collect-free-list-f-and}
({\it{x\/}} $\in$ {\rm{collect-free}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\tt{0}})\index{collect-free}) $=$ ({\it{x\/}} $\in$ {\rm{collect-free}}\,({\it{list\/}}, {\tt{1}})\index{collect-free})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-f-and \\ \index{formula-f-and}
{\rm{formula}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, {\it{symbols\/}})\index{formula} $=$ ({\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-f-and \\ \index{covering-f-and}
{\rm{covering}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{append}}\,({\rm{covering}}\,({\it{x\/}}, {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\it{y\/}}, {\it{var\/}}, {\tt{0}})\index{covering})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-f-and \\ \index{collect-free-f-and}
{\rm{collect-free}}\,({\rm{f-and}}\,({\it{y\/}}, {\it{z\/}})\index{f-and}, {\tt{0}})\index{collect-free} $=$ {\rm{append}}\,({\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free}, {\rm{collect-free}}\,({\it{z\/}}, {\tt{0}})\index{collect-free})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-f-and \\ \index{subst-f-and}
\=\+{\rm{subst}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{f-and}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-and}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-f-and \\ \index{parallel-subst-f-and}
\=\+{\rm{parallel-subst}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{f-and}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\rm{parallel-subst}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{f-and}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-f-and \\ \index{eval-f-and}
{\rm{eval}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, {\it{alist\/}})\index{eval} $=$ ({\rm{eval}}\,({\it{x\/}}, {\it{alist\/}})\index{eval} $\wedge$ {\rm{eval}}\,({\it{y\/}}, {\it{alist\/}})\index{eval})
\end{tabbing}

\noindent{\sc Event}:   Enable f-and; name this event `g0384'.\index{g0384}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{subset-list-f-and-proof} \\  
\=\+{\rm{subset-list-f-and-proof}}\,({\it{list1\/}}, {\it{list2\/}})\index{subset-list-f-and-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list2\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{list2\/}})) \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{list-f-and}}\,({\it{list1\/}})\index{list-f-and}, \\ 
{\rm{car}}\,({\it{list2\/}}))\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{list-f-and}}\,({\it{list1\/}})\index{list-f-and}, \\ 
{\rm{list-f-and}}\,({\rm{cdr}}\,({\it{list2\/}}))\index{list-f-and})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{list-f-and}}\,({\it{list1\/}})\index{list-f-and}, \\ 
{\rm{list-f-and}}\,({\it{list2\/}})\index{list-f-and})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{list-f-and-member-proof}}\,(\=\+{\rm{car}}\,({\it{list2\/}}), \\ 
{\it{list1\/}})\-\index{list-f-and-member-proof}, \\ 
{\rm{subset-list-f-and-proof}}\,(\=\+{\it{list1\/}}, \\ 
{\rm{cdr}}\,({\it{list2\/}}))\-\index{subset-list-f-and-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf else }{\rm{list-f-and-member-proof}}\,({\rm{car}}\,({\it{list2\/}}), {\it{list1\/}})\index{list-f-and-member-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subset-list-f-and-proof-proves \\ \index{subset-list-f-and-proof-proves}
\=\+(\=\+{\rm{form-list}}\,({\it{list1\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{list2\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list2\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\it{list2\/}}, {\it{list1\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{subset-list-f-and-proof}}\,({\it{list1\/}}, {\it{list2\/}})\index{subset-list-f-and-proof}, \\ 
{\rm{f-implies}}\,({\rm{list-f-and}}\,({\it{list1\/}})\index{list-f-and}, {\rm{list-f-and}}\,({\it{list2\/}})\index{list-f-and})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-f-implies} \\  
\=\+{\rm{list-f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{list-f-implies} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) $\wedge$ {\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{f-implies}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{f-implies}, {\rm{list-f-implies}}\,({\rm{cdr}}\,({\it{x\/}}), {\rm{cdr}}\,({\it{y\/}}))\index{list-f-implies}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{implies-implies-proof} \\  
\=\+{\rm{implies-implies-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{implies-implies-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{u\/}}, {\it{x\/}})\index{f-implies}, {\rm{f-implies}}\,({\it{y\/}}, {\it{v\/}})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\rm{f-implies}}\,({\it{u\/}}, {\it{v\/}})\index{f-implies})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  implies-implies-proof-proves \\ \index{implies-implies-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{u\/}}, {\it{x\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{y\/}}, {\it{v\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{implies-implies-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{implies-implies-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\rm{f-implies}}\,({\it{u\/}}, {\it{v\/}})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable implies-implies-proof; name this event `g0876'.\index{g0876}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-implies-implies-proof} \\  
\=\+{\rm{list-implies-implies-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pflist\/}})\index{list-implies-implies-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) $\wedge$ {\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }{\rm{implies-implies-proof}}\,(\=\+{\rm{car}}\,({\it{y\/}}), \\ 
{\rm{list-implies}}\,({\rm{cdr}}\,({\it{y\/}}), {\it{z\/}})\index{list-implies}, \\ 
{\rm{car}}\,({\it{x\/}}), \\ 
{\rm{list-implies}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{z\/}})\index{list-implies}, \\ 
{\rm{car}}\,({\it{pflist\/}}), \\ 
{\rm{list-implies-implies-proof}}\,(\=\+{\rm{cdr}}\,({\it{x\/}}), \\ 
{\rm{cdr}}\,({\it{y\/}}), \\ 
{\it{z\/}}, \\ 
{\rm{cdr}}\,({\it{pflist\/}}))\-\index{list-implies-implies-proof})\-\index{implies-implies-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-implies}}\,({\it{z\/}}, {\it{z\/}})\index{f-implies}, {\bf{nil}})\index{tautconseq-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-implies-implies-proof-proves \\ \index{list-implies-implies-proof-prove!s}
\=\+(\=\+({\rm{length}}\,({\it{x\/}})\index{length} $=$ {\rm{length}}\,({\it{y\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\rm{list-f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{list-f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{list-implies-implies-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pflist\/}})\index{list-implies-implies-proof}, \\ 
{\rm{f-implies}}\,({\rm{list-implies}}\,({\it{y\/}}, {\it{z\/}})\index{list-implies}, {\rm{list-implies}}\,({\it{x\/}}, {\it{z\/}})\index{list-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-implies-implies-proof; name this event `g0877'.\index{g0877}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-symmetry-proof1} \\  
\=\+{\rm{list-symmetry-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{list-symmetry-proof1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) $\wedge$ {\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{symmetry-proof1}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{symmetry-proof1}, \\ 
{\rm{list-symmetry-proof1}}\,({\rm{cdr}}\,({\it{x\/}}), {\rm{cdr}}\,({\it{y\/}}))\index{list-symmetry-proof1})\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-symmetry-proof1-proves \\ \index{list-symmetry-proof1-proves}
\=\+({\rm{term-list}}\,({\it{x\/}}, {\it{symbols\/}})\index{term-list} $\wedge$ {\rm{term-list}}\,({\it{y\/}}, {\it{symbols\/}})\index{term-list}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves-list}}\,(\=\+{\rm{list-symmetry-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{list-symmetry-proof1}, \\ 
{\rm{list-f-implies}}\,({\rm{list-f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{list-f-equal}, {\rm{list-f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{list-f-equal})\index{list-f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-list}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-symmetry-proof1; name this event `g0878'.\index{g0878}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{imp-iff-not-proof} \\  
\=\+{\rm{imp-iff-not-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf\/}})\index{imp-iff-not-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y\/}}, {\it{z\/}})\index{f-iff})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\rm{f-not}}\,({\it{y\/}})\index{f-not}, {\rm{f-not}}\,({\it{z\/}})\index{f-not})\index{f-iff})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  imp-iff-not-proof-proves \\ \index{imp-iff-not-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y\/}}, {\it{z\/}})\index{f-iff})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{imp-iff-not-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf\/}})\index{imp-iff-not-proof}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\rm{f-not}}\,({\it{y\/}})\index{f-not}, {\rm{f-not}}\,({\it{z\/}})\index{f-not})\index{f-iff})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable imp-iff-not-proof; name this event `g0879'.\index{g0879}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{imp-iff-or-proof} \\  
\=\+{\rm{imp-iff-or-proof}}\,({\it{x\/}}, {\it{y1\/}}, {\it{z1\/}}, {\it{y2\/}}, {\it{z2\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{imp-iff-or-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y1\/}}, {\it{z1\/}})\index{f-iff})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y2\/}}, {\it{z2\/}})\index{f-iff})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\rm{f-or}}\,({\it{y1\/}}, {\it{y2\/}})\index{f-or}, {\rm{f-or}}\,({\it{z1\/}}, {\it{z2\/}})\index{f-or})\index{f-iff})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  imp-iff-or-proof-proves \\ \index{imp-iff-or-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y1\/}}, {\it{z1\/}})\index{f-iff})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y2\/}}, {\it{z2\/}})\index{f-iff})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{imp-iff-or-proof}}\,({\it{x\/}}, {\it{y1\/}}, {\it{z1\/}}, {\it{y2\/}}, {\it{z2\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{imp-iff-or-proof}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\rm{f-or}}\,({\it{y1\/}}, {\it{y2\/}})\index{f-or}, {\rm{f-or}}\,({\it{z1\/}}, {\it{z2\/}})\index{f-or})\index{f-iff})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable imp-iff-or-proof; name this event `g0880'.\index{g0880}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{make-chain} \\  
\=\+{\rm{make-chain}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}})\index{make-chain} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{f-implies}}\,({\it{a\/}}, {\rm{car}}\,({\it{list\/}}))\index{f-implies}, {\rm{make-chain}}\,({\rm{car}}\,({\it{list\/}}), {\it{b\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{make-chain}) \\ 
{\bf else }{\rm{list}}\,({\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{chain-proof} \\  
\=\+{\rm{chain-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}}, {\it{pflist\/}})\index{chain-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }{\rm{trans-implies-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{car}}\,({\it{list\/}}), \\ 
{\it{b\/}}, \\ 
{\rm{car}}\,({\it{pflist\/}}), \\ 
{\rm{chain-proof}}\,(\=\+{\rm{car}}\,({\it{list\/}}), \\ 
{\it{b\/}}, \\ 
{\rm{cdr}}\,({\it{list\/}}), \\ 
{\rm{cdr}}\,({\it{pflist\/}}))\-\index{chain-proof})\-\index{trans-implies-proof} \\ 
{\bf else }{\rm{car}}\,({\it{pflist\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The chain rule below is a frequently used rule of inference.  
;;;;;;;It asserts that "A -> B" is provable if there is a chain of provable
;;;;;;;implications leading from "A" to "B".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  chain-rule \\ \index{chain-rule}
\=\+{\rm{proves-list}}\,({\it{pflist\/}}, {\rm{make-chain}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}})\index{make-chain}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{chain-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}}, {\it{pflist\/}})\index{chain-proof}, \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable chain-proof; name this event `g0396'.\index{g0396}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{make-iff-chain} \\  
\=\+{\rm{make-iff-chain}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}})\index{make-iff-chain} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }{\rm{cons}}\,({\rm{f-iff}}\,({\it{a\/}}, {\rm{car}}\,({\it{list\/}}))\index{f-iff}, {\rm{make-iff-chain}}\,({\rm{car}}\,({\it{list\/}}), {\it{b\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{make-iff-chain}) \\ 
{\bf else }{\rm{list}}\,({\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{chain-iff-proof} \\  
\=\+{\rm{chain-iff-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}}, {\it{pflist\/}})\index{chain-iff-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }{\rm{trans-f-iff-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{car}}\,({\it{list\/}}), \\ 
{\it{b\/}}, \\ 
{\rm{car}}\,({\it{pflist\/}}), \\ 
{\rm{chain-iff-proof}}\,(\=\+{\rm{car}}\,({\it{list\/}}), \\ 
{\it{b\/}}, \\ 
{\rm{cdr}}\,({\it{list\/}}), \\ 
{\rm{cdr}}\,({\it{pflist\/}}))\-\index{chain-iff-proof})\-\index{trans-f-iff-proof} \\ 
{\bf else }{\rm{car}}\,({\it{pflist\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The chain rule form "<->".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  chain-iff-proof-proves \\ \index{chain-iff-proof-proves}
\=\+{\rm{proves-list}}\,({\it{pflist\/}}, {\rm{make-iff-chain}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}})\index{make-iff-chain}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{chain-iff-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{list\/}}, {\it{pflist\/}})\index{chain-iff-proof}, \\ 
{\rm{f-iff}}\,({\it{a\/}}, {\it{b\/}})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable chain-iff-proof; name this event `g0386'.\index{g0386}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{imp-iff-forsome-proof} \\  
\=\+{\rm{imp-iff-forsome-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{var\/}}, {\it{pf\/}})\index{imp-iff-forsome-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-iff}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome}, {\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\index{f-iff}, \\ 
{\rm{list}}\,($\forall\;${\it{var\/}}$\;${\rm{f-iff}}\,({\it{y\/}}, {\it{z\/}})\index{f-iff}\index{forall}), \\ 
{\rm{list}}\,(\=\+{\rm{forall-intro-proof}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y\/}}, {\it{z\/}})\index{f-iff}, {\it{var\/}}, {\it{pf\/}})\index{forall-intro-proof}, \\ 
{\rm{forall-over-iff-proof}}\,({\it{var\/}}, {\it{y\/}}, {\it{z\/}})\index{forall-over-iff-proof})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  imp-iff-forsome-proof-proves \\ \index{imp-iff-forsome-proof-proves}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y\/}}, {\it{z\/}})\index{f-iff})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{imp-iff-forsome-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{var\/}}, {\it{pf\/}})\index{imp-iff-forsome-proof}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome}, {\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\index{f-iff})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable imp-iff-forsome-proof; name this event `g0881'.\index{g0881}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{dist-f-implies-symmetry-proof} \\  
\=\+{\rm{dist-f-implies-symmetry-proof}}\,({\it{x\/}}, {\it{list1\/}}, {\it{list2\/}}, {\it{pflist\/}})\index{dist-f-implies-symmetry-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list1\/}}) $\wedge$ {\rm{listp}}\,({\it{list2\/}}) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{chain-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-equal}}\,({\rm{car}}\,({\it{list2\/}}), {\rm{car}}\,({\it{list1\/}}))\index{f-equal}, \\ 
{\rm{list}}\,({\rm{f-equal}}\,({\rm{car}}\,({\it{list1\/}}), {\rm{car}}\,({\it{list2\/}}))\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{car}}\,({\it{pflist\/}}), \\ 
{\rm{symmetry-proof1}}\,({\rm{car}}\,({\it{list1\/}}), {\rm{car}}\,({\it{list2\/}}))\index{symmetry-proof1})\-)\-\index{chain-proof}, \\ 
{\rm{dist-f-implies-symmetry-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{cdr}}\,({\it{list1\/}}), \\ 
{\rm{cdr}}\,({\it{list2\/}}), \\ 
{\rm{cdr}}\,({\it{pflist\/}}))\-\index{dist-f-implies-symmetry-proof})\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  dist-f-implies-symmetry-proof-proves \\ \index{dist-f-implies-symmetry-proof-p!roves}
\=\+{\rm{proves-list}}\,(\=\+{\it{pflist\/}}, \\ 
{\rm{dist-f-implies}}\,({\it{x\/}}, {\rm{list-f-equal}}\,({\it{list1\/}}, {\it{list2\/}})\index{list-f-equal})\index{dist-f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-list} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves-list}}\,(\=\+{\rm{dist-f-implies-symmetry-proof}}\,({\it{x\/}}, {\it{list1\/}}, {\it{list2\/}}, {\it{pflist\/}})\index{dist-f-implies-symmetry-proof}, \\ 
{\rm{dist-f-implies}}\,({\it{x\/}}, {\rm{list-f-equal}}\,({\it{list2\/}}, {\it{list1\/}})\index{list-f-equal})\index{dist-f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-list}\-
\end{tabbing}

\noindent{\sc Event}:   Enable dist-f-implies-symmetry-proof; name this event `g0882'.\index{g0882}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{imp-iff-implies-proof} \\  
\=\+{\rm{imp-iff-implies-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{imp-iff-implies-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{y\/}}, {\it{z\/}})\index{f-implies})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{z\/}}, {\it{y\/}})\index{f-implies})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y\/}}, {\it{z\/}})\index{f-iff})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  imp-iff-implies-proof-proves \\ \index{imp-iff-implies-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{y\/}}, {\it{z\/}})\index{f-implies})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{z\/}}, {\it{y\/}})\index{f-implies})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{imp-iff-implies-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{imp-iff-implies-proof}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-iff}}\,({\it{y\/}}, {\it{z\/}})\index{f-iff})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable imp-iff-implies-proof; name this event `g0883'.\index{g0883}

\addvspace{18pt}
\begin{verbatim};;;;;;;TERM-EQUAL-PROOF is the proof-constructor for the equality theorem
;;;;;;;for terms.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{term-equal-proof} \\  
\=\+{\rm{term-equal-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\it{flg\/}})\index{term-equal-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\it{terma\/}} $=$ {\it{termb\/}} \\ 
{\bf then }{\rm{add-hyp-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{f-equal}}\,({\it{terma\/}}, {\it{terma\/}})\index{f-equal}, \\ 
{\rm{ident-axiom-proof}}\,({\it{terma\/}})\index{ident-axiom-proof})\-\index{add-hyp-proof} \\ 
{\bf elseif }{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal} $\in$ {\it{list\/}} \\ 
{\bf then }{\rm{list-f-and-member-proof}}\,({\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, {\it{list\/}})\index{list-f-and-member-proof} \\ 
{\bf elseif }{\rm{f-equal}}\,({\it{termb\/}}, {\it{terma\/}})\index{f-equal} $\in$ {\it{list\/}} \\ 
{\bf then }{\rm{chain-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{f-equal}}\,({\it{termb\/}}, {\it{terma\/}})\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{list-f-and-member-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{termb\/}}, \\ 
{\it{terma\/}})\-\index{f-equal}, \\ 
{\it{list\/}})\-\index{list-f-and-member-proof}, \\ 
{\rm{symmetry-proof1}}\,({\it{termb\/}}, {\it{terma\/}})\index{symmetry-proof1})\-)\-\index{chain-proof} \\ 
{\bf elseif }{\rm{listp}}\,({\it{terma\/}}) $\wedge$ {\rm{listp}}\,({\it{termb\/}}) \\ 
{\bf then }{\rm{elim-args-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{list-f-equal}}\,({\rm{cdr}}\,({\it{terma\/}}), {\rm{cdr}}\,({\it{termb\/}}))\index{list-f-equal}, \\ 
{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{term-equal-proof}}\,(\=\+{\rm{cdr}}\,({\it{terma\/}}), \\ 
{\rm{cdr}}\,({\it{termb\/}}), \\ 
{\it{list\/}}, \\ 
{\tt{1}})\-\index{term-equal-proof}, \\ 
{\rm{add-hyp-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{list-implies}}\,(\=\+{\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{terma\/}}), \\ 
{\rm{cdr}}\,({\it{termb\/}}))\-\index{list-f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-\index{f-equal})\-\index{list-implies}, \\ 
{\rm{equal-axiom1-proof}}\,(\=\+{\rm{car}}\,({\it{terma\/}}), \\ 
{\rm{cdr}}\,({\it{terma\/}}), \\ 
{\rm{cdr}}\,({\it{termb\/}}))\-\index{equal-axiom1-proof})\-\index{add-hyp-proof})\-\index{elim-args-proof} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{terma\/}}) $\wedge$ {\rm{listp}}\,({\it{termb\/}}) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{term-equal-proof}}\,({\rm{car}}\,({\it{terma\/}}), {\rm{car}}\,({\it{termb\/}}), {\it{list\/}}, {\tt{0}})\index{term-equal-proof}, \\ 
{\rm{term-equal-proof}}\,({\rm{cdr}}\,({\it{terma\/}}), {\rm{cdr}}\,({\it{termb\/}}), {\it{list\/}}, {\tt{1}})\index{term-equal-proof})\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;TERM-EQUAL checks if TERMA and TERMB are identical modulo the list of
;;;;;;;equalities in LIST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{term-equal} \\  
\=\+{\rm{term-equal}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\it{flg\/}})\index{term-equal} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\it{terma\/}} $=$ {\it{termb\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal} $\in$ {\it{list\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{f-equal}}\,({\it{termb\/}}, {\it{terma\/}})\index{f-equal} $\in$ {\it{list\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{listp}}\,({\it{terma\/}}) $\wedge$ {\rm{listp}}\,({\it{termb\/}}) \\ 
{\bf then }\=\+({\rm{car}}\,({\it{terma\/}}) $=$ {\rm{car}}\,({\it{termb\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{term-equal}}\,({\rm{cdr}}\,({\it{terma\/}}), {\rm{cdr}}\,({\it{termb\/}}), {\it{list\/}}, {\tt{1}})\index{term-equal}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{terma\/}}) $\wedge$ {\rm{listp}}\,({\it{termb\/}}) \\ 
{\bf then }\=\+{\rm{term-equal}}\,({\rm{car}}\,({\it{terma\/}}), {\rm{car}}\,({\it{termb\/}}), {\it{list\/}}, {\tt{0}})\index{term-equal} \\ 
$\wedge$$\;\;\;\;${\rm{term-equal}}\,({\rm{cdr}}\,({\it{terma\/}}), {\rm{cdr}}\,({\it{termb\/}}), {\it{list\/}}, {\tt{1}})\index{term-equal}\- \\ 
{\bf else }{\it{terma\/}} $=$ {\it{termb\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The equality theorem for terms:  If "a" and "b" are identical
;;;;;;;modulo the equalities "x1 = y1",..., "xn = yn", etc., then
;;;;;;;it is provable that, "((x1=y1) ^ ... ^ (xn=yn))->(a=b)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  term-equal-proof-proves \\ \index{term-equal-proof-proves}
\=\+(\=\+{\rm{flg-termp}}\,({\it{terma\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{flg-termp} \\ 
$\wedge$$\;\;\;\;${\rm{flg-termp}}\,({\it{termb\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{flg-termp} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{term-equal}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\it{flg\/}})\index{term-equal})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{flg-proves}}\,(\=\+{\rm{term-equal-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\it{flg\/}})\index{term-equal-proof}, \\ 
{\rm{flg-f-implies-equal}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\it{terma\/}}, {\it{termb\/}}, {\it{flg\/}})\index{flg-f-implies-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{flg\/}})\-\index{flg-proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable term-equal-proof; name this event `g0364'.\index{g0364}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  term-equal-proof-proves1 \\ \index{term-equal-proof-proves1}
\=\+(\=\+{\rm{termp}}\,({\it{terma\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{termb\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{term-equal}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\tt{0}})\index{term-equal})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{term-equal-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\tt{0}})\index{term-equal-proof}, \\ 
{\rm{f-implies}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-equal-proof-proves2 \\ \index{term-equal-proof-proves2}
\=\+(\=\+{\rm{term-list}}\,({\it{terma\/}}, {\it{symbols\/}})\index{term-list} \\ 
$\wedge$$\;\;\;\;${\rm{term-list}}\,({\it{termb\/}}, {\it{symbols\/}})\index{term-list} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;${\rm{term-equal}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\tt{1}})\index{term-equal} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{dist-f-implies}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\rm{list-f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{list-f-equal})\index{dist-f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves-list}}\,(\=\+{\rm{term-equal-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{list\/}}, {\tt{1}})\index{term-equal-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-list}\-
\end{tabbing}

\noindent{\sc Event}:   Enable term-equal-proof-proves; name this event `g0887'.\index{g0887}

\addvspace{18pt}
\begin{verbatim};;;;;;;FORM-EQUAL is used in the equality theorem for formulas.
;;;;;;;It checks that FORMA and FORMB are identical modulo the equalities in LIST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{form-equal} \\  
\=\+{\rm{form-equal}}\,({\it{forma\/}}, {\it{formb\/}}, {\it{list\/}})\index{form-equal} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{forma\/}} $=$ {\it{formb\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{forma\/}})\index{f-notp} $\wedge$ {\rm{f-notp}}\,({\it{formb\/}})\index{f-notp} \\ 
{\bf then }{\rm{form-equal}}\,({\rm{arg}}\,({\it{forma\/}})\index{arg}, {\rm{arg}}\,({\it{formb\/}})\index{arg}, {\it{list\/}})\index{form-equal} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{forma\/}})\index{f-orp} $\wedge$ {\rm{f-orp}}\,({\it{formb\/}})\index{f-orp} \\ 
{\bf then }\=\+{\rm{form-equal}}\,({\rm{arg1}}\,({\it{forma\/}})\index{arg1}, {\rm{arg1}}\,({\it{formb\/}})\index{arg1}, {\it{list\/}})\index{form-equal} \\ 
$\wedge$$\;\;\;\;${\rm{form-equal}}\,({\rm{arg2}}\,({\it{forma\/}})\index{arg2}, {\rm{arg2}}\,({\it{formb\/}})\index{arg2}, {\it{list\/}})\index{form-equal}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{forma\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{formb\/}})\index{forsomep} \\ 
{\bf then }\=\+({\rm{bind}}\,({\it{forma\/}})\index{bind} $=$ {\rm{bind}}\,({\it{formb\/}})\index{bind}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\rm{elim-free-terms}}\,({\rm{bind}}\,({\it{forma\/}})\index{bind}, {\it{list\/}})\index{elim-free-terms}) \\ 
$\wedge$$\;\;\;\;${\rm{form-equal}}\,(\=\+{\rm{body}}\,({\it{forma\/}}), \\ 
{\rm{body}}\,({\it{formb\/}}), \\ 
{\rm{elim-free-terms}}\,({\rm{bind}}\,({\it{forma\/}})\index{bind}, {\it{list\/}})\index{elim-free-terms})\-\index{form-equal}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{forma\/}}) $\wedge$ {\rm{listp}}\,({\it{formb\/}}) \\ 
{\bf then }\=\+({\rm{car}}\,({\it{forma\/}}) $=$ {\rm{car}}\,({\it{formb\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{term-equal}}\,({\rm{cdr}}\,({\it{forma\/}}), {\rm{cdr}}\,({\it{formb\/}}), {\it{list\/}}, {\tt{1}})\index{term-equal}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;FORM-EQUAL-PROOF is the corresponding proof-constructor.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{form-equal-proof} \\  
\=\+{\rm{form-equal-proof}}\,({\it{forma\/}}, {\it{formb\/}}, {\it{list\/}})\index{form-equal-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{forma\/}} $=$ {\it{formb\/}} \\ 
{\bf then }{\rm{add-hyp-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{f-iff}}\,({\it{forma\/}}, {\it{formb\/}})\index{f-iff}, \\ 
{\rm{eqn-form-proof}}\,({\it{forma\/}}, {\it{formb\/}}, {\bf{nil}}, {\bf{nil}})\index{eqn-form-proof})\-\index{add-hyp-proof} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{forma\/}})\index{f-notp} $\wedge$ {\rm{f-notp}}\,({\it{formb\/}})\index{f-notp} \\ 
{\bf then }{\rm{imp-iff-not-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{arg}}\,({\it{forma\/}})\index{arg}, \\ 
{\rm{arg}}\,({\it{formb\/}})\index{arg}, \\ 
{\rm{form-equal-proof}}\,({\rm{arg}}\,({\it{forma\/}})\index{arg}, {\rm{arg}}\,({\it{formb\/}})\index{arg}, {\it{list\/}})\index{form-equal-proof})\-\index{imp-iff-not-proof} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{forma\/}})\index{f-orp} $\wedge$ {\rm{f-orp}}\,({\it{formb\/}})\index{f-orp} \\ 
{\bf then }{\rm{imp-iff-or-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{arg1}}\,({\it{forma\/}})\index{arg1}, \\ 
{\rm{arg1}}\,({\it{formb\/}})\index{arg1}, \\ 
{\rm{arg2}}\,({\it{forma\/}})\index{arg2}, \\ 
{\rm{arg2}}\,({\it{formb\/}})\index{arg2}, \\ 
{\rm{form-equal-proof}}\,({\rm{arg1}}\,({\it{forma\/}})\index{arg1}, {\rm{arg1}}\,({\it{formb\/}})\index{arg1}, {\it{list\/}})\index{form-equal-proof}, \\ 
{\rm{form-equal-proof}}\,({\rm{arg2}}\,({\it{forma\/}})\index{arg2}, {\rm{arg2}}\,({\it{formb\/}})\index{arg2}, {\it{list\/}})\index{form-equal-proof})\-\index{imp-iff-or-proof} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{forma\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{formb\/}})\index{forsomep} \\ 
{\bf then }{\rm{chain-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{f-iff}}\,({\it{forma\/}}, {\it{formb\/}})\index{f-iff}, \\ 
{\rm{list}}\,({\rm{list-f-and}}\,({\rm{elim-free-terms}}\,({\rm{bind}}\,({\it{forma\/}})\index{bind}, {\it{list\/}})\index{elim-free-terms})\index{list-f-and}), \\ 
{\rm{list}}\,(\=\+{\rm{subset-list-f-and-proof}}\,(\=\+{\it{list\/}}, \\ 
{\rm{elim-free-terms}}\,(\=\+{\rm{bind}}\,({\it{forma\/}})\index{bind}, \\ 
{\it{list\/}})\-\index{elim-free-terms})\-\index{subset-list-f-and-proof}, \\ 
{\rm{imp-iff-forsome-proof}}\,(\=\+{\rm{list-f-and}}\,({\rm{elim-free-terms}}\,(\=\+{\rm{bind}}\,({\it{forma\/}})\index{bind}, \\ 
{\it{list\/}})\-\index{elim-free-terms})\index{list-f-and}, \\ 
{\rm{body}}\,({\it{forma\/}}), \\ 
{\rm{body}}\,({\it{formb\/}}), \\ 
{\rm{bind}}\,({\it{forma\/}})\index{bind}, \\ 
{\rm{form-equal-proof}}\,(\=\+{\rm{body}}\,({\it{forma\/}}), \\ 
{\rm{body}}\,({\it{formb\/}}), \\ 
{\rm{elim-free-terms}}\,(\=\+{\rm{bind}}\,({\it{forma\/}})\index{bind}, \\ 
{\it{list\/}})\-\index{elim-free-terms})\-\index{form-equal-proof})\-\index{imp-iff-forsome-proof})\-)\-\index{chain-proof} \\ 
{\bf else }{\rm{imp-iff-implies-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\it{forma\/}}, \\ 
{\it{formb\/}}, \\ 
{\rm{elim-args-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{formb\/}}))\-\index{list-f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{forma\/}}, \\ 
{\it{formb\/}})\-\index{f-implies}, \\ 
{\rm{term-equal-proof}}\,(\=\+{\rm{cdr}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{formb\/}}), \\ 
{\it{list\/}}, \\ 
{\tt{1}})\-\index{term-equal-proof}, \\ 
{\rm{add-hyp-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{list-implies}}\,(\=\+{\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{formb\/}}))\-\index{list-f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{forma\/}}, \\ 
{\it{formb\/}})\-\index{f-implies})\-\index{list-implies}, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{car}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{formb\/}}))\-\index{equal-axiom2-proof})\-\index{add-hyp-proof})\-\index{elim-args-proof}, \\ 
{\rm{elim-args-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{formb\/}}), \\ 
{\rm{cdr}}\,({\it{forma\/}}))\-\index{list-f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{formb\/}}, \\ 
{\it{forma\/}})\-\index{f-implies}, \\ 
{\rm{dist-f-implies-symmetry-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{cdr}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{formb\/}}), \\ 
{\rm{term-equal-proof}}\,(\=\+{\rm{cdr}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{formb\/}}), \\ 
{\it{list\/}}, \\ 
{\tt{1}})\-\index{term-equal-proof})\-\index{dist-f-implies-symmetry-proof}, \\ 
{\rm{add-hyp-proof}}\,(\=\+{\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, \\ 
{\rm{list-implies}}\,(\=\+{\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{formb\/}}), \\ 
{\rm{cdr}}\,({\it{forma\/}}))\-\index{list-f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{formb\/}}, \\ 
{\it{forma\/}})\-\index{f-implies})\-\index{list-implies}, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{car}}\,({\it{forma\/}}), \\ 
{\rm{cdr}}\,({\it{formb\/}}), \\ 
{\rm{cdr}}\,({\it{forma\/}}))\-\index{equal-axiom2-proof})\-\index{add-hyp-proof})\-\index{elim-args-proof})\-\index{imp-iff-implies-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-list-elim-free-terms \\ \index{form-list-elim-free-terms}
{\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} $\rightarrow$ {\rm{form-list}}\,({\rm{elim-free-terms}}\,({\it{x\/}}, {\it{list\/}})\index{elim-free-terms}, {\it{symbols\/}})\index{form-list}
\end{tabbing}
\begin{verbatim};;;;;;;The equality theorem for formulas:  If "A" is identical to "B" modulo
;;;;;;;the equalities "x1=y1", ..., "xn=yn", etc., then it is provable that
;;;;;;; "((x1=y1) ^ ... ^ (xn=yn)) -> (A <-> B)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  form-equal-proof-proves \\ \index{form-equal-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{forma\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{formb\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{form-equal}}\,({\it{forma\/}}, {\it{formb\/}}, {\it{list\/}})\index{form-equal} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-equal-proof}}\,({\it{forma\/}}, {\it{formb\/}}, {\it{list\/}})\index{form-equal-proof}, \\ 
{\rm{f-implies}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\rm{f-iff}}\,({\it{forma\/}}, {\it{formb\/}})\index{f-iff})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-equal-proof; name this event `g0893'.\index{g0893}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  get-list-f-equal \\ \index{get-list-f-equal}
\=\+({\it{exp\/}} $\not=$ {\rm{getpf}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getpf}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{f-equal}}\,({\it{exp\/}}, {\rm{getpf}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getpf})\index{f-equal} $\in$ {\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  getsub-getpf \\ \index{getsub-getpf}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{getsub}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{getsub} $=$ {\rm{getpf}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{getpf})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-equal-parallel-subst \\ \index{term-equal-parallel-subst}
\=\+{\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp} \\ 
$\rightarrow$$\;\;\;\;${\rm{term-equal}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst}, \\ 
{\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal}, \\ 
{\it{flg\/}})\-\index{term-equal}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  delete-delete \\ \index{delete-delete}
{\rm{del}}\,({\it{x\/}}, {\rm{del}}\,({\it{y\/}}, {\it{z\/}})\index{del})\index{del} $=$ {\rm{del}}\,({\it{y\/}}, {\rm{del}}\,({\it{x\/}}, {\it{z\/}})\index{del})\index{del}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  del-pairs-del-pairs \\ \index{del-pairs-del-pairs}
\=\+{\rm{del-pairs}}\,({\it{x\/}}, {\rm{del}}\,({\it{y\/}}, {\it{vars\/}})\index{del}, {\rm{del-pairs}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs})\index{del-pairs} \\ 
$=$$\;\;\;\;${\rm{del-pairs}}\,({\it{y\/}}, {\rm{del}}\,({\it{x\/}}, {\it{vars\/}})\index{del}, {\rm{del-pairs}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs})\index{del-pairs}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-del-pairs-parallel-subst \\ \index{not-free-del-pairs-parallel-sub!st}
\=\+({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{parallel-subst}}\,({\it{exp\/}}, {\rm{del}}\,({\it{x\/}}, {\it{vars\/}})\index{del}, {\rm{del-pairs}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs}, {\it{flg\/}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable f-equal; name this event `g0894'.\index{g0894}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  collect-free-f-equal \\ \index{collect-free-f-equal}
\=\+({\it{x\/}} $\in$ {\rm{collect-free}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\tt{0}})\index{collect-free}) \\ 
$=$$\;\;\;\;$(({\it{x\/}} $\in$ {\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free}) $\vee$ ({\it{x\/}} $\in$ {\rm{collect-free}}\,({\it{z\/}}, {\tt{0}})\index{collect-free}))\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-equal; name this event `g0895'.\index{g0895}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  del-pairs-elim-free \\ \index{del-pairs-elim-free}
\=\+(\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{list-f-equal}}\,({\rm{del}}\,({\it{x\/}}, {\it{vars\/}})\index{del}, {\rm{del-pairs}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs})\index{list-f-equal} \\ 
$=$$\;\;\;\;${\rm{elim-free-terms}}\,({\it{x\/}}, {\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal})\index{elim-free-terms})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  var-list-del \\ \index{var-list-del}
{\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} $\rightarrow$ {\rm{var-list}}\,({\rm{del}}\,({\it{x\/}}, {\it{vars\/}})\index{del}, {\rm{length}}\,({\rm{del}}\,({\it{x\/}}, {\it{vars\/}})\index{del})\index{length})\index{var-list}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  append-nil-one \\ \index{append-nil-one}
({\it{y\/}} $\not=$ {\bf{nil}}) $\rightarrow$ ({\rm{append}}\,({\it{x\/}}, {\it{y\/}}) $\not=$ {\bf{nil}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  append-nil-two \\ \index{append-nil-two}
({\rm{append}}\,({\it{x\/}}, {\bf{nil}}) $\not=$ {\bf{nil}}) $\rightarrow$ ({\rm{append}}\,({\it{x\/}}, {\it{y\/}}) $\not=$ {\bf{nil}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  del-pairs-collect-free \\ \index{del-pairs-collect-free}
\=\+({\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{collect-free}}\,({\rm{del-pairs}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}})\index{del-pairs}, {\tt{1}})\index{collect-free} $=$ {\bf{nil}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nlistp-term-equal \\ \index{nlistp-term-equal}
(({\it{x\/}} $\not=$ {\it{y\/}}) $\wedge$ ({\it{list\/}} $\simeq\;${{\bf{nil}}}${}$)) $\rightarrow$ ($\neg$ {\rm{term-equal}}\,({\it{x\/}}, {\it{y\/}}, {\it{list\/}}, {\it{flg\/}})\index{term-equal})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nlistp-form-equal \\ \index{nlistp-form-equal}
(({\it{x\/}} $\not=$ {\it{y\/}}) $\wedge$ ({\it{list\/}} $\simeq\;${{\bf{nil}}}${}$)) $\rightarrow$ ($\neg$ {\rm{form-equal}}\,({\it{x\/}}, {\it{y\/}}, {\it{list\/}})\index{form-equal})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-equal-parallel-subst \\ \index{form-equal-parallel-subst}
\=\+(\=\+{\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free} $=$ {\bf{nil}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{form-equal}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal})\-\index{form-equal}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-equal-subst \\ \index{term-equal-subst}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ {\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{term-equal}}\,({\it{exp\/}}, {\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\rm{list}}\,({\rm{f-equal}}\,({\it{var\/}}, {\it{term\/}})\index{f-equal}), {\it{flg\/}})\index{term-equal}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-equal-subst \\ \index{form-equal-subst}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{free-for}) \\ 
$\rightarrow$$\;\;\;\;${\rm{form-equal}}\,({\it{exp\/}}, {\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{list}}\,({\rm{f-equal}}\,({\it{var\/}}, {\it{term\/}})\index{f-equal}))\index{form-equal}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  symb-equals \\ \index{symb-equals}
{\rm{symb}}\,({\it{sym\/}}, {\it{symbols\/}})\index{symb} $=$ ({\rm{memb}}\,({\it{sym\/}}, {\it{symbols\/}})\index{memb} $\vee$ ({\it{sym\/}} $=$ {\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  index-not-equal \\ \index{index-not-equal}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{x\/}} $\not=$ {\tt{0}})) $\rightarrow$ ({\rm{p}}\,({\it{x\/}}, {\it{y\/}})\index{p} $\not=$ {\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  fn-p-not-equal \\ \index{fn-p-not-equal}
{\rm{fn}}\,({\it{x\/}}, {\it{y\/}})\index{fn} $\not=$ {\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  memb-fn \\ \index{memb-fn}
{\rm{memb}}\,({\rm{fn}}\,({\it{x\/}}, {\it{y\/}})\index{fn}, {\it{symbols\/}})\index{memb} $=$ ({\rm{fn}}\,({\it{x\/}}, {\it{y\/}})\index{fn} $\in$ {\it{symbols\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  memb-p \\ \index{memb-p}
{\rm{memb}}\,({\rm{p}}\,({\it{x\/}}, {\it{y\/}})\index{p}, {\it{symbols\/}})\index{memb} $=$ ({\rm{p}}\,({\it{x\/}}, {\it{y\/}})\index{p} $\in$ {\it{symbols\/}})
\end{tabbing}
\begin{verbatim};;;;;;;The next section contains results that follow from the Z2 set axioms.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  formula-isin \\ \index{formula-isin}
\=\+{\rm{formula}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$({\rm{memb}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\it{symbols\/}})\index{memb} $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-axiom2-isin \\ \index{equal-axiom2-isin}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ ({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-axiom2-proof}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\rm{list}}\,({\it{u\/}}, {\it{v\/}}))\index{equal-axiom2-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{u\/}}, {\it{v\/}})\index{isin})\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-isin \\ \index{collect-free-isin}
{\rm{collect-free}}\,({\rm{isin}}\,({\it{y\/}}, {\it{z\/}})\index{isin}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-isin \\ \index{covering-isin}
{\rm{covering}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\it{z\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{z\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-isin \\ \index{subst-isin}
\=\+{\rm{subst}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{isin}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{isin}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-isin \\ \index{parallel-subst-isin}
\=\+{\rm{parallel-subst}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{isin}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\rm{parallel-subst}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{isin}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isin; name this event `g0381'.\index{g0381}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{mk-bnd} \\  
{\rm{mk-bnd}}\,({\it{list\/}}, {\it{n\/}})\index{mk-bnd} $=$ (($1\;+$ {\it{n\/}}) $+$ {\rm{find-hi-index}}\,({\rm{collect-free}}\,({\it{list\/}}, {\tt{1}})\index{collect-free})\index{find-hi-index})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-mk-bnd \\ \index{lessp-mk-bnd}
{\rm{find-hi-index}}\,({\rm{collect-free}}\,({\it{list\/}}, {\tt{1}})\index{collect-free})\index{find-hi-index} $<$ {\rm{mk-bnd}}\,({\it{list\/}}, {\it{n\/}})\index{mk-bnd}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  mk-bnd-nequal \\ \index{mk-bnd-nequal}
({\rm{fix}}\,({\it{m\/}}) $\not=$ {\rm{fix}}\,({\it{n\/}})) $\rightarrow$ ({\rm{mk-bnd}}\,({\it{list\/}}, {\it{m\/}})\index{mk-bnd} $\not=$ {\rm{mk-bnd}}\,({\it{list\/}}, {\it{n\/}})\index{mk-bnd})
\end{tabbing}

\noindent{\sc Event}:   Enable mk-bnd; name this event `g0612'.\index{g0612}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  mk-bnd-not-free1 \\ \index{mk-bnd-not-free1}
{\rm{mk-bnd}}\,({\rm{list}}\,({\it{x\/}}), {\it{n\/}})\index{mk-bnd} $\not\in$ {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  mk-bnd-not-free2 \\ \index{mk-bnd-not-free2}
\=\+({\rm{mk-bnd}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{n\/}})\index{mk-bnd} $\not\in$ {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\rm{mk-bnd}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{n\/}})\index{mk-bnd} $\not\in$ {\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  mk-bnd-not-free3 \\ \index{mk-bnd-not-free3}
\=\+({\rm{mk-bnd}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\it{n\/}})\index{mk-bnd} $\not\in$ {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\rm{mk-bnd}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\it{n\/}})\index{mk-bnd} $\not\in$ {\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\rm{mk-bnd}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\it{n\/}})\index{mk-bnd} $\not\in$ {\rm{collect-free}}\,({\it{z\/}}, {\tt{0}})\index{collect-free})\-
\end{tabbing}
\begin{verbatim};;;;;;;The next few rules are usable versions of the set-theory axioms.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{ext-axiom-eq-elim1-proof} \\  
\=\+{\rm{ext-axiom-eq-elim1-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{pf\/}})\index{ext-axiom-eq-elim1-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{ext-axiom}}\,(\=\+{\rm{mk-bnd}}\,({\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), {\tt{0}})\index{mk-bnd}, \\ 
{\it{terma\/}}, \\ 
{\it{termb\/}})\-\index{ext-axiom}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{mk-bnd\/}}, (list terma termb), {\tt{0}}$\;${\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{termb\/}})\-\index{isin})\-\index{f-iff}\index{forall}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, \\ 
{\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\-\index{f-iff})\-\index{f-implies}, \\ 
{\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin})\-, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-proof}}\,(\=\+{\rm{mk-bnd}}\,({\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), {\tt{0}})\index{mk-bnd}, \\ 
{\it{terma\/}}, \\ 
{\it{termb\/}})\-\index{ext-axiom-proof}, \\ 
{\rm{subst-thm-b-proof}}\,(\=\+{\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{termb\/}})\-\index{isin})\-\index{f-iff}, \\ 
{\rm{list}}\,({\rm{mk-bnd}}\,(\=\+{\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), \\ 
{\tt{0}})\-\index{mk-bnd}), \\ 
{\rm{list}}\,({\it{termc\/}}))\-\index{subst-thm-b-proof}, \\ 
{\it{pf\/}})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-intersect-del \\ \index{nil-intersect-del}
{\rm{nil-intersect}}\,({\it{x\/}}, {\rm{del}}\,({\it{y\/}}, {\it{z\/}})\index{del})\index{nil-intersect} $\rightarrow$ {\rm{nil-intersect}}\,({\rm{del}}\,({\it{y\/}}, {\it{x\/}})\index{del}, {\it{z\/}})\index{nil-intersect}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-parallel-subst \\ \index{not-free-parallel-subst}
\=\+{\rm{nil-intersect}}\,({\it{vars\/}}, {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})\index{nil-intersect} \\ 
$\rightarrow$$\;\;\;\;$({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst} $=$ {\it{exp\/}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-f-iff \\ \index{parallel-subst-f-iff}
\=\+{\rm{parallel-subst}}\,({\rm{f-iff}}\,({\it{x\/}}, {\it{y\/}})\index{f-iff}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{f-iff}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\rm{parallel-subst}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{f-iff}\-
\end{tabbing}
\begin{verbatim};;;;;;;One case of the extensionality axiom: "(c in a)"|- "(a=b)->(c in b)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ext-axiom-eq-elim1 \\ \index{ext-axiom-eq-elim1}
\=\+(\=\+{\rm{termp}}\,({\it{termb\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-axiom-eq-elim1-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{pf\/}})\index{ext-axiom-eq-elim1-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ext-axiom-eq-elim2-proof} \\  
\=\+{\rm{ext-axiom-eq-elim2-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{pf\/}})\index{ext-axiom-eq-elim2-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{ext-axiom}}\,(\=\+{\rm{mk-bnd}}\,({\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), {\tt{0}})\index{mk-bnd}, \\ 
{\it{terma\/}}, \\ 
{\it{termb\/}})\-\index{ext-axiom}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{mk-bnd\/}}, (list terma termb), {\tt{0}}$\;${\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{termb\/}})\-\index{isin})\-\index{f-iff}\index{forall}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, \\ 
{\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\-\index{f-iff})\-\index{f-implies}, \\ 
{\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\-, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, {\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin})\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-proof}}\,(\=\+{\rm{mk-bnd}}\,({\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), {\tt{0}})\index{mk-bnd}, \\ 
{\it{terma\/}}, \\ 
{\it{termb\/}})\-\index{ext-axiom-proof}, \\ 
{\rm{subst-thm-b-proof}}\,(\=\+{\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{termb\/}})\-\index{isin})\-\index{f-iff}, \\ 
{\rm{list}}\,({\rm{mk-bnd}}\,(\=\+{\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), \\ 
{\tt{0}})\-\index{mk-bnd}), \\ 
{\rm{list}}\,({\it{termc\/}}))\-\index{subst-thm-b-proof}, \\ 
{\it{pf\/}})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Second case of the extensionality axiom: "(c in b)" |- "(a=b)->(c in a)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ext-axiom-eq-elim2 \\ \index{ext-axiom-eq-elim2}
\=\+(\=\+{\rm{termp}}\,({\it{terma\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-axiom-eq-elim2-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{pf\/}})\index{ext-axiom-eq-elim2-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, {\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-axiom-eq-elim1-proof; name this event `g0631'.\index{g0631}

\addvspace{18pt}

\noindent{\sc Event}:   Enable ext-axiom-eq-elim2-proof; name this event `g0632'.\index{g0632}

\addvspace{18pt}

\noindent{\sc Event}:   Enable f-iff-mp-proof2; name this event `g0633'.\index{g0633}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ext-axiom-eq-intro-proof} \\  
\=\+{\rm{ext-axiom-eq-intro-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}}, {\it{pf\/}})\index{ext-axiom-eq-intro-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-iff}\index{forall}, \\ 
{\rm{ext-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{ext-axiom-proof}, \\ 
{\rm{generalise-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-iff}, \\ 
{\it{pf\/}})\-\index{generalise-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;;Third case of ext. axiom:  "(x in a) <-> (x in b)" |- "(a=b)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ext-axiom-eql-intro \\ \index{ext-axiom-eql-intro}
\=\+(\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), {\tt{1}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-axiom-eq-intro-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}}, {\it{pf\/}})\index{ext-axiom-eq-intro-proof}, \\ 
{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-axiom-eq-intro-proof; name this event `g0634'.\index{g0634}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  symb-reduc \\ \index{symb-reduc}
({\it{sym\/}} $\in$ {\it{symbols\/}}) $\rightarrow$ {\rm{symb}}\,({\it{sym\/}}, {\it{symbols\/}})\index{symb}
\end{tabbing}

\noindent{\sc Event}:   Enable symb; name this event `g0635'.\index{g0635}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  equal-axiom1-z-pair \\ \index{equal-axiom1-z-pair}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ ({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-axiom1-proof}}\,({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\rm{list}}\,({\it{u\/}}, {\it{v\/}}))\index{equal-axiom1-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-axiom1-z-union \\ \index{equal-axiom1-z-union}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ ({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-axiom1-proof}}\,({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\rm{list}}\,({\it{u\/}}, {\it{v\/}}))\index{equal-axiom1-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{z-union}}\,({\it{x\/}}, {\it{y\/}})\index{z-union}, {\rm{z-union}}\,({\it{u\/}}, {\it{v\/}})\index{z-union})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-z-pair \\ \index{collect-free-z-pair}
{\rm{collect-free}}\,({\rm{z-pair}}\,({\it{y\/}}, {\it{z\/}})\index{z-pair}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-z-pair \\ \index{covering-z-pair}
{\rm{covering}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-pair \\ \index{subst-z-pair}
\=\+{\rm{subst}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{z-pair}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-pair}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-pair \\ \index{parallel-subst-z-pair}
\=\+{\rm{parallel-subst}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-pair}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\rm{parallel-subst}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{z-pair}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-z-pair \\ \index{termp-z-pair}
\=\+{\rm{termp}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$(\=\+({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-equal-z-pair \\ \index{term-equal-z-pair}
\=\+({\rm{term-equal}}\,({\it{x\/}}, {\it{u\/}}, {\it{list\/}}, {\tt{0}})\index{term-equal} $\wedge$ {\rm{term-equal}}\,({\it{y\/}}, {\it{v\/}}, {\it{list\/}}, {\tt{0}})\index{term-equal}) \\ 
$\rightarrow$$\;\;\;\;${\rm{term-equal}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair}, {\it{list\/}}, {\tt{0}})\index{term-equal}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-equal-z-union \\ \index{term-equal-z-union}
\=\+({\rm{term-equal}}\,({\it{x\/}}, {\it{u\/}}, {\it{list\/}}, {\tt{0}})\index{term-equal} $\wedge$ {\rm{term-equal}}\,({\it{y\/}}, {\it{v\/}}, {\it{list\/}}, {\tt{0}})\index{term-equal}) \\ 
$\rightarrow$$\;\;\;\;${\rm{term-equal}}\,({\rm{z-union}}\,({\it{x\/}}, {\it{y\/}})\index{z-union}, {\rm{z-union}}\,({\it{u\/}}, {\it{v\/}})\index{z-union}, {\it{list\/}}, {\tt{0}})\index{term-equal}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-pair; name this event `g0387'.\index{g0387}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-z-union \\ \index{termp-z-union}
\=\+{\rm{termp}}\,({\rm{z-union}}\,({\it{x\/}}, {\it{y\/}})\index{z-union}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$(\=\+({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-z-union \\ \index{collect-free-z-union}
{\rm{collect-free}}\,({\rm{z-union}}\,({\it{y\/}}, {\it{z\/}})\index{z-union}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-z-union \\ \index{covering-z-union}
{\rm{covering}}\,({\rm{z-union}}\,({\it{x\/}}, {\it{y\/}})\index{z-union}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-union \\ \index{subst-z-union}
\=\+{\rm{subst}}\,({\rm{z-union}}\,({\it{x\/}}, {\it{y\/}})\index{z-union}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{z-union}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-union}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-union \\ \index{parallel-subst-z-union}
\=\+{\rm{parallel-subst}}\,({\rm{z-union}}\,({\it{x\/}}, {\it{y\/}})\index{z-union}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-union}}\,(\=\+{\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\index{z-union}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-union; name this event `g0388'.\index{g0388}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-pair-elim-proof} \\  
\=\+{\rm{z-pair-elim-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{z-pair-elim-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{terma\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{termb\/}})\index{f-equal})\index{f-or})\-\index{f-iff}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{terma\/}})\index{f-equal}, {\it{exp\/}})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{termb\/}})\index{f-equal}, {\it{exp\/}})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, {\it{exp\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\rm{pair-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{pair-axiom-proof}, {\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Pair-elimination:  if |-"(x = a)->exp" and |-"(x=b)->exp", 
;;;;;;;then |-"(x in {a,b}) -> exp".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-pair-elim \\ \index{z-pair-elim}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{terma\/}})\index{f-equal}, {\it{exp\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{termb\/}})\index{f-equal}, {\it{exp\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-pair-elim-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{z-pair-elim-proof}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, {\it{exp\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-pair-elim-proof; name this event `g0643'.\index{g0643}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{isin-pair1-proof} \\  
\=\+{\rm{isin-pair1-proof}}\,({\it{terma\/}}, {\it{termb\/}})\index{isin-pair1-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,({\it{terma\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{terma\/}}, {\it{terma\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal})\index{f-or}, \\ 
{\rm{pair-axiom-proof}}\,({\it{terma\/}}, {\it{terma\/}}, {\it{termb\/}})\index{pair-axiom-proof}, \\ 
{\rm{rt-expan-proof}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{terma\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{ident-axiom-proof}}\,({\it{terma\/}})\index{ident-axiom-proof})\-\index{rt-expan-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;; |-"(x in {x, y})".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  isin-pair1-proof-proves \\ \index{isin-pair1-proof-proves}
\=\+{\rm{formula}}\,({\rm{isin}}\,({\it{terma\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isin-pair1-proof}}\,({\it{terma\/}}, {\it{termb\/}})\index{isin-pair1-proof}, \\ 
{\rm{isin}}\,({\it{terma\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isin-pair1-proof; name this event `g0391'.\index{g0391}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{isin-pair2-proof} \\  
\=\+{\rm{isin-pair2-proof}}\,({\it{terma\/}}, {\it{termb\/}})\index{isin-pair2-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,({\it{termb\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{termb\/}}, {\it{terma\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{termb\/}}, {\it{termb\/}})\index{f-equal})\index{f-or}, \\ 
{\rm{pair-axiom-proof}}\,({\it{termb\/}}, {\it{terma\/}}, {\it{termb\/}})\index{pair-axiom-proof}, \\ 
{\rm{expan-proof}}\,(\=\+{\rm{f-equal}}\,({\it{termb\/}}, {\it{terma\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{termb\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{ident-axiom-proof}}\,({\it{termb\/}})\index{ident-axiom-proof})\-\index{expan-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(y in {x,y})".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  isin-pair2-proof-proves \\ \index{isin-pair2-proof-proves}
\=\+{\rm{formula}}\,({\rm{isin}}\,({\it{termb\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isin-pair2-proof}}\,({\it{terma\/}}, {\it{termb\/}})\index{isin-pair2-proof}, \\ 
{\rm{isin}}\,({\it{termb\/}}, {\rm{z-pair}}\,({\it{terma\/}}, {\it{termb\/}})\index{z-pair})\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isin-pair2-proof; name this event `g0392'.\index{g0392}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-sing}$\;\;$
{\rm{z-sing}}\,({\it{x\/}})\index{z-sing} $=$ {\rm{z-pair}}\,({\it{x\/}}, {\it{x\/}})\index{z-pair}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{isin-sing-proof} \\  
\=\+{\rm{isin-sing-proof}}\,({\it{x\/}}, {\it{y\/}})\index{isin-sing-proof} \\ 
$=$$\;\;\;\;${\rm{chain-iff-proof}}\,(\=\+{\rm{isin}}\,({\it{y\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-or}), \\ 
{\rm{list}}\,(\=\+{\rm{pair-axiom-proof}}\,({\it{y\/}}, {\it{x\/}}, {\it{x\/}})\index{pair-axiom-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\-\index{f-or}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\-\index{f-iff}, \\ 
{\bf{nil}})\-\index{tautconseq-proof})\-)\-\index{chain-iff-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(y in {x}) <-> (y=x)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  isin-sing \\ \index{isin-sing}
\=\+{\rm{formula}}\,({\rm{f-iff}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-iff}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isin-sing-proof}}\,({\it{x\/}}, {\it{y\/}})\index{isin-sing-proof}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isin-sing-proof; name this event `g0913'.\index{g0913}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-z-sing \\ \index{termp-z-sing}
\=\+{\rm{termp}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$(({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-z-sing \\ \index{collect-free-z-sing}
{\rm{collect-free}}\,({\rm{z-sing}}\,({\it{y\/}})\index{z-sing}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{y\/}}, {\it{y\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-z-sing \\ \index{covering-z-sing}
{\rm{covering}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}, {\it{x\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-sing \\ \index{subst-z-sing}
{\rm{subst}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\rm{z-sing}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-sing}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-sing \\ \index{parallel-subst-z-sing}
\=\+{\rm{parallel-subst}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-sing}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{z-sing}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-equal-z-sing \\ \index{term-equal-z-sing}
{\rm{term-equal}}\,({\it{x\/}}, {\it{y\/}}, {\it{list\/}}, {\tt{0}})\index{term-equal} $\rightarrow$ {\rm{term-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-sing}}\,({\it{y\/}})\index{z-sing}, {\it{list\/}}, {\tt{0}})\index{term-equal}
\end{tabbing}

\noindent{\sc Event}:   Enable z-sing; name this event `g0389'.\index{g0389}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{isin-sing1-proof} \\  
\=\+{\rm{isin-sing1-proof}}\,({\it{x\/}})\index{isin-sing1-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{isin-sing-proof}}\,({\it{x\/}}, {\it{x\/}})\index{isin-sing-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\it{x\/}})\index{ident-axiom-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(x in {x})".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  isin-sing1-proof-proves \\ \index{isin-sing1-proof-proves}
\=\+{\rm{formula}}\,({\rm{isin}}\,({\it{term\/}}, {\rm{z-sing}}\,({\it{term\/}})\index{z-sing})\index{isin}, {\it{symbols\/}})\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isin-sing1-proof}}\,({\it{term\/}})\index{isin-sing1-proof}, \\ 
{\rm{isin}}\,({\it{term\/}}, {\rm{z-sing}}\,({\it{term\/}})\index{z-sing})\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isin-sing1-proof; name this event `g0393'.\index{g0393}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{del-forsome-proof} \\  
\=\+{\rm{del-forsome-proof}}\,({\it{x\/}}, {\it{exp\/}})\index{del-forsome-proof} \\ 
$=$$\;\;\;\;${\rm{iff-join-proof}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\it{exp\/}})\index{forsome}, \\ 
{\it{exp\/}}, \\ 
{\rm{forsome-intro-proof}}\,({\it{x\/}}, {\it{exp\/}}, {\it{exp\/}}, {\rm{prop-axiom-proof}}\,({\it{exp\/}})\index{prop-axiom-proof})\index{forsome-intro-proof}, \\ 
{\rm{subst-thm-a-proof}}\,({\it{exp\/}}, {\rm{list}}\,({\it{x\/}}), {\rm{list}}\,({\it{x\/}}))\index{subst-thm-a-proof})\-\index{iff-join-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If "x" not free in "exp", then |- "(forsome x exp) <-> exp".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  del-forsome-proof-proves \\ \index{del-forsome-proof-proves}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} $\wedge$ ({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{del-forsome-proof}}\,({\it{x\/}}, {\it{exp\/}})\index{del-forsome-proof}, \\ 
{\rm{f-iff}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{exp\/}})\index{forsome}, {\it{exp\/}})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable del-forsome-proof; name this event `g0597'.\index{g0597}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{del-forall-proof} \\  
\=\+{\rm{del-forall-proof}}\,({\it{x\/}}, {\it{exp\/}})\index{del-forall-proof} \\ 
$=$$\;\;\;\;${\rm{iff-join-proof}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{exp\/}}\index{forall}, \\ 
{\it{exp\/}}, \\ 
{\rm{subst-thm-b-proof}}\,({\it{exp\/}}, {\rm{list}}\,({\it{x\/}}), {\rm{list}}\,({\it{x\/}}))\index{subst-thm-b-proof}, \\ 
{\rm{forall-intro-proof}}\,({\it{exp\/}}, {\it{exp\/}}, {\it{x\/}}, {\rm{prop-axiom-proof}}\,({\it{exp\/}})\index{prop-axiom-proof})\index{forall-intro-proof})\-\index{iff-join-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If "x" not free in "exp", then |- "(forall x exp) <-> exp".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  del-forall-proof-proves \\ \index{del-forall-proof-proves}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} $\wedge$ ({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{del-forall-proof}}\,({\it{x\/}}, {\it{exp\/}})\index{del-forall-proof}, {\rm{f-iff}}\,($\forall\;${\it{x\/}}$\;${\it{exp\/}}\index{forall}, {\it{exp\/}})\index{f-iff}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable del-forall-proof; name this event `g0599'.\index{g0599}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{sing-sing-equal-proof} \\  
\=\+{\rm{sing-sing-equal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{sing-sing-equal-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-sing}}\,({\it{y\/}})\index{z-sing})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{z-sing}}\,({\it{y\/}})\index{z-sing})\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-elim1-proof}}\,(\=\+{\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, \\ 
{\rm{z-sing}}\,({\it{y\/}})\index{z-sing}, \\ 
{\it{x\/}}, \\ 
{\rm{isin-sing1-proof}}\,({\it{x\/}})\index{isin-sing1-proof})\-\index{ext-axiom-eq-elim1-proof}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-sing}}\,({\it{y\/}})\index{z-sing})\index{isin}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{isin-sing-proof}}\,({\it{y\/}}, {\it{x\/}})\index{isin-sing-proof})\-\index{iff-split-proof1})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "({x}={y}) -> (x=y)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  sing-sing-equal-proof-proves \\ \index{sing-sing-equal-proof-proves}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{sing-sing-equal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{sing-sing-equal-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-sing}}\,({\it{y\/}})\index{z-sing})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable sing-sing-equal-proof; name this event `g0652'.\index{g0652}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{sing-pair-equal-proof1} \\  
\=\+{\rm{sing-pair-equal-proof1}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}})\index{sing-pair-equal-proof1} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\it{u\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, {\rm{f-equal}}\,({\it{u\/}}, {\it{x\/}})\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-elim2-proof}}\,(\=\+{\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair}, \\ 
{\it{u\/}}, \\ 
{\rm{isin-pair1-proof}}\,({\it{u\/}}, {\it{v\/}})\index{isin-pair1-proof})\-\index{ext-axiom-eq-elim2-proof}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,({\it{u\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, \\ 
{\rm{f-equal}}\,({\it{u\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{isin-sing-proof}}\,({\it{x\/}}, {\it{u\/}})\index{isin-sing-proof})\-\index{iff-split-proof1}, \\ 
{\rm{symmetry-proof1}}\,({\it{u\/}}, {\it{x\/}})\index{symmetry-proof1})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "({x}={u,v}) -> (x=u)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  sing-pair-equal-proof1-proves \\ \index{sing-pair-equal-proof1-proves}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{sing-pair-equal-proof1}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}})\index{sing-pair-equal-proof1}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable sing-pair-equal-proof1; name this event `g0653'.\index{g0653}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{sing-pair-equal-proof2} \\  
\=\+{\rm{sing-pair-equal-proof2}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}})\index{sing-pair-equal-proof2} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\it{v\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, {\rm{f-equal}}\,({\it{v\/}}, {\it{x\/}})\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-elim2-proof}}\,(\=\+{\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair}, \\ 
{\it{v\/}}, \\ 
{\rm{isin-pair2-proof}}\,({\it{u\/}}, {\it{v\/}})\index{isin-pair2-proof})\-\index{ext-axiom-eq-elim2-proof}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,({\it{v\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, \\ 
{\rm{f-equal}}\,({\it{v\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{isin-sing-proof}}\,({\it{x\/}}, {\it{v\/}})\index{isin-sing-proof})\-\index{iff-split-proof1}, \\ 
{\rm{symmetry-proof1}}\,({\it{v\/}}, {\it{x\/}})\index{symmetry-proof1})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |-"({x}={u,v}) -> (x=v)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  sing-pair-equal-proof-proves \\ \index{sing-pair-equal-proof-proves}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{sing-pair-equal-proof2}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}})\index{sing-pair-equal-proof2}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{v\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable sing-pair-equal-proof2; name this event `g0654'.\index{g0654}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-opair}$\;\;$
{\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair} $=$ {\rm{z-pair}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair})\index{z-pair}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-z-opair \\ \index{termp-z-opair}
\=\+{\rm{termp}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$(\=\+({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-equal-z-opair \\ \index{term-equal-z-opair}
\=\+(({\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal} $\in$ {\it{list\/}}) $\wedge$ ({\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal} $\in$ {\it{list\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{term-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair}, {\it{list\/}}, {\tt{0}})\index{term-equal}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-car-unique-proof} \\  
\=\+{\rm{z-car-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{z-car-unique-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-elim1-proof}}\,(\=\+{\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair}, \\ 
{\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, \\ 
{\rm{isin-pair1-proof}}\,(\=\+{\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair})\-\index{isin-pair1-proof})\-\index{ext-axiom-eq-elim1-proof}, \\ 
{\rm{z-pair-elim-proof}}\,(\=\+{\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{sing-sing-equal-proof}}\,({\it{x\/}}, {\it{u\/}})\index{sing-sing-equal-proof}, \\ 
{\rm{sing-pair-equal-proof1}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}})\index{sing-pair-equal-proof1})\-\index{z-pair-elim-proof})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(<x y>=<u v>)-> (x=u)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-car-unique-proof-proves \\ \index{z-car-unique-proof-proves}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-car-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{z-car-unique-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-unique-proof; name this event `g0656'.\index{g0656}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-cdr-unique-step-proof} \\  
\=\+{\rm{z-cdr-unique-step-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{z-cdr-unique-step-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{u\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal})\index{f-or}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-elim1-proof}}\,(\=\+{\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair}, \\ 
{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, \\ 
{\rm{isin-pair2-proof}}\,(\=\+{\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair})\-\index{isin-pair2-proof})\-\index{ext-axiom-eq-elim1-proof}, \\ 
{\rm{z-pair-elim-proof}}\,(\=\+{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{u\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal})\index{f-or}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-pair}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{z-pair}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{u\/}})\-\index{f-equal})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-pair}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{z-pair}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\-\index{f-equal}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{u\/}})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{v\/}})\-\index{f-equal})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-pair}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{z-pair}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{u\/}})\-\index{f-equal}, \\ 
{\rm{list}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{z-pair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{u\/}}, \\ 
{\it{y\/}})\-\index{f-equal})\-, \\ 
{\rm{list}}\,(\=\+{\rm{symmetry-proof1}}\,(\=\+{\rm{z-pair}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{z-pair}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\-\index{symmetry-proof1}, \\ 
{\rm{sing-pair-equal-proof2}}\,(\=\+{\it{u\/}}, \\ 
{\it{x\/}}, \\ 
{\it{y\/}})\-\index{sing-pair-equal-proof2}, \\ 
{\rm{symmetry-proof1}}\,(\=\+{\it{u\/}}, \\ 
{\it{y\/}})\-\index{symmetry-proof1})\-)\-\index{chain-proof}))\-\index{tautconseq-proof}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\-\index{f-equal}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal})\-\index{f-or}, \\ 
{\rm{list}}\,({\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\-\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-elim1-proof}}\,(\=\+{\rm{z-pair}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{z-pair}, \\ 
{\rm{z-pair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-pair}, \\ 
{\it{y\/}}, \\ 
{\rm{isin-pair2-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{isin-pair2-proof})\-\index{ext-axiom-eq-elim1-proof}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-pair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-pair})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{u\/}})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{v\/}})\-\index{f-equal})\-\index{f-or}, \\ 
{\rm{pair-axiom-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{u\/}}, \\ 
{\it{v\/}})\-\index{pair-axiom-proof})\-\index{iff-split-proof1})\-)\-\index{chain-proof})\-\index{z-pair-elim-proof})\-)\-\index{chain-proof}\-
\end{tabbing}

\begin{verbatim};;;;;;; |- "(<x y>=<u v>) -> ((y=u) v (y=v))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-cdr-unique-step \\ \index{z-cdr-unique-step}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-cdr-unique-step-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{z-cdr-unique-step-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{u\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal})\index{f-or})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr-unique-step-proof; name this event `g0657'.\index{g0657}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{trans-equal-proof1} \\  
\=\+{\rm{trans-equal-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list-implies}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal})\-, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\-\index{list-implies}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z\/}})\index{f-equal})\-, \\ 
{\rm{list-implies}}\,(\=\+{\rm{list}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}), \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\-\index{list-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{equal-axiom2-proof}}\,({\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}, {\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\rm{list}}\,({\it{x\/}}, {\it{z\/}}))\index{equal-axiom2-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\it{z\/}})\index{ident-axiom-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |-"(y = x) -> ((y = z) -> (x = z))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  trans-equal-proof-proves \\ \index{trans-equal-proof-proves}
\=\+({\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{z\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{trans-equal-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof1}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{f-implies}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable trans-equal-proof1; name this event `g0362'.\index{g0362}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-covering \\ \index{termp-covering}
{\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp} $\rightarrow$ ({\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering} $=$ {\bf{nil}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  atomp-covering \\ \index{atomp-covering}
{\rm{atomp}}\,({\it{exp\/}}, {\it{symbols\/}})\index{atomp} $\rightarrow$ ({\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\bf{nil}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-z-opair \\ \index{collect-free-z-opair}
\=\+{\rm{collect-free}}\,({\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair}, {\tt{0}})\index{collect-free} \\ 
$=$$\;\;\;\;${\rm{collect-free}}\,({\rm{list}}\,({\rm{z-sing}}\,({\it{y\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{y\/}}, {\it{z\/}})\index{z-pair}), {\tt{1}})\index{collect-free}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-opair; name this event `g0366'.\index{g0366}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{trans-equal-proof2} \\  
\=\+{\rm{trans-equal-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{z\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal})\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{z\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{symmetry-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{symmetry-proof1}, \\ 
{\rm{symmetry-proof1}}\,({\it{z\/}}, {\it{y\/}})\index{symmetry-proof1}, \\ 
{\rm{trans-equal-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof1})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(x = y) -> ((z = y) -> (x = z))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  trans-equal-proof2-proves \\ \index{trans-equal-proof2-proves}
\=\+({\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{z\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{trans-equal-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof2}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-implies}}\,({\rm{f-equal}}\,({\it{z\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable trans-equal-proof2; name this event `g0367'.\index{g0367}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{trans-equal-proof3} \\  
\=\+{\rm{trans-equal-proof3}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof3} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{ident-axiom-proof}}\,({\it{x\/}})\index{ident-axiom-proof}, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}, \\ 
{\rm{list}}\,({\it{x\/}}, {\it{y\/}}), \\ 
{\rm{list}}\,({\it{x\/}}, {\it{z\/}}))\-\index{equal-axiom2-proof})\-\index{detach-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(x = y) -> ((y = z) -> (x = z))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  trans-equal-proof3-proves \\ \index{trans-equal-proof3-proves}
\=\+({\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{z\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{trans-equal-proof3}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof3}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-implies}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable trans-equal-proof3; name this event `g0736'.\index{g0736}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-cdr-unique-proof} \\  
\=\+{\rm{z-cdr-unique-proof}}\,({\it{x\/}}, {\it{u\/}}, {\it{y\/}}, {\it{v\/}})\index{z-cdr-unique-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{u\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{u\/}}, {\it{v\/}})\index{f-equal})\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{v\/}}, {\it{x\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{v\/}}, {\it{u\/}})\index{f-equal})\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{v\/}}, {\it{u\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{u\/}}, {\it{v\/}})\index{f-equal})\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{u\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{u\/}}, {\it{x\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{u\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{v\/}}, {\it{x\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{u\/}}, {\it{v\/}})\index{f-equal})\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{u\/}}, {\it{v\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{z-cdr-unique-step-proof}}\,({\it{x\/}}, {\it{u\/}}, {\it{y\/}}, {\it{v\/}})\index{z-cdr-unique-step-proof}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{v\/}}, {\it{x\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{v\/}}, {\it{u\/}})\index{f-equal})\index{f-or}, \\ 
{\rm{list}}\,({\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair})\-\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{symmetry-proof1}}\,(\=\+{\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair})\-\index{symmetry-proof1}, \\ 
{\rm{z-cdr-unique-step-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{v\/}}, \\ 
{\it{x\/}}, \\ 
{\it{u\/}})\-\index{z-cdr-unique-step-proof})\-)\-\index{chain-proof}, \\ 
{\rm{z-car-unique-proof}}\,({\it{x\/}}, {\it{u\/}}, {\it{y\/}}, {\it{v\/}})\index{z-car-unique-proof}, \\ 
{\rm{symmetry-proof1}}\,({\it{v\/}}, {\it{u\/}})\index{symmetry-proof1}, \\ 
{\rm{trans-equal-proof2}}\,({\it{u\/}}, {\it{y\/}}, {\it{x\/}})\index{trans-equal-proof2}, \\ 
{\rm{trans-equal-proof2}}\,({\it{u\/}}, {\it{x\/}}, {\it{v\/}})\index{trans-equal-proof2})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(<x u>=<u v>) -> (u = v)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-cdr-unique-proof-proves \\ \index{z-cdr-unique-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{u\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{v\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-cdr-unique-proof}}\,({\it{x\/}}, {\it{u\/}}, {\it{y\/}}, {\it{v\/}})\index{z-cdr-unique-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{u\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, {\rm{f-equal}}\,({\it{u\/}}, {\it{v\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr-unique-proof; name this event `g0369'.\index{g0369}

\addvspace{18pt}
\begin{verbatim};;;;;;;The next definition introduces the "if" connective which
;;;;;;;plays an important role in the proofs to follow.
;;;;;;;Def.: "(if A B C)" =  "((A ^ B) v ((not A) ^ C))".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{f-if}$\;\;$
{\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if} $=$ {\rm{f-or}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, {\rm{f-and}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{z\/}})\index{f-and})\index{f-or}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{f-if-intro-step-proof1} \\  
\=\+{\rm{f-if-intro-step-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if-intro-step-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{y\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{f-implies})\index{f-implies}, {\bf{nil}})\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(A -> (B -> (if A B C)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-if-intro-step1 \\ \index{f-if-intro-step1}
\=\+({\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{f-if-intro-step-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if-intro-step-proof1}, \\ 
{\rm{f-implies}}\,({\it{x\/}}, {\rm{f-implies}}\,({\it{y\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{f-if-intro-step-proof2} \\  
\=\+{\rm{f-if-intro-step-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if-intro-step-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-implies}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\rm{f-implies}}\,({\it{z\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{f-implies})\index{f-implies}, {\bf{nil}})\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((not A) -> (C -> (if A B C)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-if-intro-step2 \\ \index{f-if-intro-step2}
\=\+({\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{f-if-intro-step-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if-intro-step-proof2}, \\ 
{\rm{f-implies}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\rm{f-implies}}\,({\it{z\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-if-intro-step-proof1; name this event `g0663'.\index{g0663}

\addvspace{18pt}

\noindent{\sc Event}:   Enable f-if-intro-step-proof2; name this event `g0664'.\index{g0664}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{f-if-intro-proof1} \\  
\=\+{\rm{f-if-intro-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfx\/}}, {\it{pfy\/}})\index{f-if-intro-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\rm{list}}\,({\it{pfx\/}}, {\it{pfy\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "A" and |- "B", then |- "(if A B C)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-if-intro-proof1-proves \\ \index{f-if-intro-proof1-proves}
\=\+(\=\+{\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfx\/}}, {\it{x\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfy\/}}, {\it{y\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{f-if-intro-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfx\/}}, {\it{pfy\/}})\index{f-if-intro-proof1}, \\ 
{\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-if-intro-proof1; name this event `g0902'.\index{g0902}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{f-if-intro-proof2} \\  
\=\+{\rm{f-if-intro-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfnx\/}}, {\it{pfz\/}})\index{f-if-intro-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{z\/}}), {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\rm{list}}\,({\it{pfnx\/}}, {\it{pfz\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "(not A)" and |- "C", then |- "(if A B C)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-if-intro-proof2-proves \\ \index{f-if-intro-proof2-proves}
\=\+(\=\+{\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfnx\/}}, {\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfz\/}}, {\it{z\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{f-if-intro-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfnx\/}}, {\it{pfz\/}})\index{f-if-intro-proof2}, \\ 
{\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-if-intro-proof2; name this event `g0903'.\index{g0903}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{f-if-elim-proof1} \\  
\=\+{\rm{f-if-elim-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfx\/}})\index{f-if-elim-proof1} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\it{x\/}}), {\rm{f-implies}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{y\/}})\index{f-implies}, {\rm{list}}\,({\it{pfx\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "A", then |- "((if A B C) -> C)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-if-elim-proof1-proves \\ \index{f-if-elim-proof1-proves}
\=\+(\=\+{\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfx\/}}, {\it{x\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{f-if-elim-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfx\/}})\index{f-if-elim-proof1}, \\ 
{\rm{f-implies}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{y\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-if-elim-proof1; name this event `g0904'.\index{g0904}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{f-if-elim-proof2} \\  
\=\+{\rm{f-if-elim-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfnx\/}})\index{f-if-elim-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\rm{list}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}), {\rm{f-implies}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{z\/}})\index{f-implies}, {\rm{list}}\,({\it{pfnx\/}}))\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |-"(not A)", then |-"((if A B C) -> C)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  f-if-elim-proof2-proves \\ \index{f-if-elim-proof2-proves}
\=\+(\=\+{\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfnx\/}}, {\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{f-if-elim-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pfnx\/}})\index{f-if-elim-proof2}, \\ 
{\rm{f-implies}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{z\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-if-elim-proof2; name this event `g0905'.\index{g0905}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-f-if \\ \index{formula-f-if}
\=\+{\rm{formula}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$({\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-f-if \\ \index{collect-free-f-if}
\=\+{\rm{collect-free}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\tt{0}})\index{collect-free} \\ 
$=$$\;\;\;\;${\rm{collect-free}}\,({\rm{f-or}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, {\rm{f-and}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{z\/}})\index{f-and})\index{f-or}, {\tt{0}})\index{collect-free}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-f-if \\ \index{subst-f-if}
\=\+{\rm{subst}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{z\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-f-if \\ \index{parallel-subst-f-if}
\=\+{\rm{parallel-subst}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\it{z\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-f-if \\ \index{covering-f-if}
\=\+{\rm{covering}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{var\/}}, {\tt{0}})\index{covering} \\ 
$=$$\;\;\;\;${\rm{append}}\,(\=\+{\rm{append}}\,({\rm{covering}}\,({\it{x\/}}, {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\it{y\/}}, {\it{var\/}}, {\tt{0}})\index{covering}), \\ 
{\rm{append}}\,({\rm{covering}}\,({\it{x\/}}, {\it{var\/}}, {\tt{0}})\index{covering}, {\rm{covering}}\,({\it{z\/}}, {\it{var\/}}, {\tt{0}})\index{covering}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-f-if \\ \index{eval-f-if}
\=\+{\rm{eval}}\,({\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if}, {\it{alist\/}})\index{eval} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{eval}}\,({\it{x\/}}, {\it{alist\/}})\index{eval}$\;\;${\bf then }{\rm{eval}}\,({\it{y\/}}, {\it{alist\/}})\index{eval} \\ 
{\bf else }{\rm{eval}}\,({\it{z\/}}, {\it{alist\/}})\index{eval}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable f-if; name this event `g0909'.\index{g0909}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{case-split-proof} \\  
\=\+{\rm{case-split-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{case-split-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\rm{f-implies}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{y\/}})\index{f-implies}), \\ 
{\it{y\/}}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;If |- "(A -> B)" and |- "((not A) -> B)", then |- "B".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  case-split-proof-proves \\ \index{case-split-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{y\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{case-split-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{case-split-proof}, {\it{y\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable case-split-proof; name this event `g0601'.\index{g0601}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-phi \\ \index{termp-phi}
{\rm{termp}}\,({\sc{phi}}\index{phi}, {\tt{0}}, {\it{symbols\/}})\index{termp} $=$ ({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn} $\in$ {\it{symbols\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-phi \\ \index{collect-free-phi}
{\rm{collect-free}}\,({\sc{phi}}\index{phi}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-phi \\ \index{covering-phi}
{\rm{covering}}\,({\sc{phi}}\index{phi}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\bf{nil}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-phi \\ \index{subst-phi}
{\rm{subst}}\,({\sc{phi}}\index{phi}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\sc{phi}}\index{phi}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-phi \\ \index{parallel-subst-phi}
{\rm{parallel-subst}}\,({\sc{phi}}\index{phi}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} $=$ {\sc{phi}}\index{phi}
\end{tabbing}

\noindent{\sc Event}:   Enable phi; name this event `g0668'.\index{g0668}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  proves-list-nil \\ \index{proves-list-nil}
{\rm{proves-list}}\,({\bf{nil}}, {\bf{nil}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{forsome-distrib-or-proof1} \\  
\=\+{\rm{forsome-distrib-or-proof1}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-or-proof1} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, \\ 
{\rm{f-or}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-or}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\it{a\/}}, {\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\it{b\/}}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, \\ 
{\rm{f-or}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{e-intro-proof}}\,({\it{x\/}}, {\it{a\/}})\index{e-intro-proof}, \\ 
{\rm{e-intro-proof}}\,({\it{x\/}}, {\it{b\/}})\index{e-intro-proof})\-)\-\index{tautconseq-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((forsome x (A v B)) -> ((forsome x A) v (forsome x B)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forsome-distrib-or-proof-proves1 \\ \index{forsome-distrib-or-proof-proves!1}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-distrib-or-proof1}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-or-proof1}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome}, {\rm{f-or}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-or})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{forsome-distrib-step-proof} \\  
\=\+{\rm{forsome-distrib-step-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-step-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome}, \\ 
{\rm{chain-proof}}\,(\=\+{\it{b\/}}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome}, \\ 
{\rm{list}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}), \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}, \\ 
{\rm{e-intro-proof}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{e-intro-proof})\-)\-\index{chain-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(forsome x B) -> (forsome x (A v B))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forsome-distrib-step-proof-proves \\ \index{forsome-distrib-step-proof-prove!s}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-distrib-step-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-step-proof}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forsome-distrib-or-proof1; name this event `g0669'.\index{g0669}

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\noindent{\sc Event}:   Enable forsome-distrib-step-proof; name this event `g0670'.\index{g0670}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{forsome-distrib-or-proof2} \\  
\=\+{\rm{forsome-distrib-or-proof2}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-or-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-or}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-or}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{forsome-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{a\/}}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome}, \\ 
{\rm{chain-proof}}\,(\=\+{\it{a\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\-\index{forsome}, \\ 
{\rm{list}}\,({\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}), \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{a\/}}, \\ 
{\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}, \\ 
{\rm{e-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\-\index{e-intro-proof})\-)\-\index{chain-proof})\-\index{forsome-intro-proof}, \\ 
{\rm{forsome-distrib-step-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-step-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(((forsome x A) v (forsome x B)) -> (forsome x (A v B)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forsome-distrib-or-proof-proves2 \\ \index{forsome-distrib-or-proof-proves!2}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-distrib-or-proof2}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-or-proof2}, \\ 
{\rm{f-implies}}\,({\rm{f-or}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-or}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or})\index{forsome})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{forall-distrib-or-proof} \\  
\=\+{\rm{forall-distrib-or-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forall-distrib-or-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,($\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall}, $\forall\;${\it{x\/}}$\;${\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}\index{forall})\index{f-implies}, \\ 
{\rm{f-implies}}\,($\forall\;${\it{x\/}}$\;${\it{b\/}}\index{forall}, $\forall\;${\it{x\/}}$\;${\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}\index{forall})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\rm{f-or}}\,($\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall}, $\forall\;${\it{x\/}}$\;${\it{b\/}}\index{forall})\index{f-or}, $\forall\;${\it{x\/}}$\;${\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}\index{forall})\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{forall-intro-proof}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall}, \\ 
{\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, \\ 
{\it{x\/}}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall}, \\ 
{\it{a\/}})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall}, \\ 
{\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{all-elim-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{a\/}})\-\index{all-elim-proof}))\-\index{tautconseq-proof})\-\index{forall-intro-proof}, \\ 
{\rm{forall-intro-proof}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{b\/}}\index{forall}, \\ 
{\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}, \\ 
{\it{x\/}}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{b\/}}\index{forall}, \\ 
{\it{b\/}})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{b\/}}\index{forall}, \\ 
{\rm{f-or}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{all-elim-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{all-elim-proof}))\-\index{tautconseq-proof})\-\index{forall-intro-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  proves-list-reduc \\ \index{proves-list-reduc}
\=\+{\rm{proves-list}}\,({\rm{cons}}\,({\it{pf1\/}}, {\it{pf2\/}}), {\rm{cons}}\,({\it{exp1\/}}, {\it{exp2\/}}), {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$=$$\;\;\;\;$(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\it{exp1\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pf2\/}}, {\it{exp2\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list})\-\-
\end{tabbing}
\begin{verbatim};;;;;;; |-"((forall x A) v (forall x B)) -> (forall x (A v B))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forall-distrib-or \\ \index{forall-distrib-or}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forall-distrib-or-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forall-distrib-or-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-or}}\,($\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall}, $\forall\;${\it{x\/}}$\;${\it{b\/}}\index{forall})\index{f-or}, $\forall\;${\it{x\/}}$\;${\rm{f-or}}\,({\it{a\/}}, {\it{b\/}})\index{f-or}\index{forall})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forall-distrib-or-proof; name this event `g0672'.\index{g0672}

\addvspace{18pt}

\noindent{\sc Event}:   Disable forall; name this event `g0673'.\index{g0673}

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\begin{tabbing}{\sc Definition}:\index{forsome-forall-proof} \\  
\=\+{\rm{forsome-forall-proof}}\,({\it{x\/}}, {\it{a\/}})\index{forsome-forall-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-iff}}\,({\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome})\index{f-iff}), \\ 
{\rm{f-iff}}\,({\rm{f-not}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome})\index{f-not}, $\forall\;${\it{x\/}}$\;${\rm{f-not}}\,({\it{a\/}})\index{f-not}\index{forall})\index{f-iff}, \\ 
{\rm{list}}\,({\rm{eqn-form-proof}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not})\index{forsome}, \\ 
{\rm{list}}\,({\rm{f-iff}}\,({\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, {\it{a\/}})\index{f-iff}), \\ 
{\rm{list}}\,({\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{f-not}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-not}, \\ 
{\it{a\/}})\-\index{f-iff}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}))\-\index{eqn-form-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(not (forsome x A)) <-> (forall x (not A))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forsome-forall \\ \index{forsome-forall}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-forall-proof}}\,({\it{x\/}}, {\it{a\/}})\index{forsome-forall-proof}, \\ 
{\rm{f-iff}}\,({\rm{f-not}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{a\/}})\index{forsome})\index{f-not}, $\forall\;${\it{x\/}}$\;${\rm{f-not}}\,({\it{a\/}})\index{f-not}\index{forall})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{forall-forsome-proof} \\  
\=\+{\rm{forall-forsome-proof}}\,({\it{x\/}}, {\it{a\/}})\index{forall-forsome-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-iff}}\,({\rm{f-not}}\,($\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall})\index{f-not}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome})\index{f-iff}, {\bf{nil}})\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(not (forall x A)) <-> (forsome x (not A))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forall-forsome \\ \index{forall-forsome}
\=\+({\rm{variable}}\,({\it{x\/}})\index{variable} $\wedge$ {\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forall-forsome-proof}}\,({\it{x\/}}, {\it{a\/}})\index{forall-forsome-proof}, \\ 
{\rm{f-iff}}\,({\rm{f-not}}\,($\forall\;${\it{x\/}}$\;${\it{a\/}}\index{forall})\index{f-not}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forsome-forall-proof; name this event `g0674'.\index{g0674}

\addvspace{18pt}

\noindent{\sc Event}:   Enable forall-forsome-proof; name this event `g0675'.\index{g0675}

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\noindent{\sc Event}:   Enable forsome-distrib-or-proof2; name this event `g0593'.\index{g0593}

\addvspace{18pt}

\noindent{\sc Event}:   Enable forall; name this event `g0676'.\index{g0676}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{forsome-distrib-and-proof} \\  
\=\+{\rm{forsome-distrib-and-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-and-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-and}}\,({\it{a\/}}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-and}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and})\index{forsome}, \\ 
{\rm{list}}\,(\=\+{\rm{f-not}}\,({\rm{f-or}}\,({\rm{f-not}}\,({\it{a\/}})\index{f-not}, {\rm{f-not}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-not})\index{f-or})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,({\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome}, {\rm{f-not}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-not})\index{f-or})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome})\index{f-implies})\index{f-not}, \\ 
{\rm{f-not}}\,($\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\it{b\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}\index{forall})\index{f-not}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\rm{f-implies}}\,({\it{b\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies})\index{f-not})\index{forsome})\-, \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\it{a\/}}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-and}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\index{f-not})\-\index{f-or})\index{f-not})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-not})\-\index{f-or})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-not})\-\index{f-or})\index{f-not}, \\ 
{\rm{eqn-form-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{forsome}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\index{f-not})\-\index{f-or})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\index{f-not})\-\index{f-or})\index{f-not}, \\ 
{\rm{list}}\,({\rm{f-iff}}\,(\=\+{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{forsome})\-\index{f-iff}), \\ 
{\rm{list}}\,({\rm{iff-commute-proof}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{forsome}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not}, \\ 
{\rm{del-forsome-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{del-forsome-proof})\-\index{iff-commute-proof}))\-\index{eqn-form-proof})\-\index{iff-split-proof1}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-not}}\,({\rm{f-or}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{forsome}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome})\index{f-not})\-\index{f-or})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{forsome})\-\index{f-implies})\index{f-not})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}, \\ 
{\rm{contra-pos-proof1}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\it{b\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}\index{forall}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{forsome})\-\index{f-implies}, \\ 
{\rm{forall-over-implies-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{b\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{forall-over-implies-proof})\-\index{contra-pos-proof1}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{f-not}}\,($\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\it{b\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies}\index{forall})\index{f-not}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\rm{f-implies}}\,({\it{b\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies})\index{f-not})\-\index{forsome}, \\ 
{\rm{forall-forsome-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{f-implies})\-\index{forall-forsome-proof})\-\index{iff-split-proof1}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\rm{f-implies}}\,({\it{b\/}}, {\rm{f-not}}\,({\it{a\/}})\index{f-not})\index{f-implies})\index{f-not})\-\index{forsome}, \\ 
{\rm{forsome}}\,({\it{x\/}}, {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and})\index{forsome}, \\ 
{\rm{eqn-form-proof}}\,(\=\+{\rm{forsome}}\,({\it{x\/}}, {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and})\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{f-implies})\index{f-not})\-\index{forsome}, \\ 
{\rm{list}}\,({\rm{f-iff}}\,(\=\+{\rm{f-not}}\,({\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{f-implies})\index{f-not}, \\ 
{\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and})\-\index{f-iff}), \\ 
{\rm{list}}\,({\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{f-not}}\,({\rm{f-implies}}\,(\=\+{\it{b\/}}, \\ 
{\rm{f-not}}\,({\it{a\/}})\index{f-not})\-\index{f-implies})\index{f-not}, \\ 
{\rm{f-and}}\,(\=\+{\it{a\/}}, \\ 
{\it{b\/}})\-\index{f-and})\-\index{f-iff}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}))\-\index{eqn-form-proof})\-\index{iff-split-proof1})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If "x" not free in "A", then |- "(A ^ (forsome x B)) -> (forsome x (A ^ B))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forsome-distrib-and-proof-proves \\ \index{forsome-distrib-and-proof-prove!s}
\=\+(\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{a\/}}, {\tt{0}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-distrib-and-proof}}\,({\it{x\/}}, {\it{a\/}}, {\it{b\/}})\index{forsome-distrib-and-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\rm{forsome}}\,({\it{x\/}}, {\it{b\/}})\index{forsome})\index{f-and}, {\rm{forsome}}\,({\it{x\/}}, {\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and})\index{forsome})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forsome-distrib-and-proof; name this event `g0681'.\index{g0681}

\addvspace{18pt}

\noindent{\sc Event}:   Enable forsome-distrib-or-proof2; name this event `g0683'.\index{g0683}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{forsome-f-if-proof} \\  
\=\+{\rm{forsome-f-if-proof}}\,({\it{var\/}}, {\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{forsome-f-if-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-if}}\,({\it{x\/}}, {\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome}, {\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\index{f-if}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{forsome}, \\ 
{\rm{list}}\,({\rm{f-or}}\,(\=\+{\rm{forsome}}\,({\it{var\/}}, {\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-and}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{z\/}})\index{f-and})\index{forsome})\-\index{f-or}), \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\it{x\/}}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome})\-\index{f-and}, \\ 
{\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-not}}\,({\it{x\/}})\index{f-not}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\-\index{f-and}, \\ 
{\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{f-not}}\,({\it{x\/}})\index{f-not}, \\ 
{\it{z\/}})\-\index{f-and})\-\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-if}}\,(\=\+{\it{x\/}}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\-\index{f-if}, \\ 
{\rm{f-or}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{f-not}}\,({\it{x\/}})\index{f-not}, \\ 
{\it{z\/}})\-\index{f-and})\-\index{forsome})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{forsome-distrib-and-proof}}\,(\=\+{\it{var\/}}, \\ 
{\it{x\/}}, \\ 
{\it{y\/}})\-\index{forsome-distrib-and-proof}, \\ 
{\rm{forsome-distrib-and-proof}}\,(\=\+{\it{var\/}}, \\ 
{\rm{f-not}}\,({\it{x\/}})\index{f-not}, \\ 
{\it{z\/}})\-\index{forsome-distrib-and-proof})\-)\-\index{tautconseq-proof}, \\ 
{\rm{forsome-distrib-or-proof2}}\,(\=\+{\it{var\/}}, \\ 
{\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and}, \\ 
{\rm{f-and}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\it{z\/}})\index{f-and})\-\index{forsome-distrib-or-proof2})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If "v" not free in "A", then
;;;;;;; |-"(if A (forsome v B) (forsome v C)) -> (forsome v (if A B C))".
;;;;;;;This important rule justifies definition by cases using the "if" connective.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forsome-f-if-proof-proves \\ \index{forsome-f-if-proof-proves}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-f-if-proof}}\,({\it{var\/}}, {\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{forsome-f-if-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-if}}\,({\it{x\/}}, {\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome}, {\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\index{f-if}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{forsome})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forsome-f-if-proof; name this event `g0684'.\index{g0684}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{unique-f-if-proof} \\  
\=\+{\rm{unique-f-if-proof}}\,({\it{test\/}}, {\it{left1\/}}, {\it{left2\/}}, {\it{right1\/}}, {\it{right2\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{unique-f-if-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\rm{f-and}}\,({\it{left1\/}}, {\it{left2\/}})\index{f-and}, {\it{exp\/}})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{right1\/}}, {\it{right2\/}})\index{f-and}, {\it{exp\/}})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-if}}\,({\it{test\/}}, {\it{left1\/}}, {\it{right1\/}})\index{f-if}, \\ 
{\rm{f-if}}\,({\it{test\/}}, {\it{left2\/}}, {\it{right2\/}})\index{f-if})\-\index{f-and}, \\ 
{\it{exp\/}})\-\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;If |-"(B1 ^ B2) -> P" and |-"(C1 ^ C2) -> P", then 
;;;;;;; |- "(((if A B1 C1) ^ (if A B2 C2)) -> P)".
;;;;;;;This lemma justifies the uniqueness part of the admissibility of a 
;;;;;;;definition by cases using the "if" connective.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  unique-f-if-proof-proves \\ \index{unique-f-if-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{test\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\rm{f-and}}\,({\it{left1\/}}, {\it{left2\/}})\index{f-and}, {\it{exp\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\rm{f-and}}\,({\it{right1\/}}, {\it{right2\/}})\index{f-and}, {\it{exp\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{unique-f-if-proof}}\,({\it{test\/}}, {\it{left1\/}}, {\it{left2\/}}, {\it{right1\/}}, {\it{right2\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{unique-f-if-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-if}}\,({\it{test\/}}, {\it{left1\/}}, {\it{right1\/}})\index{f-if}, \\ 
{\rm{f-if}}\,({\it{test\/}}, {\it{left2\/}}, {\it{right2\/}})\index{f-if})\-\index{f-and}, \\ 
{\it{exp\/}})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable unique-f-if-proof; name this event `g0603'.\index{g0603}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{forsome-f-if-case-proof} \\  
\=\+{\rm{forsome-f-if-case-proof}}\,({\it{var\/}}, {\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{forsome-f-if-case-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\it{x\/}}, {\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-if}}\,({\it{x\/}}, {\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome}, {\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\index{f-if}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{forsome}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}, {\rm{forsome-f-if-proof}}\,({\it{var\/}}, {\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{forsome-f-if-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;If |- "(A -> (forsome v B))" and |-"((not A) -> (forsome v C))", 
;;;;;;;then |- "(forsome v (if A B C))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  forsome-f-if-case-proof-proves \\ \index{forsome-f-if-case-proof-proves}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{x\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{y\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{z\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{x\/}}, {\rm{forsome}}\,({\it{var\/}}, {\it{y\/}})\index{forsome})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not}, {\rm{forsome}}\,({\it{var\/}}, {\it{z\/}})\index{forsome})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-f-if-case-proof}}\,({\it{var\/}}, {\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{forsome-f-if-case-proof}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{f-if})\index{forsome}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable forsome-f-if-case-proof; name this event `g0687'.\index{g0687}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{exis-fun-intro-proof} \\  
\=\+{\rm{exis-fun-intro-proof}}\,({\it{var\/}}, {\it{term\/}})\index{exis-fun-intro-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,({\it{term\/}}, {\it{term\/}})\index{f-equal}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-equal}}\,({\it{var\/}}, {\it{term\/}})\index{f-equal})\index{forsome}, \\ 
{\rm{ident-axiom-proof}}\,({\it{term\/}})\index{ident-axiom-proof}, \\ 
{\rm{subst-thm-a-proof}}\,({\rm{f-equal}}\,({\it{var\/}}, {\it{term\/}})\index{f-equal}, {\rm{list}}\,({\it{var\/}}), {\rm{list}}\,({\it{term\/}}))\index{subst-thm-a-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-f-equal \\ \index{parallel-subst-f-equal}
\=\+{\rm{parallel-subst}}\,({\rm{f-equal}}\,({\it{a\/}}, {\it{b\/}})\index{f-equal}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{f-equal}}\,(\=\+{\rm{parallel-subst}}\,({\it{a\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\it{b\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\index{f-equal}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  f-equal-col-free \\ \index{f-equal-col-free}
{\rm{collect-free}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-f-equal \\ \index{covering-f-equal}
{\rm{covering}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{verbatim};;;;;;; |- "(forsome v (v = t))", "v" not free in "t".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  exis-fun-intro-proof-proves \\ \index{exis-fun-intro-proof-proves}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ {\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ ({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{exis-fun-intro-proof}}\,({\it{var\/}}, {\it{term\/}})\index{exis-fun-intro-proof}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-equal}}\,({\it{var\/}}, {\it{term\/}})\index{f-equal})\index{forsome}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable exis-fun-intro-proof; name this event `g0690'.\index{g0690}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subset-append-reduc \\ \index{subset-append-reduc}
{\rm{subset}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}), {\it{z\/}})\index{subset} $=$ ({\rm{subset}}\,({\it{x\/}}, {\it{z\/}})\index{subset} $\wedge$ {\rm{subset}}\,({\it{y\/}}, {\it{z\/}})\index{subset})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-succ-hyps} \\  
\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{f-equal}}\,({\rm{z-succ}}\,({\tt{0}})\index{z-succ}, {\rm{z-union}}\,({\rm{z-sing}}\,({\tt{0}})\index{z-sing}, {\tt{0}})\index{z-union})\index{f-equal}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\rm{rest-of}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{unique-fun-intro-proof} \\  
\=\+{\rm{unique-fun-intro-proof}}\,({\it{var1\/}}, {\it{var2\/}}, {\it{term\/}})\index{unique-fun-intro-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{var1\/}}, {\it{term\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{var2\/}}, {\it{term\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{var1\/}}, {\it{var2\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,({\it{var1\/}}, {\it{term\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{var2\/}}, {\it{term\/}})\index{f-equal})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{var1\/}}, {\it{var2\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{trans-equal-proof2}}\,({\it{var1\/}}, {\it{term\/}}, {\it{var2\/}})\index{trans-equal-proof2}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((v1 = t) ^ (v2 = t)) -> (v1 = v2)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  unique-fun-intro-proof-proves \\ \index{unique-fun-intro-proof-proves}
\=\+{\rm{termp}}\,({\rm{list}}\,({\it{var1\/}}, {\it{var2\/}}, {\it{term\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{unique-fun-intro-proof}}\,({\it{var1\/}}, {\it{var2\/}}, {\it{term\/}})\index{unique-fun-intro-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{f-equal}}\,({\it{var1\/}}, {\it{term\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{var2\/}}, {\it{term\/}})\index{f-equal})\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{var1\/}}, {\it{var2\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable unique-fun-intro-proof; name this event `g0604'.\index{g0604}

\addvspace{18pt}

\noindent{\sc Event}:   Enable rest-of; name this event `g0702'.\index{g0702}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-succ-defn-proof} \\  
\=\+{\rm{z-succ-defn-proof}}\,({\it{term\/}})\index{z-succ-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\tt{0}})\index{z-succ}, {\rm{z-union}}\,({\rm{z-sing}}\,({\tt{0}})\index{z-sing}, {\tt{0}})\index{z-union})\index{f-equal}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{f-equal}}\,({\tt{1}}, {\rm{z-union}}\,({\rm{z-sing}}\,({\tt{0}})\index{z-sing}, {\tt{0}})\index{z-union})\index{f-equal}, \\ 
{\rm{exis-fun-intro-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{z-union}}\,(\=\+{\rm{z-sing}}\,({\tt{0}})\index{z-sing}, \\ 
{\tt{0}})\-\index{z-union})\-\index{exis-fun-intro-proof}, \\ 
{\rm{unique-fun-intro-proof}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{z-union}}\,(\=\+{\rm{z-sing}}\,({\tt{0}})\index{z-sing}, \\ 
{\tt{0}})\-\index{z-union})\-\index{unique-fun-intro-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The definition of the successor operation: |- "(S t) = ({t} U t)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-succ-defn \\ \index{z-succ-defn}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-succ-defn-proof}}\,({\it{term\/}})\index{z-succ-defn-proof}, \\ 
{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{term\/}})\index{z-succ}, {\rm{z-union}}\,({\rm{z-sing}}\,({\it{term\/}})\index{z-sing}, {\it{term\/}})\index{z-union})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-succ-defn-proof; name this event `g0407'.\index{g0407}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-succ-hyps; name this event `g0677'.\index{g0677}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-z-succ \\ \index{termp-z-succ}
\=\+{\rm{termp}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$(({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-succ \\ \index{subst-z-succ}
{\rm{subst}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\rm{z-succ}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-succ}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-succ \\ \index{parallel-subst-z-succ}
\=\+{\rm{parallel-subst}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-succ}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{z-succ}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-z-succ \\ \index{covering-z-succ}
{\rm{covering}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-succ-col-free \\ \index{z-succ-col-free}
{\rm{collect-free}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-axiom1-z-succ \\ \index{equal-axiom1-z-succ}
\=\+(({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-axiom1-proof}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{list}}\,({\it{x\/}}), {\rm{list}}\,({\it{y\/}}))\index{equal-axiom1-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-succ; name this event `g0408'.\index{g0408}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ext-axiom-eq-detach-proof} \\  
\=\+{\rm{ext-axiom-eq-detach-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{pf\/}})\index{ext-axiom-eq-detach-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff}, \\ 
{\it{pf\/}}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff}, \\ 
{\rm{list}}\,($\forall\;${\it{mk-bnd\/}}, (list terma termb), {\tt{0}}$\;${\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{termb\/}})\-\index{isin})\-\index{f-iff}\index{forall}), \\ 
{\rm{list}}\,(\=\+{\rm{iff-split-proof1}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
$\forall\;${\it{mk-bnd\/}}, (list terma termb), {\tt{0}}$\;${\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{termb\/}})\-\index{isin})\-\index{f-iff}\index{forall}, \\ 
{\rm{ext-axiom-proof}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}}, \\ 
{\it{termb\/}})\-\index{ext-axiom-proof})\-\index{iff-split-proof1}, \\ 
{\rm{subst-thm-b-proof}}\,(\=\+{\rm{f-iff}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{terma\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}, \\ 
{\it{termb\/}})\-\index{isin})\-\index{f-iff}, \\ 
{\rm{list}}\,({\rm{mk-bnd}}\,(\=\+{\rm{list}}\,(\=\+{\it{terma\/}}, \\ 
{\it{termb\/}})\-, \\ 
{\tt{0}})\-\index{mk-bnd}), \\ 
{\rm{list}}\,({\it{termc\/}}))\-\index{subst-thm-b-proof})\-)\-\index{chain-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |-"(a = b)", then |-"(c in a) <-> (c in b)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ext-axiom-eq-detach \\ \index{ext-axiom-eq-detach}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-axiom-eq-detach-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{pf\/}})\index{ext-axiom-eq-detach-proof}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-axiom-eq-detach-proof; name this event `g0704'.\index{g0704}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-z-succ-proof} \\  
\=\+{\rm{in-z-succ-proof}}\,({\it{x\/}}, {\it{y\/}})\index{in-z-succ-proof} \\ 
$=$$\;\;\;\;${\rm{chain-iff-proof}}\,(\=\+{\rm{isin}}\,({\it{y\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-or}, \\ 
{\rm{list}}\,(\=\+{\rm{isin}}\,({\it{y\/}}, {\rm{z-union}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\it{x\/}})\index{z-union})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-or})\-, \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-detach-proof}}\,(\=\+{\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, \\ 
{\rm{z-union}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\it{x\/}})\index{z-union}, \\ 
{\it{y\/}}, \\ 
{\rm{z-succ-defn-proof}}\,({\it{x\/}})\index{z-succ-defn-proof})\-\index{ext-axiom-eq-detach-proof}, \\ 
{\rm{union-axiom-proof}}\,({\it{y\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\it{x\/}})\index{union-axiom-proof}, \\ 
{\rm{eqn-form-proof}}\,(\=\+{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-or}, \\ 
{\rm{f-or}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-or}, \\ 
{\rm{list}}\,({\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{y\/}}, {\rm{z-sing}}\,({\it{x\/}})\index{z-sing})\index{isin}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\-\index{f-iff}), \\ 
{\rm{list}}\,({\rm{isin-sing-proof}}\,({\it{x\/}}, {\it{y\/}})\index{isin-sing-proof}))\-\index{eqn-form-proof})\-)\-\index{chain-iff-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |-"(y in (S x)) <-> ((y = x) v (y in x))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-z-succ-proof-proves \\ \index{in-z-succ-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{in-z-succ-proof}}\,({\it{x\/}}, {\it{y\/}})\index{in-z-succ-proof}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{isin}, {\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-or})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-z-succ-proof; name this event `g0705'.\index{g0705}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{x-in-z-succ-x-proof} \\  
\=\+{\rm{x-in-z-succ-x-proof}}\,({\it{x\/}})\index{x-in-z-succ-x-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\it{x\/}}, {\it{x\/}})\index{isin})\index{f-or}, \\ 
{\rm{in-z-succ-proof}}\,({\it{x\/}}, {\it{x\/}})\index{in-z-succ-proof}, \\ 
{\rm{rt-expan-proof}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\it{x\/}})\index{isin}, \\ 
{\rm{ident-axiom-proof}}\,({\it{x\/}})\index{ident-axiom-proof})\-\index{rt-expan-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;; |-"(x in (S x))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  x-in-z-succ-x \\ \index{x-in-z-succ-x}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{x-in-z-succ-x-proof}}\,({\it{x\/}})\index{x-in-z-succ-x-proof}, {\rm{isin}}\,({\it{x\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable x-in-z-succ-x-proof; name this event `g0707'.\index{g0707}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-succ-equal-step-proof} \\  
\=\+{\rm{z-succ-equal-step-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-succ-equal-step-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin})\index{f-or}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{ext-axiom-eq-elim1-proof}}\,(\=\+{\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{x\/}}, \\ 
{\rm{x-in-z-succ-x-proof}}\,({\it{x\/}})\index{x-in-z-succ-x-proof})\-\index{ext-axiom-eq-elim1-proof}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin})\index{f-or}, \\ 
{\rm{in-z-succ-proof}}\,({\it{y\/}}, {\it{x\/}})\index{in-z-succ-proof})\-\index{iff-split-proof1})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((S x)=(S y)) -> ((x = y) v (x in y))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-succ-equal-step \\ \index{z-succ-equal-step}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-succ-equal-step-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-succ-equal-step-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin})\index{f-or})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-succ-equal-step-proof; name this event `g0708'.\index{g0708}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{equal-z-succ-proof} \\  
\=\+{\rm{equal-z-succ-proof}}\,({\it{x\/}}, {\it{y\/}})\index{equal-z-succ-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin})\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{z-succ-equal-step-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-succ-equal-step-proof}, \\ 
{\rm{symmetry-proof1}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{symmetry-proof1}, \\ 
{\rm{z-succ-equal-step-proof}}\,({\it{y\/}}, {\it{x\/}})\index{z-succ-equal-step-proof}, \\ 
{\rm{symmetry-proof1}}\,({\it{y\/}}, {\it{x\/}})\index{symmetry-proof1})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((S x)=(S y)) -> ((x = y) v ((x in y) ^ (y in x)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  equal-z-succ-proof-proves \\ \index{equal-z-succ-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-z-succ-proof}}\,({\it{x\/}}, {\it{y\/}})\index{equal-z-succ-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\index{f-or})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable equal-z-succ-proof; name this event `g0709'.\index{g0709}

\addvspace{18pt}
\begin{verbatim};;;;;;;The next series of defns and lemmas deal with the definition of "(I x)".
;;;;;;;"(I x)" holds of "x" iff "x" is transitive and linearly ordered by "in".
;;;;;;;"(Trans x)" iff "(forall y (forall z ((y in x) -> ((z in y) -> (z in x)))))".
;;;;;;;"x" is linearly ordered by "in" (i.e., "(Trich x)") iff
;;;;;;;"(forall y (forall z (((y in x) ^ (z in x)) -> ((y = z) v ((y in z) v (z in y))))))"
;;;;;;;Z-TRANS generates the transitivity assertion.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-trans1} \\  
{\rm{z-trans1}}\,({\it{x\/}}, {\it{u\/}}, {\it{v\/}})\index{z-trans1} $=$ {\rm{f-implies}}\,({\rm{f-and}}\,({\rm{isin}}\,({\it{u\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{v\/}}, {\it{u\/}})\index{isin})\index{f-and}, {\rm{isin}}\,({\it{v\/}}, {\it{x\/}})\index{isin})\index{f-implies}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-trans}$\;\;$
{\rm{z-trans}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans} $=$ $\forall\;${\it{y\/}}$\;$$\forall\;${\it{z\/}}$\;${\rm{z-trans1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans1}\index{forall}\index{forall}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-trans-instance-proof} \\  
\=\+{\rm{z-trans-instance-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-trans-instance-proof} \\ 
$=$$\;\;\;\;${\rm{subst-thm-b-proof}}\,({\rm{z-trans1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans1}, {\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\rm{list}}\,({\it{y1\/}}, {\it{z1\/}}))\index{subst-thm-b-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-trans-instance-proof-proves \\ \index{z-trans-instance-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y1\/}}, {\it{z1\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans}, {\rm{z-trans1}}\,({\it{x\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-trans1})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-trans-instance-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-trans-instance-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-trans-instance-proof; name this event `g0710'.\index{g0710}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-z-trans \\ \index{formula-z-trans}
\=\+{\rm{formula}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$(({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-z-trans \\ \index{collect-free-z-trans}
\=\+({\rm{var-list}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{2}})\index{var-list} $\wedge$ {\rm{nil-intersect}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\it{var\/}} $\in$ {\rm{collect-free}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans}, {\tt{0}})\index{collect-free}) \\ 
$=$$\;\;\;\;$({\it{var\/}} $\in$ {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-trans; name this event `g0715'.\index{g0715}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-z-trans-proof} \\  
\=\+{\rm{in-z-trans-proof}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}}, {\it{y\/}}, {\it{z\/}})\index{in-z-trans-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, {\rm{z-trans1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans1})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, {\rm{z-trans1}}\,({\it{y\/}}, {\it{x\/}}, {\it{z\/}})\index{z-trans1})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, \\ 
{\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans})\-\index{f-and}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-and}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{z\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-iff})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{z-trans-instance-proof}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans-instance-proof}, \\ 
{\rm{z-trans-instance-proof}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}}, {\it{x\/}}, {\it{z\/}})\index{z-trans-instance-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((Trans x) ^ (Trans y) ^ (x in y) ^ (y in x)) -> ((z in x) <-> (z in y))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-z-trans-proof-proves \\ \index{in-z-trans-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{v1\/}}, {\it{v2\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{v1\/}}, {\it{v2\/}}), {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{in-z-trans-proof}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}}, {\it{y\/}}, {\it{z\/}})\index{in-z-trans-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-and}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, {\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans})\index{f-and}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-and}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{z\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-iff})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-z-trans-proof; name this event `g0718'.\index{g0718}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-trans-equal-proof1} \\  
\=\+{\rm{z-trans-equal-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans-equal-proof1} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-and}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, {\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans})\index{f-and}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{list}}\,($\forall\;${\it{v1\/}}$\;${\rm{f-iff}}\,({\rm{isin}}\,({\it{v1\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{v1\/}}, {\it{y\/}})\index{isin})\index{f-iff}\index{forall}), \\ 
{\rm{list}}\,(\=\+{\rm{forall-intro-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, \\ 
{\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans})\-\index{f-and}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-and}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{v1\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{v1\/}}, {\it{y\/}})\index{isin})\index{f-iff}, \\ 
{\it{v1\/}}, \\ 
{\rm{in-z-trans-proof}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}}, {\it{y\/}}, {\it{v1\/}})\index{in-z-trans-proof})\-\index{forall-intro-proof}, \\ 
{\rm{iff-split-proof2}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
$\forall\;${\it{v1\/}}$\;${\rm{f-iff}}\,({\rm{isin}}\,({\it{v1\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{v1\/}}, {\it{y\/}})\index{isin})\index{f-iff}\index{forall}, \\ 
{\rm{ext-axiom-proof}}\,({\it{v1\/}}, {\it{x\/}}, {\it{y\/}})\index{ext-axiom-proof})\-\index{iff-split-proof2})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((Trans x) ^ (Trans y) ^ (x in y) ^ (y in x)) -> (x = y)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-trans-equal-proof1-proves \\ \index{z-trans-equal-proof1-proves}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{v1\/}}, {\it{v2\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{v1\/}}, {\it{v2\/}}), {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-trans-equal-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans-equal-proof1}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-and}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, {\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans})\index{f-and}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-trans-equal-proof1; name this event `g0719'.\index{g0719}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-succ-equal-proof} \\  
\=\+{\rm{z-succ-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-succ-equal-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, \\ 
{\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans})\-\index{f-and}, \\ 
{\rm{f-and}}\,({\rm{isin}}\,({\it{x\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-and})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{equal-z-succ-proof}}\,({\it{x\/}}, {\it{y\/}})\index{equal-z-succ-proof}, \\ 
{\rm{z-trans-equal-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans-equal-proof1})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((Trans x) -> ((Trans y) -> (((S x)=(S y)) -> (x = y))))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-succ-equal-proof-proves \\ \index{z-succ-equal-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{v1\/}}, {\it{v2\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{v1\/}}, {\it{v2\/}}), {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-succ-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-succ-equal-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans}}\,({\it{y\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trans}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-succ-equal-proof; name this event `g0720'.\index{g0720}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-trans-equal-proof1-proves; name this event `g0721'.\index{g0721}

\addvspace{18pt}

\noindent{\sc Event}:   Enable in-z-trans-proof-proves; name this event `g0722'.\index{g0722}

\addvspace{18pt}
\begin{verbatim};;;;;;;NUMERAL returns the set-theoretic ordinal corresponding to a Lisp natural number.
;;;;;;;The numeral for X will be represented as "|X|".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{numeral} \\  
\=\+{\rm{numeral}}\,({\it{x\/}})\index{numeral} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\sc{phi}}\index{phi} \\ 
{\bf else }{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-numeral \\ \index{termp-numeral}
\=\+(({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ ({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-numeral \\ \index{collect-free-numeral}
{\rm{collect-free}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{chain-or-proof} \\  
\=\+{\rm{chain-or-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{pf1\/}}, {\it{pf2\/}}, {\it{pf3\/}})\index{chain-or-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\it{a\/}}, {\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\it{b\/}}, {\it{d\/}})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\it{c\/}}, {\it{d\/}})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\it{d\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}, {\it{pf3\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "(A -> (B v C))",  |- "(B -> D)", and |- "(C -> D)", then |- "(A -> D)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  chain-or-proof-proves \\ \index{chain-or-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\rm{f-or}}\,({\it{b\/}}, {\it{c\/}})\index{f-or})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{d\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf3\/}}, {\rm{f-implies}}\,({\it{c\/}}, {\it{d\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{chain-or-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{pf1\/}}, {\it{pf2\/}}, {\it{pf3\/}})\index{chain-or-proof}, \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\it{d\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable chain-or-proof; name this event `g0605'.\index{g0605}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-trans-numeral-proof} \\  
\=\+{\rm{z-trans-numeral-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans-numeral-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{isin}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{isin})\index{f-not}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{isin}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-and}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\sc{phi}}\index{phi})\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{null-axiom-proof}}\,({\it{y\/}})\index{null-axiom-proof}))\-\index{tautconseq-proof} \\ 
{\bf else }{\rm{chain-proof}}\,(\=\+{\rm{f-and}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-and}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{isin}, \\ 
{\rm{list}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\-\index{f-and}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{isin})\-, \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\-\index{f-and}, \\ 
{\rm{f-and}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\-\index{f-and})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or}, \\ 
{\rm{in-z-succ-proof}}\,(\=\+{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral}, \\ 
{\it{y\/}})\-\index{in-z-succ-proof})\-\index{iff-split-proof1}))\-\index{tautconseq-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-\index{isin})\-\index{f-and}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\-\index{f-and}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z\/}})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{ident-axiom-proof}}\,({\it{z\/}})\index{ident-axiom-proof}, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{p}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-\index{p}, \\ 
{\rm{list}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-, \\ 
{\rm{list}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-)\-\index{equal-axiom2-proof})\-\index{detach-proof}, \\ 
{\rm{z-trans-numeral-proof}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-trans-numeral-proof})\-)\-\index{tautconseq-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{iff-split-proof2}}\,(\=\+{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or}, \\ 
{\rm{in-z-succ-proof}}\,(\=\+{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral}, \\ 
{\it{z\/}})\-\index{in-z-succ-proof})\-\index{iff-split-proof2}))\-\index{tautconseq-proof})\-)\-\index{chain-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((y in |X|) ^ (z in y)) -> (z in |X|)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-trans-numeral-proof-proves \\ \index{z-trans-numeral-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-trans-numeral-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans-numeral-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-and}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{isin})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-trans-numeral-proof; name this event `g0723'.\index{g0723}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  form-equal-proof-proves1 \\ \index{form-equal-proof-proves1}
\=\+(\=\+{\rm{formula}}\,({\it{forma\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{formb\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{form-equal}}\,({\it{forma\/}}, {\it{formb\/}}, {\it{list\/}})\index{form-equal} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{form-list}}\,({\it{list\/}}, {\it{symbols\/}})\index{form-list} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\rm{list-f-and}}\,({\it{list\/}})\index{list-f-and}, {\rm{f-iff}}\,({\it{forma\/}}, {\it{formb\/}})\index{f-iff})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{form-equal-proof}}\,({\it{forma\/}}, {\it{formb\/}}, {\it{list\/}})\index{form-equal-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-equal-proof-proves; name this event `g0724'.\index{g0724}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-del-pairs \\ \index{termp-del-pairs}
{\rm{termp}}\,({\it{x\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} $\rightarrow$ {\rm{termp}}\,({\rm{del-pairs}}\,({\it{y\/}}, {\it{vars\/}}, {\it{x\/}})\index{del-pairs}, {\tt{1}}, {\it{symbols\/}})\index{termp}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-parallel-subst \\ \index{length-parallel-subst}
{\rm{length}}\,({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{1}})\index{parallel-subst})\index{length} $=$ {\rm{length}}\,({\it{exp\/}})\index{length}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-parallel-subst \\ \index{termp-parallel-subst}
\=\+({\rm{termp}}\,({\it{term\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{parallel-subst}}\,({\it{term\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst}, {\it{flg\/}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-parallel-subst \\ \index{formula-parallel-subst}
\=\+({\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-equal-parallel-subst-again \\ \index{form-equal-parallel-subst-again}
\=\+(\=\+{\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{list\/}} $=$ {\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{form-equal}}\,({\it{exp\/}}, {\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\it{list\/}})\index{form-equal}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-equal-parallel-subst; name this event `g0729'.\index{g0729}

\addvspace{18pt}

\noindent{\sc Event}:   Enable form-equal; name this event `g0730'.\index{g0730}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-z-succ-proof1} \\  
\=\+{\rm{in-z-succ-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{in-z-succ-proof1} \\ 
$=$$\;\;\;\;${\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,({\it{y\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin})\index{f-or}, \\ 
{\rm{in-z-succ-proof}}\,({\it{x\/}}, {\it{y\/}})\index{in-z-succ-proof})\-\index{iff-split-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-trich1} \\  
\=\+{\rm{z-trich1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trich1} \\ 
$=$$\;\;\;\;${\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{isin}}\,({\it{y\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{x\/}})\index{isin})\index{f-and}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-or}}\,({\rm{isin}}\,({\it{y\/}}, {\it{z\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-or})\index{f-or})\-\index{f-implies}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ext-case1-proof} \\  
\=\+{\rm{ext-case1-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{x\/}})\index{ext-case1-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff}, \\ 
{\rm{list}}\,($\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-iff}\index{forall}), \\ 
{\rm{list}}\,(\=\+{\rm{iff-split-proof1}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
$\forall\;${\it{x\/}}$\;${\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\-\index{f-iff}\index{forall}, \\ 
{\rm{ext-axiom-proof}}\,({\it{x\/}}, {\it{terma\/}}, {\it{termb\/}})\index{ext-axiom-proof})\-\index{iff-split-proof1}, \\ 
{\rm{subst-thm-b-proof}}\,(\=\+{\rm{f-iff}}\,({\rm{isin}}\,({\it{x\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{x\/}}, {\it{termb\/}})\index{isin})\index{f-iff}, \\ 
{\rm{list}}\,({\it{x\/}}), \\ 
{\rm{list}}\,({\it{termc\/}}))\-\index{subst-thm-b-proof})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(a=b) -> ((c in a) <-> (c in b))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ext-case1 \\ \index{ext-case1}
\=\+(\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{terma\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{termb\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff})\-\index{f-implies}, \\ 
{\it{symbols\/}})\-\index{formula})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-case1-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\it{x\/}})\index{ext-case1-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-case1-proof; name this event `g0914'.\index{g0914}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ext-case-proof} \\  
\=\+{\rm{ext-case-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}})\index{ext-case-proof} \\ 
$=$$\;\;\;\;${\rm{ext-case1-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}}, {\rm{mk-bnd}}\,({\rm{list}}\,({\it{terma\/}}, {\it{termb\/}}), {\tt{0}})\index{mk-bnd})\index{ext-case1-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ext-axiom-case \\ \index{ext-axiom-case}
\=\+{\rm{formula}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff})\-\index{f-implies}, \\ 
{\it{symbols\/}})\-\index{formula} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-case-proof}}\,({\it{terma\/}}, {\it{termb\/}}, {\it{termc\/}})\index{ext-case-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{terma\/}}, {\it{termb\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{termc\/}}, {\it{terma\/}})\index{isin}, {\rm{isin}}\,({\it{termc\/}}, {\it{termb\/}})\index{isin})\index{f-iff})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-case-proof; name this event `g0915'.\index{g0915}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ext-subcase-proof2} \\  
\=\+{\rm{ext-subcase-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{ext-subcase-proof2} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-iff}}\,({\rm{isin}}\,({\it{z\/}}, {\it{x\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-iff})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{x\/}})\index{isin})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{ext-case-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{ext-case-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |-"(a=b) -> ((c in a) -> (c in b))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ext-subcase-proof2-proves \\ \index{ext-subcase-proof2-proves}
\=\+(({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-subcase-proof2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{ext-subcase-proof2}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-implies}}\,({\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{x\/}})\index{isin})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-subcase-proof2; name this event `g0737'.\index{g0737}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-trich-numeral-proof} \\  
\=\+{\rm{z-trich-numeral-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trich-numeral-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{isin}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{isin})\index{f-not}), \\ 
{\rm{z-trich1}}\,({\sc{phi}}\index{phi}, {\it{y\/}}, {\it{z\/}})\index{z-trich1}, \\ 
{\rm{list}}\,({\rm{null-axiom-proof}}\,({\it{y\/}})\index{null-axiom-proof}))\-\index{tautconseq-proof} \\ 
{\bf else }{\rm{chain-or-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,({\it{y\/}}, {\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\index{isin}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\index{isin})\-\index{f-and}, \\ 
{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{f-equal})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{isin})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{isin})\-\index{f-and}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{isin}, \\ 
{\rm{isin}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{isin})\-\index{f-and})\-\index{f-or})\-\index{f-or}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-or}}\,({\rm{isin}}\,({\it{y\/}}, {\it{z\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-or})\index{f-or}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-or})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{z-succ}}\,({\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\index{z-succ})\-\index{isin})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-and}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-and})\-\index{f-or})\-\index{f-or})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{in-z-succ-proof1}}\,(\=\+{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral}, \\ 
{\it{y\/}})\-\index{in-z-succ-proof1}, \\ 
{\rm{in-z-succ-proof1}}\,(\=\+{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral}, \\ 
{\it{z\/}})\-\index{in-z-succ-proof1})\-)\-\index{tautconseq-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, \\ 
{\rm{f-or}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-\index{isin})\-\index{f-or})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{trans-equal-proof2}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral}, \\ 
{\it{z\/}})\-\index{trans-equal-proof2}))\-\index{tautconseq-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-\index{isin})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{isin})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{f-equal}, \\ 
{\rm{f-or}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-\index{isin})\-\index{f-or})\-\index{f-or})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-and}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral})\-\index{isin})\-\index{f-and})\-\index{f-or})\-\index{f-or}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, \\ 
{\rm{f-or}}\,(\=\+{\rm{isin}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{z\/}}, \\ 
{\it{y\/}})\-\index{isin})\-\index{f-or})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{ext-subcase-proof2}}\,(\=\+{\it{y\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral}, \\ 
{\it{z\/}})\-\index{ext-subcase-proof2}, \\ 
{\rm{ext-subcase-proof2}}\,(\=\+{\it{z\/}}, \\ 
{\rm{numeral}}\,({\it{x\/}} $-\;1$)\index{numeral}, \\ 
{\it{y\/}})\-\index{ext-subcase-proof2}, \\ 
{\rm{z-trich-numeral-proof}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-trich-numeral-proof})\-)\-\index{tautconseq-proof})\-\index{chain-or-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((y in |X|) ^ (z in |X|)) -> ((y = z) ^ ((y in z) v (z in y)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-trich-numeral-proof-proves \\ \index{z-trich-numeral-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-trich-numeral-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trich-numeral-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{isin}}\,({\it{y\/}}, {\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{isin})\index{f-and}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-or}}\,({\rm{isin}}\,({\it{y\/}}, {\it{z\/}})\index{isin}, {\rm{isin}}\,({\it{z\/}}, {\it{y\/}})\index{isin})\index{f-or})\index{f-or})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-trich-numeral-proof; name this event `g0738'.\index{g0738}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ind-axiom-case-proof} \\  
\=\+{\rm{ind-axiom-case-proof}}\,({\it{form\/}}, {\it{x\/}}, {\it{pfbase\/}}, {\it{pfind\/}})\index{ind-axiom-case-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{subst}}\,({\it{form\/}}, {\it{x\/}}, {\sc{phi}}\index{phi}, {\tt{0}})\index{subst}, \\ 
$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-and}, \\ 
{\rm{subst}}\,({\it{form\/}}, {\it{x\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\tt{0}})\index{subst})\-\index{f-implies}\index{forall}, \\ 
{\rm{ind-axiom}}\,({\it{form\/}}, {\it{x\/}})\index{ind-axiom}, \\ 
{\rm{f-implies}}\,(\=\+$\forall\;${\it{x\/}}$\;${\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-implies}\index{forall}, \\ 
{\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\it{pfbase\/}}, \\ 
{\rm{generalise-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-and}, \\ 
{\rm{subst}}\,(\=\+{\it{form\/}}, \\ 
{\it{x\/}}, \\ 
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, \\ 
{\tt{0}})\-\index{subst})\-\index{f-implies}, \\ 
{\it{pfind\/}})\-\index{generalise-proof}, \\ 
{\rm{ind-axiom-proof}}\,({\it{form\/}}, {\it{x\/}})\index{ind-axiom-proof}, \\ 
{\rm{all-elim-proof}}\,({\it{x\/}}, {\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-implies})\index{all-elim-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "A[|0|]" and |- "((I x) ^ A[x]) -> A[(S x)]", then
;;;;;;; |- "(I x) -> A[x]".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ind-axiom-case \\ \index{ind-axiom-case}
\=\+(\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-implies}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfbase\/}}, {\rm{subst}}\,({\it{form\/}}, {\it{x\/}}, {\sc{phi}}\index{phi}, {\tt{0}})\index{subst}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pfind\/}}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-and}, {\rm{subst}}\,({\it{form\/}}, {\it{x\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\tt{0}})\index{subst})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ind-axiom-case-proof}}\,({\it{form\/}}, {\it{x\/}}, {\it{pfbase\/}}, {\it{pfind\/}})\index{ind-axiom-case-proof}, \\ 
{\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{form\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ind-axiom-case-proof; name this event `g0422'.\index{g0422}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  atomp-z-int \\ \index{atomp-z-int}
{\rm{atomp}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{symbols\/}})\index{atomp} $=$ (({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-z-int \\ \index{formula-z-int}
\=\+{\rm{formula}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{symbols\/}})\index{formula} \\ 
$=$$\;\;\;\;$(({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{phi-in-int-proof} \\  
\=\+{\rm{phi-in-int-proof}}\,({\it{x\/}})\index{phi-in-int-proof} \\ 
$=$$\;\;\;\;${\rm{ind-axiom-case-proof}}\,(\=\+{\rm{f-or}}\,({\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{isin})\index{f-or}, \\ 
{\it{x\/}}, \\ 
{\rm{rt-expan-proof}}\,(\=\+{\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\sc{phi}}\index{phi})\index{f-equal}, \\ 
{\rm{isin}}\,({\sc{phi}}\index{phi}, {\sc{phi}}\index{phi})\index{isin}, \\ 
{\rm{ident-axiom-proof}}\,({\sc{phi}}\index{phi})\index{ident-axiom-proof})\-\index{rt-expan-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\it{x\/}})\-\index{f-equal}, \\ 
{\rm{isin}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\-\index{isin})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\it{x\/}})\-\index{f-equal}, \\ 
{\rm{isin}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{isin})\-\index{f-or})\-\index{f-and}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\-\index{isin})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{iff-split-proof2}}\,(\=\+{\rm{isin}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\it{x\/}})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\it{x\/}})\-\index{isin})\-\index{f-or}, \\ 
{\rm{in-z-succ-proof}}\,(\=\+{\it{x\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{in-z-succ-proof})\-\index{iff-split-proof2}))\-\index{tautconseq-proof})\-\index{ind-axiom-case-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(I x) -> ((|0| = x) v (|0| in x))".  This is the only proof
;;;;;;;in this sequence of events in which the induction axiom schema of Z2 is employed.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  phi-in-int-proof-proves \\ \index{phi-in-int-proof-proves}
\=\+(\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\rm{f-or}}\,({\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{f-equal}, {\rm{isin}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{isin})\index{f-or})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{phi-in-int-proof}}\,({\it{x\/}})\index{phi-in-int-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable phi-in-int-proof; name this event `g0426'.\index{g0426}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ext-not-equal-proof} \\  
\=\+{\rm{ext-not-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{d\/}}, {\it{pfx\/}}, {\it{pfy\/}})\index{ext-not-equal-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{isin}}\,({\it{d\/}}, {\it{x\/}})\index{isin})\index{f-implies}, \\ 
{\rm{f-not}}\,({\rm{isin}}\,({\it{d\/}}, {\it{x\/}})\index{isin})\index{f-not})\-, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-not}, \\ 
{\rm{list}}\,({\rm{ext-axiom-eq-elim2-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{d\/}}, {\it{pfy\/}})\index{ext-axiom-eq-elim2-proof}, {\it{pfx\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |-"(not (d in x))" and |- "(d in y)", then |-"(not (x = y))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ext-not-equal-proof-proves \\ \index{ext-not-equal-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{d\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfx\/}}, {\rm{f-not}}\,({\rm{isin}}\,({\it{d\/}}, {\it{x\/}})\index{isin})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfy\/}}, {\rm{isin}}\,({\it{d\/}}, {\it{y\/}})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ext-not-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{d\/}}, {\it{pfx\/}}, {\it{pfy\/}})\index{ext-not-equal-proof}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ext-not-equal-proof; name this event `g0427'.\index{g0427}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-opair; name this event `g0428'.\index{g0428}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{phi-not-in-opair-proof} \\  
\=\+{\rm{phi-not-in-opair-proof}}\,({\it{u\/}}, {\it{v\/}})\index{phi-not-in-opair-proof} \\ 
$=$$\;\;\;\;${\rm{and-join-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{isin}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin})\index{f-not}, \\ 
{\rm{ext-not-equal-proof}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, \\ 
{\rm{null-axiom-proof}}\,({\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\index{null-axiom-proof}, \\ 
{\rm{isin-pair1-proof}}\,(\=\+{\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\-\index{isin-pair1-proof})\-\index{ext-not-equal-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-pair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-pair})\-\index{f-equal})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\index{f-equal})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\index{f-equal})\index{f-not})\-, \\ 
{\rm{f-not}}\,({\rm{isin}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin})\index{f-not}, \\ 
{\rm{list}}\,(\=\+{\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-opair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-opair})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-pair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-pair})\-\index{f-equal})\-\index{f-or}, \\ 
{\rm{pair-axiom-proof}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, \\ 
{\rm{z-pair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-pair})\-\index{pair-axiom-proof})\-\index{iff-split-proof1}, \\ 
{\rm{ext-not-equal-proof}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, \\ 
{\it{u\/}}, \\ 
{\rm{null-axiom-proof}}\,({\it{u\/}})\index{null-axiom-proof}, \\ 
{\rm{isin-sing1-proof}}\,({\it{u\/}})\index{isin-sing1-proof})\-\index{ext-not-equal-proof}, \\ 
{\rm{ext-not-equal-proof}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair}, \\ 
{\it{u\/}}, \\ 
{\rm{null-axiom-proof}}\,({\it{u\/}})\index{null-axiom-proof}, \\ 
{\rm{isin-pair1-proof}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{isin-pair1-proof})\-\index{ext-not-equal-proof})\-)\-\index{tautconseq-proof})\-\index{and-join-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((not (|0| in <u, v>)) ^ (not (|0| in <u, v>)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  phi-not-in-opair-proof-proves \\ \index{phi-not-in-opair-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{u\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{v\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-and}}\,(\=\+{\rm{f-not}}\,({\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{isin}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin})\index{f-not})\-\index{f-and}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{phi-not-in-opair-proof}}\,({\it{u\/}}, {\it{v\/}})\index{phi-not-in-opair-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable phi-not-in-opair-proof; name this event `g0429'.\index{g0429}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-opair; name this event `g0430'.\index{g0430}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{not-int-opair-proof} \\  
\=\+{\rm{not-int-opair-proof}}\,({\it{u\/}}, {\it{v\/}}, {\it{x\/}})\index{not-int-opair-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{z-int}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{isin}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-and}}\,(\=\+{\rm{f-not}}\,({\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{isin}}\,({\sc{phi}}\index{phi}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{isin})\index{f-not})\-\index{f-and})\-, \\ 
{\rm{f-not}}\,({\rm{z-int}}\,({\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{z-int})\index{f-not}, \\ 
{\rm{list}}\,(\=\+{\rm{subst-rule-proof}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{isin}}\,({\sc{phi}}\index{phi}, {\it{x\/}})\index{isin})\-\index{f-or})\-\index{f-implies}, \\ 
{\it{x\/}}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair}, \\ 
{\rm{phi-in-int-proof}}\,({\it{x\/}})\index{phi-in-int-proof})\-\index{subst-rule-proof}, \\ 
{\rm{phi-not-in-opair-proof}}\,({\it{u\/}}, {\it{v\/}})\index{phi-not-in-opair-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(not (I <u v>))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  not-int-opair-proof-proves \\ \index{not-int-opair-proof-proves}
\=\+(\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{u\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{v\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{not-int-opair-proof}}\,({\it{u\/}}, {\it{v\/}}, {\it{x\/}})\index{not-int-opair-proof}, \\ 
{\rm{f-not}}\,({\rm{z-int}}\,({\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{z-int})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable not-int-opair-proof; name this event `g0868'.\index{g0868}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-trich}$\;\;$
{\rm{z-trich}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trich} $=$ $\forall\;${\it{v1\/}}$\;$$\forall\;${\it{v2\/}}$\;${\rm{z-trich1}}\,({\it{x\/}}, {\it{v1\/}}, {\it{v2\/}})\index{z-trich1}\index{forall}\index{forall}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-int-defn} \\  
{\rm{z-int-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-int-defn} $=$ {\rm{f-and}}\,({\rm{z-trans}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trans}, {\rm{z-trich}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-trich})\index{f-and}
\end{tabbing}

\noindent{\sc Event}:   Enable pred-defn-proof; name this event `g0609'.\index{g0609}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subst-f-iff \\ \index{subst-f-iff}
\=\+{\rm{subst}}\,({\rm{f-iff}}\,({\it{x\/}}, {\it{y\/}})\index{f-iff}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{f-iff}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-iff}\-
\end{tabbing}

\noindent{\sc Event}:   Disable z-trans; name this event `g0872'.\index{g0872}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subset-del \\ \index{subset-del}
{\rm{variable}}\,({\it{x\/}})\index{variable} $\rightarrow$ ({\rm{subset}}\,({\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del}, {\it{z\/}})\index{subset} $=$ {\rm{subset}}\,({\it{y\/}}, {\rm{cons}}\,({\it{x\/}}, {\it{z\/}}))\index{subset})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-int-defn-proof} \\  
\=\+{\rm{z-int-defn-proof}}\,({\it{term\/}})\index{z-int-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{f-iff}}\,({\rm{z-int}}\,({\tt{0}})\index{z-int}, {\rm{z-int-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-int-defn})\index{f-iff}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{pred-defn-proof}}\,({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{list}}\,({\tt{0}}), {\rm{z-int-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-int-defn})\index{pred-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Enable not-member-varlist; name this event `g0613'.\index{g0613}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-int-hyps} \\  
\=\+{\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\rm{rest-of}}\,({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\it{symbols\/}})\index{rest-of}) \\ 
$\wedge$$\;\;\;\;$({\rm{assoc}}\,({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\it{defns\/}}) $=$ {\rm{z-int-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-int-defn}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-int-defn-proof-proves \\ \index{z-int-defn-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{1}}, {\tt{2}}), {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-int-defn-proof}}\,({\it{term\/}})\index{z-int-defn-proof}, \\ 
{\rm{f-iff}}\,({\rm{z-int}}\,({\it{term\/}})\index{z-int}, {\rm{f-and}}\,({\rm{z-trans}}\,({\it{term\/}}, {\tt{1}}, {\tt{2}})\index{z-trans}, {\rm{z-trich}}\,({\it{term\/}}, {\tt{1}}, {\tt{2}})\index{z-trich})\index{f-and})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-int-defn-proof; name this event `g0639'.\index{g0639}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  covering-numeral \\ \index{covering-numeral}
{\rm{covering}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\bf{nil}}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-int-numeral-proof} \\  
\=\+{\rm{z-int-numeral-proof}}\,({\it{x\/}})\index{z-int-numeral-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{z-int}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{z-int}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-trans}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral}, {\tt{1}}, {\tt{2}})\index{z-trans}, \\ 
{\rm{z-trich}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral}, {\tt{1}}, {\tt{2}})\index{z-trich})\-\index{f-and}, \\ 
{\rm{z-int-defn-proof}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{z-int-defn-proof}, \\ 
{\rm{and-join-proof}}\,(\=\+{\rm{z-trans}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral}, {\tt{1}}, {\tt{2}})\index{z-trans}, \\ 
{\rm{z-trich}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral}, {\tt{1}}, {\tt{2}})\index{z-trich}, \\ 
{\rm{generalise-proof}}\,(\=\+{\tt{1}}, \\ 
$\forall\;${\tt{2}}$\;${\rm{z-trans1}}\,(\=\+{\rm{numeral}}\,({\it{x\/}})\index{numeral}, \\ 
{\tt{1}}, \\ 
{\tt{2}})\-\index{z-trans1}\index{forall}, \\ 
{\rm{generalise-proof}}\,(\=\+{\tt{2}}, \\ 
{\rm{z-trans1}}\,(\=\+{\rm{numeral}}\,({\it{x\/}})\index{numeral}, \\ 
{\tt{1}}, \\ 
{\tt{2}})\-\index{z-trans1}, \\ 
{\rm{z-trans-numeral-proof}}\,(\=\+{\it{x\/}}, \\ 
{\tt{1}}, \\ 
{\tt{2}})\-\index{z-trans-numeral-proof})\-\index{generalise-proof})\-\index{generalise-proof}, \\ 
{\rm{generalise-proof}}\,(\=\+{\tt{1}}, \\ 
$\forall\;${\tt{2}}$\;${\rm{z-trich1}}\,(\=\+{\rm{numeral}}\,({\it{x\/}})\index{numeral}, \\ 
{\tt{1}}, \\ 
{\tt{2}})\-\index{z-trich1}\index{forall}, \\ 
{\rm{generalise-proof}}\,(\=\+{\tt{2}}, \\ 
{\rm{z-trich1}}\,(\=\+{\rm{numeral}}\,({\it{x\/}})\index{numeral}, \\ 
{\tt{1}}, \\ 
{\tt{2}})\-\index{z-trich1}, \\ 
{\rm{z-trich-numeral-proof}}\,(\=\+{\it{x\/}}, \\ 
{\tt{1}}, \\ 
{\tt{2}})\-\index{z-trich-numeral-proof})\-\index{generalise-proof})\-\index{generalise-proof})\-\index{and-join-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-int-defn; name this event `g0744'.\index{g0744}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-int-hyps; name this event `g0614'.\index{g0614}

\addvspace{18pt}
\begin{verbatim};;;;;;; |- "(I |X|)".  At this point we are overloading our notation.  Though |X|
;;;;;;;appears in quotes, it denotes the numeral which is the result of
;;;;;;;evaluating |X| for the given X.  So the metatheorem is that "(I |X|)" is 
;;;;;;;a Z2 theorem for any X.  
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-int-numeral-proof-proves \\ \index{z-int-numeral-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-int-numeral-proof}}\,({\it{x\/}})\index{z-int-numeral-proof}, {\rm{z-int}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{z-int}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-int-numeral-proof; name this event `g0906'.\index{g0906}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  covering-z-int \\ \index{covering-z-int}
{\rm{covering}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-z-int \\ \index{collect-z-int}
{\rm{collect-free}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-int \\ \index{subst-z-int}
{\rm{subst}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\rm{z-int}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-int}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-int \\ \index{parallel-subst-z-int}
\=\+{\rm{parallel-subst}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-int}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{parallel-subst})\index{z-int}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-int; name this event `g0907'.\index{g0907}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-car}$\;\;$
{\rm{z-car}}\,({\it{x\/}})\index{z-car} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-cdr}$\;\;$
{\rm{z-cdr}}\,({\it{x\/}})\index{z-cdr} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}

\noindent{\sc Event}:   Disable z-opair; name this event `g0628'.\index{g0628}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  covering-z-opair \\ \index{covering-z-opair}
\=\+{\rm{covering}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\it{var\/}}, {\tt{0}})\index{covering} \\ 
$=$$\;\;\;\;${\rm{covering}}\,({\rm{list}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}), {\it{var\/}}, {\tt{1}})\index{covering}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-opair \\ \index{subst-z-opair}
\=\+{\rm{subst}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{z-opair}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\rm{subst}}\,({\it{y\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-opair}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-opair \\ \index{parallel-subst-z-opair}
\=\+{\rm{parallel-subst}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-opair}}\,(\=\+{\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{parallel-subst}}\,({\it{y\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\-\index{z-opair}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-opair; name this event `g0608'.\index{g0608}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-car-unique-step1-proof} \\  
\=\+{\rm{z-car-unique-step1-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-unique-step1-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{trans-equal-proof1}}\,(\=\+{\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair}, \\ 
{\it{x\/}}, \\ 
{\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\-\index{trans-equal-proof1}, \\ 
{\rm{z-car-unique-proof}}\,({\it{y\/}}, {\it{z\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-car-unique-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(x = <y, z>) -> ((x = <y1, z1>) -> (y = y1))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-car-unique-step1-proof-proves \\ \index{z-car-unique-step1-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-car-unique-step1-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-unique-step1-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-unique-step1-proof; name this event `g0630'.\index{g0630}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-car-unique-step2-proof} \\  
\=\+{\rm{z-car-unique-step2-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-unique-step2-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{z\/}}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{z1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+$\forall\;${\it{z1\/}}$\;${\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies}\index{forall}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{list}}\,(\=\+{\rm{forall-intro-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{f-implies}, \\ 
{\it{z1\/}}, \\ 
{\rm{z-car-unique-step1-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{z-car-unique-step1-proof})\-\index{forall-intro-proof}, \\ 
{\rm{forall-over-implies-proof}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{forall-over-implies-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{forsome-intro-proof}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\index{prop-axiom-proof})\-\index{forsome-intro-proof}))\-\index{tautconseq-proof})\-)\-\index{chain-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-car-unique-step2-proof-proves \\ \index{z-car-unique-step2-proof-proves}
\=\+(\=\+{\rm{var-set}}\,({\rm{list}}\,({\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{z\/}}, {\it{z1\/}}), {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-car-unique-step2-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-unique-step2-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{z1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-unique-step2-proof; name this event `g0637'.\index{g0637}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subst-subst \\ \index{subst-subst}
\=\+(\=\+({\it{y\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\it{flg\/}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{y\/}})\index{variable})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{subst}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\it{flg\/}})\index{subst}, {\it{y\/}}, {\it{x\/}}, {\it{flg\/}})\index{subst} $=$ {\it{exp\/}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-subst \\ \index{not-free-subst}
\=\+({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free}) \\ 
$\rightarrow$$\;\;\;\;$({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{flg\/}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{rename-bndvar-proof} \\  
\=\+{\rm{rename-bndvar-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{rename-bndvar-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{subst}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst}, {\it{y\/}}, {\it{x\/}}, {\tt{0}})\index{subst}, \\ 
{\rm{forsome}}\,({\it{y\/}}, {\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst})\index{forsome}, \\ 
{\rm{subst-axiom-proof}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst}, {\it{y\/}}, {\it{x\/}})\index{subst-axiom-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rename-bndvar-proof-proves \\ \index{rename-bndvar-proof-proves}
\=\+(\=\+({\it{y\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{exp-inst\/}} $=$ {\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{rename-bndvar-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{rename-bndvar-proof}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{exp\/}})\index{forsome}, {\rm{forsome}}\,({\it{y\/}}, {\it{exp-inst\/}})\index{forsome})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable rename-bndvar-proof; name this event `g0640'.\index{g0640}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-car-uniqueness-case-proof} \\  
\=\+{\rm{z-car-uniqueness-case-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-uniqueness-case-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{z1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{z1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{rename-bndvar-proof}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\-\index{rename-bndvar-proof}, \\ 
{\rm{z-car-unique-step2-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-unique-step2-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-car-uniqueness-case-proof-proves \\ \index{z-car-uniqueness-case-proof-prove!s}
\=\+(\=\+{\rm{var-set}}\,({\rm{list}}\,({\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{z\/}}, {\it{z1\/}}), {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-car-uniqueness-case-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-uniqueness-case-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-uniqueness-case-proof; name this event `g0645'.\index{g0645}

\addvspace{18pt}
\begin{verbatim};;;;;;;The definition of the Z2 analogue of CAR.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-car-defn} \\  
\=\+{\rm{z-car-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}})\index{z-car-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{forsome}}\,({\it{y1\/}}, {\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{f-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The existence part of the admissibility proof of "(Z-car x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-car-exis-proof} \\  
\=\+{\rm{z-car-exis-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-car-exis-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-f-if-case-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{forsome}}\,({\it{y\/}}, {\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{z\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome})\-\index{forsome})\index{prop-axiom-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{forsome}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{z\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome})\-\index{forsome})\index{f-not}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{exis-fun-intro-proof}}\,(\=\+{\it{y\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{exis-fun-intro-proof}))\-\index{tautconseq-proof})\-\index{forsome-f-if-case-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-car-exis-proof-proves \\ \index{z-car-exis-proof-proves}
\=\+(\=\+{\rm{var-set}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{forsome}}\,({\it{y\/}}, {\rm{z-car-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{y\/}}, {\it{z\/}})\index{z-car-defn})\index{forsome}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-car-exis-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-car-exis-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-exis-proof; name this event `g0646'.\index{g0646}

\addvspace{18pt}
\begin{verbatim};;;;;;;The uniqueness part of the admissibility proof of "(Z-car x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-car-uniqueness-proof} \\  
\=\+{\rm{z-car-uniqueness-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-uniqueness-proof} \\ 
$=$$\;\;\;\;${\rm{unique-f-if-proof}}\,(\=\+{\rm{forsome}}\,({\it{y\/}}, {\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y1\/}}, {\sc{phi}}\index{phi})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{y1\/}})\index{f-equal}, \\ 
{\rm{z-car-uniqueness-case-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-car-uniqueness-case-proof}, \\ 
{\rm{unique-fun-intro-proof}}\,({\it{y\/}}, {\it{y1\/}}, {\sc{phi}}\index{phi})\index{unique-fun-intro-proof})\-\index{unique-f-if-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-car-defn-proof} \\  
\=\+{\rm{z-car-defn-proof}}\,({\it{term\/}})\index{z-car-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{z-car-defn}}\,({\tt{0}}, {\rm{z-car}}\,({\tt{0}})\index{z-car}, {\tt{1}}, {\tt{2}})\index{z-car-defn}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{1}}, \\ 
{\tt{3}}, \\ 
{\rm{z-car-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{1}}, {\tt{2}})\index{z-car-defn}, \\ 
{\rm{z-car-exis-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-car-exis-proof}, \\ 
{\rm{z-car-uniqueness-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{3}}, {\tt{2}}, {\tt{4}})\index{z-car-uniqueness-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Enable not-member-delete-equals; name this event `g0648'.\index{g0648}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  covering-forsome \\ \index{covering-forsome}
\=\+{\rm{variable}}\,({\it{x\/}})\index{variable} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{covering}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{y\/}})\index{forsome}, {\it{var\/}}, {\tt{0}})\index{covering} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{memb}}\,({\it{var\/}}, {\rm{collect-free}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{y\/}})\index{forsome}, {\tt{0}})\index{collect-free})\index{memb} \\ 
{\bf then }{\rm{cons}}\,({\it{x\/}}, {\rm{covering}}\,({\it{y\/}}, {\it{var\/}}, {\tt{0}})\index{covering}) \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-car-hyps} \\  
\=\+{\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$=$$\;\;\;\;$(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\rm{rest-of}}\,({\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{assoc}}\,({\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-car-defn}}\,({\tt{0}}, {\rm{z-car}}\,({\tt{0}})\index{z-car}, {\tt{1}}, {\tt{2}})\index{z-car-defn}))\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The admissibility of "(Z-car x)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-car-defn-proof-proves \\ \index{z-car-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-car-defn}}\,({\it{term\/}}, {\rm{z-car}}\,({\it{term\/}})\index{z-car}, {\tt{1}}, {\tt{2}})\index{z-car-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-car-defn-proof}}\,({\it{term\/}})\index{z-car-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-defn-proof; name this event `g0649'.\index{g0649}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-car-hyps; name this event `g0678'.\index{g0678}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-cdr-unique-step1-proof} \\  
\=\+{\rm{z-cdr-unique-step1-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-unique-step1-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{trans-equal-proof1}}\,(\=\+{\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair}, \\ 
{\it{x\/}}, \\ 
{\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\-\index{trans-equal-proof1}, \\ 
{\rm{z-cdr-unique-proof}}\,({\it{y\/}}, {\it{z\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-cdr-unique-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-cdr-unique-step1-proof-proves \\ \index{z-cdr-unique-step1-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-cdr-unique-step1-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-unique-step1-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal}, {\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr-unique-step1-proof; name this event `g0647'.\index{g0647}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-cdr-unique-step2-proof} \\  
\=\+{\rm{z-cdr-unique-step2-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-unique-step2-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{y1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+$\forall\;${\it{y1\/}}$\;${\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies}\index{forall}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{list}}\,(\=\+{\rm{forall-intro-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal})\-\index{f-implies}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-cdr-unique-step1-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{z-cdr-unique-step1-proof})\-\index{forall-intro-proof}, \\ 
{\rm{forall-over-implies-proof}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal})\-\index{forall-over-implies-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{forsome-intro-proof}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\it{z1\/}})\-\index{f-equal})\index{prop-axiom-proof})\-\index{forsome-intro-proof}))\-\index{tautconseq-proof})\-)\-\index{chain-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-cdr-unique-step2-proof-proves \\ \index{z-cdr-unique-step2-proof-proves}
\=\+(\=\+{\rm{var-set}}\,({\rm{list}}\,({\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{y\/}}, {\it{y1\/}}), {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-cdr-unique-step2-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-unique-step2-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{y1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr-unique-step2-proof; name this event `g0650'.\index{g0650}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-cdr-uniqueness-case-proof} \\  
\=\+{\rm{z-cdr-uniqueness-case-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-uniqueness-case-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{y1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y1\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y1\/}}, \\ 
{\it{z1\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{rename-bndvar-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\-\index{rename-bndvar-proof}, \\ 
{\rm{z-cdr-unique-step2-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-unique-step2-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-cdr-uniqueness-case-proof-proves \\ \index{z-cdr-uniqueness-case-proof-prove!s}
\=\+(\=\+{\rm{var-set}}\,({\rm{list}}\,({\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{y\/}}, {\it{y1\/}}), {\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-cdr-uniqueness-case-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-uniqueness-case-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;;The definition of the Z2 analogue of CDR, "(Z-cdr x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-cdr-defn} \\  
\=\+{\rm{z-cdr-defn}}\,({\it{x\/}}, {\it{z\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-cdr-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{forsome}}\,({\it{z1\/}}, {\rm{forsome}}\,({\it{y1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{y1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{f-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The existence part of the admissibility of "(Z-cdr x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-cdr-exis-proof} \\  
\=\+{\rm{z-cdr-exis-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-cdr-exis-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-f-if-case-proof}}\,(\=\+{\it{z\/}}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\sc{phi}}\index{phi})\index{f-equal}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{forsome}}\,(\=\+{\it{z\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome})\-\index{forsome})\index{prop-axiom-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{forsome}}\,(\=\+{\it{z\/}}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{forsome}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{z\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{y\/}}, \\ 
{\it{z\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome})\-\index{forsome})\index{f-not}, \\ 
{\rm{forsome}}\,(\=\+{\it{z\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{z\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{exis-fun-intro-proof}}\,(\=\+{\it{z\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{exis-fun-intro-proof}))\-\index{tautconseq-proof})\-\index{forsome-f-if-case-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-cdr-exis-proof-proves \\ \index{z-cdr-exis-proof-proves}
\=\+(\=\+{\rm{var-set}}\,({\rm{list}}\,({\it{y\/}}, {\it{z\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-cdr-exis-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-cdr-exis-proof}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{z-cdr-defn}}\,({\it{x\/}}, {\it{z\/}}, {\it{y\/}}, {\it{z\/}})\index{z-cdr-defn})\index{forsome}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr-exis-proof; name this event `g0651'.\index{g0651}

\addvspace{18pt}
\begin{verbatim};;;;;;;The uniqueness part of the admissibility of "(Z-cdr x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-cdr-uniqueness-proof} \\  
\=\+{\rm{z-cdr-uniqueness-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-uniqueness-proof} \\ 
$=$$\;\;\;\;${\rm{unique-f-if-proof}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\sc{phi}}\index{phi})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{z1\/}}, {\sc{phi}}\index{phi})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{z\/}}, {\it{z1\/}})\index{f-equal}, \\ 
{\rm{z-cdr-uniqueness-case-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}}, {\it{z1\/}})\index{z-cdr-uniqueness-case-proof}, \\ 
{\rm{unique-fun-intro-proof}}\,({\it{z\/}}, {\it{z1\/}}, {\sc{phi}}\index{phi})\index{unique-fun-intro-proof})\-\index{unique-f-if-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-cdr-defn-proof} \\  
\=\+{\rm{z-cdr-defn-proof}}\,({\it{term\/}})\index{z-cdr-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{z-cdr-defn}}\,({\tt{0}}, {\rm{z-cdr}}\,({\tt{0}})\index{z-cdr}, {\tt{1}}, {\tt{2}})\index{z-cdr-defn}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{2}}, \\ 
{\tt{4}}, \\ 
{\rm{z-cdr-defn}}\,({\tt{0}}, {\tt{2}}, {\tt{1}}, {\tt{2}})\index{z-cdr-defn}, \\ 
{\rm{z-cdr-exis-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-cdr-exis-proof}, \\ 
{\rm{z-cdr-uniqueness-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{3}}, {\tt{2}}, {\tt{4}})\index{z-cdr-uniqueness-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-cdr-hyps} \\  
\=\+{\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$=$$\;\;\;\;$(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\rm{rest-of}}\,({\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{assoc}}\,({\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-cdr-defn}}\,({\tt{0}}, {\rm{z-cdr}}\,({\tt{0}})\index{z-cdr}, {\tt{1}}, {\tt{2}})\index{z-cdr-defn}))\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The admissibility of "(Z-cdr x)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-cdr-defn-proof-proves \\ \index{z-cdr-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-cdr-defn}}\,({\it{term\/}}, {\rm{z-cdr}}\,({\it{term\/}})\index{z-cdr}, {\tt{1}}, {\tt{2}})\index{z-cdr-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-cdr-defn-proof}}\,({\it{term\/}})\index{z-cdr-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr-hyps; name this event `g0679'.\index{g0679}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-cdr-defn-proof; name this event `g0655'.\index{g0655}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-num-nequal-proof} \\  
\=\+{\rm{list-num-nequal-proof}}\,({\it{n\/}}, {\it{u\/}}, {\it{v\/}})\index{list-num-nequal-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral})\index{z-int}, \\ 
{\rm{z-int}}\,({\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{z-int})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{z-int}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral})\index{z-int}, \\ 
{\rm{f-not}}\,({\rm{z-int}}\,({\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{z-int})\index{f-not})\-, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\index{f-not}, \\ 
{\rm{list}}\,(\=\+{\rm{equal-axiom2-proof}}\,(\=\+{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{list}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral}), \\ 
{\rm{list}}\,({\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair}))\-\index{equal-axiom2-proof}, \\ 
{\rm{z-int-numeral-proof}}\,({\it{n\/}})\index{z-int-numeral-proof}, \\ 
{\rm{not-int-opair-proof}}\,({\it{u\/}}, {\it{v\/}}, {\tt{0}})\index{not-int-opair-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Disable z-int; name this event `g0659'.\index{g0659}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  equal-axiom2-z-int \\ \index{equal-axiom2-z-int}
\=\+({\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} $\wedge$ ({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-axiom2-proof}}\,({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{list}}\,({\it{x\/}}), {\rm{list}}\,({\it{y\/}}))\index{equal-axiom2-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-implies}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\rm{z-int}}\,({\it{y\/}})\index{z-int})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-int; name this event `g0660'.\index{g0660}

\addvspace{18pt}
\begin{verbatim};;;;;;;An important metatheorem: |- "(not (|N| = <u v>))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  list-num-nequal-proof-proves \\ \index{list-num-nequal-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{u\/}}, {\it{v\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-not}}\,({\rm{f-equal}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\index{f-not}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{list-num-nequal-proof}}\,({\it{n\/}}, {\it{u\/}}, {\it{v\/}})\index{list-num-nequal-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-num-nequal-proof; name this event `g0658'.\index{g0658}

\addvspace{18pt}
\begin{verbatim};;;;;;;The Goedel-encoding.  It takes Lisp data-structures built from numbers using
;;;;;;;CONS to sets in Z2 which are formed from numerals using ordered pairing.
;;;;;;;We represent the encoding of X as <X>.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{ncode} \\  
\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{z-opair}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, {\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\index{z-opair} \\ 
{\bf else }{\rm{numeral}}\,({\it{x\/}})\index{numeral}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-ncode \\ \index{termp-ncode}
\=\+(({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ ({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ ({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn} $\in$ {\it{symbols\/}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-ncode \\ \index{collect-free-ncode}
{\rm{collect-free}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ncode-equal} \\  
\=\+{\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }{\rm{ncode-equal}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{ncode-equal} $\wedge$ {\rm{ncode-equal}}\,({\rm{cdr}}\,({\it{x\/}}), {\rm{cdr}}\,({\it{y\/}}))\index{ncode-equal} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{y\/}})$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{y\/}} $\simeq {\tt{0}}$ \\ 
{\bf elseif }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf else }{\rm{ncode-equal}}\,({\it{x\/}} $-\;1$, {\it{y\/}} $-\;1$)\index{ncode-equal}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-trans; name this event `g0756'.\index{g0756}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-trich; name this event `g0757'.\index{g0757}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ncode-nequal-step-proof} \\  
\=\+{\rm{ncode-nequal-step-proof}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{x\/}}, {\it{y\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{ncode-nequal-step-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-equal}}\,({\it{x1\/}}, {\it{y1\/}})\index{f-equal})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-not}, \\ 
{\it{pf1\/}}, \\ 
{\rm{contra-pos-proof1}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x1\/}}, {\it{y1\/}})\index{f-equal}, {\it{pf2\/}})\index{contra-pos-proof1})\-\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-trans2}$\;\;$
{\rm{z-trans2}}\,({\it{x\/}})\index{z-trans2} $=$ {\rm{z-trans}}\,({\it{x\/}}, {\tt{1}}, {\tt{2}})\index{z-trans}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-trich2}$\;\;$
{\rm{z-trich2}}\,({\it{x\/}})\index{z-trich2} $=$ {\rm{z-trich}}\,({\it{x\/}}, {\tt{1}}, {\tt{2}})\index{z-trich}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ncode-nequal-proof} \\  
\=\+{\rm{ncode-nequal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-nequal-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }\=\+{\bf if }$\neg$ {\rm{ncode-equal}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{ncode-equal} \\ 
{\bf then }{\rm{ncode-nequal-step-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,(\=\+{\rm{car}}\,({\it{x\/}}), \\ 
{\rm{car}}\,({\it{y\/}}))\-\index{ncode-nequal-proof}, \\ 
{\rm{z-car-unique-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-\index{z-car-unique-proof})\-\index{ncode-nequal-step-proof} \\ 
{\bf else }{\rm{ncode-nequal-step-proof}}\,(\=\+{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,(\=\+{\rm{cdr}}\,({\it{x\/}}), \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{ncode-nequal-proof}, \\ 
{\rm{z-cdr-unique-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-\index{z-cdr-unique-proof})\-\index{ncode-nequal-step-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{ncode-nequal-step-proof}}\,(\=\+{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{list-num-nequal-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{list-num-nequal-proof}, \\ 
{\rm{symmetry-proof1}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode})\-\index{symmetry-proof1})\-\index{ncode-nequal-step-proof}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }{\rm{list-num-nequal-proof}}\,({\it{x\/}}, {\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, {\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\index{list-num-nequal-proof} \\ 
{\bf elseif }{\it{x\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{ext-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode}, \\ 
{\rm{null-axiom-proof}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{null-axiom-proof}, \\ 
{\rm{x-in-z-succ-x-proof}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{x-in-z-succ-x-proof})\-\index{ext-not-equal-proof} \\ 
{\bf elseif }{\it{y\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{ncode-nequal-step-proof}}\,(\=\+{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ext-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{null-axiom-proof}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{null-axiom-proof}, \\ 
{\rm{x-in-z-succ-x-proof}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{x-in-z-succ-x-proof})\-\index{ext-not-equal-proof}, \\ 
{\rm{symmetry-proof1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{symmetry-proof1})\-\index{ncode-nequal-step-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\-\index{f-equal})\index{f-not})\-, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal})\index{f-not}, \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{z-trich2}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-trich2})\-\index{f-and})\-\index{f-iff})\-, \\ 
{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{list}}\,(\=\+{\rm{z-int-numeral-proof}}\,({\it{x\/}} $-\;1$)\index{z-int-numeral-proof}, \\ 
{\rm{z-int-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-int-defn-proof})\-)\-\index{tautconseq-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{z-trich2}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-trich2})\-\index{f-and})\-\index{f-iff})\-, \\ 
{\rm{z-trans2}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-trans2}, \\ 
{\rm{list}}\,(\=\+{\rm{z-int-numeral-proof}}\,({\it{y\/}} $-\;1$)\index{z-int-numeral-proof}, \\ 
{\rm{z-int-defn-proof}}\,({\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode})\index{z-int-defn-proof})\-)\-\index{tautconseq-proof}, \\ 
{\rm{z-succ-equal-proof}}\,(\=\+{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}} $-\;1$)\index{ncode}, \\ 
{\tt{1}}, \\ 
{\tt{2}})\-\index{z-succ-equal-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{x\/}} $-\;1$, {\it{y\/}} $-\;1$)\index{ncode-nequal-proof})\-)\-\index{tautconseq-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;A crucial metatheorem: If X is different from Y, then |- "(not (<X>=<Y>))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ncode-nequal-proof-proves \\ \index{ncode-nequal-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ncode-nequal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-nequal-proof}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ncode-nequal-proof; name this event `g0758'.\index{g0758}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  ncode-nequal-proof-proves-again \\ \index{ncode-nequal-proof-proves-again}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-not}}\,({\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal})\index{f-not}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{ncode-nequal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-nequal-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ncode-nequal-proof-proves; name this event `g0760'.\index{g0760}

\addvspace{18pt}
\begin{verbatim};;;;;;;What we would like to do next is to demonstrate the admissibility of
;;;;;;;Z2 analogues of several Lisp functions which are used to define the Lisp
;;;;;;;interpreter.  These include the primitive Lisp functions like LISTP, CAR, CDR,
;;;;;;;CONS, ZEROP, NUMBERP, ADD1, SUB1, and EQUAL.  The definitions of these Z2
;;;;;;;functions involve the use of the logical-if connective.  For example, the
;;;;;;;definition of the analogue of CAR, "(Z-car x)", is 
;;;;;;;         "(if (forsome y (forsome z (x = <y, z>)))
;;;;;;;           then (forsome z (x = <(Z-car x), z>))
;;;;;;;           else ((Z-car x) = |0|))".
;;;;;;;The definition of the analogue of LISTP, "(Z-listp x)", is
;;;;;;;         "(if (forsome y (forsome z (x = <y, z>)))
;;;;;;;           then ((Z-car x) = |1|)
;;;;;;;           else ((Z-car x) = |0|))".
;;;;;;;Since we will be proving a number of metatheorem regarding the logical-if
;;;;;;;connective, it is convenient to have a representation for such expressions
;;;;;;;which is easily manipulated without having to expand the abbreviation "(if A B C)".
;;;;;;;The shell Z-IF is used to represent if-trees.
\end{verbatim}
\noindent{\sc Event}:   Add the shell {\it{z-if\/}}\index{z-if}, with recognizer function symbol {\it{z-ifp\/}} and 3 accessors: {\it{z-test\/}}, with type restriction (none-of) and default value zero; {\it{z-left\/}}, with type restriction (none-of) and default value zero; {\it{z-right\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}
\begin{verbatim};;;;;;;TERM-Z-IF checks if every test position of the if-tree EXP is a formula,
;;;;;;;and every leaf expression is a term in which VAR does not occur free.
;;;;;;;We will call such an if-tree a `term-if-tree'.  This is useful in representing
;;;;;;;a definition by cases where each case has the form "((f x) = exp)".
;;;;;;;The term-if-tree corresponding to the definition of "(Z-listp x)" is 
;;;;;;;(Z-IF "(forsome y (forsome z (x = <y, z>)))"  |1|  |0|).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{term-z-if} \\  
\=\+{\rm{term-z-if}}\,({\it{exp\/}}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{term-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{term-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if}\- \\ 
{\bf else }{\rm{termp}}\,({\it{exp\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ ({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;MAKE-TERM-Z-IF makes an formula out of the term-if-tree by replacing each occurrence
;;;;;;;of Z-IF by the logical-if connective, and returning "(var = exp)" for each
;;;;;;;leaf term "exp".  Thus MAKE-TERM-Z-IF applied to the term-if-tree for "Z-listp"
;;;;;;;with the variable "value" for VAR returns the definition for "Z-listp".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{make-term-z-if} \\  
\=\+{\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }{\rm{f-if}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-term-z-if}}\,({\it{var\/}}, {\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,({\it{var\/}}, {\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-term-z-if})\-\index{f-if} \\ 
{\bf else }{\rm{f-equal}}\,({\it{var\/}}, {\it{exp\/}})\index{f-equal}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{term-z-if-exis-proof} \\  
\=\+{\rm{term-z-if-exis-proof}}\,({\it{var\/}}, {\it{exp\/}})\index{term-z-if-exis-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }{\rm{forsome-f-if-case-proof}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-term-z-if}}\,({\it{var\/}}, {\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,({\it{var\/}}, {\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-term-z-if}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left})\-\index{make-term-z-if})\-\index{forsome}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left})\-\index{make-term-z-if})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{term-z-if-exis-proof}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left})\-\index{term-z-if-exis-proof}))\-\index{tautconseq-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right})\-\index{make-term-z-if})\-\index{forsome}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-not}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test})\index{f-not}, \\ 
{\rm{forsome}}\,(\=\+{\it{var\/}}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right})\-\index{make-term-z-if})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{term-z-if-exis-proof}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right})\-\index{term-z-if-exis-proof}))\-\index{tautconseq-proof})\-\index{forsome-f-if-case-proof} \\ 
{\bf else }{\rm{exis-fun-intro-proof}}\,({\it{var\/}}, {\it{exp\/}})\index{exis-fun-intro-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-make-term-z-if \\ \index{formula-make-term-z-if}
\=\+({\rm{term-z-if}}\,({\it{exp\/}}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if} $\wedge$ {\rm{variable}}\,({\it{var\/}})\index{variable}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{verbatim};;;;;;;Many of the functions are defined as formulas constructed from
;;;;;;;term-if-trees generated by the Z-IF shell.  The lemma below is used to
;;;;;;;carry out the existence part of the admissibility of such definitions.
;;;;;;;The crucial part in the proof of this lemma is the observation that
;;;;;;;if "value" is not free in TEST, then from
;;;;;;; "(if TEST (forsome value EXPL) (forsome value EXPR))",
;;;;;;;we can derive "(forsome value (if TEST EXPL EXPR))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  term-z-if-exis-proof-proves \\ \index{term-z-if-exis-proof-proves}
\=\+({\rm{variable}}\,({\it{var\/}})\index{variable} $\wedge$ {\rm{term-z-if}}\,({\it{exp\/}}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{term-z-if-exis-proof}}\,({\it{var\/}}, {\it{exp\/}})\index{term-z-if-exis-proof}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if})\index{forsome}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable term-z-if-exis-proof; name this event `g0761'.\index{g0761}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  term-z-if-exis-proof-proves1 \\ \index{term-z-if-exis-proof-proves1}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{term-z-if}}\,({\it{exp\/}}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{forsome}}\,({\it{var\/}}, {\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if})\index{forsome}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{term-z-if-exis-proof}}\,({\it{var\/}}, {\it{exp\/}})\index{term-z-if-exis-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable term-z-if-exis-proof-proves; name this event `g0762'.\index{g0762}

\addvspace{18pt}
\begin{verbatim};;;;;;;We would now like to show for example that the definition of "Z-listp"
;;;;;;;does in fact make it the Z2 analogue of the Lisp function LISTP.
;;;;;;;That is |- "((Z-listp <X>) = <(BOOL-FIX (LISTP X))>)",
;;;;;;;where BOOL-FIX converts F to 0, and T to 1.  Again, we overload the notation
;;;;;;;by permitting Lisp functions within quotes to denote their values.
;;;;;;;The predicate PROVES-Z-IF
;;;;;;;requires careful explanation.  Its first argument PF-TREE is a proof-tree
;;;;;;;that mimics the structure of the if-tree EXP.  A proof-tree is of the
;;;;;;;form:  <<cond proof1 proof2> proof-tree-left proof-tree-right>.
;;;;;;;PROVES-Z-IF checks that when EXP is of the form (Z-IF test left right),
;;;;;;;then if cond is T, proof1 is a proof of test, and proof-tree-left is
;;;;;;;the proof-tree corresponding to left; and if cond is F, then proof2 is
;;;;;;;a proof of the negation of test, and proof-tree-right is the proof-tree
;;;;;;;corresponding to right.  In the base case, PROVES-Z-IF checks that PF-TREE
;;;;;;;is a proof of "(EXP = VAL)".  Once this has been checked, we can later evaluate
;;;;;;;the term-if-tree in terms of the values of the leaf nodes by constructing the
;;;;;;;appropriate proof-tree.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{proves-z-if} \\  
\=\+{\rm{proves-z-if}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-z-if} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pf-tree\/}}) \\ 
{\bf then }\=\+{\rm{proves}}\,({\rm{cadar}}\,({\it{pf-tree\/}}), {\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-z-if}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{var\/}}, \\ 
{\it{val\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-z-if}\- \\ 
{\bf else }\=\+{\rm{proves}}\,(\=\+{\rm{caddar}}\,({\it{pf-tree\/}}), \\ 
{\rm{f-not}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-z-if}}\,(\=\+{\rm{caddr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{var\/}}, \\ 
{\it{val\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-z-if}\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{proves}}\,({\it{pf-tree\/}}, {\rm{f-equal}}\,({\it{exp\/}}, {\it{val\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-if-eval-proof} \\  
\=\+{\rm{z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}})\index{z-if-eval-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pf-tree\/}}) \\ 
{\bf then }{\rm{chain-proof}}\,(\=\+{\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if}, \\ 
{\rm{f-equal}}\,({\it{var\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{make-term-z-if}}\,({\it{var\/}}, {\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-term-z-if}), \\ 
{\rm{list}}\,(\=\+{\rm{f-if-elim-proof1}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left})\-\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right})\-\index{make-term-z-if}, \\ 
{\rm{cadar}}\,({\it{pf-tree\/}}))\-\index{f-if-elim-proof1}, \\ 
{\rm{z-if-eval-proof}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{var\/}}, \\ 
{\it{val\/}})\-\index{z-if-eval-proof})\-)\-\index{chain-proof} \\ 
{\bf else }{\rm{chain-proof}}\,(\=\+{\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if}, \\ 
{\rm{f-equal}}\,({\it{var\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{make-term-z-if}}\,({\it{var\/}}, {\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-term-z-if}), \\ 
{\rm{list}}\,(\=\+{\rm{f-if-elim-proof2}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left})\-\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,(\=\+{\it{var\/}}, \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right})\-\index{make-term-z-if}, \\ 
{\rm{caddar}}\,({\it{pf-tree\/}}))\-\index{f-if-elim-proof2}, \\ 
{\rm{z-if-eval-proof}}\,(\=\+{\rm{caddr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{var\/}}, \\ 
{\it{val\/}})\-\index{z-if-eval-proof})\-)\-\index{chain-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,({\it{exp\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{var\/}}, {\it{exp\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{var\/}}, {\it{val\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{pf-tree\/}}, \\ 
{\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,({\it{var\/}}, {\it{var\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{exp\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{var\/}}, \\ 
{\it{exp\/}})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{var\/}}, \\ 
{\it{val\/}})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{ident-axiom-proof}}\,({\it{var\/}})\index{ident-axiom-proof}, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}, \\ 
{\rm{list}}\,({\it{var\/}}, {\it{exp\/}}), \\ 
{\rm{list}}\,({\it{var\/}}, {\it{val\/}}))\-\index{equal-axiom2-proof})\-\index{detach-proof})\-\index{detach-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-make-term-z-if-again \\ \index{formula-make-term-z-if-again}
\=\+({\rm{term-z-if}}\,({\it{exp\/}}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if} $\wedge$ {\rm{termp}}\,({\it{var\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{verbatim};;;;;;;If each leaf term "exp" in EXP has VAL
;;;;;;;as its value (this is checked by PROVES-Z-IF), then the if-expression
;;;;;;;which has "(var = exp)" for each leaf term "exp" in EXP can be proved
;;;;;;;to imply "(var = VAL)".  
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-if-eval-proof-proves \\ \index{z-if-eval-proof-proves}
\=\+(\=\+{\rm{term-z-if}}\,({\it{exp\/}}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{var\/}}, {\it{val\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves-z-if}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-z-if})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}})\index{z-if-eval-proof}, \\ 
{\rm{f-implies}}\,({\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if}, {\rm{f-equal}}\,({\it{var\/}}, {\it{val\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-if-eval-proof; name this event `g0765'.\index{g0765}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-if-eval-proof1} \\  
\=\+{\rm{z-if-eval-proof1}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}}, {\it{pf\/}})\index{z-if-eval-proof1} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if}, \\ 
{\rm{f-equal}}\,({\it{var\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\it{pf\/}}, \\ 
{\rm{z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}})\index{z-if-eval-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-if-eval-proof-proves1 \\ \index{z-if-eval-proof-proves1}
\=\+(\=\+{\rm{term-z-if}}\,({\it{exp\/}}, {\it{var\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{var\/}}, {\it{val\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves-z-if}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{make-term-z-if}}\,({\it{var\/}}, {\it{exp\/}})\index{make-term-z-if}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-if-eval-proof1}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{val\/}}, {\it{pf\/}})\index{z-if-eval-proof1}, \\ 
{\rm{f-equal}}\,({\it{var\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-if-eval-proof1; name this event `g0766'.\index{g0766}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{stack-and-implies} \\  
\=\+{\rm{stack-and-implies}}\,({\it{exp\/}}, {\it{hyps\/}})\index{stack-and-implies} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{hyps\/}})$\;\;${\bf then }{\rm{f-implies}}\,({\rm{list-f-and}}\,({\it{hyps\/}})\index{list-f-and}, {\it{exp\/}})\index{f-implies} \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-z-if-exis-step-proof} \\  
\=\+{\rm{form-z-if-exis-step-proof}}\,({\it{test\/}}, {\it{left\/}}, {\it{right\/}}, {\it{var\/}}, {\it{tests\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{form-z-if-exis-step-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{stack-and-implies}}\,(\=\+{\rm{forsome}}\,({\it{var\/}}, {\it{left\/}})\index{forsome}, \\ 
{\rm{cons}}\,({\it{test\/}}, {\it{tests\/}}))\-\index{stack-and-implies}, \\ 
{\rm{stack-and-implies}}\,(\=\+{\rm{forsome}}\,({\it{var\/}}, {\it{right\/}})\index{forsome}, \\ 
{\rm{cons}}\,({\rm{f-not}}\,({\it{test\/}})\index{f-not}, {\it{tests\/}}))\-\index{stack-and-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-if}}\,(\=\+{\it{test\/}}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{left\/}})\index{forsome}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\it{right\/}})\index{forsome})\-\index{f-if}, \\ 
{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{test\/}}, {\it{left\/}}, {\it{right\/}})\index{f-if})\index{forsome})\-\index{f-implies})\-, \\ 
{\rm{stack-and-implies}}\,(\=\+{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{test\/}}, {\it{left\/}}, {\it{right\/}})\index{f-if})\index{forsome}, \\ 
{\it{tests\/}})\-\index{stack-and-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}, {\rm{forsome-f-if-proof}}\,({\it{var\/}}, {\it{test\/}}, {\it{left\/}}, {\it{right\/}})\index{forsome-f-if-proof}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;In the definitions which use term-if-trees, the existence and uniqueness of the
;;;;;;;object being defined follows from that of the leaf terms.  There are other
;;;;;;;definitions where the leaf nodes of the if-trees are formulas (a `form-if-tree'), 
;;;;;;;and the existence/uniqueness of the defined object in each leaf node case requires
;;;;;;;the conjunction of the tests leading up to it.  The definition of "(Z-car x)",
;;;;;;;               "(if (forsome y (forsome z (x = <y z>)))  
;;;;;;;                 then (forsome z (x = <(Z-car x), z>)) 
;;;;;;;                 else ((Z-car x) = |0|))".
;;;;;;;is a good example of this.  The lemma below combines the
;;;;;;;existence proofs in each of the left/right cases to form the existence proof
;;;;;;;for the if-subtree rooted at the corresponding test node.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  form-z-if-exis-step-proof-proves \\ \index{form-z-if-exis-step-proof-prove!s}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{test\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{stack-and-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{left\/}})\index{forsome}, {\rm{cons}}\,({\it{test\/}}, {\it{tests\/}}))\index{stack-and-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{stack-and-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{right\/}})\index{forsome}, {\rm{cons}}\,({\rm{f-not}}\,({\it{test\/}})\index{f-not}, {\it{tests\/}}))\index{stack-and-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{stack-and-implies}}\,(\=\+{\rm{forsome}}\,({\it{var\/}}, {\rm{f-if}}\,({\it{test\/}}, {\it{left\/}}, {\it{right\/}})\index{f-if})\index{forsome}, \\ 
{\it{tests\/}})\-\index{stack-and-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-exis-step-proof}}\,({\it{test\/}}, {\it{left\/}}, {\it{right\/}}, {\it{var\/}}, {\it{tests\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{form-z-if-exis-step-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-exis-step-proof; name this event `g0769'.\index{g0769}

\addvspace{18pt}
\begin{verbatim};;;;;;;FORM-Z-IF checks if EXP is a form-if-tree.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{form-z-if} \\  
\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right}, {\it{symbols\/}})\index{form-z-if}\- \\ 
{\bf else }{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;MAKE-FORM-Z-IF returns the if-expression represented by the form-if-tree EXP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{make-form-z-if} \\  
\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }{\rm{f-if}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if})\-\index{f-if} \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;PROVES-FORM-Z-IF-EXIS checks that for every leaf formula in the form-if-tree EXP,
;;;;;;;PF-TREE contains a corresponding proof that the conjunction of the tests leading
;;;;;;;to that leaf node "exp" implies "(forsome var exp)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{proves-form-z-if-exis} \\  
\=\+{\rm{proves-form-z-if-exis}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{tests\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-exis} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-exis}}\,(\=\+{\rm{car}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{var\/}}, \\ 
{\rm{cons}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{tests\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-form-z-if-exis} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-exis}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{var\/}}, \\ 
{\rm{cons}}\,({\rm{f-not}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test})\index{f-not}, {\it{tests\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-form-z-if-exis}\- \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{pf-tree\/}}, \\ 
{\rm{stack-and-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{exp\/}})\index{forsome}, {\it{tests\/}})\index{stack-and-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-z-if-exis-proof} \\  
\=\+{\rm{form-z-if-exis-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{tests\/}})\index{form-z-if-exis-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }{\rm{form-z-if-exis-step-proof}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\it{var\/}}, \\ 
{\it{tests\/}}, \\ 
{\rm{form-z-if-exis-proof}}\,(\=\+{\rm{car}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{var\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\it{tests\/}})\-)\-\index{form-z-if-exis-proof}, \\ 
{\rm{form-z-if-exis-proof}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{var\/}}, \\ 
{\rm{cons}}\,(\=\+{\rm{f-not}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test})\index{f-not}, \\ 
{\it{tests\/}})\-)\-\index{form-z-if-exis-proof})\-\index{form-z-if-exis-step-proof} \\ 
{\bf else }{\it{pf-tree\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-make-form \\ \index{formula-make-form}
{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} $\rightarrow$ {\rm{formula}}\,({\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, {\it{symbols\/}})\index{formula}
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below demonstrates that the existence proofs for the leaf nodes
;;;;;;;can be combined to give the existence proof for the if-expression represented
;;;;;;;by the form-if-tree EXP.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  form-z-if-exis-proof-proves \\ \index{form-z-if-exis-proof-proves}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-exis}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{tests\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-exis})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-exis-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{tests\/}})\index{form-z-if-exis-proof}, \\ 
{\rm{stack-and-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if})\index{forsome}, {\it{tests\/}})\index{stack-and-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-exis-proof; name this event `g0770'.\index{g0770}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  form-z-if-exis-proof-proves1 \\ \index{form-z-if-exis-proof-proves1}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-exis}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{tests\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-exis} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{stack-and-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if})\index{forsome}, {\it{tests\/}})\index{stack-and-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-exis-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{tests\/}})\index{form-z-if-exis-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-exis-proof-proves; name this event `g0784'.\index{g0784}

\addvspace{18pt}
\begin{verbatim};;;;;;;Next we show how definitions represented by form-if-trees are evaluated.
;;;;;;;PROVES-FORM-Z-IF-EVAL is the form-if-tree analogue of PROVES-Z-IF.
;;;;;;;The only difference is that in the base case, PROVES-FORM-Z-IF-EVAL
;;;;;;;checks if PF-TREE is a proof of "(EXP -> (LHS = VAL))".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{proves-form-z-if-eval} \\  
\=\+{\rm{proves-form-z-if-eval}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-eval} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pf-tree\/}}) \\ 
{\bf then }\=\+{\rm{proves}}\,({\rm{cadar}}\,({\it{pf-tree\/}}), {\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-eval}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{lhs\/}}, \\ 
{\it{val\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-form-z-if-eval}\- \\ 
{\bf else }\=\+{\rm{proves}}\,(\=\+{\rm{caddar}}\,({\it{pf-tree\/}}), \\ 
{\rm{f-not}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-eval}}\,(\=\+{\rm{caddr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{lhs\/}}, \\ 
{\it{val\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-form-z-if-eval}\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{pf-tree\/}}, \\ 
{\rm{f-implies}}\,({\it{exp\/}}, {\rm{f-equal}}\,({\it{lhs\/}}, {\it{val\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-z-if-eval-proof} \\  
\=\+{\rm{form-z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}})\index{form-z-if-eval-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pf-tree\/}}) \\ 
{\bf then }{\rm{chain-proof}}\,(\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, \\ 
{\rm{f-equal}}\,({\it{lhs\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}), \\ 
{\rm{list}}\,(\=\+{\rm{f-if-elim-proof1}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\rm{cadar}}\,({\it{pf-tree\/}}))\-\index{f-if-elim-proof1}, \\ 
{\rm{form-z-if-eval-proof}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{lhs\/}}, \\ 
{\it{val\/}})\-\index{form-z-if-eval-proof})\-)\-\index{chain-proof} \\ 
{\bf else }{\rm{chain-proof}}\,(\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, \\ 
{\rm{f-equal}}\,({\it{lhs\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\rm{list}}\,({\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}), \\ 
{\rm{list}}\,(\=\+{\rm{f-if-elim-proof2}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\rm{caddar}}\,({\it{pf-tree\/}}))\-\index{f-if-elim-proof2}, \\ 
{\rm{form-z-if-eval-proof}}\,(\=\+{\rm{caddr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{lhs\/}}, \\ 
{\it{val\/}})\-\index{form-z-if-eval-proof})\-)\-\index{chain-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\it{pf-tree\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-z-if-eval-proof-proves \\ \index{form-z-if-eval-proof-proves}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{lhs\/}}, {\it{val\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-eval}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-eval})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}})\index{form-z-if-eval-proof}, \\ 
{\rm{f-implies}}\,({\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, {\rm{f-equal}}\,({\it{lhs\/}}, {\it{val\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-eval-proof; name this event `g0785'.\index{g0785}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{form-z-if-eval-proof1} \\  
\=\+{\rm{form-z-if-eval-proof1}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}}, {\it{pf\/}})\index{form-z-if-eval-proof1} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, \\ 
{\rm{f-equal}}\,({\it{lhs\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\it{pf\/}}, \\ 
{\rm{form-z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}})\index{form-z-if-eval-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Now we can show that if we have a proof-tree satisfying PROVES-FORM-Z-IF-EVAL,
;;;;;;;then we can prove that the function can be evaluated as desired, e.g.,
;;;;;;;we can show that "((Z-car <X>) = <(CAR X)>)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  form-z-if-eval-proof-proves1 \\ \index{form-z-if-eval-proof-proves1}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{lhs\/}}, {\it{val\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-eval}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-eval} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-eval-proof1}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}}, {\it{pf\/}})\index{form-z-if-eval-proof1}, \\ 
{\rm{f-equal}}\,({\it{lhs\/}}, {\it{val\/}})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-eval-proof1; name this event `g0786'.\index{g0786}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  nlistp-nil-intersect \\ \index{nlistp-nil-intersect}
({\it{y\/}} $\simeq\;${{\bf{nil}}}${}$) $\rightarrow$ {\rm{nil-intersect}}\,({\it{x\/}}, {\it{y\/}})\index{nil-intersect}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-free-free-for \\ \index{not-free-free-for}
({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $\simeq\;${{\bf{nil}}}${}$) $\rightarrow$ {\rm{free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{free-for}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  listp-append \\ \index{listp-append}
{\rm{listp}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}})) $=$ ({\rm{listp}}\,({\it{x\/}}) $\vee$ {\rm{listp}}\,({\it{y\/}}))
\end{tabbing}

\noindent{\sc Event}:   Enable free-for; name this event `g0666'.\index{g0666}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  not-free-for-list \\ \index{not-free-for-list}
({\rm{collect-free}}\,({\it{termlist\/}}, {\tt{1}})\index{collect-free} $\simeq\;${{\bf{nil}}}${}$) $\rightarrow$ {\rm{free-for-list}}\,({\it{exp\/}}, {\it{varlist\/}}, {\it{termlist\/}})\index{free-for-list}
\end{tabbing}

\noindent{\sc Event}:   Enable free-for-list; name this event `g0671'.\index{g0671}

\addvspace{18pt}

\noindent{\sc Event}:   Enable listp-append; name this event `g0680'.\index{g0680}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{listp-cond-proof1} \\  
\=\+{\rm{listp-cond-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{listp-cond-proof1} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{forsome}}\,({\tt{2}}, {\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{forsome}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair})\index{ident-axiom-proof}, \\ 
{\rm{subst-thm-a-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}), \\ 
{\rm{list}}\,({\it{x\/}}, {\it{y\/}}))\-\index{subst-thm-a-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  listp-cond-proof1-proves \\ \index{listp-cond-proof1-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{listp-cond-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{listp-cond-proof1}, \\ 
{\rm{forsome}}\,({\tt{1}}, {\rm{forsome}}\,({\tt{2}}, {\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable listp-cond-proof1; name this event `g0685'.\index{g0685}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{numeral-not-listp-proof} \\  
\=\+{\rm{numeral-not-listp-proof}}\,({\it{n\/}}, {\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{numeral-not-listp-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{x\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\-\index{forsome})\-\index{forsome})\index{f-not}, \\ 
$\forall\;${\it{x\/}}$\;${\rm{f-not}}\,({\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\rm{numeral}}\,({\it{n\/}})\index{numeral}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{f-not}\index{forall}, \\ 
{\rm{forsome-forall-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{numeral}}\,({\it{n\/}})\index{numeral}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\-\index{f-equal})\-\index{forsome})\-\index{forsome-forall-proof}, \\ 
{\rm{generalise-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{numeral}}\,({\it{n\/}})\index{numeral}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\-\index{f-equal})\-\index{forsome})\index{f-not}, \\ 
{\rm{f-iff-mp-proof2}}\,(\=\+{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{numeral}}\,({\it{n\/}})\index{numeral}, \\ 
{\rm{z-opair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome})\index{f-not}, \\ 
$\forall\;${\it{y\/}}$\;${\rm{f-not}}\,({\rm{f-equal}}\,(\=\+{\rm{numeral}}\,({\it{n\/}})\index{numeral}, \\ 
{\rm{z-opair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-opair})\-\index{f-equal})\index{f-not}\index{forall}, \\ 
{\rm{forsome-forall-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{numeral}}\,({\it{n\/}})\index{numeral}, \\ 
{\rm{z-opair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-opair})\-\index{f-equal})\-\index{forsome-forall-proof}, \\ 
{\rm{generalise-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,(\=\+{\rm{numeral}}\,({\it{n\/}})\index{numeral}, \\ 
{\rm{z-opair}}\,(\=\+{\it{u\/}}, \\ 
{\it{v\/}})\-\index{z-opair})\-\index{f-equal})\index{f-not}, \\ 
{\rm{list-num-nequal-proof}}\,(\=\+{\it{n\/}}, \\ 
{\it{u\/}}, \\ 
{\it{v\/}})\-\index{list-num-nequal-proof})\-\index{generalise-proof})\-\index{f-iff-mp-proof2})\-\index{generalise-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;;No numeral is an ordered pair.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  numeral-not-listp-proof-proves \\ \index{numeral-not-listp-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}}), {\tt{4}})\index{var-list} \\ 
$\wedge$$\;\;\;\;$({\it{n-bar\/}} $=$ {\rm{numeral}}\,({\it{n\/}})\index{numeral}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{numeral-not-listp-proof}}\,({\it{n\/}}, {\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{numeral-not-listp-proof}, \\ 
{\rm{f-not}}\,({\rm{forsome}}\,({\it{x\/}}, {\rm{forsome}}\,({\it{y\/}}, {\rm{f-equal}}\,({\it{n-bar\/}}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable numeral-not-listp-proof; name this event `g0693'.\index{g0693}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-car-ncode-proof} \\  
\=\+{\rm{z-car-ncode-proof}}\,({\it{x\/}})\index{z-car-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{form-z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{listp}}\,({\it{x\/}}), \\ 
{\rm{listp-cond-proof1}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{listp-cond-proof1}, \\ 
{\rm{numeral-not-listp-proof}}\,({\it{x\/}}, {\tt{1}}, {\tt{2}}, {\tt{1}}, {\tt{2}})\index{numeral-not-listp-proof})\-, \\ 
{\rm{forsome-intro-proof}}\,(\=\+{\tt{2}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{z-opair}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, \\ 
{\tt{2}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{z-opair}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, \\ 
{\tt{2}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{list}}\,({\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car})\-\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{z-car-unique-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, \\ 
{\tt{2}})\-\index{z-car-unique-proof}, \\ 
{\rm{symmetry-proof1}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car})\-\index{symmetry-proof1})\-)\-\index{chain-proof})\-\index{forsome-intro-proof}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-equal}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\index{prop-axiom-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{forsome}}\,(\=\+{\tt{2}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\-\index{f-equal})\-\index{forsome})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\tt{2}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, {\tt{2}})\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{z-if}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{z-car-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car-defn-proof})\-\index{form-z-if-eval-proof1}\-
\end{tabbing}
\begin{verbatim};;;;;;;FORM-Z-IF-EVAL-PROOF-PROVES1 is used to prove that "(Z-car x)" is
;;;;;;;the Z2 analogue of CAR, in the lemma below.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-car-ncode-proof-proves \\ \index{z-car-ncode-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-car}, {\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-car-ncode-proof}}\,({\it{x\/}})\index{z-car-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-ncode-proof; name this event `g0691'.\index{g0691}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{listp-cond-proof2} \\  
\=\+{\rm{listp-cond-proof2}}\,({\it{x\/}}, {\it{y\/}})\index{listp-cond-proof2} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair})\index{f-equal}, \\ 
{\rm{forsome}}\,(\=\+{\tt{2}}, \\ 
{\rm{forsome}}\,({\tt{1}}, {\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{forsome}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair})\index{ident-axiom-proof}, \\ 
{\rm{subst-thm-a-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal}, \\ 
{\rm{list}}\,({\tt{2}}, {\tt{1}}), \\ 
{\rm{list}}\,({\it{y\/}}, {\it{x\/}}))\-\index{subst-thm-a-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  listp-cond-proof2-proves \\ \index{listp-cond-proof2-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{listp-cond-proof2}}\,({\it{x\/}}, {\it{y\/}})\index{listp-cond-proof2}, \\ 
{\rm{forsome}}\,({\tt{2}}, {\rm{forsome}}\,({\tt{1}}, {\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable listp-cond-proof2; name this event `g0692'.\index{g0692}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-cdr-ncode-proof} \\  
\=\+{\rm{z-cdr-ncode-proof}}\,({\it{x\/}})\index{z-cdr-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{form-z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{listp}}\,({\it{x\/}}), \\ 
{\rm{listp-cond-proof2}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{listp-cond-proof2}, \\ 
{\rm{numeral-not-listp-proof}}\,({\it{x\/}}, {\tt{2}}, {\tt{1}}, {\tt{1}}, {\tt{2}})\index{numeral-not-listp-proof})\-, \\ 
{\rm{forsome-intro-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{z-opair}, \\ 
{\rm{z-opair}}\,(\=\+{\tt{1}}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{z-opair}, \\ 
{\rm{z-opair}}\,(\=\+{\tt{1}}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{list}}\,({\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr})\-\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{z-cdr-unique-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\tt{1}}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr})\-\index{z-cdr-unique-proof}, \\ 
{\rm{symmetry-proof1}}\,(\=\+{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr})\-\index{symmetry-proof1})\-)\-\index{chain-proof})\-\index{forsome-intro-proof}, \\ 
{\rm{prop-axiom-proof}}\,({\rm{f-equal}}\,(\=\+{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\index{prop-axiom-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\tt{2}}, \\ 
{\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\-\index{f-equal})\-\index{forsome})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\tt{1}}, {\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr})\index{z-opair})\-\index{f-equal})\-\index{forsome}, \\ 
{\rm{f-equal}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{z-if}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{z-cdr-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr-defn-proof})\-\index{form-z-if-eval-proof1}\-
\end{tabbing}
\begin{verbatim};;;;;;;"(Z-cdr x)" is the Z2 analogue of CDR.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-cdr-ncode-proof-proves \\ \index{z-cdr-ncode-proof-proves}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-cdr}, {\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-cdr-ncode-proof}}\,({\it{x\/}})\index{z-cdr-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr-ncode-proof; name this event `g0694'.\index{g0694}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  col-free-z-car \\ \index{col-free-z-car}
{\rm{collect-free}}\,({\rm{z-car}}\,({\it{x\/}})\index{z-car}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-z-car \\ \index{covering-z-car}
{\rm{covering}}\,({\rm{z-car}}\,({\it{x\/}})\index{z-car}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-car \\ \index{subst-z-car}
{\rm{subst}}\,({\rm{z-car}}\,({\it{x\/}})\index{z-car}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\rm{z-car}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-car}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-car \\ \index{parallel-subst-z-car}
\=\+{\rm{parallel-subst}}\,({\rm{z-car}}\,({\it{x\/}})\index{z-car}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-car}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{z-car}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-z-car \\ \index{termp-z-car}
\=\+{\rm{termp}}\,({\rm{z-car}}\,({\it{x\/}})\index{z-car}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$(({\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car; name this event `g0696'.\index{g0696}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  col-free-z-cdr \\ \index{col-free-z-cdr}
{\rm{collect-free}}\,({\rm{z-cdr}}\,({\it{x\/}})\index{z-cdr}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-z-cdr \\ \index{covering-z-cdr}
{\rm{covering}}\,({\rm{z-cdr}}\,({\it{x\/}})\index{z-cdr}, {\it{var\/}}, {\tt{0}})\index{covering} $=$ {\rm{covering}}\,({\rm{list}}\,({\it{x\/}}), {\it{var\/}}, {\tt{1}})\index{covering}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-z-cdr \\ \index{subst-z-cdr}
{\rm{subst}}\,({\rm{z-cdr}}\,({\it{x\/}})\index{z-cdr}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\rm{z-cdr}}\,({\rm{subst}}\,({\it{x\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{z-cdr}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-cdr \\ \index{parallel-subst-z-cdr}
\=\+{\rm{parallel-subst}}\,({\rm{z-cdr}}\,({\it{x\/}})\index{z-cdr}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-cdr}}\,({\rm{parallel-subst}}\,({\it{x\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst})\index{z-cdr}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-z-cdr \\ \index{termp-z-cdr}
\=\+{\rm{termp}}\,({\rm{z-cdr}}\,({\it{x\/}})\index{z-cdr}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$(({\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cdr; name this event `g0697'.\index{g0697}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{term-z-if-unique-proof} \\  
\=\+{\rm{term-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{term-z-if-unique-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }{\rm{unique-f-if-proof}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-term-z-if}}\,({\it{x\/}}, {\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,({\it{y\/}}, {\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,({\it{x\/}}, {\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,({\it{y\/}}, {\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-term-z-if}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{term-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{term-z-if-unique-proof}, \\ 
{\rm{term-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{term-z-if-unique-proof})\-\index{unique-f-if-proof} \\ 
{\bf else }{\rm{unique-fun-intro-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{unique-fun-intro-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The uniqueness part of the admissibility proofs for definitions constructed
;;;;;;;by term-if-trees follows from the uniqueness proofs of the leaf nodes.
;;;;;;;The lemma below proves that any definition constructed by a term-if-tree
;;;;;;;automatically has the uniqueness property, since terms represent unique values.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  term-z-if-unique-proof-proves \\ \index{term-z-if-unique-proof-proves}
\=\+(\=\+{\rm{term-z-if}}\,({\it{exp\/}}, {\it{x\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{term-z-if}}\,({\it{exp\/}}, {\it{y\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{2}})\index{var-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{term-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{term-z-if-unique-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\rm{make-term-z-if}}\,({\it{x\/}}, {\it{exp\/}})\index{make-term-z-if}, {\rm{make-term-z-if}}\,({\it{y\/}}, {\it{exp\/}})\index{make-term-z-if})\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable term-z-if-unique-proof; name this event `g0695'.\index{g0695}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  term-z-if-unique-proof-proves1 \\ \index{term-z-if-unique-proof-proves1}
\=\+(\=\+{\rm{term-z-if}}\,({\it{exp\/}}, {\it{x\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{term-z-if}}\,({\it{exp\/}}, {\it{y\/}}, {\it{symbols\/}})\index{term-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{2}})\index{var-list} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{make-term-z-if}}\,({\it{x\/}}, {\it{exp\/}})\index{make-term-z-if}, \\ 
{\rm{make-term-z-if}}\,({\it{y\/}}, {\it{exp\/}})\index{make-term-z-if})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{term-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{term-z-if-unique-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-listp-exis-proof} \\  
\=\+{\rm{z-listp-exis-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-listp-exis-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-exis-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{forsome}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{z-if})\-\index{term-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-listp-unique-proof} \\  
\=\+{\rm{z-listp-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}})\index{z-listp-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{forsome}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{z-if})\-\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The definition for "(Z-listp x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-listp-defn} \\  
\=\+{\rm{z-listp-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-listp-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{forsome}}\,({\it{y1\/}}, {\rm{forsome}}\,({\it{z1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{bool-fix} \\  
\=\+{\rm{bool-fix}}\,({\it{x\/}})\index{bool-fix} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}}$\;\;${\bf then }{\tt{1}} \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The expression representing "(Z-listp x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-listp}$\;\;$
{\rm{z-listp}}\,({\it{x\/}})\index{z-listp} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}
\begin{verbatim};;;;;;;The admissibility proof for "(Z-listp x)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-listp-defn-proof} \\  
\=\+{\rm{z-listp-defn-proof}}\,({\it{term\/}})\index{z-listp-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{z-listp-defn}}\,({\tt{0}}, {\rm{z-listp}}\,({\tt{0}})\index{z-listp}, {\tt{1}}, {\tt{2}})\index{z-listp-defn}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{1}}, \\ 
{\tt{3}}, \\ 
{\rm{z-listp-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{1}}, {\tt{2}})\index{z-listp-defn}, \\ 
{\rm{z-listp-exis-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-listp-exis-proof}, \\ 
{\rm{z-listp-unique-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{3}}, {\tt{2}})\index{z-listp-unique-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;DEFNS must contain the definition of "Z-listp" associated with the
;;;;;;;function symbol for "Z-listp" and the functions used to define
;;;;;;;"Z-listp" must follow "Z-listp" in SYMBOLS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-listp-hyps} \\  
\=\+{\rm{z-listp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-listp-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-listp-defn}}\,({\tt{0}}, {\rm{z-listp}}\,({\tt{0}})\index{z-listp}, {\tt{1}}, {\tt{2}})\index{z-listp-defn}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable free-for; name this event `g0699'.\index{g0699}

\addvspace{18pt}
\begin{verbatim};;;;;;;The definition for "Z-listp" is proved to be admissible in the lemma below.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-listp-defn-proof-proves \\ \index{z-listp-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-listp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-listp-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-listp-defn}}\,({\it{term\/}}, {\rm{z-listp}}\,({\it{term\/}})\index{z-listp}, {\tt{1}}, {\tt{2}})\index{z-listp-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-listp-defn-proof}}\,({\it{term\/}})\index{z-listp-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-listp-defn-proof; name this event `g0703'.\index{g0703}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-listp-hyps; name this event `g0706'.\index{g0706}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-listp-ncode-proof} \\  
\=\+{\rm{z-listp-ncode-proof}}\,({\it{x\/}})\index{z-listp-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{listp}}\,({\it{x\/}}), \\ 
{\rm{listp-cond-proof1}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{listp-cond-proof1}, \\ 
{\rm{numeral-not-listp-proof}}\,({\it{x\/}}, {\tt{1}}, {\tt{2}}, {\tt{1}}, {\tt{2}})\index{numeral-not-listp-proof})\-, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\sc{phi}}\index{phi})\index{ident-axiom-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{forsome}}\,(\=\+{\tt{2}}, \\ 
{\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal})\-\index{forsome})\-\index{forsome}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{z-if}, \\ 
{\rm{z-listp}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-listp}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{listp}}\,({\it{x\/}}))\index{bool-fix})\index{ncode}, \\ 
{\rm{z-listp-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-listp-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{verbatim};;;;;;;|- "((Z-listp <X>) = <(BOOL-FIX (LISTP X))>)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-listp-ncode-proof-proves \\ \index{z-listp-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-listp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-listp-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-listp}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-listp}, {\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{listp}}\,({\it{x\/}}))\index{bool-fix})\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-listp-ncode-proof}}\,({\it{x\/}})\index{z-listp-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-listp-ncode-proof; name this event `g0711'.\index{g0711}

\addvspace{18pt}
\begin{verbatim};;;;;;;A similar sequence of events for "(Z-add1 x)", the Z2 analogue of ADD1.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-add1-exis-proof} \\  
\=\+{\rm{z-add1-exis-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-add1-exis-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-exis-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{forsome}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\-\index{z-if})\-\index{term-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-add1-unique-proof} \\  
\=\+{\rm{z-add1-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z\/}})\index{z-add1-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{forsome}}\,({\it{z\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y\/}}, {\it{z\/}})\index{z-opair})\index{f-equal})\index{forsome})\-\index{forsome}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\-\index{z-if})\-\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-add1-defn} \\  
\=\+{\rm{z-add1-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}}, {\it{z1\/}})\index{z-add1-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{forsome}}\,({\it{y1\/}}, {\rm{forsome}}\,({\it{z1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\rm{z-opair}}\,({\it{y1\/}}, {\it{z1\/}})\index{z-opair})\index{f-equal})\index{forsome})\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{f-equal})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-add1}$\;\;$
{\rm{z-add1}}\,({\it{x\/}})\index{z-add1} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-add1-defn-proof} \\  
\=\+{\rm{z-add1-defn-proof}}\,({\it{term\/}})\index{z-add1-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{z-add1-defn}}\,({\tt{0}}, {\rm{z-add1}}\,({\tt{0}})\index{z-add1}, {\tt{1}}, {\tt{2}})\index{z-add1-defn}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{1}}, \\ 
{\tt{3}}, \\ 
{\rm{z-add1-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{1}}, {\tt{2}})\index{z-add1-defn}, \\ 
{\rm{z-add1-exis-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-add1-exis-proof}, \\ 
{\rm{z-add1-unique-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{3}}, {\tt{2}})\index{z-add1-unique-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-add1-hyps} \\  
\=\+{\rm{z-add1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-add1-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-add1-defn}}\,({\tt{0}}, {\rm{z-add1}}\,({\tt{0}})\index{z-add1}, {\tt{1}}, {\tt{2}})\index{z-add1-defn}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-add1-defn-proof-proves \\ \index{z-add1-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-add1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-add1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-add1-defn}}\,({\it{term\/}}, {\rm{z-add1}}\,({\it{term\/}})\index{z-add1}, {\tt{1}}, {\tt{2}})\index{z-add1-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-add1-defn-proof}}\,({\it{term\/}})\index{z-add1-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-add1-defn-proof; name this event `g0712'.\index{g0712}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-add1-hyps; name this event `g0713'.\index{g0713}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-add1-ncode-proof} \\  
\=\+{\rm{z-add1-ncode-proof}}\,({\it{x\/}})\index{z-add1-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{listp}}\,({\it{x\/}}), \\ 
{\rm{listp-cond-proof1}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode})\-\index{listp-cond-proof1}, \\ 
{\rm{numeral-not-listp-proof}}\,({\it{x\/}}, {\tt{1}}, {\tt{2}}, {\tt{1}}, {\tt{2}})\index{numeral-not-listp-proof})\-, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-succ})\index{ident-axiom-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{forsome}}\,(\=\+{\tt{2}}, \\ 
{\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{z-opair}}\,({\tt{1}}, {\tt{2}})\index{z-opair})\index{f-equal})\-\index{forsome})\-\index{forsome}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-succ})\-\index{z-if}, \\ 
{\rm{z-add1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-add1}, \\ 
{\rm{ncode}}\,($1\;+$ {\it{x\/}})\index{ncode}, \\ 
{\rm{z-add1-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-add1-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-add1-ncode-proof-proves \\ \index{z-add1-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-add1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-add1-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-add1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-add1}, {\rm{ncode}}\,($1\;+$ {\it{x\/}})\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-add1-ncode-proof}}\,({\it{x\/}})\index{z-add1-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-add1-ncode-proof; name this event `g0714'.\index{g0714}

\addvspace{18pt}
\begin{verbatim};;;;;;;The sequence of events introducing "(Z-numberp x)", the Z2 analogue of NUMBERP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-numberp-defn} \\  
\=\+{\rm{z-numberp-defn}}\,({\it{x\/}}, {\it{y\/}})\index{z-numberp-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\rm{f-equal}}\,({\it{y\/}}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-numberp-exis-proof} \\  
\=\+{\rm{z-numberp-exis-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-numberp-exis-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-exis-proof}}\,({\it{y\/}}, {\rm{z-if}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, {\sc{phi}}\index{phi})\index{z-if})\index{term-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-numberp-unique-proof} \\  
\=\+{\rm{z-numberp-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}})\index{z-numberp-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,({\it{y\/}}, {\it{y1\/}}, {\rm{z-if}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, {\sc{phi}}\index{phi})\index{z-if})\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-numberp}$\;\;$
{\rm{z-numberp}}\,({\it{x\/}})\index{z-numberp} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-numberp-defn-proof} \\  
\=\+{\rm{z-numberp-defn-proof}}\,({\it{term\/}})\index{z-numberp-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{z-numberp-defn}}\,({\tt{0}}, {\rm{z-numberp}}\,({\tt{0}})\index{z-numberp})\index{z-numberp-defn}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{z-numberp-defn}}\,({\tt{0}}, {\tt{1}})\index{z-numberp-defn}, \\ 
{\rm{z-numberp-exis-proof}}\,({\tt{0}}, {\tt{1}})\index{z-numberp-exis-proof}, \\ 
{\rm{z-numberp-unique-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-numberp-unique-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-numberp-hyps} \\  
\=\+{\rm{z-numberp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-numberp-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-numberp-defn}}\,({\tt{0}}, {\rm{z-numberp}}\,({\tt{0}})\index{z-numberp})\index{z-numberp-defn}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable z-int; name this event `g0716'.\index{g0716}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  collect-free-z-int \\ \index{collect-free-z-int}
{\rm{collect-free}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int}, {\tt{0}})\index{collect-free} $=$ {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}), {\tt{1}})\index{collect-free}
\end{tabbing}

\noindent{\sc Event}:   Enable z-int; name this event `g0717'.\index{g0717}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-numberp-defn-proof-proves \\ \index{z-numberp-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-numberp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-numberp-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-numberp-defn}}\,({\it{term\/}}, {\rm{z-numberp}}\,({\it{term\/}})\index{z-numberp})\index{z-numberp-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-numberp-defn-proof}}\,({\it{term\/}})\index{z-numberp-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-numberp-defn-proof; name this event `g0725'.\index{g0725}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-numberp-hyps; name this event `g0726'.\index{g0726}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-numberp-ncode-proof} \\  
\=\+{\rm{z-numberp-ncode-proof}}\,({\it{x\/}})\index{z-numberp-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+$\neg$ {\rm{listp}}\,({\it{x\/}}), \\ 
{\rm{z-int-numeral-proof}}\,({\it{x\/}})\index{z-int-numeral-proof}, \\ 
{\rm{not-int-opair-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\tt{0}})\-\index{not-int-opair-proof})\-, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\sc{phi}}\index{phi})\index{ident-axiom-proof})\-, \\ 
{\rm{z-if}}\,({\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-int}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, {\sc{phi}}\index{phi})\index{z-if}, \\ 
{\rm{z-numberp}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-numberp}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,($\neg$ {\rm{listp}}\,({\it{x\/}}))\index{bool-fix})\index{ncode}, \\ 
{\rm{z-numberp-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-numberp-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-int-numeral-proof-proves-again \\ \index{z-int-numeral-proof-proves-agai!n}
\=\+(\=\+{\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{x-bar\/}} $=$ {\rm{numeral}}\,({\it{x\/}})\index{numeral}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-int-numeral-proof}}\,({\it{x\/}})\index{z-int-numeral-proof}, {\rm{z-int}}\,({\it{x-bar\/}})\index{z-int}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-int-numeral-proof-proves; name this event `g0727'.\index{g0727}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-numberp-ncode-proof-proves \\ \index{z-numberp-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-numberp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-numberp-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-numberp}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-numberp}, {\rm{ncode}}\,({\rm{bool-fix}}\,($\neg$ {\rm{listp}}\,({\it{x\/}}))\index{bool-fix})\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-numberp-ncode-proof}}\,({\it{x\/}})\index{z-numberp-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-numberp-ncode-proof; name this event `g0728'.\index{g0728}

\addvspace{18pt}
\begin{verbatim};;;;;;;Sequence of events introducing "(Z-zerop x)", the Z2 analogue of ZEROP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-zerop-exis-proof} \\  
\=\+{\rm{z-zerop-exis-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-zerop-exis-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-exis-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{z-if}}\,({\rm{f-equal}}\,({\it{x\/}}, {\sc{phi}}\index{phi})\index{f-equal}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, {\sc{phi}}\index{phi})\index{z-if}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\-\index{z-if})\-\index{term-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-zerop-unique-proof} \\  
\=\+{\rm{z-zerop-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}})\index{z-zerop-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{z-if}}\,({\rm{f-equal}}\,({\it{x\/}}, {\sc{phi}}\index{phi})\index{f-equal}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, {\sc{phi}}\index{phi})\index{z-if}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\-\index{z-if})\-\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-zerop-defn} \\  
\=\+{\rm{z-zerop-defn}}\,({\it{x\/}}, {\it{y\/}})\index{z-zerop-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-if}}\,({\rm{f-equal}}\,({\it{x\/}}, {\sc{phi}}\index{phi})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\index{f-if}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{f-equal})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-zerop}$\;\;$
{\rm{z-zerop}}\,({\it{x\/}})\index{z-zerop} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-zerop-defn-proof} \\  
\=\+{\rm{z-zerop-defn-proof}}\,({\it{term\/}})\index{z-zerop-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{z-zerop-defn}}\,({\tt{0}}, {\rm{z-zerop}}\,({\tt{0}})\index{z-zerop})\index{z-zerop-defn}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{z-zerop-defn}}\,({\tt{0}}, {\tt{1}})\index{z-zerop-defn}, \\ 
{\rm{z-zerop-exis-proof}}\,({\tt{0}}, {\tt{1}})\index{z-zerop-exis-proof}, \\ 
{\rm{z-zerop-unique-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-zerop-unique-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-zerop-hyps} \\  
\=\+{\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-zerop-defn}}\,({\tt{0}}, {\rm{z-zerop}}\,({\tt{0}})\index{z-zerop})\index{z-zerop-defn}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-zerop-defn-proof-proves \\ \index{z-zerop-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-zerop-defn}}\,({\it{term\/}}, {\rm{z-zerop}}\,({\it{term\/}})\index{z-zerop})\index{z-zerop-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-zerop-defn-proof}}\,({\it{term\/}})\index{z-zerop-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-zerop-defn-proof; name this event `g0763'.\index{g0763}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-zerop-hyps; name this event `g0764'.\index{g0764}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-zerop-ncode-proof} \\  
\=\+{\rm{z-zerop-ncode-proof}}\,({\it{x\/}})\index{z-zerop-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+$\neg$ {\rm{listp}}\,({\it{x\/}}), \\ 
{\rm{z-int-numeral-proof}}\,({\it{x\/}})\index{z-int-numeral-proof}, \\ 
{\rm{not-int-opair-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\tt{0}})\-\index{not-int-opair-proof})\-, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{x\/}} $\simeq {\tt{0}}$, \\ 
{\rm{ident-axiom-proof}}\,({\sc{phi}}\index{phi})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{x\/}}, {\tt{0}})\index{ncode-nequal-proof})\-, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\sc{phi}}\index{phi})\index{ident-axiom-proof})\-, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{ident-axiom-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-int}, \\ 
{\rm{z-if}}\,({\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\sc{phi}}\index{phi})\index{f-equal}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, {\sc{phi}}\index{phi})\index{z-if}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\-\index{z-if}, \\ 
{\rm{z-zerop}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-zerop}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,({\it{x\/}} $\simeq {\tt{0}}$)\index{bool-fix})\index{ncode}, \\ 
{\rm{z-zerop-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-zerop-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-zerop-ncode-proof-proves \\ \index{z-zerop-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-zerop}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-zerop}, {\rm{ncode}}\,({\rm{bool-fix}}\,({\it{x\/}} $\simeq {\tt{0}}$)\index{bool-fix})\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-zerop-ncode-proof}}\,({\it{x\/}})\index{z-zerop-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-zerop-ncode-proof; name this event `g0767'.\index{g0767}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{form-z-if-unique-proves} \\  
\=\+{\rm{form-z-if-unique-proves}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pftree\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{form-z-if-unique-proves} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+({\it{x\/}} $\not\in$ {\rm{collect-free}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $\not\in$ {\rm{collect-free}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{form-z-if-unique-proves}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\rm{car}}\,({\it{pftree\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{form-z-if-unique-proves} \\ 
$\wedge$$\;\;\;\;${\rm{form-z-if-unique-proves}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\rm{cdr}}\,({\it{pftree\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{form-z-if-unique-proves}\- \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{pftree\/}}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{exp\/}}, {\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst})\index{f-and}, {\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-z-if-unique-proof} \\  
\=\+{\rm{form-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pftree\/}})\index{form-z-if-unique-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }{\rm{unique-f-if-proof}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{subst}}\,({\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\rm{subst}}\,({\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{form-z-if-unique-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\rm{car}}\,({\it{pftree\/}}))\-\index{form-z-if-unique-proof}, \\ 
{\rm{form-z-if-unique-proof}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\rm{cdr}}\,({\it{pftree\/}}))\-\index{form-z-if-unique-proof})\-\index{unique-f-if-proof} \\ 
{\bf else }{\it{pftree\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-z-if-unique-proof-proves \\ \index{form-z-if-unique-proof-proves}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{2}})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{form-z-if-unique-proves}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pftree\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{form-z-if-unique-proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pftree\/}})\index{form-z-if-unique-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, \\ 
{\rm{subst}}\,({\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-eval-proof; name this event `g0700'.\index{g0700}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  form-z-if-unique-proof-proves1 \\ \index{form-z-if-unique-proof-proves1}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{2}})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{form-z-if-unique-proves}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pftree\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{form-z-if-unique-proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, \\ 
{\rm{subst}}\,({\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}}, {\it{pftree\/}})\index{form-z-if-unique-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-unique-proof-proves; name this event `g0701'.\index{g0701}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{trans-not-equal-proof} \\  
\=\+{\rm{trans-not-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{trans-not-equal-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-not}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal})\-, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-not}, \\ 
{\rm{list}}\,({\rm{trans-equal-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{trans-equal-proof1}, {\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  trans-not-equal-proof-proves \\ \index{trans-not-equal-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-not}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{z\/}})\index{f-equal})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-equal}}\,({\it{y\/}}, {\it{z\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{trans-not-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{trans-not-equal-proof}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable trans-not-equal-proof; name this event `g0840'.\index{g0840}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{flip-equals-proof} \\  
\=\+{\rm{flip-equals-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pf\/}})\index{flip-equals-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\it{pf\/}}, {\rm{symmetry-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{symmetry-proof1})\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  flip-equals-proof-proves \\ \index{flip-equals-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{flip-equals-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pf\/}})\index{flip-equals-proof}, {\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable flip-equals-proof; name this event `g0841'.\index{g0841}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{rename-forall-proof} \\  
\=\+{\rm{rename-forall-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{rename-forall-proof} \\ 
$=$$\;\;\;\;${\rm{forall-intro-proof}}\,(\=\+$\forall\;${\it{x\/}}$\;${\it{exp\/}}\index{forall}, \\ 
{\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst}, \\ 
{\it{y\/}}, \\ 
{\rm{subst-thm-b-proof}}\,({\it{exp\/}}, {\rm{list}}\,({\it{x\/}}), {\rm{list}}\,({\it{y\/}}))\index{subst-thm-b-proof})\-\index{forall-intro-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Disable free-for-list; name this event `g0731'.\index{g0731}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  parallel-subst-subst-case \\ \index{parallel-subst-subst-case}
{\rm{parallel-subst}}\,({\it{exp\/}}, {\rm{list}}\,({\it{x\/}}), {\rm{list}}\,({\it{term\/}}), {\it{flg\/}})\index{parallel-subst} $=$ {\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rename-forall-proof-proves \\ \index{rename-forall-proof-proves}
\=\+(\=\+({\it{y\/}} $\not\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{free-for}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{exp-inst\/}} $=$ {\rm{subst}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\tt{0}})\index{subst}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{rename-forall-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp\/}})\index{rename-forall-proof}, \\ 
{\rm{f-implies}}\,($\forall\;${\it{x\/}}$\;${\it{exp\/}}\index{forall}, $\forall\;${\it{y\/}}$\;${\it{exp-inst\/}}\index{forall})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable free-for-list; name this event `g0732'.\index{g0732}

\addvspace{18pt}

\noindent{\sc Event}:   Enable parallel-subst-subst-case; name this event `g0733'.\index{g0733}

\addvspace{18pt}

\noindent{\sc Event}:   Enable form-z-if-unique-proof; name this event `g0734'.\index{g0734}

\addvspace{18pt}
\begin{verbatim};;;;;;;Sequence of events introducing "(Z-sub1 x)", the Z2 analogue of SUB1.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-sub1}$\;\;$
{\rm{z-sub1}}\,({\it{x\/}})\index{z-sub1} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-sub1-defn} \\  
\=\+{\rm{z-sub1-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-sub1-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-if}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{f-and}}\,({\rm{z-int}}\,({\it{z\/}})\index{z-int}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{z\/}})\index{z-succ}, {\it{x\/}})\index{f-equal})\index{f-and})\index{forsome}, \\ 
{\rm{f-and}}\,({\rm{z-int}}\,({\it{y\/}})\index{z-int}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\it{x\/}})\index{f-equal})\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{f-if}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-sub1-expr} \\  
\=\+{\rm{z-sub1-expr}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-sub1-expr} \\ 
$=$$\;\;\;\;${\rm{z-if}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{z-if}}\,(\=\+{\rm{forsome}}\,({\it{z\/}}, {\rm{f-and}}\,({\rm{z-int}}\,({\it{z\/}})\index{z-int}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{z\/}})\index{z-succ}, {\it{x\/}})\index{f-equal})\index{f-and})\index{forsome}, \\ 
{\rm{f-and}}\,({\rm{z-int}}\,({\it{y\/}})\index{z-int}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\it{x\/}})\index{f-equal})\index{f-and}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{z-if}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{z-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-sub1-exis-proof} \\  
\=\+{\rm{z-sub1-exis-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-sub1-exis-proof} \\ 
$=$$\;\;\;\;${\rm{form-z-if-exis-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{x\/}})\-\index{f-equal})\-\index{f-and})\-\index{forsome}, \\ 
{\rm{z-int}}\,({\it{x\/}})\index{z-int})\-\index{f-and}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{x\/}})\-\index{f-equal})\-\index{f-and})\-\index{forsome})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{forsome}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{x\/}})\-\index{f-equal})\-\index{f-and})\-\index{forsome})\index{f-not}, \\ 
{\rm{z-int}}\,({\it{x\/}})\index{z-int})\-\index{f-and}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{exis-fun-intro-proof}}\,(\=\+{\it{y\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{exis-fun-intro-proof}))\-\index{tautconseq-proof})\-, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{forsome}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-not}}\,({\rm{z-int}}\,({\it{x\/}})\index{z-int})\index{f-not}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{forsome})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{exis-fun-intro-proof}}\,(\=\+{\it{y\/}}, \\ 
{\sc{phi}}\index{phi})\-\index{exis-fun-intro-proof}))\-\index{tautconseq-proof})\-, \\ 
{\rm{z-sub1-expr}}\,({\it{x\/}}, {\it{y\/}}, {\it{y\/}})\index{z-sub1-expr}, \\ 
{\it{y\/}}, \\ 
{\bf{nil}})\-\index{form-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-int-z-succ-equal-proof} \\  
\=\+{\rm{z-int-z-succ-equal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-int-z-succ-equal-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-and}}\,({\rm{z-trans2}}\,({\it{x\/}})\index{z-trans2}, {\rm{z-trich2}}\,({\it{x\/}})\index{z-trich2})\index{f-and})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-and}}\,({\rm{z-trans2}}\,({\it{y\/}})\index{z-trans2}, {\rm{z-trich2}}\,({\it{y\/}})\index{z-trich2})\index{f-and})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans2}}\,({\it{x\/}})\index{z-trans2}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-trans2}}\,({\it{y\/}})\index{z-trans2}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{iff-split-proof1}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-and}}\,({\rm{z-trans2}}\,({\it{x\/}})\index{z-trans2}, {\rm{z-trich2}}\,({\it{x\/}})\index{z-trich2})\index{f-and}, \\ 
{\rm{z-int-defn-proof}}\,({\it{x\/}})\index{z-int-defn-proof})\-\index{iff-split-proof1}, \\ 
{\rm{iff-split-proof1}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-and}}\,({\rm{z-trans2}}\,({\it{y\/}})\index{z-trans2}, {\rm{z-trich2}}\,({\it{y\/}})\index{z-trich2})\index{f-and}, \\ 
{\rm{z-int-defn-proof}}\,({\it{y\/}})\index{z-int-defn-proof})\-\index{iff-split-proof1}, \\ 
{\rm{z-succ-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\tt{1}}, {\tt{2}})\index{z-succ-equal-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-int-z-succ-equal-proof-proves \\ \index{z-int-z-succ-equal-proof-proves}
\=\+(\=\+{\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{1}}, {\tt{2}}), {\rm{collect-free}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}})\index{collect-free})\index{nil-intersect})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-int-z-succ-equal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-int-z-succ-equal-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{x\/}})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ}, {\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-int-z-succ-equal-proof; name this event `g0739'.\index{g0739}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-sub1-unique-proof} \\  
\=\+{\rm{z-sub1-unique-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{y1\/}})\index{z-sub1-unique-proof} \\ 
$=$$\;\;\;\;${\rm{form-z-if-unique-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-sub1-expr}}\,({\it{x\/}}, {\it{y\/}}, {\it{y\/}})\index{z-sub1-expr}, \\ 
{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{x\/}})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y1\/}})\index{z-succ}, \\ 
{\it{x\/}})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{y1\/}})\index{z-succ})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\it{y1\/}})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\it{y1\/}})\index{z-succ})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{x\/}})\-\index{f-equal})\-\index{f-and}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y1\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y1\/}})\index{z-succ}, \\ 
{\it{x\/}})\-\index{f-equal})\-\index{f-and})\-\index{f-and}, \\ 
{\rm{f-equal}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{trans-equal-proof2}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{x\/}}, \\ 
{\rm{z-succ}}\,({\it{y1\/}})\index{z-succ})\-\index{trans-equal-proof2}, \\ 
{\rm{z-int-z-succ-equal-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}})\-\index{z-int-z-succ-equal-proof})\-)\-\index{tautconseq-proof}, \\ 
{\rm{unique-fun-intro-proof}}\,({\it{y\/}}, {\it{y1\/}}, {\sc{phi}}\index{phi})\index{unique-fun-intro-proof})\-, \\ 
{\rm{unique-fun-intro-proof}}\,({\it{y\/}}, {\it{y1\/}}, {\sc{phi}}\index{phi})\index{unique-fun-intro-proof})\-)\-\index{form-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-sub1-defn-proof} \\  
\=\+{\rm{z-sub1-defn-proof}}\,({\it{term\/}})\index{z-sub1-defn-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{z-sub1-defn}}\,({\tt{0}}, {\rm{z-sub1}}\,({\tt{0}})\index{z-sub1}, {\tt{3}})\index{z-sub1-defn}, \\ 
{\tt{0}}, \\ 
{\it{term\/}}, \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{3}}, \\ 
{\tt{4}}, \\ 
{\rm{z-sub1-defn}}\,({\tt{0}}, {\tt{3}}, {\tt{3}})\index{z-sub1-defn}, \\ 
{\rm{z-sub1-exis-proof}}\,({\tt{0}}, {\tt{3}})\index{z-sub1-exis-proof}, \\ 
{\rm{z-sub1-unique-proof}}\,({\tt{0}}, {\tt{3}}, {\tt{4}})\index{z-sub1-unique-proof})\-\index{fun-defn-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-sub1-hyps} \\  
\=\+{\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-sub1-defn}}\,({\tt{0}}, {\rm{z-sub1}}\,({\tt{0}})\index{z-sub1}, {\tt{3}})\index{z-sub1-defn}) \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\rm{rest-of}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\rm{rest-of}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-sub1-defn-proof-proves \\ \index{z-sub1-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{term\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{3}}, {\tt{4}}), {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-sub1-defn}}\,({\it{term\/}}, {\rm{z-sub1}}\,({\it{term\/}})\index{z-sub1}, {\tt{3}})\index{z-sub1-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-sub1-defn-proof}}\,({\it{term\/}})\index{z-sub1-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-sub1-defn-proof; name this event `g0787'.\index{g0787}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-sub1-hyps; name this event `g0788'.\index{g0788}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{enable-nzerop-proof} \\  
\=\+{\rm{enable-nzerop-proof}}\,({\it{x\/}}, {\it{y\/}})\index{enable-nzerop-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-equal}}\,({\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ}, {\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ})\index{f-equal})\-\index{f-and}, \\ 
{\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ})\index{f-equal})\-\index{f-and})\-\index{forsome}, \\ 
{\rm{and-join-proof}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ})\-\index{f-equal}, \\ 
{\rm{z-int-numeral-proof}}\,({\it{x\/}} $-\;1$)\index{z-int-numeral-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ})\index{ident-axiom-proof})\-\index{and-join-proof}, \\ 
{\rm{subst-thm-a-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ})\-\index{f-equal})\-\index{f-and}, \\ 
{\rm{list}}\,({\it{y\/}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}))\-\index{subst-thm-a-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  enable-nzerop-proof-proves \\ \index{enable-nzerop-proof-proves}
\=\+(\=\+({\it{x\/}} $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{y\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{forsome}}\,({\it{y\/}}, {\rm{f-and}}\,({\rm{z-int}}\,({\it{y\/}})\index{z-int}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal})\index{f-and})\index{forsome}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{enable-nzerop-proof}}\,({\it{x\/}}, {\it{y\/}})\index{enable-nzerop-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable enable-nzerop-proof; name this event `g0790'.\index{g0790}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{disable-nzerop-proof} \\  
\=\+{\rm{disable-nzerop-proof}}\,({\it{x\/}}, {\it{y\/}})\index{disable-nzerop-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal})\-\index{f-and})\-\index{forsome})\index{f-not}, \\ 
$\forall\;${\it{y\/}}$\;${\rm{f-not}}\,({\rm{f-and}}\,({\rm{z-int}}\,({\it{y\/}})\index{z-int}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal})\index{f-and})\index{f-not}\index{forall}, \\ 
{\rm{forsome-forall-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal})\-\index{f-and})\-\index{forsome-forall-proof}, \\ 
{\rm{generalise-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-not}}\,({\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode})\-\index{f-equal})\-\index{f-and})\index{f-not}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-succ}}\,({\it{y\/}})\index{z-succ})\-\index{f-equal})\index{f-not})\-, \\ 
{\rm{f-not}}\,({\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\it{y\/}})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{f-and})\index{f-not}, \\ 
{\rm{list}}\,(\=\+{\rm{symmetry-proof1}}\,(\=\+{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{symmetry-proof1}, \\ 
{\rm{ext-not-equal-proof}}\,(\=\+{\sc{phi}}\index{phi}, \\ 
{\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, \\ 
{\it{y\/}}, \\ 
{\rm{null-axiom-proof}}\,({\it{y\/}})\index{null-axiom-proof}, \\ 
{\rm{x-in-z-succ-x-proof}}\,({\it{y\/}})\index{x-in-z-succ-x-proof})\-\index{ext-not-equal-proof})\-)\-\index{tautconseq-proof})\-\index{generalise-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  disable-nzerop-proof-proves \\ \index{disable-nzerop-proof-proves}
\=\+(\=\+($\neg$ {\rm{listp}}\,({\it{x\/}})) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{y\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-not}}\,({\rm{forsome}}\,(\=\+{\it{y\/}}, \\ 
{\rm{f-and}}\,({\rm{z-int}}\,({\it{y\/}})\index{z-int}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\it{y\/}})\index{z-succ}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal})\index{f-and})\-\index{forsome})\index{f-not}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{disable-nzerop-proof}}\,({\it{x\/}}, {\it{y\/}})\index{disable-nzerop-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable disable-nzerop-proof; name this event `g0791'.\index{g0791}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{nzerop-z-sub1-ncode-proof} \\  
\=\+{\rm{nzerop-z-sub1-ncode-proof}}\,({\it{x\/}})\index{nzerop-z-sub1-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{z-int}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-int}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{z-int}}\,({\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1})\index{z-int}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-succ}}\,({\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1})\index{z-succ}, \\ 
{\rm{z-succ}}\,({\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{z-succ})\-\index{f-equal})\-\index{f-and}, \\ 
{\rm{f-equal}}\,({\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, {\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{z-int-numeral-proof}}\,({\it{x\/}} $-\;1$)\index{z-int-numeral-proof}, \\ 
{\rm{z-int-z-succ-equal-proof}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\-\index{z-int-z-succ-equal-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-sub1-ncode-proof} \\  
\=\+{\rm{z-sub1-ncode-proof}}\,({\it{x\/}})\index{z-sub1-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{form-z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{x\/}} $\simeq\;${{\bf{nil}}}${}$, \\ 
{\rm{z-int-numeral-proof}}\,({\it{x\/}})\index{z-int-numeral-proof}, \\ 
{\rm{not-int-opair-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode}, \\ 
{\tt{0}})\-\index{not-int-opair-proof})\-, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{x\/}} $\not\simeq {\tt{0}}$, \\ 
{\rm{enable-nzerop-proof}}\,({\it{x\/}}, {\tt{3}})\index{enable-nzerop-proof}, \\ 
{\rm{disable-nzerop-proof}}\,({\it{x\/}}, {\tt{3}})\index{disable-nzerop-proof})\-, \\ 
{\rm{nzerop-z-sub1-ncode-proof}}\,({\it{x\/}})\index{nzerop-z-sub1-ncode-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof})\-, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\sc{phi}}\index{phi})\-\index{f-equal})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof})\-, \\ 
{\rm{z-sub1-expr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, {\tt{3}})\index{z-sub1-expr}, \\ 
{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{z-sub1-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1-defn-proof})\-\index{form-z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-sub1-ncode-proof-proves \\ \index{z-sub1-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, {\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-sub1-ncode-proof}}\,({\it{x\/}})\index{z-sub1-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-sub1-ncode-proof; name this event `g0793'.\index{g0793}

\addvspace{18pt}
\begin{verbatim};;;;;;;Sequence of events introducing "(Z-equal x y)", the Z2 analogue of EQUAL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-equal}$\;\;$
{\rm{z-equal}}\,({\it{x\/}}, {\it{y\/}})\index{z-equal} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-equal-defn} \\  
\=\+{\rm{z-equal-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-equal-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{z\/}}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{f-equal}, {\rm{f-equal}}\,({\it{z\/}}, {\sc{phi}}\index{phi})\index{f-equal})\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-equal-defn-proof} \\  
\=\+{\rm{z-equal-defn-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-equal-defn-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{z-equal-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-equal}}\,({\tt{0}}, {\tt{1}})\index{z-equal})\index{z-equal-defn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\rm{list}}\,({\it{x\/}}, {\it{y\/}}), \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\rm{z-equal-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-equal-defn}, \\ 
{\rm{term-z-if-exis-proof}}\,(\=\+{\tt{2}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\tt{0}}, {\tt{1}})\index{f-equal}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{z-if})\-\index{term-z-if-exis-proof}, \\ 
{\rm{term-z-if-unique-proof}}\,(\=\+{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,(\=\+{\tt{0}}, \\ 
{\tt{1}})\-\index{f-equal}, \\ 
{\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, \\ 
{\sc{phi}}\index{phi})\-\index{z-if})\-\index{term-z-if-unique-proof})\-\index{fun-defn-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-equal-hyps} \\  
\=\+{\rm{z-equal-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-equal-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-equal-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-equal}}\,({\tt{0}}, {\tt{1}})\index{z-equal})\index{z-equal-defn}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\rm{rest-of}}\,({\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, {\it{symbols\/}})\index{rest-of})\index{subset})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable free-for-list; name this event `g0740'.\index{g0740}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-equal-defn-proof-proves \\ \index{z-equal-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-equal-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-equal-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-equal-defn}}\,({\it{x\/}}, {\it{y\/}}, {\rm{z-equal}}\,({\it{x\/}}, {\it{y\/}})\index{z-equal})\index{z-equal-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-equal-defn-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-equal-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-equal-defn-proof; name this event `g0794'.\index{g0794}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-equal-hyps; name this event `g0795'.\index{g0795}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  ncode-equal-ncode-equal \\ \index{ncode-equal-ncode-equal}
{\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal} $\rightarrow$ ({\rm{ncode}}\,({\it{y\/}})\index{ncode} $=$ {\rm{ncode}}\,({\it{x\/}})\index{ncode})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-equal-ncode-proof} \\  
\=\+{\rm{z-equal-ncode-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-equal-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-nequal-proof})\-, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\sc{phi}}\index{phi})\index{ident-axiom-proof})\-, \\ 
{\rm{z-if}}\,({\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal}, {\rm{z-succ}}\,({\sc{phi}}\index{phi})\index{z-succ}, {\sc{phi}}\index{phi})\index{z-if}, \\ 
{\rm{z-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-equal}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal})\index{bool-fix})\index{ncode}, \\ 
{\rm{z-equal-defn-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-equal-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-equal-ncode-proof-proves \\ \index{z-equal-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-equal-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-equal-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,(\=\+{\rm{z-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-equal}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal})\index{bool-fix})\index{ncode})\-\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-equal-ncode-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-equal-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-equal-ncode-proof; name this event `g0796'.\index{g0796}

\addvspace{18pt}
\begin{verbatim};;;;;;;Sequence of events introducing "(Z-cons x y)", the Z2 analogue of CONS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-cons}$\;\;$
{\rm{z-cons}}\,({\it{x\/}}, {\it{y\/}})\index{z-cons} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-cons-defn}$\;\;$
{\rm{z-cons-defn}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{z-cons-defn} $=$ {\rm{f-equal}}\,({\it{z\/}}, {\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair})\index{f-equal}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-cons-defn-proof} \\  
\=\+{\rm{z-cons-defn-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-cons-defn-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{z-cons-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-cons}}\,({\tt{0}}, {\tt{1}})\index{z-cons})\index{z-cons-defn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\rm{list}}\,({\it{x\/}}, {\it{y\/}}), \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\rm{z-cons-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-cons-defn}, \\ 
{\rm{exis-fun-intro-proof}}\,({\tt{2}}, {\rm{z-opair}}\,({\tt{0}}, {\tt{1}})\index{z-opair})\index{exis-fun-intro-proof}, \\ 
{\rm{unique-fun-intro-proof}}\,(\=\+{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\rm{z-opair}}\,({\tt{0}}, {\tt{1}})\index{z-opair})\-\index{unique-fun-intro-proof})\-\index{fun-defn-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-cons-hyps} \\  
\=\+{\rm{z-cons-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cons-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-cons-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-cons}}\,({\tt{0}}, {\tt{1}})\index{z-cons})\index{z-cons-defn}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\rm{rest-of}}\,({\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, {\it{symbols\/}})\index{rest-of}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-cons-defn-proof-proves \\ \index{z-cons-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-cons-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cons-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{x\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{y\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-cons-defn}}\,({\it{x\/}}, {\it{y\/}}, {\rm{z-cons}}\,({\it{x\/}}, {\it{y\/}})\index{z-cons})\index{z-cons-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-cons-defn-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-cons-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-cons-defn-proof; name this event `g0800'.\index{g0800}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-cons-hyps; name this event `g0801'.\index{g0801}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-cons-defn-proof; name this event `g0741'.\index{g0741}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-cons-hyps; name this event `g0742'.\index{g0742}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  nzerop-ncode-equal \\ \index{nzerop-ncode-equal}
(({\it{x\/}} $\not\simeq {\tt{0}}$) $\vee$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) $\rightarrow$ ({\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal} $=$ ({\it{x\/}} $=$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{list-and-join-proof} \\  
\=\+{\rm{list-and-join-proof}}\,({\it{explist\/}}, {\it{pflist\/}})\index{list-and-join-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{explist\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{explist\/}})) \\ 
{\bf then }{\rm{and-join-proof}}\,(\=\+{\rm{car}}\,({\it{explist\/}}), \\ 
{\rm{list-f-and}}\,({\rm{cdr}}\,({\it{explist\/}}))\index{list-f-and}, \\ 
{\rm{car}}\,({\it{pflist\/}}), \\ 
{\rm{list-and-join-proof}}\,({\rm{cdr}}\,({\it{explist\/}}), {\rm{cdr}}\,({\it{pflist\/}}))\index{list-and-join-proof})\-\index{and-join-proof} \\ 
{\bf else }{\rm{car}}\,({\it{pflist\/}})$\;${\bf  endif}\- \\ 
{\bf else }{\it{pflist\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-and-join-proof-proves \\ \index{list-and-join-proof-proves}
\=\+({\rm{listp}}\,({\it{explist\/}}) $\wedge$ {\rm{proves-list}}\,({\it{pflist\/}}, {\it{explist\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{list-and-join-proof}}\,({\it{explist\/}}, {\it{pflist\/}})\index{list-and-join-proof}, \\ 
{\rm{list-f-and}}\,({\it{explist\/}})\index{list-f-and}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-and-join-proof; name this event `g0804'.\index{g0804}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{compose-proof} \\  
\=\+{\rm{compose-proof}}\,({\it{fn\/}}, {\it{val\/}}, {\it{args\/}}, {\it{vals\/}}, {\it{pflist\/}}, {\it{pf\/}})\index{compose-proof} \\ 
$=$$\;\;\;\;${\rm{list-detach-proof}}\,(\=\+{\rm{list-f-equal}}\,(\=\+{\rm{list}}\,(\=\+{\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), \\ 
{\rm{cons}}\,({\it{fn\/}}, {\it{vals\/}}), \\ 
{\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}))\-, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\it{val\/}}, {\rm{cons}}\,({\it{fn\/}}, {\it{vals\/}})))\-\index{list-f-equal}, \\ 
{\rm{f-equal}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\it{val\/}})\index{f-equal}, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}))\index{ident-axiom-proof}, \\ 
{\it{pf\/}}, \\ 
{\rm{list-detach-proof}}\,(\=\+{\rm{list-f-equal}}\,({\it{args\/}}, {\it{vals\/}})\index{list-f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), \\ 
{\rm{cons}}\,({\it{fn\/}}, {\it{vals\/}}))\-\index{f-equal}, \\ 
{\it{pflist\/}}, \\ 
{\rm{equal-axiom1-proof}}\,(\=\+{\it{fn\/}}, \\ 
{\it{args\/}}, \\ 
{\it{vals\/}})\-\index{equal-axiom1-proof})\-\index{list-detach-proof})\-, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{p}}\,({\tt{0}}, {\tt{2}})\index{p}, \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\rm{cons}}\,({\it{fn\/}}, {\it{vals\/}})), \\ 
{\rm{list}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\it{val\/}}))\-\index{equal-axiom2-proof})\-\index{list-detach-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;So far we have seen how to introduce functions defined by cases.  Another
;;;;;;;important way of defining functions is by composition, i.e., "(f x1 ... xn)"
;;;;;;;can be defined as "(g (h1 x1...xn) ... (hn x1...xn))".  (It is worth noting
;;;;;;;that it is exactly because we make heavy use of function composition that
;;;;;;;we introduced function symbols corresponding to the Lisp primitives.
;;;;;;;When we were working with the predicates representing the definitions,
;;;;;;;composition was extremely tedious to simulate.)  We can of course
;;;;;;;combine definition by cases and by composition.  The remaining important
;;;;;;;mode of definition is recursion, and we will see how this is carried out, later.
;;;;;;;The lemma below asserts that if |- "(arg1=val1)", . . ., |- "(argn=valn)",
;;;;;;;and |- "((f val1 ... valn) = val)", then |- "((f arg1 ... argn) = val)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  compose-proof-proves \\ \index{compose-proof-proves}
\=\+(\=\+{\rm{function}}\,({\it{fn\/}})\index{function} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{fn\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{args\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{vals\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{args\/}})\index{length} $=$ {\rm{degree}}\,({\it{fn\/}})\index{degree}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{vals\/}})\index{length} $=$ {\rm{degree}}\,({\it{fn\/}})\index{degree}) \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\rm{list-f-equal}}\,({\it{args\/}}, {\it{vals\/}})\index{list-f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{vals\/}}), {\it{val\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\it{val\/}})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{compose-proof}}\,({\it{fn\/}}, {\it{val\/}}, {\it{args\/}}, {\it{vals\/}}, {\it{pflist\/}}, {\it{pf\/}})\index{compose-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable compose-proof; name this event `g0805'.\index{g0805}

\addvspace{18pt}
\begin{verbatim};;;;;;;Perhaps the most commonly used composition is the iterated use of CAR/CDR.
;;;;;;;The function CAR-CDR recurses down LIST and if (CAR LIST) is 1, it applies
;;;;;;;CAR to the result of (CAR-CDR X (CDR LIST)).  Otherwise, it returns 
;;;;;;;(CDR (CAR-CDR X (CDR LIST))).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{car-cdr} \\  
\=\+{\rm{car-cdr}}\,({\it{x\/}}, {\it{list\/}})\index{car-cdr} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{list\/}}) $=$ {\tt{1}}$\;\;${\bf then }{\rm{car}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{car-cdr}) \\ 
{\bf else }{\rm{cdr}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{car-cdr})$\;${\bf  endif}\- \\ 
{\bf else }{\it{x\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;Z-CAR-CDR composes "Z-car" and "Z-cdr" in a manner similar to CAR-CDR.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-car-cdr} \\  
\=\+{\rm{z-car-cdr}}\,({\it{x\/}}, {\it{list\/}})\index{z-car-cdr} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{list\/}}) $=$ {\tt{1}}$\;\;${\bf then }{\rm{z-car}}\,({\rm{z-car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{z-car-cdr})\index{z-car} \\ 
{\bf else }{\rm{z-cdr}}\,({\rm{z-car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{z-car-cdr})\index{z-cdr}$\;${\bf  endif}\- \\ 
{\bf else }{\it{x\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-z-car-cdr \\ \index{termp-z-car-cdr}
\=\+({\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{z-car-cdr}}\,({\it{x\/}}, {\it{list\/}})\index{z-car-cdr}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-car-cdr-proof} \\  
\=\+{\rm{z-car-cdr-proof}}\,({\it{x\/}}, {\it{list\/}})\index{z-car-cdr-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{list\/}}) $=$ {\tt{1}} \\ 
{\bf then }{\rm{compose-proof}}\,(\=\+{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\it{list\/}})\index{car-cdr})\index{ncode}, \\ 
{\rm{list}}\,({\rm{z-car-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{cdr}}\,({\it{list\/}}))\index{z-car-cdr}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{car-cdr})\index{ncode}), \\ 
{\rm{list}}\,({\rm{z-car-cdr-proof}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{z-car-cdr-proof}), \\ 
{\rm{z-car-ncode-proof}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{car-cdr})\index{z-car-ncode-proof})\-\index{compose-proof} \\ 
{\bf else }{\rm{compose-proof}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\it{list\/}})\index{car-cdr})\index{ncode}, \\ 
{\rm{list}}\,({\rm{z-car-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{cdr}}\,({\it{list\/}}))\index{z-car-cdr}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{car-cdr})\index{ncode}), \\ 
{\rm{list}}\,({\rm{z-car-cdr-proof}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{z-car-cdr-proof}), \\ 
{\rm{z-cdr-ncode-proof}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{car-cdr})\index{z-cdr-ncode-proof})\-\index{compose-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{ident-axiom-proof}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable z-car; name this event `g0806'.\index{g0806}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-cdr; name this event `g0807'.\index{g0807}

\addvspace{18pt}
\begin{verbatim};;;;;;;Z-CAR-CDR is the Z2 analogue of CAR-CDR.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  z-car-cdr-proof-proves \\ \index{z-car-cdr-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-car-cdr}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\it{list\/}})\index{z-car-cdr}, {\rm{ncode}}\,({\rm{car-cdr}}\,({\it{x\/}}, {\it{list\/}})\index{car-cdr})\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-car-cdr-proof}}\,({\it{x\/}}, {\it{list\/}})\index{z-car-cdr-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-car-cdr-proof; name this event `g0808'.\index{g0808}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{compose-car-proof} \\  
\=\+{\rm{compose-car-proof}}\,({\it{fn\/}}, {\it{val\/}}, {\it{arg\/}}, {\it{pfval\/}})\index{compose-car-proof} \\ 
$=$$\;\;\;\;${\rm{compose-proof}}\,(\=\+{\it{fn\/}}, \\ 
{\it{val\/}}, \\ 
{\rm{list}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{arg\/}})\index{ncode})\index{z-car}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{arg\/}}))\index{ncode}), \\ 
{\rm{list}}\,({\rm{z-car-ncode-proof}}\,({\it{arg\/}})\index{z-car-ncode-proof}), \\ 
{\it{pfval\/}})\-\index{compose-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  compose-car-proof-proves \\ \index{compose-car-proof-proves}
\=\+(\=\+{\rm{function}}\,({\it{fn\/}})\index{function} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{fn\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;$({\rm{degree}}\,({\it{fn\/}})\index{degree} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pfval\/}}, \\ 
{\rm{f-equal}}\,({\rm{list}}\,({\it{fn\/}}, {\rm{ncode}}\,({\rm{car}}\,({\it{arg\/}}))\index{ncode}), {\it{val\/}})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{list}}\,({\it{fn\/}}, {\rm{z-car}}\,({\rm{ncode}}\,({\it{arg\/}})\index{ncode})\index{z-car}), {\it{val\/}})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{compose-car-proof}}\,({\it{fn\/}}, {\it{val\/}}, {\it{arg\/}}, {\it{pfval\/}})\index{compose-car-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable compose-car-proof; name this event `g0821'.\index{g0821}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{compose-car-cadr-proof} \\  
\=\+{\rm{compose-car-cadr-proof}}\,({\it{fn\/}}, {\it{val\/}}, {\it{arg\/}}, {\it{pfval\/}})\index{compose-car-cadr-proof} \\ 
$=$$\;\;\;\;${\rm{compose-proof}}\,(\=\+{\it{fn\/}}, \\ 
{\it{val\/}}, \\ 
{\rm{list}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{arg\/}})\index{ncode})\index{z-car}, {\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{arg\/}})\index{ncode})\index{z-cdr})\index{z-car}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{arg\/}}))\index{ncode}, {\rm{ncode}}\,({\rm{cadr}}\,({\it{arg\/}}))\index{ncode}), \\ 
{\rm{list}}\,(\=\+{\rm{z-car-ncode-proof}}\,({\it{arg\/}})\index{z-car-ncode-proof}, \\ 
{\rm{z-car-cdr-proof}}\,({\it{arg\/}}, {\rm{list}}\,({\tt{1}}, {\tt{2}}))\index{z-car-cdr-proof})\-, \\ 
{\it{pfval\/}})\-\index{compose-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  compose-car-cadr-proof-proves \\ \index{compose-car-cadr-proof-proves}
\=\+(\=\+{\rm{function}}\,({\it{fn\/}})\index{function} \\ 
$\wedge$$\;\;\;\;${\rm{symb}}\,({\it{fn\/}}, {\it{symbols\/}})\index{symb} \\ 
$\wedge$$\;\;\;\;$({\rm{degree}}\,({\it{fn\/}})\index{degree} $=$ {\tt{2}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pfval\/}}, \\ 
{\rm{f-equal}}\,({\rm{list}}\,({\it{fn\/}}, {\rm{ncode}}\,({\rm{car}}\,({\it{arg\/}}))\index{ncode}, {\rm{ncode}}\,({\rm{cadr}}\,({\it{arg\/}}))\index{ncode}), {\it{val\/}})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{arg\/}})\index{ncode})\index{z-car}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{arg\/}})\index{ncode})\index{z-cdr})\index{z-car})\-, \\ 
{\it{val\/}})\-\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{compose-car-cadr-proof}}\,({\it{fn\/}}, {\it{val\/}}, {\it{arg\/}}, {\it{pfval\/}})\index{compose-car-cadr-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable compose-car-cadr-proof; name this event `g0822'.\index{g0822}

\addvspace{18pt}
\begin{verbatim};;;;;;;Next we would like to define the Z2 analogue of APPLY-SUBR, the part
;;;;;;;of the Lisp interpreter that evaluates the Lisp primitives.  This is
;;;;;;;done in three parts to make the proof manageable.  In the first part
;;;;;;;we represent APPLY-SUBR1 which evaluates LISTP, CONS, CAR, and CDR.
;;;;;;;These operations are represented by the symbols 6, 7, 8, and 9, respectively,
;;;;;;;to the Lisp interpreter.  APPLY-SUBR1 is represented by "(Z-apply-subr1 f args)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-apply-subr1}$\;\;$
{\rm{z-apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, {\it{fn\/}}, {\it{args\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr1-defn} \\  
\=\+{\rm{z-apply-subr1-defn}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}})\index{z-apply-subr1-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{6}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-listp}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-listp})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{7}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-cons}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car}, {\rm{z-car}}\,({\rm{z-cdr}}\,({\it{args\/}})\index{z-cdr})\index{z-car})\index{z-cons})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{8}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-car}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-car})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-cdr}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-cdr})\index{f-equal})\-\index{f-if})\-\index{f-if})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr1-exis-proof} \\  
\=\+{\rm{z-apply-subr1-exis-proof}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}})\index{z-apply-subr1-exis-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-exis-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{6}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-listp}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-listp}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{7}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-cons}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car}, {\rm{z-car}}\,({\rm{z-cdr}}\,({\it{args\/}})\index{z-cdr})\index{z-car})\index{z-cons}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{8}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-car}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-car}, \\ 
{\rm{z-cdr}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-cdr})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{term-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr1-unique-proof} \\  
\=\+{\rm{z-apply-subr1-unique-proof}}\,({\it{y\/}}, {\it{y1\/}}, {\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{6}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-listp}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-listp}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{7}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-cons}}\,(\=\+{\rm{z-car}}\,({\it{args\/}})\index{z-car}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\it{args\/}})\index{z-cdr})\index{z-car})\-\index{z-cons}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{8}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-car}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-car}, \\ 
{\rm{z-cdr}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-cdr})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr1-defn-proof} \\  
\=\+{\rm{z-apply-subr1-defn-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1-defn-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{z-apply-subr1-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr1}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr1})\index{z-apply-subr1-defn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\rm{list}}\,({\it{fn\/}}, {\it{args\/}}), \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\rm{z-apply-subr1-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-apply-subr1-defn}, \\ 
{\rm{z-apply-subr1-exis-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-apply-subr1-exis-proof}, \\ 
{\rm{z-apply-subr1-unique-proof}}\,({\tt{2}}, {\tt{3}}, {\tt{0}}, {\tt{1}})\index{z-apply-subr1-unique-proof})\-\index{fun-defn-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr1-hyps} \\  
\=\+{\rm{z-apply-subr1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr1-hyps} \\ 
$=$$\;\;\;\;$(\=\+(\=\+{\rm{assoc}}\,({\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, {\it{defns\/}}) \\ 
$=$$\;\;\;\;${\rm{z-apply-subr1-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr1}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr1})\index{z-apply-subr1-defn})\- \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn})\-, \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr1-defn-proof-proves \\ \index{z-apply-subr1-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-apply-subr1-defn}}\,({\it{x\/}}, {\it{y\/}}, {\rm{z-apply-subr1}}\,({\it{x\/}}, {\it{y\/}})\index{z-apply-subr1})\index{z-apply-subr1-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-apply-subr1-defn-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-apply-subr1-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-apply-subr1-defn-proof; name this event `g0810'.\index{g0810}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-apply-subr1-hyps; name this event `g0811'.\index{g0811}

\addvspace{18pt}
\begin{verbatim};;;;;;;APPLY-SUBR2 evaluates the Lisp primitives ZEROP, ADD1, SUB1, and those
;;;;;;;evaluated by APPLY-SUBR1.  It is represented by "(Z-apply-subr2 f args)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-apply-subr2}$\;\;$
{\rm{z-apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, {\it{fn\/}}, {\it{args\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr2-defn} \\  
\=\+{\rm{z-apply-subr2-defn}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}})\index{z-apply-subr2-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{3}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-zerop}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-zerop})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{4}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-add1}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-add1})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{5}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-sub1}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-sub1})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1})\index{f-equal})\-\index{f-if})\-\index{f-if})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr2-exis-proof} \\  
\=\+{\rm{z-apply-subr2-exis-proof}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}})\index{z-apply-subr2-exis-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-exis-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{3}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-zerop}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-zerop}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{4}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-add1}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-add1}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{5}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-sub1}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-sub1}, \\ 
{\rm{z-apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{term-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr2-unique-proof} \\  
\=\+{\rm{z-apply-subr2-unique-proof}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}}, {\it{y1\/}})\index{z-apply-subr2-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{3}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-zerop}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-zerop}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{4}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-add1}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-add1}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{5}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-sub1}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-sub1}, \\ 
{\rm{z-apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr2-defn-proof} \\  
\=\+{\rm{z-apply-subr2-defn-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2-defn-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{z-apply-subr2-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr2}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr2})\index{z-apply-subr2-defn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\rm{list}}\,({\it{fn\/}}, {\it{args\/}}), \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\rm{z-apply-subr2-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-apply-subr2-defn}, \\ 
{\rm{z-apply-subr2-exis-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-apply-subr2-exis-proof}, \\ 
{\rm{z-apply-subr2-unique-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}}, {\tt{3}})\index{z-apply-subr2-unique-proof})\-\index{fun-defn-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr2-hyps} \\  
\=\+{\rm{z-apply-subr2-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr2-hyps} \\ 
$=$$\;\;\;\;$(\=\+(\=\+{\rm{assoc}}\,({\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, {\it{defns\/}}) \\ 
$=$$\;\;\;\;${\rm{z-apply-subr2-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr2}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr2})\index{z-apply-subr2-defn})\- \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn})\-, \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr2-defn-proof-proves \\ \index{z-apply-subr2-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr2-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr2-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-apply-subr2-defn}}\,({\it{x\/}}, {\it{y\/}}, {\rm{z-apply-subr2}}\,({\it{x\/}}, {\it{y\/}})\index{z-apply-subr2})\index{z-apply-subr2-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-apply-subr2-defn-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-apply-subr2-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-apply-subr2-defn-proof; name this event `g0812'.\index{g0812}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-apply-subr2-hyps; name this event `g0813'.\index{g0813}

\addvspace{18pt}
\begin{verbatim};;;;;;;Finally, APPLY-SUBR evaluates the Lisp primitives EQUAL, NUMBERP,
;;;;;;;and those evaluated by APPLY-SUBR2.  It is represented by
;;;;;;;"(Z-apply-subr f args)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-apply-subr}$\;\;$
{\rm{z-apply-subr}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, {\it{fn\/}}, {\it{args\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr-defn} \\  
\=\+{\rm{z-apply-subr-defn}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}})\index{z-apply-subr-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-equal}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car}, {\rm{z-car}}\,({\rm{z-cdr}}\,({\it{args\/}})\index{z-cdr})\index{z-car})\index{z-equal})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{2}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-numberp}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-numberp})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{z-apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2})\index{f-equal})\-\index{f-if})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr-exis-proof} \\  
\=\+{\rm{z-apply-subr-exis-proof}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}})\index{z-apply-subr-exis-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-exis-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-equal}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car}, {\rm{z-car}}\,({\rm{z-cdr}}\,({\it{args\/}})\index{z-cdr})\index{z-car})\index{z-equal}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{2}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-numberp}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-numberp}, \\ 
{\rm{z-apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2})\-\index{z-if})\-\index{z-if})\-\index{term-z-if-exis-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr-unique-proof} \\  
\=\+{\rm{z-apply-subr-unique-proof}}\,({\it{fn\/}}, {\it{args\/}}, {\it{y\/}}, {\it{y1\/}})\index{z-apply-subr-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{y1\/}}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-equal}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car}, {\rm{z-car}}\,({\rm{z-cdr}}\,({\it{args\/}})\index{z-cdr})\index{z-car})\index{z-equal}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{2}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-numberp}}\,({\rm{z-car}}\,({\it{args\/}})\index{z-car})\index{z-numberp}, \\ 
{\rm{z-apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2})\-\index{z-if})\-\index{z-if})\-\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr-defn-proof} \\  
\=\+{\rm{z-apply-subr-defn-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr-defn-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{z-apply-subr-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr})\index{z-apply-subr-defn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\rm{list}}\,({\it{fn\/}}, {\it{args\/}}), \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}, {\tt{1}}), \\ 
{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\rm{z-apply-subr-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-apply-subr-defn}, \\ 
{\rm{z-apply-subr-exis-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-apply-subr-exis-proof}, \\ 
{\rm{z-apply-subr-unique-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}}, {\tt{3}})\index{z-apply-subr-unique-proof})\-\index{fun-defn-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr-hyps} \\  
\=\+{\rm{z-apply-subr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-hyps} \\ 
$=$$\;\;\;\;$(\=\+(\=\+{\rm{assoc}}\,({\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, {\it{defns\/}}) \\ 
$=$$\;\;\;\;${\rm{z-apply-subr-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr})\index{z-apply-subr-defn})\- \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn})\-, \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr-defn-proof-proves \\ \index{z-apply-subr-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-apply-subr-defn}}\,({\it{x\/}}, {\it{y\/}}, {\rm{z-apply-subr}}\,({\it{x\/}}, {\it{y\/}})\index{z-apply-subr})\index{z-apply-subr-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-apply-subr-defn-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-apply-subr-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-apply-subr-defn-proof; name this event `g0814'.\index{g0814}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-apply-subr-hyps; name this event `g0815'.\index{g0815}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{apply-subr1} \\  
\=\+{\rm{apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{fn\/}} $=$ {\tt{6}}$\;\;${\bf then }{\rm{bool-fix}}\,({\rm{listp}}\,({\rm{car}}\,({\it{args\/}})))\index{bool-fix} \\ 
{\bf elseif }{\it{fn\/}} $=$ {\tt{7}}$\;\;${\bf then }{\rm{cons}}\,({\rm{car}}\,({\it{args\/}}), {\rm{cadr}}\,({\it{args\/}})) \\ 
{\bf elseif }{\it{fn\/}} $=$ {\tt{8}}$\;\;${\bf then }{\rm{car}}\,({\rm{car}}\,({\it{args\/}})) \\ 
{\bf else }{\rm{cdr}}\,({\rm{car}}\,({\it{args\/}}))$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr1-ncode-proof} \\  
\=\+{\rm{z-apply-subr1-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{6}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{6}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{6}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{listp}}\,({\rm{car}}\,({\it{args\/}})))\index{bool-fix})\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-listp-ncode-proof}}\,({\rm{car}}\,({\it{args\/}}))\index{z-listp-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{7}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{7}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{7}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-cadr-proof}}\,(\=\+{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{cons}}\,(\=\+{\rm{car}}\,({\it{args\/}}), \\ 
{\rm{cadr}}\,({\it{args\/}}))\-)\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-cons-defn-proof}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{args\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{args\/}}))\index{ncode})\-\index{z-cons-defn-proof})\-\index{compose-car-cadr-proof}, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{8}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{8}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{8}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\rm{car}}\,({\it{args\/}})))\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-car-ncode-proof}}\,({\rm{car}}\,({\it{args\/}}))\index{z-car-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\rm{car}}\,({\it{args\/}})))\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\rm{car}}\,({\it{args\/}}))\index{z-cdr-ncode-proof})\-\index{compose-car-proof})\-)\-)\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{6}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-listp}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car})\index{z-listp}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{7}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-cons}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-cdr})\index{z-car})\-\index{z-cons}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{8}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-car}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car})\index{z-car}, \\ 
{\rm{z-cdr}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car})\index{z-cdr})\-\index{z-if})\-\index{z-if})\-\index{z-if}, \\ 
{\rm{z-apply-subr1}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr1}, \\ 
{\rm{ncode}}\,({\rm{apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr1})\index{ncode}, \\ 
{\rm{z-apply-subr1-defn-proof}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr1-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr1-ncode-proof-proves-step \\ \index{z-apply-subr1-ncode-proof-prove!s-step}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cons-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cons-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-listp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-listp-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,(\=\+{\rm{z-apply-subr1}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr1}, \\ 
{\rm{ncode}}\,({\rm{apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr1})\index{ncode})\-\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-apply-subr1-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-apply-subr1-ncode-proof; name this event `g0820'.\index{g0820}

\addvspace{18pt}

\noindent{\sc Event}:   Enable apply-subr1; name this event `g0825'.\index{g0825}

\addvspace{18pt}

\noindent{\sc Event}:   Enable sub1; name this event `g0826'.\index{g0826}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{apply-subr2} \\  
\=\+{\rm{apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{fn\/}} $=$ {\tt{3}}$\;\;${\bf then }{\rm{bool-fix}}\,({\rm{car}}\,({\it{args\/}}) $\simeq {\tt{0}}$)\index{bool-fix} \\ 
{\bf elseif }{\it{fn\/}} $=$ {\tt{4}}$\;\;${\bf then }$1\;+$ {\rm{car}}\,({\it{args\/}}) \\ 
{\bf elseif }{\it{fn\/}} $=$ {\tt{5}}$\;\;${\bf then }{\rm{car}}\,({\it{args\/}}) $-\;1$ \\ 
{\bf else }{\rm{apply-subr1}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr1}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr2-ncode-proof} \\  
\=\+{\rm{z-apply-subr2-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{3}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{3}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{3}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{car}}\,({\it{args\/}}) $\simeq {\tt{0}}$)\index{bool-fix})\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-zerop-ncode-proof}}\,({\rm{car}}\,({\it{args\/}}))\index{z-zerop-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{4}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{4}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{4}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,($1\;+$ {\rm{car}}\,({\it{args\/}}))\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-add1-ncode-proof}}\,({\rm{car}}\,({\it{args\/}}))\index{z-add1-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{5}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{5}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{5}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{args\/}}) $-\;1$)\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-sub1-ncode-proof}}\,({\rm{car}}\,({\it{args\/}}))\index{z-sub1-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{z-apply-subr1-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr1-ncode-proof})\-)\-)\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{3}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-zerop}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car})\index{z-zerop}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{4}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-add1}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car})\index{z-add1}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{5}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-sub1}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car})\index{z-sub1}, \\ 
{\rm{z-apply-subr1}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr1})\-\index{z-if})\-\index{z-if})\-\index{z-if}, \\ 
{\rm{z-apply-subr2}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr2}, \\ 
{\rm{ncode}}\,({\rm{apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr2})\index{ncode}, \\ 
{\rm{z-apply-subr2-defn-proof}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr2-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr2-ncode-proof-proves-step \\ \index{z-apply-subr2-ncode-proof-prove!s-step}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr2-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr2-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cons-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cons-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-listp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-listp-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-add1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-add1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,(\=\+{\rm{z-apply-subr2}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr2}, \\ 
{\rm{ncode}}\,({\rm{apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr2})\index{ncode})\-\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-apply-subr2-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-apply-subr2-ncode-proof; name this event `g0827'.\index{g0827}

\addvspace{18pt}

\noindent{\sc Event}:   Enable apply-subr2; name this event `g0828'.\index{g0828}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{apply-subr} \\  
\=\+{\rm{apply-subr}}\,({\it{fn\/}}, {\it{args\/}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{fn\/}} $=$ {\tt{1}}$\;\;${\bf then }{\rm{bool-fix}}\,({\rm{ncode-equal}}\,({\rm{car}}\,({\it{args\/}}), {\rm{cadr}}\,({\it{args\/}}))\index{ncode-equal})\index{bool-fix} \\ 
{\bf elseif }{\it{fn\/}} $=$ {\tt{2}}$\;\;${\bf then }{\rm{bool-fix}}\,($\neg$ {\rm{listp}}\,({\rm{car}}\,({\it{args\/}})))\index{bool-fix} \\ 
{\bf else }{\rm{apply-subr2}}\,({\it{fn\/}}, {\it{args\/}})\index{apply-subr2}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr-ncode-proof} \\  
\=\+{\rm{z-apply-subr-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr-ncode-proof} \\ 
$=$$\;\;\;\;${\rm{z-if-eval-proof1}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{1}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{1}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{1}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-cadr-proof}}\,(\=\+{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{ncode-equal}}\,(\=\+{\rm{car}}\,({\it{args\/}}), \\ 
{\rm{cadr}}\,({\it{args\/}}))\-\index{ncode-equal})\index{bool-fix})\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-equal-ncode-proof}}\,(\=\+{\rm{car}}\,({\it{args\/}}), \\ 
{\rm{cadr}}\,({\it{args\/}}))\-\index{z-equal-ncode-proof})\-\index{compose-car-cadr-proof}, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{fn\/}} $=$ {\tt{2}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{2}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{fn\/}}, {\tt{2}})\index{ncode-nequal-proof})\-, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{bool-fix}}\,($\neg$ {\rm{listp}}\,({\rm{car}}\,({\it{args\/}})))\index{bool-fix})\index{ncode}, \\ 
{\it{args\/}}, \\ 
{\rm{z-numberp-ncode-proof}}\,({\rm{car}}\,({\it{args\/}}))\index{z-numberp-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{z-apply-subr2-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr2-ncode-proof})\-)\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-equal}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-cdr})\index{z-car})\-\index{z-equal}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\tt{2}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-numberp}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-car})\index{z-numberp}, \\ 
{\rm{z-apply-subr2}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr2})\-\index{z-if})\-\index{z-if}, \\ 
{\rm{z-apply-subr}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,({\it{fn\/}}, {\it{args\/}}))\index{ncode}, \\ 
{\rm{z-apply-subr-defn-proof}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr-defn-proof})\-\index{z-if-eval-proof1}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-apply-subr-all-hyps} \\  
\=\+{\rm{z-apply-subr-all-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-all-hyps} \\ 
$=$$\;\;\;\;$(\=\+{\rm{z-apply-subr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr2-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr2-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cons-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cons-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-listp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-listp-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-add1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-add1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-equal-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-equal-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-numberp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-numberp-hyps})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr-ncode-proof-proves \\ \index{z-apply-subr-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-all-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-all-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,(\=\+{\rm{z-apply-subr}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode}, {\rm{ncode}}\,({\it{args\/}})\index{ncode})\index{z-apply-subr}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,({\it{fn\/}}, {\it{args\/}}))\index{ncode})\-\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-apply-subr-ncode-proof}}\,({\it{fn\/}}, {\it{args\/}})\index{z-apply-subr-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-apply-subr-ncode-proof; name this event `g0829'.\index{g0829}

\addvspace{18pt}

\noindent{\sc Event}:   Enable apply-subr; name this event `g0830'.\index{g0830}

\addvspace{18pt}
\begin{verbatim};;;;;;;Thus far we have shown how we can derive Z2 analogues of the Lisp primitives
;;;;;;;and the function APPLY-SUBR.  These functions will be used in demonstrating
;;;;;;;the representability of the Lisp interpreter.  A major obstacle that has
;;;;;;;to be overcome is the representability (in Z2) of recursively defined Lisp functions.
;;;;;;;These will be represented by specifying the partial graph (the set of value/argument
;;;;;;;ordered pairs used in the computation of the value of the function at
;;;;;;;a specific argument).  This partial graph will first be constructed as a 
;;;;;;;list of pairs for the computation of the Lisp function and then translated
;;;;;;;into the corresponding set.  The specification of the partial graph in Z2 will be
;;;;;;;shown to hold of this translation.  It will also be shown that any set
;;;;;;;satisfying the specification.
;;;;;;;The function GRAPH-TRANS below translates a list into the corresponding set
;;;;;;;so that the list (X1 . (X2 . ... (Xn . NIL))) is converted to the set
;;;;;;; "{<X1>, <X2>, ..., <Xn>}".  The translation of list X will be represented as [X].
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{graph-trans} \\  
\=\+{\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{z-union}}\,({\rm{z-sing}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode})\index{z-sing}, {\rm{graph-trans}}\,({\rm{cdr}}\,({\it{x\/}}))\index{graph-trans})\index{z-union} \\ 
{\bf else }{\sc{phi}}\index{phi}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{in-graph-proof} \\  
\=\+{\rm{in-graph-proof}}\,({\it{x\/}}, {\it{graph\/}})\index{in-graph-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{graph\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $=$ {\rm{car}}\,({\it{graph\/}}) \\ 
{\bf then }{\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-union}}\,(\=\+{\rm{z-sing}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sing}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\-\index{z-union})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{isin}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{z-sing}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sing})\index{isin}, \\ 
{\rm{isin}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\index{isin})\-\index{f-or}, \\ 
{\rm{union-axiom-proof}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-sing}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sing}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\-\index{union-axiom-proof}, \\ 
{\rm{rt-expan-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-sing}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sing})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin-sing1-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{isin-sing1-proof})\-\index{rt-expan-proof})\-\index{f-iff-mp-proof2} \\ 
{\bf else }{\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-union}}\,(\=\+{\rm{z-sing}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{graph\/}}))\index{ncode})\index{z-sing}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\-\index{z-union})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-sing}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{graph\/}}))\index{ncode})\index{z-sing})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\-\index{isin})\-\index{f-or}, \\ 
{\rm{union-axiom-proof}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-sing}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{graph\/}}))\index{ncode})\index{z-sing}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\-\index{union-axiom-proof}, \\ 
{\rm{expan-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{z-sing}}\,({\rm{ncode}}\,({\rm{car}}\,({\it{graph\/}}))\index{ncode})\index{z-sing})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{graph\/}}))\index{graph-trans})\-\index{isin}, \\ 
{\rm{in-graph-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{cdr}}\,({\it{graph\/}}))\-\index{in-graph-proof})\-\index{expan-proof})\-\index{f-iff-mp-proof2}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-graph-trans \\ \index{termp-graph-trans}
\=\+(\=\+({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{verbatim};;;;;;;If X is in GRAPH, then |- "(<X> in [GRAPH])".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-graph-proof-proves \\ \index{in-graph-proof-proves}
\=\+(\=\+({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\in$ {\it{graph\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{x-bar\/}} $=$ {\rm{ncode}}\,({\it{x\/}})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $=$ {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{in-graph-proof}}\,({\it{x\/}}, {\it{graph\/}})\index{in-graph-proof}, {\rm{isin}}\,({\it{x-bar\/}}, {\it{trace\/}})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-graph-proof; name this event `g0831'.\index{g0831}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{opair-equal-proof} \\  
\=\+{\rm{opair-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{opair-equal-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-sing}}\,({\it{u\/}})\index{z-sing})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}, \\ 
{\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, \\ 
{\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{equal-axiom1-proof}}\,({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\rm{list}}\,({\it{u\/}}, {\it{v\/}}))\index{equal-axiom1-proof}, \\ 
{\rm{equal-axiom1-proof}}\,({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{list}}\,({\it{x\/}}, {\it{x\/}}), {\rm{list}}\,({\it{u\/}}, {\it{u\/}}))\index{equal-axiom1-proof}, \\ 
{\rm{equal-axiom1-proof}}\,(\=\+{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{list}}\,({\rm{z-sing}}\,({\it{x\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair}), \\ 
{\rm{list}}\,({\rm{z-sing}}\,({\it{u\/}})\index{z-sing}, {\rm{z-pair}}\,({\it{u\/}}, {\it{v\/}})\index{z-pair}))\-\index{equal-axiom1-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Disable z-opair; name this event `g0832'.\index{g0832}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-sing; name this event `g0833'.\index{g0833}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  opair-equal-proof-proves \\ \index{opair-equal-proof-proves}
\=\+(\=\+({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{u\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{v\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{opair-equal-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{opair-equal-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\it{u\/}})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{v\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\rm{z-opair}}\,({\it{u\/}}, {\it{v\/}})\index{z-opair})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-sing; name this event `g0834'.\index{g0834}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-opair; name this event `g0835'.\index{g0835}

\addvspace{18pt}

\noindent{\sc Event}:   Enable opair-equal-proof; name this event `g0836'.\index{g0836}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-graph-trans-proof} \\  
\=\+{\rm{in-graph-trans-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{in-graph-trans-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{z\/}}))\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-sing}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-sing})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{z\/}})\index{graph-trans})\index{isin})\-\index{f-or})\-\index{f-iff}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{z-sing}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-sing})\index{isin}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal})\-\index{f-iff})\-, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{z\/}}))\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{z\/}})\index{graph-trans})\index{isin})\-\index{f-or})\-\index{f-iff}, \\ 
{\rm{list}}\,(\=\+{\rm{union-axiom-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{z-sing}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-sing}, \\ 
{\rm{graph-trans}}\,({\it{z\/}})\index{graph-trans})\-\index{union-axiom-proof}, \\ 
{\rm{isin-sing-proof}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\it{x\/}})\index{isin-sing-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{isin-graph-trans-proof} \\  
\=\+{\rm{isin-graph-trans-proof}}\,({\it{x\/}}, {\it{y\/}})\index{isin-graph-trans-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin})\-\index{f-or}, \\ 
{\rm{in-graph-trans-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\it{x\/}}, {\it{y\/}})\index{in-graph-trans-proof}, \\ 
{\rm{rt-expan-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{ident-axiom-proof})\-\index{rt-expan-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{graph-trans-merge-proof} \\  
\=\+{\rm{graph-trans-merge-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp1\/}}, {\it{exp2\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{graph-trans-merge-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{exp1\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin})\-\index{f-or})\-\index{f-iff}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{exp1\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, {\it{exp2\/}})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}, {\it{exp2\/}})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans})\index{isin}, {\it{exp2\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\rm{in-graph-trans-proof}}\,({\it{exp1\/}}, {\it{x\/}}, {\it{y\/}})\index{in-graph-trans-proof}, {\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((x in [(CONS Y Z)]) <-> ((x = <Y>) v (x in [Z])))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-graph-trans-proof-proves \\ \index{in-graph-trans-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{y-bar\/}} $=$ {\rm{ncode}}\,({\it{y\/}})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{z-bar\/}} $=$ {\rm{graph-trans}}\,({\it{z\/}})\index{graph-trans}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{in-graph-trans-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{in-graph-trans-proof}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{z\/}}))\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y-bar\/}})\index{f-equal}, {\rm{isin}}\,({\it{x\/}}, {\it{z-bar\/}})\index{isin})\index{f-or})\-\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-graph-trans-proof; name this event `g0488'.\index{g0488}

\addvspace{18pt}
\begin{verbatim};;;;;;; |- "(<X> in [(CONS X Y)])".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  isin-graph-trans-proof-proves \\ \index{isin-graph-trans-proof-proves}
\=\+(\=\+({\it{x-bar\/}} $=$ {\rm{ncode}}\,({\it{x\/}})\index{ncode}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isin-graph-trans-proof}}\,({\it{x\/}}, {\it{y\/}})\index{isin-graph-trans-proof}, \\ 
{\rm{isin}}\,({\it{x-bar\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans})\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "((a = <X>) -> B)", and |- "((a in [Y]) -> B)", then |- "((a in [CONS X Y]) -> B)".
;;;;;;;This lemma is really important because we prove that every member of the
;;;;;;;partial graph satisfies the specification, by induction on the list from which
;;;;;;;the graph is constructed.  This lemma permits the case-split required to carry
;;;;;;;out such an induction.  This will be clarified later.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  graph-trans-merge \\ \index{graph-trans-merge}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{exp1\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp2\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{exp1\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, {\it{exp2\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}, {\it{exp2\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{graph-trans-merge-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp1\/}}, {\it{exp2\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{graph-trans-merge-proof}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans})\index{isin}, {\it{exp2\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isin-graph-trans-proof; name this event `g0490'.\index{g0490}

\addvspace{18pt}

\noindent{\sc Event}:   Enable graph-trans-merge-proof; name this event `g0491'.\index{g0491}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  nlistp-graph-trans \\ \index{nlistp-graph-trans}
({\it{x\/}} $\simeq\;${{\bf{nil}}}${}$) $\rightarrow$ ({\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans} $=$ {\sc{phi}}\index{phi})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-graph-trans \\ \index{collect-free-graph-trans}
{\rm{collect-free}}\,({\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}
\end{tabbing}

\noindent{\sc Event}:   Enable graph-trans; name this event `g0459'.\index{g0459}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-graph-step-proof} \\  
\=\+{\rm{in-graph-step-proof}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\it{pf\/}})\index{in-graph-step-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{isin}}\,({\it{exp\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,({\rm{f-equal}}\,({\it{exp\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, {\rm{isin}}\,({\it{exp\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin})\index{f-or}, \\ 
{\rm{in-graph-trans-proof}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}})\index{in-graph-trans-proof}, \\ 
{\rm{expan-proof}}\,(\=\+{\rm{f-equal}}\,({\it{exp\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{exp\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}, \\ 
{\it{pf\/}})\-\index{expan-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "(exp in [Y])", then |- "(exp in [(CONS X Y)])".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-graph-step-proof-proves \\ \index{in-graph-step-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{exp\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{isin}}\,({\it{exp\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $=$ {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{in-graph-step-proof}}\,({\it{exp\/}}, {\it{x\/}}, {\it{y\/}}, {\it{pf\/}})\index{in-graph-step-proof}, \\ 
{\rm{isin}}\,({\it{exp\/}}, {\it{trace\/}})\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-graph-step-proof; name this event `g0837'.\index{g0837}

\addvspace{18pt}
\begin{verbatim};;;;;;;Z-LIST converts a list of the form (X1. (... (Xn . NIL))) to <<X1>, ..., <Xn>>.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-list} \\  
\=\+{\rm{z-list}}\,({\it{exp\/}})\index{z-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{exp\/}}))$\;\;${\bf then }{\rm{z-opair}}\,({\rm{car}}\,({\it{exp\/}}), {\rm{z-list}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{z-list})\index{z-opair} \\ 
{\bf else }{\rm{car}}\,({\it{exp\/}})$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-z-list \\ \index{termp-z-list}
\=\+(({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{listp}}\,({\it{list\/}}) $\wedge$ {\rm{termp}}\,({\it{list\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{z-list}}\,({\it{list\/}})\index{z-list}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-list-equal-proof} \\  
\=\+{\rm{z-list-equal-proof}}\,({\it{list1\/}}, {\it{list2\/}}, {\it{pf\/}})\index{z-list-equal-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list1\/}}) $\wedge$ {\rm{listp}}\,({\it{list2\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{list1\/}})) $\wedge$ {\rm{listp}}\,({\rm{cdr}}\,({\it{list2\/}})) \\ 
{\bf then }{\rm{and-join-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{car}}\,({\it{list1\/}}), {\rm{car}}\,({\it{list2\/}}))\index{f-equal}, \\ 
{\rm{list-f-and}}\,({\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{list1\/}}), \\ 
{\rm{cdr}}\,({\it{list2\/}}))\-\index{list-f-equal})\index{list-f-and}, \\ 
{\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-list}}\,({\it{list1\/}})\index{z-list}, \\ 
{\rm{z-list}}\,({\it{list2\/}})\index{z-list})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{car}}\,({\it{list1\/}}), {\rm{car}}\,({\it{list2\/}}))\index{f-equal}, \\ 
{\it{pf\/}}, \\ 
{\rm{z-car-unique-proof}}\,(\=\+{\rm{car}}\,({\it{list1\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{list1\/}}))\index{z-list}, \\ 
{\rm{car}}\,({\it{list2\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{list2\/}}))\index{z-list})\-\index{z-car-unique-proof})\-\index{detach-proof}, \\ 
{\rm{z-list-equal-proof}}\,(\=\+{\rm{cdr}}\,({\it{list1\/}}), \\ 
{\rm{cdr}}\,({\it{list2\/}}), \\ 
{\rm{detach-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-list}}\,({\it{list1\/}})\index{z-list}, \\ 
{\rm{z-list}}\,({\it{list2\/}})\index{z-list})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-list}}\,({\rm{cdr}}\,({\it{list1\/}}))\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{list2\/}}))\index{z-list})\-\index{f-equal}, \\ 
{\it{pf\/}}, \\ 
{\rm{z-cdr-unique-proof}}\,(\=\+{\rm{car}}\,({\it{list1\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{list1\/}}))\index{z-list}, \\ 
{\rm{car}}\,({\it{list2\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{list2\/}}))\index{z-list})\-\index{z-cdr-unique-proof})\-\index{detach-proof})\-\index{z-list-equal-proof})\-\index{and-join-proof} \\ 
{\bf else }{\it{pf\/}}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{len-equal} \\  
\=\+{\rm{len-equal}}\,({\it{x\/}}, {\it{y\/}})\index{len-equal} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) $\wedge$ {\rm{listp}}\,({\it{y\/}})$\;\;${\bf then }{\rm{len-equal}}\,({\rm{cdr}}\,({\it{x\/}}), {\rm{cdr}}\,({\it{y\/}}))\index{len-equal} \\ 
{\bf else }({\it{x\/}} $\simeq\;${{\bf{nil}}}${}$) $\wedge$ ({\it{y\/}} $\simeq\;${{\bf{nil}}}${}$)$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-list-equal-proof-proves \\ \index{z-list-equal-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{list1\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{list2\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list2\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{len-equal}}\,({\it{list1\/}}, {\it{list2\/}})\index{len-equal} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\rm{z-list}}\,({\it{list1\/}})\index{z-list}, {\rm{z-list}}\,({\it{list2\/}})\index{z-list})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-list-equal-proof}}\,({\it{list1\/}}, {\it{list2\/}}, {\it{pf\/}})\index{z-list-equal-proof}, \\ 
{\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{list1\/}}, {\it{list2\/}})\index{list-f-equal})\index{list-f-and}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-list-equal-proof; name this event `g0838'.\index{g0838}

\addvspace{18pt}
\begin{verbatim};;;;;;;PROVES-FORM-Z-IF-OK is similar to PROVES-Z-IF except that in the base case,
;;;;;;;it merely checks if PFTREE is a proof of EXP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{proves-form-z-if-ok} \\  
\=\+{\rm{proves-form-z-if-ok}}\,({\it{pftree\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-ok} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pftree\/}}) \\ 
{\bf then }\=\+{\rm{proves}}\,({\rm{cadar}}\,({\it{pftree\/}}), {\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-ok}}\,(\=\+{\rm{cadr}}\,({\it{pftree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-form-z-if-ok}\- \\ 
{\bf else }\=\+{\rm{proves}}\,(\=\+{\rm{caddar}}\,({\it{pftree\/}}), \\ 
{\rm{f-not}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-ok}}\,(\=\+{\rm{caddr}}\,({\it{pftree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-form-z-if-ok}\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{proves}}\,({\it{pftree\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-z-if-ok-proof} \\  
\=\+{\rm{form-z-if-ok-proof}}\,({\it{pftree\/}}, {\it{exp\/}})\index{form-z-if-ok-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pftree\/}}) \\ 
{\bf then }{\rm{f-if-intro-proof1}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\rm{cadar}}\,({\it{pftree\/}}), \\ 
{\rm{form-z-if-ok-proof}}\,(\=\+{\rm{cadr}}\,({\it{pftree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left})\-\index{form-z-if-ok-proof})\-\index{f-if-intro-proof1} \\ 
{\bf else }{\rm{f-if-intro-proof2}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\rm{caddar}}\,({\it{pftree\/}}), \\ 
{\rm{form-z-if-ok-proof}}\,(\=\+{\rm{caddr}}\,({\it{pftree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right})\-\index{form-z-if-ok-proof})\-\index{f-if-intro-proof2}$\;${\bf  endif}\- \\ 
{\bf else }{\it{pftree\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;From a proof-tree containing proofs of each of the test nodes of an form-if-tree
;;;;;;;and the leaf node that they lead upto, we can construct a proof of the entire
;;;;;;;if-expression represented by the form-if-tree.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  form-z-if-ok-proof-proves \\ \index{form-z-if-ok-proof-proves}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-ok}}\,({\it{pftree\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-ok} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{form-z-if-ok-proof}}\,({\it{pftree\/}}, {\it{exp\/}})\index{form-z-if-ok-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-ok-proof; name this event `g0839'.\index{g0839}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{equal-z-list-proof} \\  
\=\+{\rm{equal-z-list-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pflist\/}})\index{equal-z-list-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) $\wedge$ {\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{x\/}})) $\wedge$ {\rm{listp}}\,({\rm{cdr}}\,({\it{y\/}})) \\ 
{\bf then }{\rm{list-detach-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-equal}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-list}}\,({\rm{cdr}}\,({\it{x\/}}))\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{y\/}}))\index{z-list})\-\index{f-equal})\-, \\ 
{\rm{f-equal}}\,({\rm{z-list}}\,({\it{x\/}})\index{z-list}, {\rm{z-list}}\,({\it{y\/}})\index{z-list})\index{f-equal}, \\ 
{\rm{list}}\,(\=\+{\rm{car}}\,({\it{pflist\/}}), \\ 
{\rm{equal-z-list-proof}}\,(\=\+{\rm{cdr}}\,({\it{x\/}}), \\ 
{\rm{cdr}}\,({\it{y\/}}), \\ 
{\rm{cdr}}\,({\it{pflist\/}}))\-\index{equal-z-list-proof})\-, \\ 
{\rm{opair-equal-proof}}\,(\=\+{\rm{car}}\,({\it{x\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{x\/}}))\index{z-list}, \\ 
{\rm{car}}\,({\it{y\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{y\/}}))\index{z-list})\-\index{opair-equal-proof})\-\index{list-detach-proof} \\ 
{\bf else }{\rm{car}}\,({\it{pflist\/}})$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-z-list-proof-proves \\ \index{equal-z-list-proof-proves}
\=\+(\=\+({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{x\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{y\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{len-equal}}\,({\it{x\/}}, {\it{y\/}})\index{len-equal} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\rm{list-f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{list-f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{equal-z-list-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pflist\/}})\index{equal-z-list-proof}, \\ 
{\rm{f-equal}}\,({\rm{z-list}}\,({\it{x\/}})\index{z-list}, {\rm{z-list}}\,({\it{y\/}})\index{z-list})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable equal-z-list-proof; name this event `g0842'.\index{g0842}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  equal-z-list-proof-proves1 \\ \index{equal-z-list-proof-proves1}
\=\+(\=\+({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{x\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{y\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{len-equal}}\,({\it{x\/}}, {\it{y\/}})\index{len-equal} \\ 
$\wedge$$\;\;\;\;${\rm{proves-list}}\,({\it{pflist\/}}, {\rm{list-f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{list-f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-list} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-list}}\,({\it{x\/}})\index{z-list}, {\rm{z-list}}\,({\it{y\/}})\index{z-list})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{equal-z-list-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pflist\/}})\index{equal-z-list-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable equal-z-list-proof-proves; name this event `g0843'.\index{g0843}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  nlistp-ncode \\ \index{nlistp-ncode}
({\it{x\/}} $\simeq\;${{\bf{nil}}}${}$) $\rightarrow$ ({\rm{ncode}}\,({\it{x\/}})\index{ncode} $=$ {\rm{numeral}}\,({\it{x\/}})\index{numeral})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{flip-not-equals-proof} \\  
\=\+{\rm{flip-not-equals-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{pf\/}})\index{flip-not-equals-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{f-not}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{f-not}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal})\index{f-not}, \\ 
{\it{pf\/}}, \\ 
{\rm{contra-pos-proof1}}\,(\=\+{\rm{f-equal}}\,({\it{y\/}}, {\it{x\/}})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, \\ 
{\rm{symmetry-proof1}}\,({\it{y\/}}, {\it{x\/}})\index{symmetry-proof1})\-\index{contra-pos-proof1})\-\index{detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-list-var-list \\ \index{term-list-var-list}
{\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} $\rightarrow$ {\rm{term-list}}\,({\it{vars\/}}, {\it{symbols\/}})\index{term-list}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-equal-case-proof} \\  
\=\+{\rm{form-equal-case-proof}}\,({\it{vars\/}}, {\it{terms\/}}, {\it{exp\/}}, {\it{pf\/}})\index{form-equal-case-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal})\index{list-f-and}, \\ 
{\rm{f-iff}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{parallel-subst}}\,(\=\+{\it{exp\/}}, \\ 
{\it{vars\/}}, \\ 
{\it{terms\/}}, \\ 
{\tt{0}})\-\index{parallel-subst})\-\index{f-iff})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal})\index{list-f-and}, {\it{exp\/}})\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{form-equal-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{parallel-subst}}\,(\=\+{\it{exp\/}}, \\ 
{\it{vars\/}}, \\ 
{\it{terms\/}}, \\ 
{\tt{0}})\-\index{parallel-subst}, \\ 
{\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal})\-\index{form-equal-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-list-termp \\ \index{term-list-termp}
{\rm{term-list}}\,({\it{terms\/}}, {\it{symbols\/}})\index{term-list} $\rightarrow$ {\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp}
\end{tabbing}
\begin{verbatim};;;;;;;If |- "[a1/x1, ..., an/xn]A", then |- "(((x1=a1) ^ ... ^ (xn=an)) -> A)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  form-equal-case-proof-proves \\ \index{form-equal-case-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{term-list}}\,({\it{terms\/}}, {\it{symbols\/}})\index{term-list} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal}) \\ 
$\wedge$$\;\;\;\;$({\rm{collect-free}}\,({\it{terms\/}}, {\tt{1}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{vars\/}}, {\it{terms\/}})\index{list-f-equal})\index{list-f-and}, {\it{exp\/}})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-equal-case-proof}}\,({\it{vars\/}}, {\it{terms\/}}, {\it{exp\/}}, {\it{pf\/}})\index{form-equal-case-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-equal-case-proof; name this event `g0931'.\index{g0931}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z-list-equal-imp-proof} \\  
\=\+{\rm{z-list-equal-imp-proof}}\,({\it{x\/}}, {\it{y\/}})\index{z-list-equal-imp-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) $\wedge$ {\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\rm{cdr}}\,({\it{x\/}})) $\wedge$ {\rm{listp}}\,({\rm{cdr}}\,({\it{y\/}})) \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-list}}\,({\it{x\/}})\index{z-list}, {\rm{z-list}}\,({\it{y\/}})\index{z-list})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{f-equal})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-list}}\,({\it{x\/}})\index{z-list}, {\rm{z-list}}\,({\it{y\/}})\index{z-list})\index{f-equal}, \\ 
{\rm{list-f-and}}\,({\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{x\/}}), \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{list-f-equal})\index{list-f-and})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-list}}\,({\it{x\/}})\index{z-list}, {\rm{z-list}}\,({\it{y\/}})\index{z-list})\index{f-equal}, \\ 
{\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{list-f-equal})\index{list-f-and})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{z-car-unique-proof}}\,(\=\+{\rm{car}}\,({\it{x\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{x\/}}))\index{z-list}, \\ 
{\rm{car}}\,({\it{y\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{y\/}}))\index{z-list})\-\index{z-car-unique-proof}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-list}}\,({\it{x\/}})\index{z-list}, \\ 
{\rm{z-list}}\,({\it{y\/}})\index{z-list})\-\index{f-equal}, \\ 
{\rm{list-f-and}}\,({\rm{list-f-equal}}\,(\=\+{\rm{cdr}}\,({\it{x\/}}), \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{list-f-equal})\index{list-f-and}, \\ 
{\rm{list}}\,({\rm{f-equal}}\,(\=\+{\rm{z-list}}\,({\rm{cdr}}\,({\it{x\/}}))\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{y\/}}))\index{z-list})\-\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{z-cdr-unique-proof}}\,(\=\+{\rm{car}}\,({\it{x\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{x\/}}))\index{z-list}, \\ 
{\rm{car}}\,({\it{y\/}}), \\ 
{\rm{z-list}}\,({\rm{cdr}}\,({\it{y\/}}))\index{z-list})\-\index{z-cdr-unique-proof}, \\ 
{\rm{z-list-equal-imp-proof}}\,(\=\+{\rm{cdr}}\,({\it{x\/}}), \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{z-list-equal-imp-proof})\-)\-\index{chain-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{car}}\,({\it{x\/}}), {\rm{car}}\,({\it{y\/}}))\index{f-equal})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  termp-car-cdr \\ \index{termp-car-cdr}
\=\+({\rm{listp}}\,({\it{x\/}}) $\wedge$ {\rm{termp}}\,({\it{x\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{termp}}\,({\rm{car}}\,({\it{x\/}}), {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\rm{cdr}}\,({\it{x\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-list-equal-imp-proof-proves \\ \index{z-list-equal-imp-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{list1\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{list2\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list2\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{len-equal}}\,({\it{list1\/}}, {\it{list2\/}})\index{len-equal})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{z-list-equal-imp-proof}}\,({\it{list1\/}}, {\it{list2\/}})\index{z-list-equal-imp-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-list}}\,({\it{list1\/}})\index{z-list}, {\rm{z-list}}\,({\it{list2\/}})\index{z-list})\index{f-equal}, \\ 
{\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{list1\/}}, {\it{list2\/}})\index{list-f-equal})\index{list-f-and})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-list-equal-imp-proof; name this event `g0845'.\index{g0845}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-list-equal-imp-proof-proves1 \\ \index{z-list-equal-imp-proof-proves1}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{list1\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{list2\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list1\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{list2\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{len-equal}}\,({\it{list1\/}}, {\it{list2\/}})\index{len-equal} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\rm{z-list}}\,({\it{list1\/}})\index{z-list}, {\rm{z-list}}\,({\it{list2\/}})\index{z-list})\index{f-equal}, \\ 
{\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{list1\/}}, {\it{list2\/}})\index{list-f-equal})\index{list-f-and})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-list-equal-imp-proof}}\,({\it{list1\/}}, {\it{list2\/}})\index{z-list-equal-imp-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-list-equal-imp-proof-proves; name this event `g0846'.\index{g0846}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{equal-isin-proof} \\  
\=\+{\rm{equal-isin-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{equal-isin-proof} \\ 
$=$$\;\;\;\;${\rm{list-detach-proof}}\,(\=\+{\rm{list}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{z\/}}, {\it{z\/}})\index{f-equal}, {\rm{isin}}\,({\it{x\/}}, {\it{z\/}})\index{isin}), \\ 
{\rm{isin}}\,({\it{y\/}}, {\it{z\/}})\index{isin}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\rm{ident-axiom-proof}}\,({\it{z\/}})\index{ident-axiom-proof}, {\it{pf2\/}}), \\ 
{\rm{equal-axiom2-proof}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{list}}\,({\it{x\/}}, {\it{z\/}}), {\rm{list}}\,({\it{y\/}}, {\it{z\/}}))\index{equal-axiom2-proof})\-\index{list-detach-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;If |- "(x = y)", and |- "(x in z)", then |- "(y in z)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  equal-isin-proof-proves \\ \index{equal-isin-proof-proves}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{isin}}\,({\it{x\/}}, {\it{z\/}})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{equal-isin-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{equal-isin-proof}, {\rm{isin}}\,({\it{y\/}}, {\it{z\/}})\index{isin}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;;As it turns out, in order to represent the Lisp interpreter in Z2, we only need
;;;;;;;represent two recursive functions, the function GET below, and the Lisp interpreter
;;;;;;;EV itself.  The Lisp interpreter is a Lisp function which evaluates quoted
;;;;;;;Lisp objects which represent pure-Lisp expressions.  These e-expressions are
;;;;;;;either e-variables (represented by numbers) or e-function symbols (also represented
;;;;;;;by numbers) applied to a list of e-expressions.  The interpreter takes as
;;;;;;;its arguments: the e-expression to be evaluated, the bindings of values to the
;;;;;;;e-variables, and the bindings of definitions to the defined e-function symbols.
;;;;;;;Since e-variables are numbers, their bindings are stored in a list so that the
;;;;;;;binding of the variable N is the N+1th element in the list.  GET does exactly
;;;;;;;this by returning the X+1th element in the list Y.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{get} \\  
\=\+{\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{car}}\,({\it{y\/}}) \\ 
{\bf else }{\rm{get}}\,({\it{x\/}} $-\;1$, {\rm{cdr}}\,({\it{y\/}}))\index{get}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;We said earlier that the partial graph corresponding to a computation is first
;;;;;;;constructed as a list of value/argument pairs which is then translated into the
;;;;;;;corresponding set.  GET-GRAPH constructs this list of value/argument pairs
;;;;;;;corresponding to the computation of (GET X Y).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{get-graph} \\  
\=\+{\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{list}}\,({\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))) \\ 
{\bf else }{\rm{cons}}\,({\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}})), {\rm{get-graph}}\,({\it{x\/}} $-\;1$, {\rm{cdr}}\,({\it{y\/}}))\index{get-graph})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-get-graph \\ \index{member-get-graph}
{\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}})) $\in$ {\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  in-graph-cdr-get-graph \\ \index{in-graph-cdr-get-graph}
({\it{x\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ ({\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}} $-\;1$, {\rm{cdr}}\,({\it{y\/}}))) $\in$ {\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-get-defn-part-ok-proof1} \\  
\=\+{\rm{a-get-defn-part-ok-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{a-get-defn-part-ok-proof1} \\ 
$=$$\;\;\;\;${\rm{form-z-if-ok-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{x\/}} $\simeq {\tt{0}}$, \\ 
{\rm{z-zerop-ncode-proof}}\,({\it{x\/}})\index{z-zerop-ncode-proof}, \\ 
{\rm{trans-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-zerop}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-zerop}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,(\=\+{\tt{1}}, \\ 
{\tt{0}})\-\index{ncode-nequal-proof}, \\ 
{\rm{z-zerop-ncode-proof}}\,({\it{x\/}})\index{z-zerop-ncode-proof})\-\index{trans-not-equal-proof})\-, \\ 
{\rm{flip-equals-proof}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, \\ 
{\rm{z-car-ncode-proof}}\,({\it{y\/}})\index{z-car-ncode-proof})\-\index{flip-equals-proof}, \\ 
{\rm{equal-isin-proof}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-)\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, \\ 
{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans}, \\ 
{\rm{equal-z-list-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get})\index{ncode}, \\ 
{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr})\-, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\rm{get}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{flip-equals-proof}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{z-sub1-ncode-proof}}\,({\it{x\/}})\index{z-sub1-ncode-proof})\-\index{flip-equals-proof}, \\ 
{\rm{flip-equals-proof}}\,(\=\+{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{y\/}})\index{z-cdr-ncode-proof})\-\index{flip-equals-proof})\-)\-\index{equal-z-list-proof}, \\ 
{\rm{in-graph-proof}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, \\ 
{\rm{cons}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\rm{cdr}}\,({\it{y\/}}))\-)\-, \\ 
{\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\-\index{in-graph-proof})\-\index{equal-isin-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-zerop}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-zerop}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, {\rm{z-car}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-car})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr})\-\index{z-opair})\-\index{z-opair}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans})\-\index{isin})\-\index{z-if})\-\index{form-z-if-ok-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;At this point we present the Z2 formula which represents the Lisp function GET.
;;;;;;;The formula representing the assertion (GET X Y) = VAL asserts the existence
;;;;;;;of a partial graph "trace" to which the tuple <<VAL>, <X>, <Y>> belongs, such
;;;;;;;that all of its members satisfy the Z2 analogue of the definition of GET.
;;;;;;;More formally, 
;;;;;;;     "(forsome trace ((<val, x, y> in trace)
;;;;;;;                      ^
;;;;;;;                      (forall v1, x1, y1
;;;;;;;                        ((<v1, x1, y1> in trace)
;;;;;;;                         -> 
;;;;;;;                         (if ((Z-zerop x1) = <1>)
;;;;;;;                          then (v1 = (Z-car y1))
;;;;;;;                          else (<v1, (Z-sub1 x1), (Z-cdr y1)> in trace))))))".
;;;;;;;
;;;;;;;Notice how in the second part where we specify that each 3-tuple "<v1,x1,y1>"
;;;;;;;in "trace" satisfies the Z2 analogue of the definition of GET, the analogue of
;;;;;;;the recursive call (GET (SUB1 X)(CDR Y)) is the membership check
;;;;;;;"(<v1, (Z-sub1 x1), (Z-cdr y1)> in trace)".  A-GET-DEFN-PART below constructs
;;;;;;;the definition (second) part of the above formula.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-get-defn-part} \\  
\=\+{\rm{a-get-defn-part}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-get-defn-part} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-zerop}}\,({\it{x\/}})\index{z-zerop}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-car}}\,({\it{y\/}})\index{z-car})\index{f-equal}, \\ 
{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\it{val\/}}, {\rm{z-sub1}}\,({\it{x\/}})\index{z-sub1}, {\rm{z-cdr}}\,({\it{y\/}})\index{z-cdr}))\index{z-list}, {\it{trace\/}})\index{isin})\-\index{f-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;Next we wish to show that the partial graph given by (GET-GRAPH X Y)
;;;;;;;satisfies the formula above.  By this we show that the representing formula
;;;;;;;is `OK'.  This is done in many steps.  One part
;;;;;;;of this is to show that each element of this partial graph satisfies the
;;;;;;;A-GET-DEFN-PART.  This will be done by induction.  The first part is
;;;;;;;to show that "<<(GET X Y)>, <X>, <Y>>" satisfies the definition part
;;;;;;;with (GRAPH-TRANS (GET-GRAPH X Y)) for "trace".  This is carried out below.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-get-defn-part-ok-proof1-proves \\ \index{a-get-defn-part-ok-proof1-prove!s}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-get-defn-part}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{a-get-defn-part-ok-proof1}}\,({\it{x\/}}, {\it{y\/}})\index{a-get-defn-part-ok-proof1}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-defn-part-ok-proof1; name this event `g0847'.\index{g0847}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-graph-cdr-proof} \\  
\=\+{\rm{in-graph-cdr-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{graph\/}})\index{in-graph-cdr-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{graph\/}}))\index{graph-trans})\index{isin})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{graph\/}}))\index{graph-trans})\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{iff-split-proof2}}\,(\=\+{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{graph\/}}))\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{isin})\-\index{f-or}, \\ 
{\rm{in-graph-trans-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{graph\/}})\index{in-graph-trans-proof})\-\index{iff-split-proof2}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((x in [GRAPH]) -> (x in [(CONS Y GRAPH)]))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-graph-cdr-proof-proves \\ \index{in-graph-cdr-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{in-graph-cdr-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{graph\/}})\index{in-graph-cdr-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{graph\/}}))\index{graph-trans})\index{isin})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-graph-cdr-proof; name this event `g0850'.\index{g0850}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-get-defn-part-step-proof} \\  
\=\+{\rm{a-get-defn-part-step-proof}}\,({\it{y\/}}, {\it{u\/}}, {\it{v\/}}, {\it{v1\/}}, {\it{pf\/}})\index{a-get-defn-part-step-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\it{v\/}}, {\it{v1\/}})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\rm{f-if}}\,({\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{f-if}, {\rm{f-if}}\,({\it{y\/}}, {\it{u\/}}, {\it{v1\/}})\index{f-if})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;If |- "(C -> C1)", then |- "((if A then B else C) -> (if A then B else C1))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-get-defn-part-step-proof-proves \\ \index{a-get-defn-part-step-proof-prove!s}
\=\+(\=\+{\rm{formula}}\,({\rm{f-if}}\,({\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{f-if}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{v1\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{v\/}}, {\it{v1\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-defn-part-step-proof}}\,({\it{y\/}}, {\it{u\/}}, {\it{v\/}}, {\it{v1\/}}, {\it{pf\/}})\index{a-get-defn-part-step-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-if}}\,({\it{y\/}}, {\it{u\/}}, {\it{v\/}})\index{f-if}, {\rm{f-if}}\,({\it{y\/}}, {\it{u\/}}, {\it{v1\/}})\index{f-if})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-defn-part-step-proof; name this event `g0851'.\index{g0851}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{inst-formals-proof} \\  
\=\+{\rm{inst-formals-proof}}\,({\it{formals\/}}, {\it{actuals\/}}, {\it{exp\/}}, {\it{pf\/}})\index{inst-formals-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\rm{z-list}}\,({\it{formals\/}})\index{z-list}, {\rm{z-list}}\,({\it{actuals\/}})\index{z-list})\index{f-equal}, \\ 
{\it{exp\/}}, \\ 
{\rm{list}}\,({\rm{list-f-and}}\,({\rm{list-f-equal}}\,({\it{formals\/}}, {\it{actuals\/}})\index{list-f-equal})\index{list-f-and}), \\ 
{\rm{list}}\,(\=\+{\rm{z-list-equal-imp-proof}}\,({\it{formals\/}}, {\it{actuals\/}})\index{z-list-equal-imp-proof}, \\ 
{\rm{form-equal-case-proof}}\,({\it{formals\/}}, {\it{actuals\/}}, {\it{exp\/}}, {\it{pf\/}})\index{form-equal-case-proof})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-get-defn-ok-proof} \\  
\=\+{\rm{a-get-defn-ok-proof}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{x\/}}, {\it{y\/}})\index{a-get-defn-ok-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{graph-trans-merge-proof}}\,(\=\+{\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}})), \\ 
{\bf{nil}}, \\ 
{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}))\index{z-list}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part}, \\ 
{\rm{inst-formals-proof}}\,(\=\+{\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}), \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode})\-, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part}, \\ 
{\rm{a-get-defn-part-ok-proof1}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{a-get-defn-part-ok-proof1})\-\index{inst-formals-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\sc{phi}}\index{phi})\-\index{isin})\index{f-not}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\sc{phi}}\index{phi})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{null-axiom-proof}}\,({\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list})\index{null-axiom-proof}))\-\index{tautconseq-proof})\-\index{graph-trans-merge-proof} \\ 
{\bf else }{\rm{graph-trans-merge-proof}}\,(\=\+{\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}})), \\ 
{\rm{get-graph}}\,({\it{x\/}} $-\;1$, {\rm{cdr}}\,({\it{y\/}}))\index{get-graph}, \\ 
{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}))\index{z-list}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part}, \\ 
{\rm{inst-formals-proof}}\,(\=\+{\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}), \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode})\-, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part}, \\ 
{\rm{a-get-defn-part-ok-proof1}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{a-get-defn-part-ok-proof1})\-\index{inst-formals-proof}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{get-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part}, \\ 
{\rm{list}}\,({\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part}), \\ 
{\rm{list}}\,(\=\+{\rm{a-get-defn-ok-proof}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{x\/}} $-\;1$, \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{a-get-defn-ok-proof}, \\ 
{\rm{a-get-defn-part-step-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-zerop}}\,({\it{x1\/}})\index{z-zerop}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-car}}\,({\it{y1\/}})\index{z-car})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-sub1}}\,({\it{x1\/}})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\it{y1\/}})\index{z-cdr})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{get-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-sub1}}\,({\it{x1\/}})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\it{y1\/}})\index{z-cdr})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{in-graph-cdr-proof}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-sub1}}\,({\it{x1\/}})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\it{y1\/}})\index{z-cdr})\-)\index{z-list}, \\ 
{\rm{cons}}\,(\=\+{\rm{get}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get}, \\ 
{\rm{cons}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-)\-, \\ 
{\rm{get-graph}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{get-graph})\-\index{in-graph-cdr-proof})\-\index{a-get-defn-part-step-proof})\-)\-\index{chain-proof})\-\index{graph-trans-merge-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;In the lemma below, we carry out the induction to prove that every element of
;;;;;;;the partial graph satisfies the analogue of the definition of GET, i.e.,
;;;;;;; |- "((<v1,x1,y1> in [(GET-GRAPH X Y)]) -> (A-GET-DEFN-PART x1 y1 v1 [(GET-GRAPH X Y)]))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-get-defn-ok-proof-proves \\ \index{a-get-defn-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}), {\tt{3}})\index{var-set})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-defn-ok-proof}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{x\/}}, {\it{y\/}})\index{a-get-defn-ok-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}))\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-defn-ok-proof; name this event `g0852'.\index{g0852}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{list-gen-proof} \\  
\=\+{\rm{list-gen-proof}}\,({\it{vars\/}}, {\it{exp\/}}, {\it{pf\/}})\index{list-gen-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) \\ 
{\bf then }{\rm{generalise-proof}}\,(\=\+{\rm{car}}\,({\it{vars\/}}), \\ 
{\rm{list-all-quantify}}\,({\rm{cdr}}\,({\it{vars\/}}), {\it{exp\/}})\index{list-all-quantify}, \\ 
{\rm{list-gen-proof}}\,({\rm{cdr}}\,({\it{vars\/}}), {\it{exp\/}}, {\it{pf\/}})\index{list-gen-proof})\-\index{generalise-proof} \\ 
{\bf else }{\it{pf\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-gen-proof-proves \\ \index{list-gen-proof-proves}
\=\+({\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} $\wedge$ {\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{list-gen-proof}}\,({\it{vars\/}}, {\it{exp\/}}, {\it{pf\/}})\index{list-gen-proof}, \\ 
{\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{exp\/}})\index{list-all-quantify}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable list-gen-proof; name this event `g0853'.\index{g0853}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-get-defn-part; name this event `g0854'.\index{g0854}

\addvspace{18pt}
\begin{verbatim};;;;;;;A-GET combines the first and second parts of the specification of "trace".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-get} \\  
\=\+{\rm{a-get}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get} \\ 
$=$$\;\;\;\;${\rm{f-and}}\,(\=\+{\rm{isin}}\,({\rm{z-opair}}\,({\it{val\/}}, {\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair})\index{z-opair}, {\it{trace\/}})\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}))\index{z-list}, {\it{trace\/}})\index{isin}, \\ 
{\rm{a-get-defn-part}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify})\-\index{f-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-get-ok-proof1} \\  
\=\+{\rm{a-get-ok-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}})\index{a-get-ok-proof1} \\ 
$=$$\;\;\;\;${\rm{and-join-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}})))\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{in-graph-proof}}\,({\rm{cons}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}})), {\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{in-graph-proof}, \\ 
{\rm{list-gen-proof}}\,(\=\+{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}))\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-defn-part})\-\index{f-implies}, \\ 
{\rm{a-get-defn-ok-proof}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{x\/}}, {\it{y\/}})\index{a-get-defn-ok-proof})\-\index{list-gen-proof})\-\index{and-join-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;Finally we show that 
;;;;;;;|- "[<X>/x, <Y>/y, <(GET X Y)>/val, [(GET-GRAPH X Y)]/trace]A-get".
;;;;;;;This completes the first part of the representability proof for GET.
;;;;;;;The second part involves showing that any "trace" satisfying the
;;;;;;;specification A-GET must encode only evaluations of GET.  By this we
;;;;;;;show that the representing formula is `GOOD'.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-get-ok-proof-proves \\ \index{a-get-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}), {\tt{3}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{x-bar\/}} $=$ {\rm{ncode}}\,({\it{x\/}})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{y-bar\/}} $=$ {\rm{ncode}}\,({\it{y\/}})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{val-bar\/}} $=$ {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $=$ {\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-ok-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}})\index{a-get-ok-proof1}, \\ 
{\rm{a-get}}\,({\it{x-bar\/}}, {\it{y-bar\/}}, {\it{val-bar\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-ok-proof1; name this event `g0856'.\index{g0856}

\addvspace{18pt}

\noindent{\sc Event}:   Disable a-get-defn-part; name this event `g0857'.\index{g0857}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{equal-isin-imp-proof} \\  
\=\+{\rm{equal-isin-imp-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{pf\/}})\index{equal-isin-imp-proof} \\ 
$=$$\;\;\;\;${\rm{list-detach-proof}}\,(\=\+{\rm{list}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal}, {\rm{f-equal}}\,({\it{z\/}}, {\it{z\/}})\index{f-equal}), \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{x\/}}, {\it{z\/}})\index{isin}, {\rm{isin}}\,({\it{y\/}}, {\it{z\/}})\index{isin})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf\/}}, {\rm{ident-axiom-proof}}\,({\it{z\/}})\index{ident-axiom-proof}), \\ 
{\rm{equal-axiom2-proof}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{list}}\,({\it{x\/}}, {\it{z\/}}), {\rm{list}}\,({\it{y\/}}, {\it{z\/}}))\index{equal-axiom2-proof})\-\index{list-detach-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-z-if-eval-proof-proves-again \\ \index{form-z-if-eval-proof-proves-agai!n}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{lhs\/}}, {\it{val\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves-form-z-if-eval}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-form-z-if-eval} \\ 
$\wedge$$\;\;\;\;$({\it{form\/}} $=$ {\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{form-z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{lhs\/}}, {\it{val\/}})\index{form-z-if-eval-proof}, \\ 
{\rm{f-implies}}\,({\it{form\/}}, {\rm{f-equal}}\,({\it{lhs\/}}, {\it{val\/}})\index{f-equal})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable form-z-if-eval-proof-proves; name this event `g0858'.\index{g0858}

\addvspace{18pt}
\begin{verbatim};;;;;;;PROVES-REC-FORM-Z-IF-EVAL is similar to PROVES-Z-IF, except that
;;;;;;;in the base case it checks if PF-TREE is a proof of "(EXP -> CONCL)".
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{proves-rec-form-z-if-eval} \\  
\=\+{\rm{proves-rec-form-z-if-eval}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-rec-form-z-if-eval} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pf-tree\/}}) \\ 
{\bf then }\=\+{\rm{proves}}\,({\rm{cadar}}\,({\it{pf-tree\/}}), {\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-rec-form-z-if-eval}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-rec-form-z-if-eval}\- \\ 
{\bf else }\=\+{\rm{proves}}\,(\=\+{\rm{caddar}}\,({\it{pf-tree\/}}), \\ 
{\rm{f-not}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves-rec-form-z-if-eval}}\,(\=\+{\rm{caddr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves-rec-form-z-if-eval}\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{proves}}\,({\it{pf-tree\/}}, {\rm{f-implies}}\,({\it{exp\/}}, {\it{concl\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{rec-form-z-if-eval-proof} \\  
\=\+{\rm{rec-form-z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{concl\/}})\index{rec-form-z-if-eval-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\bf if }{\rm{caar}}\,({\it{pf-tree\/}}) \\ 
{\bf then }{\rm{chain-proof}}\,(\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, \\ 
{\it{concl\/}}, \\ 
{\rm{list}}\,({\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}), \\ 
{\rm{list}}\,(\=\+{\rm{f-if-elim-proof1}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\rm{cadar}}\,({\it{pf-tree\/}}))\-\index{f-if-elim-proof1}, \\ 
{\rm{rec-form-z-if-eval-proof}}\,(\=\+{\rm{cadr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-left}}\,({\it{exp\/}})\index{z-left}, \\ 
{\it{concl\/}})\-\index{rec-form-z-if-eval-proof})\-)\-\index{chain-proof} \\ 
{\bf else }{\rm{chain-proof}}\,(\=\+{\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, \\ 
{\it{concl\/}}, \\ 
{\rm{list}}\,({\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}), \\ 
{\rm{list}}\,(\=\+{\rm{f-if-elim-proof2}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-z-if}, \\ 
{\rm{make-form-z-if}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-z-if}, \\ 
{\rm{caddar}}\,({\it{pf-tree\/}}))\-\index{f-if-elim-proof2}, \\ 
{\rm{rec-form-z-if-eval-proof}}\,(\=\+{\rm{caddr}}\,({\it{pf-tree\/}}), \\ 
{\rm{z-right}}\,({\it{exp\/}})\index{z-right}, \\ 
{\it{concl\/}})\-\index{rec-form-z-if-eval-proof})\-)\-\index{chain-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\it{pf-tree\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below proves that if each of the leaf nodes of an form-if-tree
;;;;;;;implies a certain conclusion CONCL, then the if-expression represented by
;;;;;;;the if-tree implies CONCL.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  rec-form-z-if-eval-proof-proves \\ \index{rec-form-z-if-eval-proof-proves}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{proves-rec-form-z-if-eval}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-rec-form-z-if-eval})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{rec-form-z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{concl\/}})\index{rec-form-z-if-eval-proof}, \\ 
{\rm{f-implies}}\,({\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}, {\it{concl\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable rec-form-z-if-eval-proof; name this event `g0743'.\index{g0743}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  rec-form-z-if-eval-proof-proves-again \\ \index{rec-form-z-if-eval-proof-proves!-again}
\=\+(\=\+{\rm{form-z-if}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-z-if} \\ 
$\wedge$$\;\;\;\;${\rm{proves-rec-form-z-if-eval}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves-rec-form-z-if-eval} \\ 
$\wedge$$\;\;\;\;$({\it{form\/}} $=$ {\rm{make-form-z-if}}\,({\it{exp\/}})\index{make-form-z-if}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{rec-form-z-if-eval-proof}}\,({\it{pf-tree\/}}, {\it{exp\/}}, {\it{concl\/}})\index{rec-form-z-if-eval-proof}, \\ 
{\rm{f-implies}}\,({\it{form\/}}, {\it{concl\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable rec-form-z-if-eval-proof-proves; name this event `g0745'.\index{g0745}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-if \\ \index{termp-if}
\=\+{\rm{termp}}\,(\=\+\=\+{\bf if }{\it{x\/}}$\;\;${\bf then }{\it{y\/}} \\ 
{\bf else }{\it{z\/}}$\;${\bf  endif}\-, \\ 
{\it{flg\/}}, \\ 
{\it{symbols\/}})\-\index{termp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}}$\;\;${\bf then }{\rm{termp}}\,({\it{y\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp} \\ 
{\bf else }{\rm{termp}}\,({\it{z\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-get-defn-part-good-proof1} \\  
\=\+{\rm{a-get-defn-part-good-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{trace\/}}, {\it{exp\/}}, {\it{pf\/}})\index{a-get-defn-part-good-proof1} \\ 
$=$$\;\;\;\;${\rm{rec-form-z-if-eval-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{x\/}} $\simeq {\tt{0}}$, \\ 
{\rm{z-zerop-ncode-proof}}\,({\it{x\/}})\index{z-zerop-ncode-proof}, \\ 
{\rm{trans-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-zerop}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-zerop}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,(\=\+{\tt{1}}, \\ 
{\tt{0}})\-\index{ncode-nequal-proof}, \\ 
{\rm{z-zerop-ncode-proof}}\,({\it{x\/}})\index{z-zerop-ncode-proof})\-\index{trans-not-equal-proof})\-, \\ 
{\rm{trans-implies-proof}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-car})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list-detach-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\it{val\/}})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-car})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\it{val\/}})\index{ident-axiom-proof}, \\ 
{\rm{z-car-ncode-proof}}\,({\it{y\/}})\index{z-car-ncode-proof})\-, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{p}}\,(\=\+{\tt{0}}, \\ 
{\tt{2}})\-\index{p}, \\ 
{\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-car})\-, \\ 
{\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-)\-\index{equal-axiom2-proof})\-\index{list-detach-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof})\-\index{trans-implies-proof}, \\ 
{\rm{trans-implies-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{equal-isin-imp-proof}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr})\-)\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}}, \\ 
{\rm{equal-z-list-proof}}\,(\=\+{\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr})\-, \\ 
{\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\it{val\/}})\index{ident-axiom-proof}, \\ 
{\rm{z-sub1-ncode-proof}}\,({\it{x\/}})\index{z-sub1-ncode-proof}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{y\/}})\index{z-cdr-ncode-proof})\-)\-\index{equal-z-list-proof})\-\index{equal-isin-imp-proof}, \\ 
{\it{pf\/}})\-\index{trans-implies-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-zerop}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-zerop}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-car}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-car})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-opair}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{z-sub1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{z-sub1}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-cdr})\-\index{z-opair})\-\index{z-opair}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{z-if}, \\ 
{\rm{f-implies}}\,({\it{exp\/}}, {\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode})\index{f-equal})\index{f-implies})\-\index{rec-form-z-if-eval-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The second part involves showing that A-GET is `GOOD', i.e., that
;;;;;;;any "trace" satisfying A-GET encodes a computation of GET. 
;;;;;;;The proof is somewhat tricky but the essential idea is this:
;;;;;;;we want to prove "A-get(<X>, <Y>, val, trace) -> (val = <(GET X Y)>)".
;;;;;;;We start with the left-hand side of this implication, it implies that
;;;;;;; "(<<X>, <Y>, val> in trace)" and hence by the second part of "A-get"
;;;;;;;that "<<X>, <Y>, val>" satisfies the analogue of the definition of GET.
;;;;;;;If (ZEROP X), then "((Z-zerop <X>) = <1>)" and hence "(val = (Z-car <Y>))"
;;;;;;;which by the lemma about "Z-car" is "(val = <(CAR Y)>)" which is the
;;;;;;;desired conclusion.  In the case when (ZEROP X) is F, we have
;;;;;;;"(A-get(<X>, <Y>, val, trace) -> (<val, (Z-sub1 <X>), (Z-cdr <Y>)> in trace))" 
;;;;;;;(call it [1]), the r.h.s. of which is "(<val, <(SUB1 X)>, <(CDR Y)>> in trace)" 
;;;;;;;By the induction hypothesis,
;;;;;;;"(A-get(<(SUB1 X)>, <(CDR Y)>, val, trace) -> (val = <(GET (SUB1 X)(CDR Y))>))" [2].
;;;;;;;But "A-get(<X>, <Y>, val, trace)" and  "(<val, <(SUB1 X)>, <(CDR Y)>> in trace)" imply 
;;;;;;;"A-get(<(SUB1 X)>, <(CDR Y)>, val, trace)".  By [1], [2], and the definition of GET,
;;;;;;;we have the desired conclusion.  The lemma below is one portion of the argument 
;;;;;;;above where we prove
;;;;;;; |- "(A-get-defn-part(<X>, <Y>, val, trace) -> (EXP -> (val = <(GET X Y)>)))", 
;;;;;;;provided when (ZEROP X) is F,
;;;;;;; |- "((<val, <(SUB1 X)>, <(CDR Y)>> in trace) -> (EXP -> (val = <(GET X Y)>)))".
;;;;;;;
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-get-defn-part-good-proof1-proves \\ \index{a-get-defn-part-good-proof1-prove!s}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{trace\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{val-bar\/}} $=$ {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{form\/}} $=$ {\rm{a-get-defn-part}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\it{val\/}}, {\it{trace\/}})\index{a-get-defn-part}) \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{exp\/}}, {\rm{f-equal}}\,({\it{val\/}}, {\it{val-bar\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-defn-part-good-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{trace\/}}, {\it{exp\/}}, {\it{pf\/}})\index{a-get-defn-part-good-proof1}, \\ 
{\rm{f-implies}}\,({\it{form\/}}, {\rm{f-implies}}\,({\it{exp\/}}, {\rm{f-equal}}\,({\it{val\/}}, {\it{val-bar\/}})\index{f-equal})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-defn-part-good-proof1; name this event `g0615'.\index{g0615}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{good-frame1-proof} \\  
\=\+{\rm{good-frame1-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}})\index{good-frame1-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-iff}}\,(\=\+{\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{c\/}})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\it{b\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{c\/}})\index{f-implies})\index{f-implies})\-\index{f-iff}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  good-frame1-proof-proves \\ \index{good-frame1-proof-proves}
\=\+({\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula} $\wedge$ {\rm{formula}}\,({\it{c\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{good-frame1-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}})\index{good-frame1-proof}, \\ 
{\rm{f-iff}}\,({\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{c\/}})\index{f-implies}, {\rm{f-implies}}\,({\it{b\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{c\/}})\index{f-implies})\index{f-implies})\index{f-iff}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable good-frame1-proof; name this event `g0746'.\index{g0746}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-get-reduc-proof} \\  
\=\+{\rm{a-get-reduc-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get-reduc-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get}, \\ 
{\rm{a-get-defn-part}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-get-defn-part})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\it{val\/}}, {\it{x\/}}, {\it{y\/}}))\index{z-list}, {\it{trace\/}})\index{isin}, \\ 
{\rm{a-get-defn-part}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-get-defn-part})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{good-frame1-proof}}\,(\=\+{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\it{val\/}}, {\it{x\/}}, {\it{y\/}}))\index{z-list}, {\it{trace\/}})\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{a-get-defn-part}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-get-defn-part})\-\index{good-frame1-proof}, \\ 
{\rm{subst-thm-b-proof}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies}, \\ 
{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}))\-\index{subst-thm-b-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "(A-get(x, y, val, trace) -> A-get-defn-part(x, y, val, trace))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-get-reduc-proof-proves \\ \index{a-get-reduc-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-get}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get}, \\ 
{\rm{a-get-defn-part}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-get-defn-part})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-reduc-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get-reduc-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-reduc-proof; name this event `g0644'.\index{g0644}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{good-frame2-proof} \\  
\=\+{\rm{good-frame2-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{good-frame2-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{c\/}})\index{f-implies}, \\ 
{\rm{f-implies}}\,({\it{c\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{d\/}})\index{f-implies})\index{f-implies})\-, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{d\/}})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf1\/}}, {\it{pf2\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  good-frame2-proof-proves \\ \index{good-frame2-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{a\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{b\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{c\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{d\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{c\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{c\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{d\/}})\index{f-implies})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{good-frame2-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{good-frame2-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{d\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable good-frame2-proof; name this event `g0747'.\index{g0747}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{good-frame3-proof} \\  
\=\+{\rm{good-frame3-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{good-frame3-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{c\/}})\index{f-implies}), \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{c\/}})\index{f-implies})\index{f-implies}, \\ 
{\rm{list}}\,({\it{pf\/}}))\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  good-frame3-proof-proves \\ \index{good-frame3-proof-proves}
\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{c\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{good-frame3-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{pf\/}})\index{good-frame3-proof}, \\ 
{\rm{f-implies}}\,({\it{a\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{c\/}})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable good-frame3-proof; name this event `g0748'.\index{g0748}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-get-good-proof} \\  
\=\+{\rm{a-get-good-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get-good-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\it{val\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}))\index{z-list}, {\it{trace\/}})\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{a-get-defn-part}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\it{val\/}}, {\it{trace\/}})\index{a-get-defn-part}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode})\index{f-equal}, \\ 
{\rm{a-get-reduc-proof}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\it{val\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-reduc-proof}, \\ 
{\rm{a-get-defn-part-good-proof1}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\bf{nil}})\-\index{a-get-defn-part-good-proof1})\-\index{good-frame2-proof} \\ 
{\bf else }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\it{val\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}))\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode})\index{f-equal}, \\ 
{\rm{a-get-reduc-proof}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\it{val\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-reduc-proof}, \\ 
{\rm{a-get-defn-part-good-proof1}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{x\/}} $-\;1$)\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{y\/}}))\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-get-defn-part}}\,(\=\+{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-defn-part})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}})\-\index{get})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-get-good-proof}}\,(\=\+{\it{x\/}} $-\;1$, \\ 
{\rm{cdr}}\,({\it{y\/}}), \\ 
{\it{val\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get-good-proof})\-\index{good-frame3-proof})\-\index{a-get-defn-part-good-proof1})\-\index{good-frame2-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The desired `GOOD' part:
;;;;;;; |- "(A-get(<X>, <Y>, val, trace) -> (val = <(GET X Y)>))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-get-good-proof-proves \\ \index{a-get-good-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{4}})\index{var-set})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-good-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get-good-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-good-proof; name this event `g0749'.\index{g0749}

\addvspace{18pt}

\noindent{\sc Event}:   Enable ncode; name this event `g0750'.\index{g0750}

\addvspace{18pt}
\begin{verbatim};;;;;;;SUBRP is a function used in the Lisp interpreter which checks if a given
;;;;;;;e-function symbol, i.e., a function symbol in an expression to be interpreted
;;;;;;;by the Lisp interpreter.  The events below introduce the Z2 analogue of this
;;;;;;;function.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-subrp}$\;\;$
{\rm{z-subrp}}\,({\it{fn\/}})\index{z-subrp} $=$ {\rm{list}}\,({\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, {\it{fn\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-subrp-defn} \\  
\=\+{\rm{z-subrp-defn}}\,({\it{fn\/}}, {\it{y\/}})\index{z-subrp-defn} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{2}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{3}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{4}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{5}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{6}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{7}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{8}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{9}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{y\/}}, {\sc{phi}}\index{phi})\index{f-equal})\-\index{f-if})\-\index{f-if})\-\index{f-if})\-\index{f-if})\-\index{f-if})\-\index{f-if})\-\index{f-if})\-\index{f-if})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-subrp-expr} \\  
\=\+{\rm{z-subrp-expr}}\,({\it{fn\/}})\index{z-subrp-expr} \\ 
$=$$\;\;\;\;${\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{2}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{3}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{4}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{5}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{6}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{7}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{8}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\it{fn\/}}, {\rm{ncode}}\,({\tt{9}})\index{ncode})\index{f-equal}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\sc{phi}}\index{phi})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{z-if})\-\index{z-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-subrp-exis-proof} \\  
{\rm{z-subrp-exis-proof}}\,({\it{fn\/}}, {\it{y\/}})\index{z-subrp-exis-proof} $=$ {\rm{term-z-if-exis-proof}}\,({\it{y\/}}, {\rm{z-subrp-expr}}\,({\it{fn\/}})\index{z-subrp-expr})\index{term-z-if-exis-proof}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-subrp-unique-proof} \\  
\=\+{\rm{z-subrp-unique-proof}}\,({\it{fn\/}}, {\it{y\/}}, {\it{y1\/}})\index{z-subrp-unique-proof} \\ 
$=$$\;\;\;\;${\rm{term-z-if-unique-proof}}\,({\it{y\/}}, {\it{y1\/}}, {\rm{z-subrp-expr}}\,({\it{fn\/}})\index{z-subrp-expr})\index{term-z-if-unique-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-subrp-defn-proof} \\  
\=\+{\rm{z-subrp-defn-proof}}\,({\it{fn\/}})\index{z-subrp-defn-proof} \\ 
$=$$\;\;\;\;${\rm{instance-proof}}\,(\=\+{\rm{z-subrp-defn}}\,({\tt{0}}, {\rm{z-subrp}}\,({\tt{0}})\index{z-subrp})\index{z-subrp-defn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\rm{list}}\,({\it{fn\/}}), \\ 
{\rm{fun-defn-proof}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{list}}\,({\tt{0}}), \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{z-subrp-defn}}\,({\tt{0}}, {\tt{1}})\index{z-subrp-defn}, \\ 
{\rm{z-subrp-exis-proof}}\,({\tt{0}}, {\tt{1}})\index{z-subrp-exis-proof}, \\ 
{\rm{z-subrp-unique-proof}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-subrp-unique-proof})\-\index{fun-defn-proof})\-\index{instance-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-subrp-hyps} \\  
\=\+{\rm{z-subrp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-subrp-hyps} \\ 
$=$$\;\;\;\;$(\=\+({\rm{assoc}}\,({\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, {\it{defns\/}}) $=$ {\rm{z-subrp-defn}}\,({\tt{0}}, {\rm{z-subrp}}\,({\tt{0}})\index{z-subrp})\index{z-subrp-defn}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\rm{rest-of}}\,({\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, {\it{symbols\/}})\index{rest-of})\-\index{subset})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-subrp-defn-proof-proves \\ \index{z-subrp-defn-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-subrp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-subrp-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-subrp-defn}}\,({\it{x\/}}, {\rm{z-subrp}}\,({\it{x\/}})\index{z-subrp})\index{z-subrp-defn}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-subrp-defn-proof}}\,({\it{x\/}})\index{z-subrp-defn-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-subrp-defn-proof; name this event `g0751'.\index{g0751}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-subrp-hyps; name this event `g0752'.\index{g0752}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{subrp} \\  
\=\+{\rm{subrp}}\,({\it{fn\/}}) \\ 
$=$$\;\;\;\;$(\=\+({\it{fn\/}} $=$ {\tt{1}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{2}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{3}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{4}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{5}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{6}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{7}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{8}}) \\ 
$\vee$$\;\;\;\;$({\it{fn\/}} $=$ {\tt{9}}))\-\-
\end{tabbing}
%\item 
\begin{verbatim}

(DEFN
 Z-SUBRP-NCODE-PROOF
 (FN)
 (Z-IF-EVAL-PROOF1
  (LIST
   (LIST (EQUAL FN 1) (IDENT-AXIOM-PROOF (NCODE 1)) (NCODE-NEQUAL-PROOF FN 1))
   (IDENT-AXIOM-PROOF (NCODE 1))
   (LIST (LIST (EQUAL FN 2) (IDENT-AXIOM-PROOF (NCODE 2)) (NCODE-NEQUAL-PROOF FN 2))
         (IDENT-AXIOM-PROOF (NCODE 1))
         (LIST (LIST (EQUAL FN 3) (IDENT-AXIOM-PROOF (NCODE 3)) (NCODE-NEQUAL-PROOF FN 3))
               (IDENT-AXIOM-PROOF (NCODE 1))
               (LIST (LIST (EQUAL FN 4)
                           (IDENT-AXIOM-PROOF (NCODE 4))
                           (NCODE-NEQUAL-PROOF FN 4))
                     (IDENT-AXIOM-PROOF (NCODE 1))
                     (LIST (LIST (EQUAL FN 5)
                                 (IDENT-AXIOM-PROOF (NCODE 5))
                                 (NCODE-NEQUAL-PROOF FN 5))
                           (IDENT-AXIOM-PROOF (NCODE 1))
                           (LIST (LIST (EQUAL FN 6)
                                       (IDENT-AXIOM-PROOF (NCODE 6))
                                       (NCODE-NEQUAL-PROOF FN 6))
                                 (IDENT-AXIOM-PROOF (NCODE 1))
                                 (LIST (LIST (EQUAL FN 7)
                                             (IDENT-AXIOM-PROOF (NCODE 7))
                                             (NCODE-NEQUAL-PROOF FN 7))
                                       (IDENT-AXIOM-PROOF (NCODE 1))
                                       (LIST (LIST (EQUAL FN 8)
                                                   (IDENT-AXIOM-PROOF (NCODE 8))
                                                   (NCODE-NEQUAL-PROOF FN 8))
                                             (IDENT-AXIOM-PROOF (NCODE 1))
                                             (LIST (LIST (EQUAL FN 9)
                                                         (IDENT-AXIOM-PROOF (NCODE 9))
                                                         (NCODE-NEQUAL-PROOF FN 9))
                                                   (IDENT-AXIOM-PROOF (NCODE 1))
                                                   (IDENT-AXIOM-PROOF (PHI)))))))))))
  (Z-SUBRP-EXPR (NCODE FN))
  (Z-SUBRP (NCODE FN))
  (NCODE (BOOL-FIX (SUBRP FN)))
  (Z-SUBRP-DEFN-PROOF (NCODE FN))))
\end{verbatim}

\begin{tabbing}{\sc Theorem}:  z-subrp-ncode-proof-proves \\ \index{z-subrp-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-subrp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-subrp-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-subrp}}\,({\rm{ncode}}\,({\it{fn\/}})\index{ncode})\index{z-subrp}, {\rm{ncode}}\,({\rm{bool-fix}}\,({\rm{subrp}}\,({\it{fn\/}}))\index{bool-fix})\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-subrp-ncode-proof}}\,({\it{fn\/}})\index{z-subrp-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-subrp-ncode-proof; name this event `g0753'.\index{g0753}

\addvspace{18pt}

\noindent{\sc Event}:   Enable subrp; name this event `g0754'.\index{g0754}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{eg-proof} \\  
\=\+{\rm{eg-proof}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{pf\/}})\index{eg-proof} \\ 
$=$$\;\;\;\;${\rm{detach-proof}}\,(\=\+{\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, \\ 
{\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{exp\/}})\index{list-exis-quantify}, \\ 
{\it{pf\/}}, \\ 
{\rm{subst-thm-a-proof}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{subst-thm-a-proof})\-\index{detach-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;; If |- "[a1/x1,..., an/xn]A", then |- "(forsome x1,...,xn A)".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  eg-proof-proves \\ \index{eg-proof-proves}
\=\+(\=\+{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{terms\/}}, {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{vars\/}})\index{length} $=$ {\rm{length}}\,({\it{terms\/}})\index{length}) \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{list-exis-quantify}}\,({\it{vars\/}}, {\it{exp\/}})\index{list-exis-quantify}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{eg-proof}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{pf\/}})\index{eg-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable eg-proof; name this event `g0479'.\index{g0479}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  a-get-ok-proof-proves-again \\ \index{a-get-ok-proof-proves-again}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}), {\tt{3}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-get}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans})\-\index{a-get}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{a-get-ok-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}})\index{a-get-ok-proof1}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-ok-proof-proves; name this event `g0773'.\index{g0773}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-get1} \\  
\=\+{\rm{a-get1}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get1} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,({\it{trace\/}}, {\rm{a-get}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get})\index{forsome}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-get1-ok-proof} \\  
\=\+{\rm{a-get1-ok-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get1-ok-proof} \\ 
$=$$\;\;\;\;${\rm{eg-proof}}\,(\=\+{\rm{a-get}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get}, \\ 
{\rm{list}}\,({\it{trace\/}}), \\ 
{\rm{list}}\,({\rm{graph-trans}}\,({\rm{get-graph}}\,({\it{x\/}}, {\it{y\/}})\index{get-graph})\index{graph-trans}), \\ 
{\rm{a-get-ok-proof1}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}})\index{a-get-ok-proof1})\-\index{eg-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-ncode \\ \index{parallel-subst-ncode}
{\rm{parallel-subst}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} $=$ {\rm{ncode}}\,({\it{x\/}})\index{ncode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-ncode \\ \index{subst-ncode}
{\rm{subst}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\rm{ncode}}\,({\it{x\/}})\index{ncode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-graph-trans \\ \index{parallel-subst-graph-trans}
{\rm{parallel-subst}}\,({\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} $=$ {\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subst-graph-trans \\ \index{subst-graph-trans}
{\rm{subst}}\,({\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst} $=$ {\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans}
\end{tabbing}

\noindent{\sc Event}:   Enable not-free-parallel-subst; name this event `g0755'.\index{g0755}

\addvspace{18pt}

\noindent{\sc Event}:   Enable not-free-del-pairs-parallel-subst; name this event `g0759'.\index{g0759}

\addvspace{18pt}

\noindent{\sc Event}:   Enable parallel-subst-nil; name this event `g0768'.\index{g0768}

\addvspace{18pt}

\noindent{\sc Event}:   Enable parallel-subst-not-free-var; name this event `g0771'.\index{g0771}

\addvspace{18pt}

\noindent{\sc Event}:   Enable not-member-del-pairs; name this event `g0772'.\index{g0772}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  free-for-nil \\ \index{free-for-nil}
({\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) $\rightarrow$ {\rm{free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{free-for}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{nil-list-collect-free} \\  
\=\+{\rm{nil-list-collect-free}}\,({\it{x\/}})\index{nil-list-collect-free} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) \\ 
{\bf then }\=\+({\rm{collect-free}}\,({\rm{car}}\,({\it{x\/}}), {\tt{0}})\index{collect-free} $=$ {\bf{nil}}) \\ 
$\wedge$$\;\;\;\;${\rm{nil-list-collect-free}}\,({\rm{cdr}}\,({\it{x\/}}))\index{nil-list-collect-free}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  free-for-list-nil-list \\ \index{free-for-list-nil-list}
{\rm{nil-list-collect-free}}\,({\it{terms\/}})\index{nil-list-collect-free} $\rightarrow$ {\rm{free-for-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list}
\end{tabbing}

\noindent{\sc Event}:   Enable free-for-list; name this event `g0774'.\index{g0774}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  listp-parallel-subst \\ \index{listp-parallel-subst}
{\rm{listp}}\,({\it{exp\/}}) $\rightarrow$ {\rm{listp}}\,({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\it{flg\/}})\index{parallel-subst})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-list \\ \index{parallel-subst-z-list}
\=\+{\rm{parallel-subst}}\,({\rm{z-list}}\,({\it{list\/}})\index{z-list}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-list}}\,({\rm{parallel-subst}}\,({\it{list\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{1}})\index{parallel-subst})\index{z-list}\-
\end{tabbing}

\noindent{\sc Event}:   Enable listp-parallel-subst; name this event `g0775'.\index{g0775}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  a-get1-ok-proof-proves \\ \index{a-get1-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{val1\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{trace\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-get1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get1}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get1-ok-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get1-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get1-ok-proof; name this event `g0776'.\index{g0776}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  a-get-good-proof-proves-again \\ \index{a-get-good-proof-proves-again}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-get}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode})\index{f-equal})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-good-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-good-proof-proves; name this event `g0777'.\index{g0777}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-get1-good-proof} \\  
\=\+{\rm{a-get1-good-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get1-good-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{trace\/}}, \\ 
{\rm{body}}\,({\rm{a-get1}}\,(\=\+{\rm{ncode}}\,({\it{x\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{y\/}})\index{ncode}, \\ 
{\it{val\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-get1}), \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode})\index{f-equal}, \\ 
{\rm{a-get-good-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get-good-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-get1-good-proof-proves \\ \index{a-get1-good-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $\not\in$ {\rm{collect-free}}\,({\it{val\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-get1}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get1}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\rm{get}}\,({\it{x\/}}, {\it{y\/}})\index{get})\index{ncode})\index{f-equal})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get1-good-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{val\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-get1-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get1-good-proof; name this event `g0778'.\index{g0778}

\addvspace{18pt}
\begin{verbatim};;;;;;;The next several pages cover the proof of the representability of the
;;;;;;;Lisp interpreter EV.  Apart from the e-expression to be evaluated,
;;;;;;;the e-variable bindings list, and the e-function symbol bindings list, 
;;;;;;;the interpreter takes a counter N which is the maximum permissible
;;;;;;;depth of the evaluation stack.  If this bound is exceeded before the
;;;;;;;expression is fully evaluated, EV returns a bottom object (BTM).
;;;;;;;This object is added by as a shell by the command below.
\end{verbatim}
\noindent{\sc Event}:   Add the shell {\it{btm\/}}\index{btm}, with recognizer function symbol {\it{btmp\/}} and no accessors.

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{equal-z-list-isin-proof} \\  
\=\+{\rm{equal-z-list-isin-proof}}\,({\it{list1\/}}, {\it{list2\/}}, {\it{exp\/}}, {\it{pflist\/}}, {\it{pf\/}})\index{equal-z-list-isin-proof} \\ 
$=$$\;\;\;\;${\rm{equal-isin-proof}}\,(\=\+{\rm{z-list}}\,({\it{list1\/}})\index{z-list}, \\ 
{\rm{z-list}}\,({\it{list2\/}})\index{z-list}, \\ 
{\it{exp\/}}, \\ 
{\rm{equal-z-list-proof}}\,({\it{list1\/}}, {\it{list2\/}}, {\it{pflist\/}})\index{equal-z-list-proof}, \\ 
{\it{pf\/}})\-\index{equal-isin-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{equal-z-list-isin-imp-proof} \\  
\=\+{\rm{equal-z-list-isin-imp-proof}}\,({\it{list1\/}}, {\it{list2\/}}, {\it{exp\/}}, {\it{pflist\/}})\index{equal-z-list-isin-imp-proof} \\ 
$=$$\;\;\;\;${\rm{equal-isin-imp-proof}}\,(\=\+{\rm{z-list}}\,({\it{list1\/}})\index{z-list}, \\ 
{\rm{z-list}}\,({\it{list2\/}})\index{z-list}, \\ 
{\it{exp\/}}, \\ 
{\rm{equal-z-list-proof}}\,({\it{list1\/}}, {\it{list2\/}}, {\it{pflist\/}})\index{equal-z-list-proof})\-\index{equal-isin-imp-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ncode-cons \\ \index{ncode-cons}
{\rm{ncode}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{ncode} $=$ {\rm{z-opair}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{z-opair}
\end{tabbing}

\noindent{\sc Event}:   Enable proves-is-formula-again; name this event `g0781'.\index{g0781}

\addvspace{18pt}

\noindent{\sc Event}:   Enable termp-term-list; name this event `g0782'.\index{g0782}

\addvspace{18pt}

\noindent{\sc Event}:   Enable term-list-var-list; name this event `g0783'.\index{g0783}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ev-equal-z-list-proof} \\  
\=\+{\rm{ev-equal-z-list-proof}}\,({\it{val\/}}, {\it{flg\/}}, {\it{exp\/}}, {\it{expr\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{pf\/}})\index{ev-equal-z-list-proof} \\ 
$=$$\;\;\;\;${\rm{equal-z-list-proof}}\,(\=\+{\rm{list}}\,({\it{val\/}}, {\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\rm{ncode}}\,({\it{fa\/}})\index{ncode}), \\ 
{\rm{list}}\,({\it{val\/}}, {\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\it{expr\/}}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\rm{ncode}}\,({\it{fa\/}})\index{ncode}), \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\it{val\/}})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{flg\/}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{flip-equals-proof}}\,({\it{expr\/}}, {\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\it{pf\/}})\index{flip-equals-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{va\/}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{fa\/}})\index{ncode})\index{ident-axiom-proof})\-)\-\index{equal-z-list-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ev-equal-z-list-isin-proof} \\  
\=\+{\rm{ev-equal-z-list-isin-proof}}\,({\it{val\/}}, {\it{flg\/}}, {\it{exp\/}}, {\it{expr\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{trace\/}}, {\it{pf\/}}, {\it{in-pf\/}})\index{ev-equal-z-list-isin-proof} \\ 
$=$$\;\;\;\;${\rm{equal-z-list-isin-proof}}\,(\=\+{\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-, \\ 
{\rm{list}}\,({\it{val\/}}, {\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\it{expr\/}}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\rm{ncode}}\,({\it{fa\/}})\index{ncode}), \\ 
{\it{trace\/}}, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\it{val\/}})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{flg\/}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{flip-equals-proof}}\,({\it{expr\/}}, {\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\it{pf\/}})\index{flip-equals-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{va\/}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{fa\/}})\index{ncode})\index{ident-axiom-proof})\-, \\ 
{\it{in-pf\/}})\-\index{equal-z-list-isin-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-equal-z-list-proof-proves \\ \index{ev-equal-z-list-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\it{expr\/}}, {\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\it{expr\/}}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list})\-\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ev-equal-z-list-proof}}\,({\it{val\/}}, {\it{flg\/}}, {\it{exp\/}}, {\it{expr\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{pf\/}})\index{ev-equal-z-list-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ev-equal-z-list-proof; name this event `g0789'.\index{g0789}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  ev-equal-z-list-isin-proof-proves \\ \index{ev-equal-z-list-isin-proof-prove!s}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{val\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{trace\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{in-pf\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\it{expr\/}}, {\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\it{val\/}}, {\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\it{expr\/}}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\rm{ncode}}\,({\it{fa\/}})\index{ncode}))\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\it{val\/}}, \\ 
{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{expr\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{pf\/}}, \\ 
{\it{in-pf\/}})\-\index{ev-equal-z-list-isin-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable ev-equal-z-list-isin-proof; name this event `g0792'.\index{g0792}

\addvspace{18pt}
\begin{verbatim};;;;;;;The skeleton of the representability proof for EV is similar to that of GET, but 
;;;;;;;the details are enormously more complicated since EV is an extremely complicated 
;;;;;;;function.  In the first `OK' part of the proof, we prove each section of EV to
;;;;;;;be `OK' separately.  Below, the part of EV which evaluates Lisp if-expressions
;;;;;;;is proved `OK'.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-ev-defn-if1-ok-proof} \\  
\=\+{\rm{a-ev-defn-if1-ok-proof}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{test\/}}, {\it{trace\/}}, {\it{in-pf2\/}}, {\it{in-pf3\/}})\index{a-ev-defn-if1-ok-proof} \\ 
$=$$\;\;\;\;${\rm{form-z-if-ok-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{ncode-equal}}\,({\it{test\/}}, {\tt{0}})\index{ncode-equal}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{test\/}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{test\/}}, {\tt{0}})\index{ncode-nequal-proof})\-, \\ 
{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\it{val\/}}, \\ 
{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{z-car-cdr-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr-proof}, \\ 
{\it{in-pf2\/}})\-\index{ev-equal-z-list-isin-proof}, \\ 
{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\it{val\/}}, \\ 
{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{z-car-cdr-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr-proof}, \\ 
{\it{in-pf3\/}})\-\index{ev-equal-z-list-isin-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{test\/}})\index{ncode}, {\rm{ncode}}\,({\tt{0}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}, {\tt{2}}))\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}))\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{z-if})\-\index{form-z-if-ok-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Enable termp-car-cdr; name this event `g0797'.\index{g0797}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-if1-ok-proof-proves \\ \index{a-ev-defn-if1-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{trace\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\rm{ncode-equal}}\,({\it{test\/}}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{proves}}\,(\=\+{\it{in-pf2\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{in-pf3\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{caddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{test\/}})\index{ncode}, {\rm{ncode}}\,({\tt{0}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}, {\tt{2}}))\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}))\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{f-if}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-if1-ok-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\it{test\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{in-pf2\/}}, \\ 
{\it{in-pf3\/}})\-\index{a-ev-defn-if1-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-if1-ok-proof; name this event `g0798'.\index{g0798}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-if-part} \\  
\=\+{\rm{a-ev-if-part}}\,({\it{exp-bar\/}}, {\it{va-bar\/}}, {\it{fa-bar\/}}, {\it{val-bar\/}}, {\it{trace\/}}, {\it{rvar1\/}})\index{a-ev-if-part} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\it{exp-bar\/}})\index{z-cdr})\index{z-car}, \\ 
{\it{va-bar\/}}, \\ 
{\it{fa-bar\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{rvar1\/}}, {\rm{ncode}}\,({\tt{0}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val-bar\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,({\it{exp-bar\/}}, {\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}, {\tt{2}}))\index{z-car-cdr}, \\ 
{\it{va-bar\/}}, \\ 
{\it{fa-bar\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val-bar\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,({\it{exp-bar\/}}, {\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}))\index{z-car-cdr}, \\ 
{\it{va-bar\/}}, \\ 
{\it{fa-bar\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{f-if})\-\index{f-and})\-\index{forsome}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-if-ok-proof} \\  
\=\+{\rm{a-ev-defn-if-ok-proof}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{test\/}}, {\it{rvar1\/}}, {\it{graph\/}}, {\it{in-pf1\/}}, {\it{in-pf2\/}}, {\it{in-pf3\/}})\index{a-ev-defn-if-ok-proof} \\ 
$=$$\;\;\;\;${\rm{eg-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{rvar1\/}}, {\rm{ncode}}\,({\tt{0}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}, {\tt{2}}))\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}))\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin})\-\index{f-if})\-\index{f-and}, \\ 
{\rm{list}}\,({\it{rvar1\/}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\it{test\/}})\index{ncode}), \\ 
{\rm{and-join-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{test\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{test\/}})\index{ncode}, {\rm{ncode}}\,({\tt{0}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}, {\tt{2}}))\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin})\-\index{f-if}, \\ 
{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\rm{ncode}}\,({\it{test\/}})\index{ncode}, \\ 
{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr})\index{z-car}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}, \\ 
{\rm{z-car-cdr-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr-proof}, \\ 
{\it{in-pf1\/}})\-\index{ev-equal-z-list-isin-proof}, \\ 
{\rm{a-ev-defn-if1-ok-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\it{test\/}}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}, \\ 
{\it{in-pf2\/}}, \\ 
{\it{in-pf3\/}})\-\index{a-ev-defn-if1-ok-proof})\-\index{and-join-proof})\-\index{eg-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-if-ok-proof-proves \\ \index{a-ev-defn-if-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{rvar1\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{in-pf1\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{test\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\rm{ncode-equal}}\,({\it{test\/}}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{proves}}\,(\=\+{\it{in-pf2\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{in-pf3\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{caddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-if-part}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}, \\ 
{\it{rvar1\/}})\-\index{a-ev-if-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-if-ok-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val\/}}, \\ 
{\it{test\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{graph\/}}, \\ 
{\it{in-pf1\/}}, \\ 
{\it{in-pf2\/}}, \\ 
{\it{in-pf3\/}})\-\index{a-ev-defn-if-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-if-ok-proof; name this event `g0799'.\index{g0799}

\addvspace{18pt}
\begin{verbatim};;;;;;;The Z2 representation of the part of EV which evaluates Lisp primitives is proved 
;;;;;;;`OK' below.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-ev-subr-part} \\  
\=\+{\rm{a-ev-subr-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{rvar\/}}, {\it{trace\/}})\index{a-ev-subr-part} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,(\=\+{\it{rvar\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-apply-subr}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\it{rvar\/}})\index{z-apply-subr})\index{f-equal})\-\index{f-and})\-\index{forsome}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-subr-ok-proof} \\  
\=\+{\rm{a-ev-defn-subr-ok-proof}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{rval\/}}, {\it{rvar\/}}, {\it{graph\/}}, {\it{in-pf\/}})\index{a-ev-defn-subr-ok-proof} \\ 
$=$$\;\;\;\;${\rm{eg-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-apply-subr}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\it{rvar\/}})\index{z-apply-subr})\index{f-equal})\-\index{f-and}, \\ 
{\rm{list}}\,({\it{rvar\/}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\it{rval\/}})\index{ncode}), \\ 
{\rm{and-join-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode})\-\index{z-apply-subr})\-\index{f-equal}, \\ 
{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{exp\/}})\index{z-cdr-ncode-proof}, \\ 
{\it{in-pf\/}})\-\index{ev-equal-z-list-isin-proof}, \\ 
{\rm{flip-equals-proof}}\,(\=\+{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode})\-\index{z-apply-subr}, \\ 
{\it{val\/}}, \\ 
{\rm{compose-proof}}\,(\=\+{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\it{val\/}}, \\ 
{\rm{list}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{rval\/}})\index{ncode})\index{ident-axiom-proof})\-, \\ 
{\rm{z-apply-subr-ncode-proof}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval\/}})\-\index{z-apply-subr-ncode-proof})\-\index{compose-proof})\-\index{flip-equals-proof})\-\index{and-join-proof})\-\index{eg-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-subr-ok-proof-proves \\ \index{a-ev-defn-subr-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-all-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-all-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{rvar\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{in-pf\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{val\/}} $=$ {\rm{ncode}}\,({\rm{apply-subr}}\,({\rm{car}}\,({\it{exp\/}}), {\it{rval\/}}))\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-subr-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{rvar\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{a-ev-subr-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-subr-ok-proof}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{rval\/}}, {\it{rvar\/}}, {\it{graph\/}}, {\it{in-pf\/}})\index{a-ev-defn-subr-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-subr-ok-proof; name this event `g0802'.\index{g0802}

\addvspace{18pt}

\noindent{\sc Event}:   Enable get; name this event `g0803'.\index{g0803}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pairlist; name this event `g0809'.\index{g0809}

\addvspace{18pt}

\noindent{\sc Event}:   Enable not-free-covering; name this event `g0816'.\index{g0816}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-functions symbols 0 is used for the Lisp function IF, the symbols 1-9
;;;;;;;are used for the Lisp primitives, 10 is used for the QUOTE function.  The defined
;;;;;;;e-function symbols therefore start from 11.  Since the first definition in the
;;;;;;;function definition list corresponds to 11, we have to subtract 11 from the symbol
;;;;;;;before we apply GET to the definition list.  the function SUB1-N subtracts N
;;;;;;;from EXP.  It is represented in Z2 by an iterated application of Z-SUB1.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-sub1-n} \\  
\=\+{\rm{z-sub1-n}}\,({\it{exp\/}}, {\it{n\/}})\index{z-sub1-n} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{exp\/}} \\ 
{\bf else }{\rm{z-sub1}}\,({\rm{z-sub1-n}}\,({\it{exp\/}}, {\it{n\/}} $-\;1$)\index{z-sub1-n})\index{z-sub1}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{sub1-n} \\  
\=\+{\rm{sub1-n}}\,({\it{exp\/}}, {\it{n\/}})\index{sub1-n} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{exp\/}} \\ 
{\bf else }{\rm{sub1-n}}\,({\it{exp\/}}, {\it{n\/}} $-\;1$)\index{sub1-n} $-\;1$$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{z-sub1-n-ncode-proof} \\  
\=\+{\rm{z-sub1-n-ncode-proof}}\,({\it{exp\/}}, {\it{n\/}}, {\it{val\/}}, {\it{pf\/}})\index{z-sub1-n-ncode-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{pf\/}} \\ 
{\bf else }{\rm{compose-proof}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\rm{sub1-n}}\,({\it{val\/}}, {\it{n\/}})\index{sub1-n})\index{ncode}, \\ 
{\rm{list}}\,({\rm{z-sub1-n}}\,({\it{exp\/}}, {\it{n\/}} $-\;1$)\index{z-sub1-n}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\rm{sub1-n}}\,({\it{val\/}}, {\it{n\/}} $-\;1$)\index{sub1-n})\index{ncode}), \\ 
{\rm{list}}\,({\rm{z-sub1-n-ncode-proof}}\,({\it{exp\/}}, {\it{n\/}} $-\;1$, {\it{val\/}}, {\it{pf\/}})\index{z-sub1-n-ncode-proof}), \\ 
{\rm{z-sub1-ncode-proof}}\,({\rm{sub1-n}}\,({\it{val\/}}, {\it{n\/}} $-\;1$)\index{sub1-n})\index{z-sub1-ncode-proof})\-\index{compose-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-z-sub1-n \\ \index{term-z-sub1-n}
\=\+(({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn} $\in$ {\it{symbols\/}}) $\wedge$ {\rm{termp}}\,({\it{exp\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}) \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{z-sub1-n}}\,({\it{exp\/}}, {\it{n\/}})\index{z-sub1-n}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-sub1-n-ncode-proof-proves \\ \index{z-sub1-n-ncode-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-equal}}\,({\it{exp\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-equal}}\,({\rm{z-sub1-n}}\,({\it{exp\/}}, {\it{n\/}})\index{z-sub1-n}, {\rm{ncode}}\,({\rm{sub1-n}}\,({\it{val\/}}, {\it{n\/}})\index{sub1-n})\index{ncode})\index{f-equal}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{z-sub1-n-ncode-proof}}\,({\it{exp\/}}, {\it{n\/}}, {\it{val\/}}, {\it{pf\/}})\index{z-sub1-n-ncode-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable z-sub1-n-ncode-proof; name this event `g0817'.\index{g0817}

\addvspace{18pt}

\noindent{\sc Event}:   Enable sub1-n; name this event `g0818'.\index{g0818}

\addvspace{18pt}
\begin{verbatim};;;;;;;Next we prove the representation of that part of EV which evaluates defined Lisp
;;;;;;;functions, to be `OK'.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-ev-fun-part} \\  
\=\+{\rm{a-ev-fun-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-fun-part} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,(\=\+{\tt{11}}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\tt{15}}, {\tt{16}}), \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{f-implies})\-\index{f-and})\-\index{list-all-quantify})\-\index{forsome}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-list-all-quantify \\ \index{formula-list-all-quantify}
\=\+({\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} $\wedge$ {\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{exp\/}})\index{list-all-quantify}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  list-gen-proof-proves-again \\ \index{list-gen-proof-proves-again}
\=\+(\=\+{\rm{var-list}}\,({\it{vars\/}}, {\rm{length}}\,({\it{vars\/}})\index{length})\index{var-list} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{list-all-quantify}}\,({\it{vars\/}}, {\it{exp\/}})\index{list-all-quantify}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{list-gen-proof}}\,({\it{vars\/}}, {\it{exp\/}}, {\it{pf\/}})\index{list-gen-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-get-ev-reduc-proof} \\  
\=\+{\rm{a-get-ev-reduc-proof}}\,({\it{exp\/}}, {\it{fa\/}}, {\it{getval\/}}, {\it{n\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{tr1\/}})\index{a-get-ev-reduc-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\it{n\/}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{tr1\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{ncode}}\,({\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{n\/}})\-\index{sub1-n})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{tr1\/}})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{ncode}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\it{n\/}})\index{sub1-n})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{getval\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{tr1\/}})\-\index{a-get}, \\ 
{\rm{f-equal}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\it{n\/}})\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\it{n\/}})\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{getval\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{tr1\/}})\-\index{a-get}, \\ 
{\rm{f-equal}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\it{n\/}})\index{sub1-n}, {\it{fa\/}})\index{get})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{equal-z-list-isin-imp-proof}}\,(\=\+{\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\it{n\/}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{ncode}}\,({\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{n\/}})\-\index{sub1-n})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-, \\ 
{\it{tr1\/}}, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\it{getval\/}})\index{ident-axiom-proof}, \\ 
{\rm{z-sub1-n-ncode-proof}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\it{n\/}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof})\-\index{z-sub1-n-ncode-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{fa\/}})\index{ncode})\index{ident-axiom-proof})\-)\-\index{equal-z-list-isin-imp-proof}, \\ 
{\rm{a-get-good-proof}}\,(\=\+{\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\it{n\/}})\index{sub1-n}, \\ 
{\it{fa\/}}, \\ 
{\it{getval\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{tr1\/}})\-\index{a-get-good-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-fun-ok-proof} \\  
\=\+{\rm{a-ev-defn-fun-ok-proof}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{rval\/}}, {\it{graph\/}}, {\it{in-pf1\/}}, {\it{in-pf2\/}})\index{a-ev-defn-fun-ok-proof} \\ 
$=$$\;\;\;\;${\rm{eg-proof}}\,(\=\+{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\tt{15}}, {\tt{16}}), \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin})\-\index{f-implies})\-\index{f-and})\-\index{list-all-quantify}, \\ 
{\rm{list}}\,({\tt{11}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\it{rval\/}})\index{ncode}), \\ 
{\rm{list-gen-proof}}\,(\=\+{\rm{list}}\,({\tt{15}}, {\tt{16}}), \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin})\-\index{f-implies})\-\index{f-and}, \\ 
{\rm{and-join-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{exp\/}})\index{z-cdr-ncode-proof}, \\ 
{\it{in-pf1\/}})\-\index{ev-equal-z-list-isin-proof}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{list}}\,({\rm{f-equal}}\,(\=\+{\tt{15}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode})\-\index{f-equal}), \\ 
{\rm{list}}\,(\=\+{\rm{a-get-ev-reduc-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{fa\/}}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get-ev-reduc-proof}, \\ 
{\rm{form-equal-case-proof}}\,(\=\+{\rm{list}}\,({\tt{15}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}), \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{in-pf2\/}})\-\index{form-equal-case-proof})\-)\-\index{chain-proof})\-\index{and-join-proof})\-\index{list-gen-proof})\-\index{eg-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-z-sub1-n \\ \index{parallel-subst-z-sub1-n}
\=\+{\rm{parallel-subst}}\,({\rm{z-sub1-n}}\,({\it{exp\/}}, {\it{n\/}})\index{z-sub1-n}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{z-sub1-n}}\,({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst}, {\it{n\/}})\index{z-sub1-n}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-fun-ok-proof-proves \\ \index{a-ev-defn-fun-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{in-pf1\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{in-pf2\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-fun-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{a-ev-fun-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-fun-ok-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\it{rval\/}}, \\ 
{\it{graph\/}}, \\ 
{\it{in-pf1\/}}, \\ 
{\it{in-pf2\/}})\-\index{a-ev-defn-fun-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-fun-ok-proof; name this event `g0819'.\index{g0819}

\addvspace{18pt}
\begin{verbatim};;;;;;;Finally we prove the representation of the part of EV which evaluates lists
;;;;;;;of e-expressions to be `OK'.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-ev-defn-list-ok-proof} \\  
\=\+{\rm{a-ev-defn-list-ok-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{rval2\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rvar2\/}}, \\ 
{\it{graph\/}}, \\ 
{\it{in-pf1\/}}, \\ 
{\it{in-pf2\/}})\-\index{a-ev-defn-list-ok-proof} \\ 
$=$$\;\;\;\;${\rm{eg-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\rm{cons}}\,({\it{rval1\/}}, {\it{rval2\/}}))\index{ncode}, \\ 
{\rm{z-opair}}\,({\it{rvar1\/}}, {\it{rvar2\/}})\index{z-opair})\-\index{f-equal})\-\index{f-and})\-\index{f-and}, \\ 
{\rm{list}}\,({\it{rvar1\/}}, {\it{rvar2\/}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, {\rm{ncode}}\,({\it{rval2\/}})\index{ncode}), \\ 
{\rm{and-join-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval2\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\rm{cons}}\,({\it{rval1\/}}, {\it{rval2\/}}))\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval2\/}})\index{ncode})\-\index{z-opair})\-\index{f-equal})\-\index{f-and}, \\ 
{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}, \\ 
{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof}, \\ 
{\it{in-pf1\/}})\-\index{ev-equal-z-list-isin-proof}, \\ 
{\rm{and-join-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval2\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\rm{cons}}\,({\it{rval1\/}}, {\it{rval2\/}}))\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval2\/}})\index{ncode})\-\index{z-opair})\-\index{f-equal}, \\ 
{\rm{ev-equal-z-list-isin-proof}}\,(\=\+{\rm{ncode}}\,({\it{rval2\/}})\index{ncode}, \\ 
{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{exp\/}})\index{z-cdr-ncode-proof}, \\ 
{\it{in-pf2\/}})\-\index{ev-equal-z-list-isin-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval2\/}})\index{ncode})\-\index{z-opair})\index{ident-axiom-proof})\-\index{and-join-proof})\-\index{and-join-proof})\-\index{eg-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-list-part} \\  
\=\+{\rm{a-ev-list-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{rvar1\/}}, {\it{rvar2\/}}, {\it{trace\/}})\index{a-ev-list-part} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-opair}}\,({\it{rvar1\/}}, {\it{rvar2\/}})\index{z-opair})\index{f-equal})\-\index{f-and})\-\index{f-and})\-\index{forsome})\-\index{forsome}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-list-ok-proof-proves \\ \index{a-ev-defn-list-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{rvar1\/}}, {\it{rvar2\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{in-pf1\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{in-pf2\/}}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{rval2\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-list-part}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{cons}}\,({\it{rval1\/}}, {\it{rval2\/}}))\index{ncode}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rvar2\/}}, \\ 
{\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\-\index{a-ev-list-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-list-ok-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{rval2\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rvar2\/}}, \\ 
{\it{graph\/}}, \\ 
{\it{in-pf1\/}}, \\ 
{\it{in-pf2\/}})\-\index{a-ev-defn-list-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-list-ok-proof; name this event `g0823'.\index{g0823}

\addvspace{18pt}
\begin{verbatim};;;;;;;EV is defined below.  It takes a FLG to control the recursion, an e-expression EXP,
;;;;;;;an e-variable bindings list VA, a e-function symbols definition list FA, and a stack
;;;;;;;bound N.  If the FLG is 1, it is evaluating a list of e-expressions - if any of them
;;;;;;;evaluates to (BTM), the whole list evaluates to (BTM).  Otherwise, it returns the 
;;;;;;;list of values.  If the FLG is not 1, then if EXP is not a list, it must be an
;;;;;;;e-variable and EV returns (GET EXP VA).  Otherwise, (CAR EXP) is an e-function
;;;;;;;symbol.  If the symbol is 0, then the e-expression is evaluated as a Lisp
;;;;;;;IF-expression.  If the symbol is 10, then the e-expression is evaluated as a
;;;;;;;Lisp QUOTE-expression.  If the symbol is between 1 and 9, then the appropriate
;;;;;;;Lisp primitive is applied to the recursively evaluated arguments.  In the remaining
;;;;;;;case, the e-function symbol is a defined one, and the corresponding definition is
;;;;;;;evaluated with a new e-variable-bindings list consisting of the recursively evaluated
;;;;;;;arguments.  Since there is no guarantee in the last case that the size of the 
;;;;;;;definition is going to be smaller than the e-expression being evaluated, the counter
;;;;;;;N is decremented by one.  If either the counter is exhausted or one of the 
;;;;;;;evaluated sub-expressions evaluates to (BTM), then EV returns (BTM).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{ev} \\  
\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $=$ {\tt{1}} \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\bf if }\=\+{\rm{btmp}}\,({\rm{ev}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}\-$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\-$\;${\bf  endif}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{btmp}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{ev}}\,({\tt{0}}, {\rm{cadddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
{\bf else }{\rm{ev}}\,({\tt{0}}, {\rm{caddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{exp\/}}) $=$ {\tt{10}}$\;\;${\bf then }{\rm{cadr}}\,({\it{exp\/}}) \\ 
{\bf elseif }{\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
{\bf then }{\rm{apply-subr}}\,({\rm{car}}\,({\it{exp\/}}), {\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}) \\ 
{\bf elseif }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{get}}\,({\it{exp\/}}, {\it{va\/}})\index{get}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-if-reduc \\ \index{ev-if-reduc}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{ev}}\,({\tt{0}}, {\rm{cadddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
{\bf else }{\rm{ev}}\,({\tt{0}}, {\rm{caddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}$\;${\bf  endif}\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-subr-reduc \\ \index{ev-subr-reduc}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;${\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{apply-subr}}\,({\rm{car}}\,({\it{exp\/}}), {\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}))\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-fun-reduc \\ \index{ev-fun-reduc}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}}))) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-list-reduc \\ \index{ev-list-reduc}
\=\+(({\it{flg\/}} $=$ {\tt{1}}) $\wedge$ {\rm{listp}}\,({\it{exp\/}}) $\wedge$ ($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{cons}}\,({\rm{ev}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}))\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The list of value/argument pairs generated in a computation of EV is constructed
;;;;;;;by EV-GRAPH below.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{ev-graph} \\  
\=\+{\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $=$ {\tt{1}} \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-, \\ 
{\rm{append}}\,(\=\+{\rm{ev-graph}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-)\- \\ 
{\bf else }{\rm{list}}\,({\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-)$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-, \\ 
{\rm{append}}\,(\=\+{\rm{ev-graph}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{0}}, {\rm{cadddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-)\- \\ 
{\bf else }{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-, \\ 
{\rm{append}}\,(\=\+{\rm{ev-graph}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{0}}, {\rm{caddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-)\-$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{exp\/}}) $=$ {\tt{10}} \\ 
{\bf then }{\rm{list}}\,({\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-) \\ 
{\bf elseif }{\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
{\bf then }{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-, \\ 
{\rm{ev-graph}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\- \\ 
{\bf elseif }{\it{n\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{list}}\,({\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-) \\ 
{\bf else }{\rm{cons}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-, \\ 
{\rm{append}}\,(\=\+{\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-)\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{list}}\,({\rm{cons}}\,({\rm{get}}\,({\it{exp\/}}, {\it{va\/}})\index{get}, {\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))))$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-ev-graph1 \\ \index{member-ev-graph1}
\=\+{\rm{cons}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))) \\ 
$\in$$\;\;\;\;${\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  subr-member-graph \\ \index{subr-member-graph}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;${\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{1}}, {\rm{cons}}\,({\rm{cdr}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{x\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  fun-ev-graph \\ \index{fun-ev-graph}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}}))) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph} \\ 
$=$$\;\;\;\;${\rm{cons}}\,(\=\+{\rm{cons}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))), \\ 
{\rm{append}}\,(\=\+{\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-)\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-fun-graph \\ \index{member-fun-graph}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}}))) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{1}}, {\rm{cons}}\,({\rm{cdr}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\tt{0}}, \\ 
{\rm{cons}}\,(\=\+{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{cons}}\,({\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\it{fa\/}}))\-)\-)\-) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\it{x\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  if-ev-graph1 \\ \index{if-ev-graph1}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;${\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph} \\ 
$=$$\;\;\;\;${\rm{cons}}\,(\=\+{\rm{cons}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))), \\ 
{\rm{append}}\,(\=\+{\rm{ev-graph}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{0}}, {\rm{cadddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-)\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  if-ev-graph2 \\ \index{if-ev-graph2}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph} \\ 
$=$$\;\;\;\;${\rm{cons}}\,(\=\+{\rm{cons}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))), \\ 
{\rm{append}}\,(\=\+{\rm{ev-graph}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{0}}, {\rm{caddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-)\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-if-graph1 \\ \index{member-if-graph1}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;${\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal} \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{0}}, {\rm{cons}}\,({\rm{cadr}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{cadddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{0}}, {\rm{cons}}\,({\rm{cadddr}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\it{x\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-if-graph2 \\ \index{member-if-graph2}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{0}}, {\rm{cons}}\,({\rm{cadr}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{caddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{0}}, {\rm{cons}}\,({\rm{caddr}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\it{x\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-list-graph \\ \index{member-list-graph}
\=\+(\=\+{\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{flg\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{0}}, {\rm{cons}}\,({\rm{car}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $=$ {\rm{cons}}\,(\=\+{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\tt{1}}, {\rm{cons}}\,({\rm{cdr}}\,({\it{exp\/}}), {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\it{x\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $\in$ {\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable fun-ev-graph; name this event `g0824'.\index{g0824}

\addvspace{18pt}

\noindent{\sc Event}:   Enable if-ev-graph1; name this event `g0844'.\index{g0844}

\addvspace{18pt}

\noindent{\sc Event}:   Enable if-ev-graph2; name this event `g0848'.\index{g0848}

\addvspace{18pt}

\noindent{\sc Event}:   Disable proves-is-formula-again; name this event `g0849'.\index{g0849}

\addvspace{18pt}

\noindent{\sc Event}:   Enable nlistp-ncode; name this event `g0855'.\index{g0855}

\addvspace{18pt}
\begin{verbatim};;;;;;;Below is the Z2 analogue of the definition of EV.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{z-ev-defn-part} \\  
\=\+{\rm{z-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{z-ev-defn-part} \\ 
$=$$\;\;\;\;${\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-listp}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-listp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-list-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\tt{11}}, {\tt{12}}, {\it{trace\/}})\index{a-ev-list-part}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\bf{nil}})\index{ncode})\index{f-equal})\-\index{z-if}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-listp}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-listp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-zerop}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car})\index{z-zerop}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-if-part}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}})\-\index{a-ev-if-part}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\rm{ncode}}\,({\tt{10}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-car-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\rm{list}}\,({\tt{1}}, {\tt{2}}))\index{z-car-cdr})\index{f-equal}, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-subrp}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car})\index{z-subrp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-subr-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\tt{11}}, {\it{trace\/}})\index{a-ev-subr-part}, \\ 
{\rm{a-ev-fun-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-fun-part})\-\index{z-if})\-\index{z-if})\-\index{z-if}, \\ 
{\rm{a-get1}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\it{val\/}}, {\tt{11}}, {\tt{13}}, {\tt{14}}, {\tt{16}})\index{a-get1})\-\index{z-if})\-\index{z-if}\-
\end{tabbing}
%\item 
\begin{verbatim}

(DEFN
 A-EV-DEFN-PART-OK-PFTREE
 (FLG EXP VA FA N)
 (LIST
  (LIST (EQUAL FLG 1) (IDENT-AXIOM-PROOF (NCODE 1)) (NCODE-NEQUAL-PROOF FLG 1))
  (LIST (LIST (LISTP EXP)
              (Z-LISTP-NCODE-PROOF EXP)
              (TRANS-NOT-EQUAL-PROOF (NCODE 1)
                                     (Z-LISTP (NCODE EXP))
                                     (NCODE 0)
                                     (NCODE-NEQUAL-PROOF 1 0)
                                     (Z-LISTP-NCODE-PROOF EXP)))
        (A-EV-DEFN-LIST-OK-PROOF EXP
                                 VA
                                 FA
                                 (EV 0 (CAR EXP) VA FA N)
                                 (EV 1 (CDR EXP) VA FA N)
                                 11
                                 12
                                 (EV-GRAPH FLG EXP VA FA N)
                                 (IN-GRAPH-PROOF (CONS (EV 0 (CAR EXP) VA FA N)
                                                       (CONS 0 (CONS (CAR EXP) (CONS VA FA))))
                                                 (EV-GRAPH FLG EXP VA FA N))
                                 (IN-GRAPH-PROOF (CONS (EV 1 (CDR EXP) VA FA N)
                                                       (CONS 1 (CONS (CDR EXP) (CONS VA FA))))
                                                 (EV-GRAPH FLG EXP VA FA N)))
        (IDENT-AXIOM-PROOF (NCODE NIL)))
  (LIST
   (LIST (LISTP EXP)
         (Z-LISTP-NCODE-PROOF EXP)
         (TRANS-NOT-EQUAL-PROOF (NCODE 1)
                                (Z-LISTP (NCODE EXP))
                                (NCODE 0)
                                (NCODE-NEQUAL-PROOF 1 0)
                                (Z-LISTP-NCODE-PROOF EXP)))
   (LIST
    (LIST (ZEROP (CAR EXP))
          (COMPOSE-CAR-PROOF (FN 9 1) (NCODE 1) EXP (Z-ZEROP-NCODE-PROOF (CAR EXP)))
          (TRANS-NOT-EQUAL-PROOF (NCODE 1)
                                 (Z-ZEROP (Z-CAR (NCODE EXP)))
                                 (NCODE 0)
                                 (NCODE-NEQUAL-PROOF 1 0)
                                 (COMPOSE-CAR-PROOF (FN 9 1)
                                                    (NCODE 0)
                                                    EXP
                                                    (Z-ZEROP-NCODE-PROOF (CAR EXP)))))
    (A-EV-DEFN-IF-OK-PROOF EXP
                           VA
                           FA
                           (EV FLG EXP VA FA N)
                           (EV 0 (CADR EXP) VA FA N)
                           11
                           (EV-GRAPH FLG EXP VA FA N)
                           (IN-GRAPH-PROOF (CONS (EV 0 (CADR EXP) VA FA N)
                                                 (CONS 0 (CONS (CADR EXP) (CONS VA FA))))
                                           (EV-GRAPH FLG EXP VA FA N))
                           (IN-GRAPH-PROOF (CONS (EV 0 (CADDDR EXP) VA FA N)
                                                 (CONS 0 (CONS (CADDDR EXP) (CONS VA FA))))
                                           (EV-GRAPH FLG EXP VA FA N))
                           (IN-GRAPH-PROOF (CONS (EV 0 (CADDR EXP) VA FA N)
                                                 (CONS 0 (CONS (CADDR EXP) (CONS VA FA))))
                                           (EV-GRAPH FLG EXP VA FA N)))
    (LIST
     (LIST (EQUAL (CAR EXP) 10)
           (Z-CAR-NCODE-PROOF EXP)
           (TRANS-NOT-EQUAL-PROOF (NCODE 10)
                                  (Z-CAR (NCODE EXP))
                                  (NCODE (CAR EXP))
                                  (NCODE-NEQUAL-PROOF 10 (CAR EXP))
                                  (Z-CAR-NCODE-PROOF EXP)))
     (FLIP-EQUALS-PROOF (Z-CAR-CDR (NCODE EXP) (LIST 1 2))
                        (NCODE (CADR EXP))
                        (Z-CAR-CDR-PROOF EXP (LIST 1 2)))
     (LIST (LIST (SUBRP (CAR EXP))
                 (COMPOSE-CAR-PROOF (FN 16 1) (NCODE 1) EXP (Z-SUBRP-NCODE-PROOF (CAR EXP)))
                 (TRANS-NOT-EQUAL-PROOF (NCODE 1)
                                        (Z-SUBRP (Z-CAR (NCODE EXP)))
                                        (NCODE 0)
                                        (NCODE-NEQUAL-PROOF 1 0)
                                        (COMPOSE-CAR-PROOF (FN 16 1)
                                                           (NCODE 0)
                                                           EXP
                                                           (Z-SUBRP-NCODE-PROOF (CAR EXP)))))
           (A-EV-DEFN-SUBR-OK-PROOF EXP
                                    VA
                                    FA
                                    (NCODE (EV FLG EXP VA FA N))
                                    (EV 1 (CDR EXP) VA FA N)
                                    11
                                    (EV-GRAPH FLG EXP VA FA N)
                                    (IN-GRAPH-PROOF (CONS (EV 1 (CDR EXP) VA FA N)
                                                          (CONS 1
                                                                (CONS (CDR EXP)
                                                                      (CONS VA FA))))
                                                    (EV-GRAPH FLG EXP VA FA N)))
           (A-EV-DEFN-FUN-OK-PROOF EXP
                                   VA
                                   FA
                                   (NCODE (EV FLG EXP VA FA N))
                                   (EV 1 (CDR EXP) VA FA N)
                                   (EV-GRAPH FLG EXP VA FA N)
                                   (IN-GRAPH-PROOF (CONS (EV 1 (CDR EXP) VA FA N)
                                                         (CONS 1
                                                               (CONS (CDR EXP) (CONS VA FA))))
                                                   (EV-GRAPH FLG EXP VA FA N))
                                   (IN-GRAPH-PROOF (CONS (EV FLG EXP VA FA N)
                                                         (CONS 0
                                                               (CONS (GET (SUB1-N (CAR EXP)
                                                                                  11)
                                                                          FA)
                                                                     (CONS (EV 1
                                                                               (CDR EXP)
                                                                               VA
                                                                               FA
                                                                               N)
                                                                           FA))))
                                                   (EV-GRAPH FLG EXP VA FA N))))))
   (A-GET1-OK-PROOF EXP VA 11 13 14 16))))
\end{verbatim}

\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part-ok-proof} \\  
\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof} \\ 
$=$$\;\;\;\;${\rm{form-z-if-ok-proof}}\,(\=\+{\rm{a-ev-defn-part-ok-pftree}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-pftree}, \\ 
{\rm{z-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{z-ev-defn-part})\-\index{form-z-if-ok-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-z-if-z-ev-defn-part \\ \index{form-z-if-z-ev-defn-part}
\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\rightarrow$$\;\;\;\;${\rm{form-z-if}}\,(\=\+{\rm{z-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{z-ev-defn-part}, \\ 
{\it{symbols\/}})\-\index{form-z-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part} \\  
\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-defn-part} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-listp}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-listp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-list-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\tt{11}}, {\tt{12}}, {\it{trace\/}})\index{a-ev-list-part}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\bf{nil}})\index{ncode})\index{f-equal})\-\index{f-if}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-listp}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-listp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-zerop}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car})\index{z-zerop}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-if-part}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}})\-\index{a-ev-if-part}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\rm{ncode}}\,({\tt{10}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-car-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\rm{list}}\,({\tt{1}}, {\tt{2}}))\index{z-car-cdr})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-subrp}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car})\index{z-subrp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-subr-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\tt{11}}, {\it{trace\/}})\index{a-ev-subr-part}, \\ 
{\rm{a-ev-fun-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-fun-part})\-\index{f-if})\-\index{f-if})\-\index{f-if}, \\ 
{\rm{a-get1}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\it{val\/}}, {\tt{11}}, {\tt{13}}, {\tt{14}}, {\tt{16}})\index{a-get1})\-\index{f-if})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ev-hyps} \\  
\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$=$$\;\;\;\;$(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-all-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-all-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-subrp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-subrp-hyps})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-quote-case \\ \index{a-ev-defn-part-ok-proof-proves-q!uote-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-z-if-z-ev-defn-part-again \\ \index{form-z-if-z-ev-defn-part-again}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{z-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{z-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{form-z-if}}\,({\it{concl\/}}, {\it{symbols\/}})\index{form-z-if}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-list-case \\ \index{a-ev-defn-part-ok-proof-proves-!list-case}
\=\+(\=\+({\it{flg\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $=$ {\tt{1}}$\;\;${\bf then }$\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp} \\ 
{\bf else }$\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;${\bf  endif}\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-nlist-case \\ \index{a-ev-defn-part-ok-proof-proves-!nlist-case}
\=\+(\=\+({\it{flg\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{exp\/}} $\simeq\;${{\bf{nil}}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $=$ {\tt{1}}$\;\;${\bf then }$\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp} \\ 
{\bf else }$\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;${\bf  endif}\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ncode-nequal-proof-proves-case \\ \index{ncode-nequal-proof-proves-case}
\=\+(\=\+({\tt{0}} $<$ {\it{y\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\not=$ {\it{y\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}), \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;$({\it{x-bar\/}} $=$ {\rm{ncode}}\,({\it{x\/}})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{y-bar\/}} $=$ {\rm{ncode}}\,({\it{y\/}})\index{ncode}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ncode-nequal-proof}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-nequal-proof}, \\ 
{\rm{f-not}}\,({\rm{f-equal}}\,({\it{x-bar\/}}, {\it{y-bar\/}})\index{f-equal})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-if-case1 \\ \index{a-ev-defn-part-ok-proof-proves-i!f-case1}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;${\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal} \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-if-case2 \\ \index{a-ev-defn-part-ok-proof-proves-i!f-case2}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal}) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-subr-case \\ \index{a-ev-defn-part-ok-proof-proves-!subr-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;${\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-fun-case \\ \index{a-ev-defn-part-ok-proof-proves-!fun-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}}))) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-ok-proof-proves-var-case \\ \index{a-ev-defn-part-ok-proof-proves-v!ar-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{exp\/}} $\simeq\;${{\bf{nil}}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $=$ {\tt{1}}$\;\;${\bf then }$\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp} \\ 
{\bf else }$\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;${\bf  endif}\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-var-list \\ \index{collect-free-var-list}
\=\+{\rm{var-list}}\,({\it{vlist\/}}, {\rm{length}}\,({\it{vlist\/}})\index{length})\index{var-list} \\ 
$\rightarrow$$\;\;\;\;$({\rm{collect-free}}\,({\it{vlist\/}}, {\tt{1}})\index{collect-free} $=$ {\rm{append}}\,({\it{vlist\/}}, {\bf{nil}}))\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{graph-trans-append-proof} \\  
\=\+{\rm{graph-trans-append-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{graph-trans-append-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }{\rm{graph-trans-merge-proof}}\,(\=\+{\rm{car}}\,({\it{y\/}}), \\ 
{\rm{append}}\,({\rm{cdr}}\,({\it{y\/}}), {\it{z\/}}), \\ 
{\it{x\/}}, \\ 
{\it{exp\/}}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\index{f-equal}, \\ 
{\it{exp\/}}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\it{x\/}})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\it{x\/}})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{symmetry-proof1}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{symmetry-proof1}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{ident-axiom-proof}, \\ 
{\rm{in-graph-proof}}\,(\=\+{\rm{car}}\,({\it{y\/}}), \\ 
{\it{y\/}})\-\index{in-graph-proof}, \\ 
{\rm{equal-axiom2-proof}}\,(\=\+{\rm{p}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-\index{p}, \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-, \\ 
{\rm{list}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-)\-\index{equal-axiom2-proof})\-)\-\index{tautconseq-proof}, \\ 
{\it{pf1\/}})\-)\-\index{chain-proof}, \\ 
{\rm{graph-trans-append-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{cdr}}\,({\it{y\/}}), \\ 
{\it{z\/}}, \\ 
{\it{exp\/}}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{y\/}}))\index{graph-trans})\-\index{isin}, \\ 
{\it{exp\/}}, \\ 
{\rm{list}}\,({\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{y\/}}))\index{graph-trans})\-\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{y\/}}))\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{iff-split-proof2}}\,(\=\+{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{y\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{y\/}}))\index{graph-trans})\-\index{isin})\-\index{f-or}, \\ 
{\rm{in-graph-trans-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{car}}\,({\it{y\/}}), \\ 
{\rm{cdr}}\,({\it{y\/}}))\-\index{in-graph-trans-proof})\-\index{iff-split-proof2}))\-\index{tautconseq-proof}, \\ 
{\it{pf1\/}})\-)\-\index{chain-proof}, \\ 
{\it{pf2\/}})\-\index{graph-trans-append-proof})\-\index{graph-trans-merge-proof} \\ 
{\bf else }{\it{pf2\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  graph-trans-append-proof-proves \\ \index{graph-trans-append-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}, {\it{exp\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{z\/}})\index{graph-trans})\index{isin}, {\it{exp\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $=$ {\rm{graph-trans}}\,({\rm{append}}\,({\it{y\/}}, {\it{z\/}}))\index{graph-trans}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{graph-trans-append-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{exp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{graph-trans-append-proof}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{x\/}}, {\it{trace\/}})\index{isin}, {\it{exp\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable graph-trans-append-proof; name this event `g0888'.\index{g0888}

\addvspace{18pt}
\begin{verbatim};;;;;;;Since we will be manipulating large if-trees with expressions containing
;;;;;;; "in", "forall", "^", "->", and so on, so we introduce symbols (shells) for them.
\end{verbatim}
\noindent{\sc Event}:   Add the shell {\it{s-isin\/}}\index{s-isin}, with recognizer function symbol {\it{s-isinp\/}} and 2 accessors: {\it{s-elem\/}}, with type restriction (none-of) and default value zero; {\it{s-set\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}

\noindent{\sc Event}:   Add the shell {\it{s-and\/}}\index{s-and}, with recognizer function symbol {\it{s-andp\/}} and 2 accessors: {\it{s-conj1\/}}, with type restriction (none-of) and default value zero; {\it{s-conj2\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}

\noindent{\sc Event}:   Add the shell {\it{s-implies\/}}\index{s-implies}, with recognizer function symbol {\it{s-impliesp\/}} and 2 accessors: {\it{s-prem\/}}, with type restriction (none-of) and default value zero; {\it{s-conc\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}

\noindent{\sc Event}:   Add the shell {\it{s-forall\/}}\index{s-forall}, with recognizer function symbol {\it{s-forallp\/}} and 2 accessors: {\it{s-bind\/}}, with type restriction (one-of numberp) and default value zero; {\it{s-body\/}}, with type restriction (none-of) and default value zero.

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{weaken-isin} \\  
\=\+{\rm{weaken-isin}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{form1\/}})\index{z-ifp} $\wedge$ {\rm{z-ifp}}\,({\it{form2\/}})\index{z-ifp} \\ 
{\bf then }\=\+({\rm{z-test}}\,({\it{form1\/}})\index{z-test} $=$ {\rm{z-test}}\,({\it{form2\/}})\index{z-test}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\rm{z-left}}\,({\it{form1\/}})\index{z-left}, {\rm{z-left}}\,({\it{form2\/}})\index{z-left}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\rm{z-right}}\,({\it{form1\/}})\index{z-right}, {\rm{z-right}}\,({\it{form2\/}})\index{z-right}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{form1\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{form2\/}})\index{forsomep} \\ 
{\bf then }\=\+({\rm{bind}}\,({\it{form1\/}})\index{bind} $=$ {\rm{bind}}\,({\it{form2\/}})\index{bind}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\rm{body}}\,({\it{form1\/}}), {\rm{body}}\,({\it{form2\/}}), {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin}\- \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{form1\/}})\index{s-forallp} $\wedge$ {\rm{s-forallp}}\,({\it{form2\/}})\index{s-forallp} \\ 
{\bf then }\=\+({\rm{s-bind}}\,({\it{form1\/}})\index{s-bind} $=$ {\rm{s-bind}}\,({\it{form2\/}})\index{s-bind}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\rm{s-body}}\,({\it{form1\/}})\index{s-body}, {\rm{s-body}}\,({\it{form2\/}})\index{s-body}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin}\- \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{form1\/}})\index{s-andp} $\wedge$ {\rm{s-andp}}\,({\it{form2\/}})\index{s-andp} \\ 
{\bf then }\=\+{\rm{weaken-isin}}\,({\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1}, {\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2}, {\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin}\- \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{form1\/}})\index{s-impliesp} $\wedge$ {\rm{s-impliesp}}\,({\it{form2\/}})\index{s-impliesp} \\ 
{\bf then }\=\+({\rm{s-prem}}\,({\it{form1\/}})\index{s-prem} $=$ {\rm{s-prem}}\,({\it{form2\/}})\index{s-prem}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\rm{s-conc}}\,({\it{form1\/}})\index{s-conc}, {\rm{s-conc}}\,({\it{form2\/}})\index{s-conc}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin}\- \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{form1\/}})\index{s-isinp} $\wedge$ {\rm{s-isinp}}\,({\it{form2\/}})\index{s-isinp} \\ 
{\bf then }\=\+({\rm{s-elem}}\,({\it{form1\/}})\index{s-elem} $=$ {\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}) \\ 
$\wedge$$\;\;\;\;$({\rm{s-set}}\,({\it{form1\/}})\index{s-set} $=$ {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans}) \\ 
$\wedge$$\;\;\;\;$({\rm{s-set}}\,({\it{form2\/}})\index{s-set} $=$ {\rm{graph-trans}}\,({\rm{cons}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\- \\ 
{\bf else }{\it{form1\/}} $=$ {\it{form2\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{make-form-symb} \\  
\=\+{\rm{make-form-symb}}\,({\it{exp\/}})\index{make-form-symb} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }{\rm{f-if}}\,(\=\+{\rm{z-test}}\,({\it{exp\/}})\index{z-test}, \\ 
{\rm{make-form-symb}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right})\index{make-form-symb})\-\index{f-if} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep} \\ 
{\bf then }{\rm{forsome}}\,({\rm{bind}}\,({\it{exp\/}})\index{bind}, {\rm{make-form-symb}}\,({\rm{body}}\,({\it{exp\/}}))\index{make-form-symb})\index{forsome} \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{exp\/}})\index{s-forallp}$\;\;${\bf then }$\forall\;${\it{s-bind\/}}, {\it{exp\/}}$\;${\rm{make-form-symb}}\,({\rm{s-body}}\,({\it{exp\/}})\index{s-body})\index{make-form-symb}\index{forall} \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{exp\/}})\index{s-andp} \\ 
{\bf then }{\rm{f-and}}\,({\rm{make-form-symb}}\,({\rm{s-conj1}}\,({\it{exp\/}})\index{s-conj1})\index{make-form-symb}, {\rm{make-form-symb}}\,({\rm{s-conj2}}\,({\it{exp\/}})\index{s-conj2})\index{make-form-symb})\index{f-and} \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{exp\/}})\index{s-impliesp} \\ 
{\bf then }{\rm{f-implies}}\,({\rm{s-prem}}\,({\it{exp\/}})\index{s-prem}, {\rm{make-form-symb}}\,({\rm{s-conc}}\,({\it{exp\/}})\index{s-conc})\index{make-form-symb})\index{f-implies} \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{exp\/}})\index{s-isinp}$\;\;${\bf then }{\rm{isin}}\,({\rm{s-elem}}\,({\it{exp\/}})\index{s-elem}, {\rm{s-set}}\,({\it{exp\/}})\index{s-set})\index{isin} \\ 
{\bf else }{\it{exp\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{weaken-isin-proof} \\  
\=\+{\rm{weaken-isin-proof}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{form1\/}})\index{z-ifp} $\wedge$ {\rm{z-ifp}}\,({\it{form2\/}})\index{z-ifp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{z-left}}\,({\it{form1\/}})\index{z-left})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{z-left}}\,({\it{form2\/}})\index{z-left})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{z-right}}\,({\it{form1\/}})\index{z-right})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{z-right}}\,({\it{form2\/}})\index{z-right})\index{make-form-symb})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{weaken-isin-proof}}\,(\=\+{\rm{z-left}}\,({\it{form1\/}})\index{z-left}, \\ 
{\rm{z-left}}\,({\it{form2\/}})\index{z-left}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-isin-proof}, \\ 
{\rm{weaken-isin-proof}}\,(\=\+{\rm{z-right}}\,({\it{form1\/}})\index{z-right}, \\ 
{\rm{z-right}}\,({\it{form2\/}})\index{z-right}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-isin-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{form1\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{form2\/}})\index{forsomep} \\ 
{\bf then }{\rm{distrib-rule-proof-a}}\,(\=\+{\rm{bind}}\,({\it{form2\/}})\index{bind}, \\ 
{\rm{make-form-symb}}\,({\rm{body}}\,({\it{form1\/}}))\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{body}}\,({\it{form2\/}}))\index{make-form-symb}, \\ 
{\rm{weaken-isin-proof}}\,(\=\+{\rm{body}}\,({\it{form1\/}}), \\ 
{\rm{body}}\,({\it{form2\/}}), \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-isin-proof})\-\index{distrib-rule-proof-a} \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{form1\/}})\index{s-forallp} $\wedge$ {\rm{s-forallp}}\,({\it{form2\/}})\index{s-forallp} \\ 
{\bf then }{\rm{distrib-rule-proof-b}}\,(\=\+{\rm{s-bind}}\,({\it{form2\/}})\index{s-bind}, \\ 
{\rm{make-form-symb}}\,({\rm{s-body}}\,({\it{form1\/}})\index{s-body})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-body}}\,({\it{form2\/}})\index{s-body})\index{make-form-symb}, \\ 
{\rm{weaken-isin-proof}}\,(\=\+{\rm{s-body}}\,({\it{form1\/}})\index{s-body}, \\ 
{\rm{s-body}}\,({\it{form2\/}})\index{s-body}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-isin-proof})\-\index{distrib-rule-proof-b} \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{form1\/}})\index{s-andp} $\wedge$ {\rm{s-andp}}\,({\it{form2\/}})\index{s-andp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2})\index{make-form-symb})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{weaken-isin-proof}}\,(\=\+{\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1}, \\ 
{\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-isin-proof}, \\ 
{\rm{weaken-isin-proof}}\,(\=\+{\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2}, \\ 
{\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-isin-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{form1\/}})\index{s-impliesp} $\wedge$ {\rm{s-impliesp}}\,({\it{form2\/}})\index{s-impliesp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conc}}\,({\it{form1\/}})\index{s-conc})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conc}}\,({\it{form2\/}})\index{s-conc})\index{make-form-symb})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{weaken-isin-proof}}\,(\=\+{\rm{s-conc}}\,({\it{form1\/}})\index{s-conc}, \\ 
{\rm{s-conc}}\,({\it{form2\/}})\index{s-conc}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-isin-proof}))\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{form1\/}})\index{s-isinp} $\wedge$ {\rm{s-isinp}}\,({\it{form2\/}})\index{s-isinp} \\ 
{\bf then }{\rm{in-graph-cdr-proof}}\,({\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, {\it{gr1\/}}, {\it{gr2\/}})\index{in-graph-cdr-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-implies}}\,({\it{form1\/}}, {\it{form2\/}})\index{f-implies}, {\bf{nil}})\index{tautconseq-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{form-symb1} \\  
\=\+{\rm{form-symb1}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{exp\/}})\index{z-ifp} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{z-test}}\,({\it{exp\/}})\index{z-test}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\rm{z-left}}\,({\it{exp\/}})\index{z-left}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\rm{z-right}}\,({\it{exp\/}})\index{z-right}, {\it{symbols\/}})\index{form-symb1}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep}$\;\;${\bf then }{\rm{form-symb1}}\,({\rm{body}}\,({\it{exp\/}}), {\it{symbols\/}})\index{form-symb1} \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{exp\/}})\index{s-forallp}$\;\;${\bf then }{\rm{form-symb1}}\,({\rm{s-body}}\,({\it{exp\/}})\index{s-body}, {\it{symbols\/}})\index{form-symb1} \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{exp\/}})\index{s-andp} \\ 
{\bf then }\=\+{\rm{form-symb1}}\,({\rm{s-conj1}}\,({\it{exp\/}})\index{s-conj1}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\rm{s-conj2}}\,({\it{exp\/}})\index{s-conj2}, {\it{symbols\/}})\index{form-symb1}\- \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{exp\/}})\index{s-impliesp} \\ 
{\bf then }\=\+{\rm{formula}}\,({\rm{s-prem}}\,({\it{exp\/}})\index{s-prem}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\rm{s-conc}}\,({\it{exp\/}})\index{s-conc}, {\it{symbols\/}})\index{form-symb1}\- \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{exp\/}})\index{s-isinp} \\ 
{\bf then }\=\+({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{s-elem}}\,({\it{exp\/}})\index{s-elem}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{s-set}}\,({\it{exp\/}})\index{s-set}, {\tt{0}}, {\it{symbols\/}})\index{termp}\- \\ 
{\bf else }{\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-make-form-symb \\ \index{formula-make-form-symb}
{\rm{form-symb1}}\,({\it{exp\/}}, {\it{symbols\/}})\index{form-symb1} $\rightarrow$ {\rm{formula}}\,({\rm{make-form-symb}}\,({\it{exp\/}})\index{make-form-symb}, {\it{symbols\/}})\index{formula}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  weaken-isin-proof-proves \\ \index{weaken-isin-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{form1\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{form2\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, {\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{weaken-isin-proof}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-isin-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable weaken-isin-proof; name this event `g0859'.\index{g0859}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-append-graph-proof1} \\  
\=\+{\rm{in-append-graph-proof1}}\,({\it{x\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{in-append-graph-proof1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{gr1\/}}) \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{ncode}}\,({\rm{car}}\,({\it{gr1\/}}))\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{gr1\/}}))\index{graph-trans})\-\index{isin})\-\index{f-or})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{x\/}}, {\rm{ncode}}\,({\rm{car}}\,({\it{gr1\/}}))\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{cdr}}\,({\it{gr1\/}}))\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\-\index{isin})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{iff-split-proof1}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{gr1\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cdr}}\,({\it{gr1\/}}))\index{graph-trans})\-\index{isin})\-\index{f-or}, \\ 
{\rm{in-graph-trans-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{car}}\,({\it{gr1\/}}), \\ 
{\rm{cdr}}\,({\it{gr1\/}}))\-\index{in-graph-trans-proof})\-\index{iff-split-proof1}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{gr1\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\index{graph-trans})\-\index{isin})\-\index{f-or}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,(\=\+{\rm{car}}\,({\it{gr1\/}}), \\ 
{\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\-)\index{graph-trans})\-\index{isin})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{gr1\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,(\=\+{\rm{car}}\,({\it{gr1\/}}), \\ 
{\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\-)\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{iff-split-proof2}}\,(\=\+{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,(\=\+{\rm{car}}\,({\it{gr1\/}}), \\ 
{\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\-)\index{graph-trans})\-\index{isin}, \\ 
{\rm{f-or}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{gr1\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\index{graph-trans})\-\index{isin})\-\index{f-or}, \\ 
{\rm{in-graph-trans-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{car}}\,({\it{gr1\/}}), \\ 
{\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\-\index{in-graph-trans-proof})\-\index{iff-split-proof2}))\-\index{tautconseq-proof}, \\ 
{\rm{in-append-graph-proof1}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{gr1\/}}), {\it{gr2\/}})\index{in-append-graph-proof1}, \\ 
{\rm{in-graph-cdr-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{car}}\,({\it{gr1\/}}), \\ 
{\rm{append}}\,({\rm{cdr}}\,({\it{gr1\/}}), {\it{gr2\/}}))\-\index{in-graph-cdr-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin})\index{f-not}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{null-axiom-proof}}\,({\it{x\/}})\index{null-axiom-proof}))\-\index{tautconseq-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;Let "(X|Y)" denote (APPEND X Y).  Then |- "((x in Y) -> (x in (Y|Z)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-append-graph-proof1-proves \\ \index{in-append-graph-proof1-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{in-append-graph-proof1}}\,({\it{x\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{in-append-graph-proof1}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\index{isin})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-append-graph-proof1; name this event `g0860'.\index{g0860}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{in-append-graph-proof2} \\  
\=\+{\rm{in-append-graph-proof2}}\,({\it{x\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{in-append-graph-proof2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{gr1\/}}) \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,(\=\+{\rm{cdr}}\,({\it{gr1\/}}), \\ 
{\it{gr2\/}})\-)\index{graph-trans})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\-\index{isin})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\index{isin})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{in-append-graph-proof2}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{gr1\/}}), {\it{gr2\/}})\index{in-append-graph-proof2}, \\ 
{\rm{in-graph-cdr-proof}}\,(\=\+{\it{x\/}}, \\ 
{\rm{car}}\,({\it{gr1\/}}), \\ 
{\rm{append}}\,({\rm{cdr}}\,({\it{gr1\/}}), {\it{gr2\/}}))\-\index{in-graph-cdr-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\it{x\/}}, \\ 
{\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\-\index{isin})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;; |- "((x in Z) -> (x in (Y|Z)))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  in-append-graph-proof2-proves \\ \index{in-append-graph-proof2-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{in-append-graph-proof2}}\,({\it{x\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{in-append-graph-proof2}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,({\it{x\/}}, {\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\index{isin})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable in-append-graph-proof2; name this event `g0861'.\index{g0861}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{weaken-append1} \\  
\=\+{\rm{weaken-append1}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{form1\/}})\index{z-ifp} $\wedge$ {\rm{z-ifp}}\,({\it{form2\/}})\index{z-ifp} \\ 
{\bf then }\=\+({\rm{z-test}}\,({\it{form1\/}})\index{z-test} $=$ {\rm{z-test}}\,({\it{form2\/}})\index{z-test}) \\ 
$\wedge$$\;\;\;\;$({\rm{z-test}}\,({\it{form2\/}})\index{z-test} $=$ {\rm{z-test}}\,({\it{form2\/}})\index{z-test}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\rm{z-left}}\,({\it{form1\/}})\index{z-left}, {\rm{z-left}}\,({\it{form2\/}})\index{z-left}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\rm{z-right}}\,({\it{form1\/}})\index{z-right}, {\rm{z-right}}\,({\it{form2\/}})\index{z-right}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{form1\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{form2\/}})\index{forsomep} \\ 
{\bf then }\=\+({\rm{bind}}\,({\it{form1\/}})\index{bind} $=$ {\rm{bind}}\,({\it{form2\/}})\index{bind}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\rm{body}}\,({\it{form1\/}}), {\rm{body}}\,({\it{form2\/}}), {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1}\- \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{form1\/}})\index{s-forallp} $\wedge$ {\rm{s-forallp}}\,({\it{form2\/}})\index{s-forallp} \\ 
{\bf then }\=\+({\rm{s-bind}}\,({\it{form1\/}})\index{s-bind} $=$ {\rm{s-bind}}\,({\it{form2\/}})\index{s-bind}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\rm{s-body}}\,({\it{form1\/}})\index{s-body}, {\rm{s-body}}\,({\it{form2\/}})\index{s-body}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1}\- \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{form1\/}})\index{s-andp} $\wedge$ {\rm{s-andp}}\,({\it{form2\/}})\index{s-andp} \\ 
{\bf then }\=\+{\rm{weaken-append1}}\,({\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1}, {\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2}, {\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1}\- \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{form1\/}})\index{s-impliesp} $\wedge$ {\rm{s-impliesp}}\,({\it{form2\/}})\index{s-impliesp} \\ 
{\bf then }\=\+({\rm{s-prem}}\,({\it{form1\/}})\index{s-prem} $=$ {\rm{s-prem}}\,({\it{form2\/}})\index{s-prem}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\rm{s-conc}}\,({\it{form1\/}})\index{s-conc}, {\rm{s-conc}}\,({\it{form2\/}})\index{s-conc}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1}\- \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{form1\/}})\index{s-isinp} $\wedge$ {\rm{s-isinp}}\,({\it{form2\/}})\index{s-isinp} \\ 
{\bf then }\=\+({\rm{s-elem}}\,({\it{form1\/}})\index{s-elem} $=$ {\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}) \\ 
$\wedge$$\;\;\;\;$({\rm{s-set}}\,({\it{form1\/}})\index{s-set} $=$ {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans}) \\ 
$\wedge$$\;\;\;\;$({\rm{s-set}}\,({\it{form2\/}})\index{s-set} $=$ {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}})))\index{graph-trans})\- \\ 
{\bf else }{\it{form1\/}} $=$ {\it{form2\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{weaken-append2} \\  
\=\+{\rm{weaken-append2}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{form1\/}})\index{z-ifp} $\wedge$ {\rm{z-ifp}}\,({\it{form2\/}})\index{z-ifp} \\ 
{\bf then }\=\+({\rm{z-test}}\,({\it{form1\/}})\index{z-test} $=$ {\rm{z-test}}\,({\it{form2\/}})\index{z-test}) \\ 
$\wedge$$\;\;\;\;$({\rm{z-test}}\,({\it{form2\/}})\index{z-test} $=$ {\rm{z-test}}\,({\it{form2\/}})\index{z-test}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\rm{z-left}}\,({\it{form1\/}})\index{z-left}, {\rm{z-left}}\,({\it{form2\/}})\index{z-left}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\rm{z-right}}\,({\it{form1\/}})\index{z-right}, {\rm{z-right}}\,({\it{form2\/}})\index{z-right}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{form1\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{form2\/}})\index{forsomep} \\ 
{\bf then }\=\+({\rm{bind}}\,({\it{form1\/}})\index{bind} $=$ {\rm{bind}}\,({\it{form2\/}})\index{bind}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\rm{body}}\,({\it{form1\/}}), {\rm{body}}\,({\it{form2\/}}), {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2}\- \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{form1\/}})\index{s-forallp} $\wedge$ {\rm{s-forallp}}\,({\it{form2\/}})\index{s-forallp} \\ 
{\bf then }\=\+({\rm{s-bind}}\,({\it{form1\/}})\index{s-bind} $=$ {\rm{s-bind}}\,({\it{form2\/}})\index{s-bind}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\rm{s-body}}\,({\it{form1\/}})\index{s-body}, {\rm{s-body}}\,({\it{form2\/}})\index{s-body}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2}\- \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{form1\/}})\index{s-andp} $\wedge$ {\rm{s-andp}}\,({\it{form2\/}})\index{s-andp} \\ 
{\bf then }\=\+{\rm{weaken-append2}}\,({\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1}, {\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2}, {\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2}\- \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{form1\/}})\index{s-impliesp} $\wedge$ {\rm{s-impliesp}}\,({\it{form2\/}})\index{s-impliesp} \\ 
{\bf then }\=\+({\rm{s-prem}}\,({\it{form1\/}})\index{s-prem} $=$ {\rm{s-prem}}\,({\it{form2\/}})\index{s-prem}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\rm{s-conc}}\,({\it{form1\/}})\index{s-conc}, {\rm{s-conc}}\,({\it{form2\/}})\index{s-conc}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2}\- \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{form1\/}})\index{s-isinp} $\wedge$ {\rm{s-isinp}}\,({\it{form2\/}})\index{s-isinp} \\ 
{\bf then }\=\+({\rm{s-elem}}\,({\it{form1\/}})\index{s-elem} $=$ {\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}) \\ 
$\wedge$$\;\;\;\;$({\rm{s-set}}\,({\it{form1\/}})\index{s-set} $=$ {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans}) \\ 
$\wedge$$\;\;\;\;$({\rm{s-set}}\,({\it{form2\/}})\index{s-set} $=$ {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}})))\index{graph-trans})\- \\ 
{\bf else }{\it{form1\/}} $=$ {\it{form2\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{weaken-append1-proof} \\  
\=\+{\rm{weaken-append1-proof}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{form1\/}})\index{z-ifp} $\wedge$ {\rm{z-ifp}}\,({\it{form2\/}})\index{z-ifp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{z-left}}\,({\it{form1\/}})\index{z-left})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{z-left}}\,({\it{form2\/}})\index{z-left})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{z-right}}\,({\it{form1\/}})\index{z-right})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{z-right}}\,({\it{form2\/}})\index{z-right})\index{make-form-symb})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{weaken-append1-proof}}\,(\=\+{\rm{z-left}}\,({\it{form1\/}})\index{z-left}, \\ 
{\rm{z-left}}\,({\it{form2\/}})\index{z-left}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof}, \\ 
{\rm{weaken-append1-proof}}\,(\=\+{\rm{z-right}}\,({\it{form1\/}})\index{z-right}, \\ 
{\rm{z-right}}\,({\it{form2\/}})\index{z-right}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{form1\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{form2\/}})\index{forsomep} \\ 
{\bf then }{\rm{distrib-rule-proof-a}}\,(\=\+{\rm{bind}}\,({\it{form2\/}})\index{bind}, \\ 
{\rm{make-form-symb}}\,({\rm{body}}\,({\it{form1\/}}))\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{body}}\,({\it{form2\/}}))\index{make-form-symb}, \\ 
{\rm{weaken-append1-proof}}\,(\=\+{\rm{body}}\,({\it{form1\/}}), \\ 
{\rm{body}}\,({\it{form2\/}}), \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof})\-\index{distrib-rule-proof-a} \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{form1\/}})\index{s-forallp} $\wedge$ {\rm{s-forallp}}\,({\it{form2\/}})\index{s-forallp} \\ 
{\bf then }{\rm{distrib-rule-proof-b}}\,(\=\+{\rm{s-bind}}\,({\it{form2\/}})\index{s-bind}, \\ 
{\rm{make-form-symb}}\,({\rm{s-body}}\,({\it{form1\/}})\index{s-body})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-body}}\,({\it{form2\/}})\index{s-body})\index{make-form-symb}, \\ 
{\rm{weaken-append1-proof}}\,(\=\+{\rm{s-body}}\,({\it{form1\/}})\index{s-body}, \\ 
{\rm{s-body}}\,({\it{form2\/}})\index{s-body}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof})\-\index{distrib-rule-proof-b} \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{form1\/}})\index{s-andp} $\wedge$ {\rm{s-andp}}\,({\it{form2\/}})\index{s-andp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2})\index{make-form-symb})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{weaken-append1-proof}}\,(\=\+{\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1}, \\ 
{\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof}, \\ 
{\rm{weaken-append1-proof}}\,(\=\+{\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2}, \\ 
{\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{form1\/}})\index{s-impliesp} $\wedge$ {\rm{s-impliesp}}\,({\it{form2\/}})\index{s-impliesp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conc}}\,({\it{form1\/}})\index{s-conc})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conc}}\,({\it{form2\/}})\index{s-conc})\index{make-form-symb})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{weaken-append1-proof}}\,(\=\+{\rm{s-conc}}\,({\it{form1\/}})\index{s-conc}, \\ 
{\rm{s-conc}}\,({\it{form2\/}})\index{s-conc}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof}))\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{form1\/}})\index{s-isinp} $\wedge$ {\rm{s-isinp}}\,({\it{form2\/}})\index{s-isinp} \\ 
{\bf then }{\rm{chain-proof}}\,(\=\+{\rm{isin}}\,({\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}})))\index{graph-trans})\-\index{isin}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, {\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{in-append-graph-proof1}}\,({\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, {\it{gr1\/}}, {\it{gr2\/}})\index{in-append-graph-proof1}, \\ 
{\rm{in-graph-cdr-proof}}\,(\=\+{\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, \\ 
{\it{y\/}}, \\ 
{\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\-\index{in-graph-cdr-proof})\-)\-\index{chain-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-implies}}\,({\it{form1\/}}, {\it{form2\/}})\index{f-implies}, {\bf{nil}})\index{tautconseq-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{weaken-append2-proof} \\  
\=\+{\rm{weaken-append2-proof}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{z-ifp}}\,({\it{form1\/}})\index{z-ifp} $\wedge$ {\rm{z-ifp}}\,({\it{form2\/}})\index{z-ifp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{z-left}}\,({\it{form1\/}})\index{z-left})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{z-left}}\,({\it{form2\/}})\index{z-left})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{z-right}}\,({\it{form1\/}})\index{z-right})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{z-right}}\,({\it{form2\/}})\index{z-right})\index{make-form-symb})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{weaken-append2-proof}}\,(\=\+{\rm{z-left}}\,({\it{form1\/}})\index{z-left}, \\ 
{\rm{z-left}}\,({\it{form2\/}})\index{z-left}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof}, \\ 
{\rm{weaken-append2-proof}}\,(\=\+{\rm{z-right}}\,({\it{form1\/}})\index{z-right}, \\ 
{\rm{z-right}}\,({\it{form2\/}})\index{z-right}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{form1\/}})\index{forsomep} $\wedge$ {\rm{forsomep}}\,({\it{form2\/}})\index{forsomep} \\ 
{\bf then }{\rm{distrib-rule-proof-a}}\,(\=\+{\rm{bind}}\,({\it{form2\/}})\index{bind}, \\ 
{\rm{make-form-symb}}\,({\rm{body}}\,({\it{form1\/}}))\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{body}}\,({\it{form2\/}}))\index{make-form-symb}, \\ 
{\rm{weaken-append2-proof}}\,(\=\+{\rm{body}}\,({\it{form1\/}}), \\ 
{\rm{body}}\,({\it{form2\/}}), \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof})\-\index{distrib-rule-proof-a} \\ 
{\bf elseif }{\rm{s-forallp}}\,({\it{form1\/}})\index{s-forallp} $\wedge$ {\rm{s-forallp}}\,({\it{form2\/}})\index{s-forallp} \\ 
{\bf then }{\rm{distrib-rule-proof-b}}\,(\=\+{\rm{s-bind}}\,({\it{form2\/}})\index{s-bind}, \\ 
{\rm{make-form-symb}}\,({\rm{s-body}}\,({\it{form1\/}})\index{s-body})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-body}}\,({\it{form2\/}})\index{s-body})\index{make-form-symb}, \\ 
{\rm{weaken-append2-proof}}\,(\=\+{\rm{s-body}}\,({\it{form1\/}})\index{s-body}, \\ 
{\rm{s-body}}\,({\it{form2\/}})\index{s-body}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof})\-\index{distrib-rule-proof-b} \\ 
{\bf elseif }{\rm{s-andp}}\,({\it{form1\/}})\index{s-andp} $\wedge$ {\rm{s-andp}}\,({\it{form2\/}})\index{s-andp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2})\index{make-form-symb})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{weaken-append2-proof}}\,(\=\+{\rm{s-conj1}}\,({\it{form1\/}})\index{s-conj1}, \\ 
{\rm{s-conj1}}\,({\it{form2\/}})\index{s-conj1}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof}, \\ 
{\rm{weaken-append2-proof}}\,(\=\+{\rm{s-conj2}}\,({\it{form1\/}})\index{s-conj2}, \\ 
{\rm{s-conj2}}\,({\it{form2\/}})\index{s-conj2}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof})\-)\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{s-impliesp}}\,({\it{form1\/}})\index{s-impliesp} $\wedge$ {\rm{s-impliesp}}\,({\it{form2\/}})\index{s-impliesp} \\ 
{\bf then }{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\rm{s-conc}}\,({\it{form1\/}})\index{s-conc})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\rm{s-conc}}\,({\it{form2\/}})\index{s-conc})\index{make-form-symb})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, \\ 
{\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{weaken-append2-proof}}\,(\=\+{\rm{s-conc}}\,({\it{form1\/}})\index{s-conc}, \\ 
{\rm{s-conc}}\,({\it{form2\/}})\index{s-conc}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof}))\-\index{tautconseq-proof} \\ 
{\bf elseif }{\rm{s-isinp}}\,({\it{form1\/}})\index{s-isinp} $\wedge$ {\rm{s-isinp}}\,({\it{form2\/}})\index{s-isinp} \\ 
{\bf then }{\rm{chain-proof}}\,(\=\+{\rm{isin}}\,({\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}})))\index{graph-trans})\-\index{isin}, \\ 
{\rm{list}}\,({\rm{isin}}\,({\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, {\rm{graph-trans}}\,({\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\index{graph-trans})\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{in-append-graph-proof2}}\,({\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, {\it{gr1\/}}, {\it{gr2\/}})\index{in-append-graph-proof2}, \\ 
{\rm{in-graph-cdr-proof}}\,(\=\+{\rm{s-elem}}\,({\it{form2\/}})\index{s-elem}, \\ 
{\it{y\/}}, \\ 
{\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))\-\index{in-graph-cdr-proof})\-)\-\index{chain-proof} \\ 
{\bf else }{\rm{tautconseq-proof}}\,({\bf{nil}}, {\rm{f-implies}}\,({\it{form1\/}}, {\it{form2\/}})\index{f-implies}, {\bf{nil}})\index{tautconseq-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  weaken-append1-proof-proves \\ \index{weaken-append1-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{form1\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{form2\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, {\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{weaken-append1-proof}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable weaken-append1-proof; name this event `g0862'.\index{g0862}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  weaken-append2-proof-proves \\ \index{weaken-append2-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{form1\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{form2\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,({\rm{make-form-symb}}\,({\it{form1\/}})\index{make-form-symb}, {\rm{make-form-symb}}\,({\it{form2\/}})\index{make-form-symb})\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{weaken-append2-proof}}\,({\it{form1\/}}, {\it{form2\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable weaken-append2-proof; name this event `g0863'.\index{g0863}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-ok-frame-proof2} \\  
\=\+{\rm{a-ev-defn-ok-frame-proof2}}\,(\=\+{\it{formals\/}}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}}, \\ 
{\it{defn-part\/}}, \\ 
{\it{defn-part-form\/}}, \\ 
{\it{rec1-defn-part\/}}, \\ 
{\it{rec1-form\/}}, \\ 
{\it{rec2-defn-part\/}}, \\ 
{\it{rec2-form\/}}, \\ 
{\it{defn-pf\/}}, \\ 
{\it{recpf1\/}}, \\ 
{\it{recpf2\/}})\-\index{a-ev-defn-ok-frame-proof2} \\ 
$=$$\;\;\;\;${\rm{graph-trans-merge-proof}}\,(\=\+{\it{y\/}}, \\ 
{\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}), \\ 
{\it{formals\/}}, \\ 
{\it{defn-part\/}}, \\ 
{\it{defn-pf\/}}, \\ 
{\rm{graph-trans-append-proof}}\,(\=\+{\it{formals\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}}, \\ 
{\it{defn-part\/}}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\it{formals\/}}, \\ 
{\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{defn-part\/}}, \\ 
{\rm{list}}\,({\it{rec1-defn-part\/}}), \\ 
{\rm{list}}\,(\=\+{\it{recpf1\/}}, \\ 
{\rm{weaken-append1-proof}}\,(\=\+{\it{rec1-form\/}}, \\ 
{\it{defn-part-form\/}}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1-proof})\-)\-\index{chain-proof}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\it{formals\/}}, \\ 
{\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\-\index{isin}, \\ 
{\it{defn-part\/}}, \\ 
{\rm{list}}\,({\it{rec2-defn-part\/}}), \\ 
{\rm{list}}\,(\=\+{\it{recpf2\/}}, \\ 
{\rm{weaken-append2-proof}}\,(\=\+{\it{rec2-form\/}}, \\ 
{\it{defn-part-form\/}}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2-proof})\-)\-\index{chain-proof})\-\index{graph-trans-append-proof})\-\index{graph-trans-merge-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-ok-frame-proof2-proves \\ \index{a-ev-defn-ok-frame-proof2-prove!s}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\it{graph\/}} $=$ {\rm{cons}}\,({\it{y\/}}, {\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}}))) \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{defn-part-form\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{rec1-form\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{rec2-form\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;$({\it{defn-part\/}} $=$ {\rm{make-form-symb}}\,({\it{defn-part-form\/}})\index{make-form-symb}) \\ 
$\wedge$$\;\;\;\;$({\it{rec1-defn-part\/}} $=$ {\rm{make-form-symb}}\,({\it{rec1-form\/}})\index{make-form-symb}) \\ 
$\wedge$$\;\;\;\;$({\it{rec2-defn-part\/}} $=$ {\rm{make-form-symb}}\,({\it{rec2-form\/}})\index{make-form-symb}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append1}}\,({\it{rec1-form\/}}, {\it{defn-part-form\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append1} \\ 
$\wedge$$\;\;\;\;${\rm{weaken-append2}}\,({\it{rec2-form\/}}, {\it{defn-part-form\/}}, {\it{y\/}}, {\it{gr1\/}}, {\it{gr2\/}})\index{weaken-append2} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{defn-pf\/}}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{formals\/}}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal}, {\it{defn-part\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{recpf1\/}}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{formals\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{isin}, {\it{rec1-defn-part\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{recpf2\/}}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{formals\/}}, {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\index{isin}, {\it{rec2-defn-part\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-ok-frame-proof2}}\,(\=\+{\it{formals\/}}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}}, \\ 
{\it{defn-part\/}}, \\ 
{\it{defn-part-form\/}}, \\ 
{\it{rec1-defn-part\/}}, \\ 
{\it{rec1-form\/}}, \\ 
{\it{rec2-defn-part\/}}, \\ 
{\it{rec2-form\/}}, \\ 
{\it{defn-pf\/}}, \\ 
{\it{recpf1\/}}, \\ 
{\it{recpf2\/}})\-\index{a-ev-defn-ok-frame-proof2}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{formals\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{isin}, {\it{defn-part\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-ok-frame-proof2; name this event `g0864'.\index{g0864}

\addvspace{18pt}
%\item 
\begin{verbatim}

(DEFN
 A-EV-SYMB-FORM
 (FLG EXP VA FA VAL TRACE)
 (Z-IF
  (F-EQUAL FLG (NCODE 1))
  (Z-IF (F-EQUAL (Z-LISTP EXP) (NCODE 1))
        (FORSOME 11
                 (FORSOME 12
                          (S-AND (S-ISIN (Z-LIST (LIST 11 (NCODE 0) (Z-CAR EXP) VA FA)) TRACE)
                                 (S-AND (S-ISIN (Z-LIST (LIST 12 (NCODE 1) (Z-CDR EXP) VA FA))
                                                TRACE)
                                        (F-EQUAL VAL (Z-OPAIR 11 12))))))
        (F-EQUAL VAL (NCODE NIL)))
  (Z-IF
   (F-EQUAL (Z-LISTP EXP) (NCODE 1))
   (Z-IF
    (F-EQUAL (Z-ZEROP (Z-CAR EXP)) (NCODE 1))
    (FORSOME 11
             (S-AND (S-ISIN (Z-LIST (LIST 11 (NCODE 0) (Z-CAR (Z-CDR EXP)) VA FA)) TRACE)
                    (Z-IF (F-EQUAL 11 (NCODE 0))
                          (S-ISIN (Z-LIST (LIST VAL
                                                (NCODE 0)
                                                (Z-CAR-CDR EXP (LIST 1 2 2 2))
                                                VA
                                                FA))
                                  TRACE)
                          (S-ISIN (Z-LIST (LIST VAL
                                                (NCODE 0)
                                                (Z-CAR-CDR EXP (LIST 1 2 2))
                                                VA
                                                FA))
                                  TRACE))))
    (Z-IF
     (F-EQUAL (Z-CAR EXP) (NCODE 10))
     (F-EQUAL VAL (Z-CAR-CDR EXP (LIST 1 2)))
     (Z-IF
      (F-EQUAL (Z-SUBRP (Z-CAR EXP)) (NCODE 1))
      (FORSOME 11
               (S-AND (S-ISIN (Z-LIST (LIST 11 (NCODE 1) (Z-CDR EXP) VA FA)) TRACE)
                      (F-EQUAL VAL (Z-APPLY-SUBR (Z-CAR EXP) 11))))
      (FORSOME 11
               (S-FORALL 15
                         (S-FORALL 16
                                   (S-AND (S-ISIN (Z-LIST (LIST 11
                                                                (NCODE 1)
                                                                (Z-CDR EXP)
                                                                VA
                                                                FA))
                                                  TRACE)
                                          (S-IMPLIES (A-GET (Z-SUB1-N (Z-CAR EXP) 11)
                                                            FA
                                                            15
                                                            11
                                                            13
                                                            14
                                                            16)
                                                     (S-ISIN (Z-LIST (LIST VAL
                                                                           (NCODE 0)
                                                                           15
                                                                           11
                                                                           FA))
                                                             TRACE)))))))))
   (A-GET1 EXP VA VAL 11 13 14 16))))
\end{verbatim}

\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part1} \\  
\=\+{\rm{a-ev-defn-part1}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-defn-part1} \\ 
$=$$\;\;\;\;${\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\it{flg\/}}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-listp}}\,({\it{exp\/}})\index{z-listp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{forsome}}\,(\=\+{\tt{11}}, \\ 
{\rm{forsome}}\,(\=\+{\tt{12}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\it{exp\/}})\index{z-car}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{12}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\it{exp\/}})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-opair}}\,({\tt{11}}, {\tt{12}})\index{z-opair})\index{f-equal})\-\index{f-and})\-\index{f-and})\-\index{forsome})\-\index{forsome}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{ncode}}\,({\bf{nil}})\index{ncode})\index{f-equal})\-\index{f-if}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-listp}}\,({\it{exp\/}})\index{z-listp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-zerop}}\,({\rm{z-car}}\,({\it{exp\/}})\index{z-car})\index{z-zerop}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{forsome}}\,(\=\+{\tt{11}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\it{exp\/}})\index{z-cdr})\index{z-car}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\tt{11}}, {\rm{ncode}}\,({\tt{0}})\index{ncode})\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{f-if})\-\index{f-and})\-\index{forsome}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-car}}\,({\it{exp\/}})\index{z-car}, {\rm{ncode}}\,({\tt{10}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-equal}}\,({\it{val\/}}, {\rm{z-car-cdr}}\,({\it{exp\/}}, {\rm{list}}\,({\tt{1}}, {\tt{2}}))\index{z-car-cdr})\index{f-equal}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,({\rm{z-subrp}}\,({\rm{z-car}}\,({\it{exp\/}})\index{z-car})\index{z-subrp}, {\rm{ncode}}\,({\tt{1}})\index{ncode})\index{f-equal}, \\ 
{\rm{forsome}}\,(\=\+{\tt{11}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\it{exp\/}})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val\/}}, \\ 
{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\it{exp\/}})\index{z-car}, \\ 
{\tt{11}})\-\index{z-apply-subr})\-\index{f-equal})\-\index{f-and})\-\index{forsome}, \\ 
{\rm{forsome}}\,(\=\+{\tt{11}}, \\ 
$\forall\;${\tt{15}}$\;$$\forall\;${\tt{16}}$\;${\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\it{exp\/}})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\it{exp\/}})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\it{fa\/}}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\tt{15}}, \\ 
{\tt{11}}, \\ 
{\it{fa\/}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{f-implies})\-\index{f-and}\index{forall}\index{forall})\-\index{forsome})\-\index{f-if})\-\index{f-if})\-\index{f-if}, \\ 
{\rm{a-get1}}\,({\it{exp\/}}, {\it{va\/}}, {\it{val\/}}, {\tt{11}}, {\tt{13}}, {\tt{14}}, {\tt{16}})\index{a-get1})\-\index{f-if})\-\index{f-if}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-symb-form-a-ev-defn-part \\ \index{a-ev-symb-form-a-ev-defn-part}
\=\+{\rm{make-form-symb}}\,({\rm{a-ev-symb-form}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-symb-form})\index{make-form-symb} \\ 
$=$$\;\;\;\;${\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-defn-part}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  weaken-append2-a-ev-defn-part \\ \index{weaken-append2-a-ev-defn-part}
{\rm{weaken-append2}}\,(\=\+{\rm{a-ev-symb-form}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\rm{graph-trans}}\,({\it{gr2\/}})\index{graph-trans})\index{a-ev-symb-form}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}})))\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append2}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  weaken-append1-a-ev-defn-part \\ \index{weaken-append1-a-ev-defn-part}
{\rm{weaken-append1}}\,(\=\+{\rm{a-ev-symb-form}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\rm{graph-trans}}\,({\it{gr1\/}})\index{graph-trans})\index{a-ev-symb-form}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val\/}}, \\ 
{\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\rm{append}}\,({\it{gr1\/}}, {\it{gr2\/}})))\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\it{y\/}}, \\ 
{\it{gr1\/}}, \\ 
{\it{gr2\/}})\-\index{weaken-append1}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-inst-formals-proof} \\  
\=\+{\rm{a-ev-inst-formals-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-inst-formals-proof} \\ 
$=$$\;\;\;\;${\rm{inst-formals-proof}}\,(\=\+{\rm{list}}\,({\tt{10}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}), \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-defn-part-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-defn-part-ok-proof})\-\index{inst-formals-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-symb1-a-ev-symb-form \\ \index{form-symb1-a-ev-symb-form}
\=\+(\=\+{\rm{termp}}\,({\rm{list}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{form-symb1}}\,({\rm{a-ev-symb-form}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-symb-form}, {\it{symbols\/}})\index{form-symb1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-a-ev-defn-part1 \\ \index{formula-a-ev-defn-part1}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{a-ev-defn-part1}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-defn-part1}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-a-ev-defn-part \\ \index{formula-a-ev-defn-part}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{val\/}}, {\it{trace\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-defn-part}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-a-ev-defn-part1 \\ \index{parallel-subst-a-ev-defn-part1}
\=\+{\rm{parallel-subst}}\,(\=\+{\rm{a-ev-defn-part1}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{a-ev-defn-part1}, \\ 
{\rm{list}}\,({\tt{10}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}), \\ 
{\rm{list}}\,({\it{val\/}}, {\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\rm{ncode}}\,({\it{exp\/}})\index{ncode}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\rm{ncode}}\,({\it{fa\/}})\index{ncode}), \\ 
{\tt{0}})\-\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{a-ev-defn-part}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part1; name this event `g0865'.\index{g0865}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev-defn-part; name this event `g0866'.\index{g0866}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  termp-z-opair-case \\ \index{termp-z-opair-case}
\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{termp}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$=$$\;\;\;\;$({\rm{termp}}\,({\it{x\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} $\wedge$ {\rm{termp}}\,({\it{y\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  graph-trans-merge-again \\ \index{graph-trans-merge-again}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{exp1\/}}, {\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{f-equal}, {\it{exp2\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\it{y\/}})\index{graph-trans})\index{isin}, {\it{exp2\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{graph-trans-merge-proof}}\,({\it{x\/}}, {\it{y\/}}, {\it{exp1\/}}, {\it{exp2\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{graph-trans-merge-proof}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{exp1\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{graph-trans})\index{isin}, {\it{exp2\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable graph-trans-merge; name this event `g0867'.\index{g0867}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-ok-frame-proof3} \\  
\=\+{\rm{a-ev-defn-ok-frame-proof3}}\,(\=\+{\it{formals\/}}, \\ 
{\it{y\/}}, \\ 
{\it{gr\/}}, \\ 
{\it{defn-part\/}}, \\ 
{\it{defn-part-form\/}}, \\ 
{\it{rec-defn-part\/}}, \\ 
{\it{rec-form\/}}, \\ 
{\it{defn-pf\/}}, \\ 
{\it{recpf\/}})\-\index{a-ev-defn-ok-frame-proof3} \\ 
$=$$\;\;\;\;${\rm{graph-trans-merge-proof}}\,(\=\+{\it{y\/}}, \\ 
{\it{gr\/}}, \\ 
{\it{formals\/}}, \\ 
{\it{defn-part\/}}, \\ 
{\it{defn-pf\/}}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{isin}}\,({\it{formals\/}}, {\rm{graph-trans}}\,({\it{gr\/}})\index{graph-trans})\index{isin}, \\ 
{\it{defn-part\/}}, \\ 
{\rm{list}}\,({\it{rec-defn-part\/}}), \\ 
{\rm{list}}\,(\=\+{\it{recpf\/}}, \\ 
{\rm{weaken-isin-proof}}\,(\=\+{\it{rec-form\/}}, \\ 
{\it{defn-part-form\/}}, \\ 
{\it{y\/}}, \\ 
{\it{gr\/}})\-\index{weaken-isin-proof})\-)\-\index{chain-proof})\-\index{graph-trans-merge-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-ok-frame-proof3-proves \\ \index{a-ev-defn-ok-frame-proof3-prove!s}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\it{graph\/}} $=$ {\rm{cons}}\,({\it{y\/}}, {\it{gr\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{defn-part-form\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;${\rm{form-symb1}}\,({\it{rec-form\/}}, {\it{symbols\/}})\index{form-symb1} \\ 
$\wedge$$\;\;\;\;$({\it{defn-part\/}} $=$ {\rm{make-form-symb}}\,({\it{defn-part-form\/}})\index{make-form-symb}) \\ 
$\wedge$$\;\;\;\;$({\it{rec-defn-part\/}} $=$ {\rm{make-form-symb}}\,({\it{rec-form\/}})\index{make-form-symb}) \\ 
$\wedge$$\;\;\;\;${\rm{weaken-isin}}\,({\it{rec-form\/}}, {\it{defn-part-form\/}}, {\it{y\/}}, {\it{gr\/}})\index{weaken-isin} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{defn-pf\/}}, \\ 
{\rm{f-implies}}\,({\rm{f-equal}}\,({\it{formals\/}}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal}, {\it{defn-part\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{recpf\/}}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{formals\/}}, {\rm{graph-trans}}\,({\it{gr\/}})\index{graph-trans})\index{isin}, {\it{rec-defn-part\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-ok-frame-proof3}}\,(\=\+{\it{formals\/}}, \\ 
{\it{y\/}}, \\ 
{\it{gr\/}}, \\ 
{\it{defn-part\/}}, \\ 
{\it{defn-part-form\/}}, \\ 
{\it{rec-defn-part\/}}, \\ 
{\it{rec-form\/}}, \\ 
{\it{defn-pf\/}}, \\ 
{\it{recpf\/}})\-\index{a-ev-defn-ok-frame-proof3}, \\ 
{\rm{f-implies}}\,({\rm{isin}}\,({\it{formals\/}}, {\rm{graph-trans}}\,({\it{graph\/}})\index{graph-trans})\index{isin}, {\it{defn-part\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-ok-frame-proof3; name this event `g0869'.\index{g0869}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  weaken-isin-a-ev-defn-part \\ \index{weaken-isin-a-ev-defn-part}
{\rm{weaken-isin}}\,(\=\+{\rm{a-ev-symb-form}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\rm{graph-trans}}\,({\it{gr\/}})\index{graph-trans})\index{a-ev-symb-form}, \\ 
{\rm{a-ev-symb-form}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\rm{graph-trans}}\,({\rm{cons}}\,({\it{y\/}}, {\it{gr\/}}))\index{graph-trans})\index{a-ev-symb-form}, \\ 
{\it{y\/}}, \\ 
{\it{gr\/}})\-\index{weaken-isin}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-ncode-case \\ \index{term-ncode-case}
\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  term-graph-trans-case \\ \index{term-graph-trans-case}
\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\rightarrow$$\;\;\;\;${\rm{termp}}\,({\rm{graph-trans}}\,({\it{x\/}})\index{graph-trans}, {\tt{0}}, {\it{symbols\/}})\index{termp}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-symb-form-a-ev-defn-part1 \\ \index{a-ev-symb-form-a-ev-defn-part1}
\=\+{\rm{make-form-symb}}\,({\rm{a-ev-symb-form}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-symb-form})\index{make-form-symb} \\ 
$=$$\;\;\;\;${\rm{a-ev-defn-part1}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev-defn-part1}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-symb-form; name this event `g0890'.\index{g0890}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev-defn-part-ok-proof; name this event `g0891'.\index{g0891}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-ok-proof} \\  
\=\+{\rm{a-ev-defn-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{flg1\/}}, {\it{exp1\/}}, {\it{va1\/}}, {\it{fa1\/}}, {\it{val1\/}})\index{a-ev-defn-ok-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $=$ {\tt{1}} \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{a-ev-defn-ok-frame-proof2}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\it{flg\/}}, \\ 
{\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))\-)\-, \\ 
{\rm{ev-graph}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{ev-graph}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof})\-\index{a-ev-defn-ok-frame-proof2} \\ 
{\bf else }{\rm{graph-trans-merge-proof}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\it{flg\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))\-)\-)\-, \\ 
{\bf{nil}}, \\ 
{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\bf{nil}})\index{graph-trans})\-\index{isin})\index{f-not}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\bf{nil}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{null-axiom-proof}}\,({\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list})\index{null-axiom-proof}))\-\index{tautconseq-proof})\-\index{graph-trans-merge-proof}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{a-ev-defn-ok-frame-proof2}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\it{flg\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))\-)\-)\-, \\ 
{\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph}, \\ 
{\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof})\-\index{a-ev-defn-ok-frame-proof2} \\ 
{\bf else }{\rm{a-ev-defn-ok-frame-proof2}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{cons}}\,(\=\+{\rm{ev}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\it{flg\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{va\/}}, \\ 
{\it{fa\/}})\-)\-)\-)\-, \\ 
{\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph}, \\ 
{\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof})\-\index{a-ev-defn-ok-frame-proof2}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{exp\/}}) $=$ {\tt{10}} \\ 
{\bf then }{\rm{graph-trans-merge-proof}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-, \\ 
{\bf{nil}}, \\ 
{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{flg1\/}}, {\it{exp1\/}}, {\it{va1\/}}, {\it{fa1\/}}))\index{z-list}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\bf{nil}})\index{graph-trans})\-\index{isin})\index{f-not}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\bf{nil}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{null-axiom-proof}}\,({\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list})\index{null-axiom-proof}))\-\index{tautconseq-proof})\-\index{graph-trans-merge-proof} \\ 
{\bf elseif }{\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
{\bf then }{\rm{a-ev-defn-ok-frame-proof3}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\it{flg\/}}, \\ 
{\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))\-)\-, \\ 
{\rm{ev-graph}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof})\-\index{a-ev-defn-ok-frame-proof3} \\ 
{\bf elseif }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{a-ev-defn-ok-frame-proof2}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\it{flg\/}}, \\ 
{\rm{cons}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))\-)\-)\-, \\ 
{\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev-graph}, \\ 
{\rm{ev-graph}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-symb-form}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-symb-form}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}})\-\index{a-ev-defn-ok-proof})\-\index{a-ev-defn-ok-frame-proof2}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{graph-trans-merge-proof}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,(\=\+{\it{flg\/}}, \\ 
{\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}})))\-)\-, \\ 
{\bf{nil}}, \\ 
{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{flg1\/}}, {\it{exp1\/}}, {\it{va1\/}}, {\it{fa1\/}}))\index{z-list}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1}, \\ 
{\rm{a-ev-inst-formals-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev-inst-formals-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-not}}\,({\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\bf{nil}})\index{graph-trans})\-\index{isin})\index{f-not}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\bf{nil}})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{null-axiom-proof}}\,({\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg1\/}}, \\ 
{\it{exp1\/}}, \\ 
{\it{va1\/}}, \\ 
{\it{fa1\/}})\-)\index{z-list})\index{null-axiom-proof}))\-\index{tautconseq-proof})\-\index{graph-trans-merge-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The definition part of "A-ev" is `OK'.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-ev-defn-ok-proof-proves \\ \index{a-ev-defn-ok-proof-proves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p} $\in$ {\it{symbols\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-all-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-all-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-subrp-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-subrp-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}})\index{a-ev-defn-ok-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\tt{10}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}))\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-ok-proof; name this event `g0896'.\index{g0896}

\addvspace{18pt}

\noindent{\sc Event}:   Enable ev-graph; name this event `g0897'.\index{g0897}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev} \\  
\=\+{\rm{a-ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{trace\/}})\index{a-ev} \\ 
$=$$\;\;\;\;${\rm{f-and}}\,(\=\+{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\it{val\/}}, {\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}))\index{z-list}, {\it{trace\/}})\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\tt{10}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}))\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part1})\-\index{f-implies})\-\index{list-all-quantify})\-\index{f-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-ok-proof} \\  
\=\+{\rm{a-ev-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-ok-proof} \\ 
$=$$\;\;\;\;${\rm{and-join-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}})\-)\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{in-graph-proof}}\,(\=\+{\rm{cons}}\,(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{cons}}\,({\it{flg\/}}, {\rm{cons}}\,({\it{exp\/}}, {\rm{cons}}\,({\it{va\/}}, {\it{fa\/}}))))\-, \\ 
{\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\-\index{in-graph-proof}, \\ 
{\rm{list-gen-proof}}\,(\=\+{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\tt{10}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}))\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1})\-\index{f-implies}, \\ 
{\rm{a-ev-defn-ok-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}})\-\index{a-ev-defn-ok-proof})\-\index{list-gen-proof})\-\index{and-join-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-ok-proof-proves-again \\ \index{a-ev-defn-ok-proof-proves-again}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\tt{10}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}))\index{z-list}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev-graph})\index{graph-trans})\-\index{a-ev-defn-part1})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}})\index{a-ev-defn-ok-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-ok-proof-proves; name this event `g0898'.\index{g0898}

\addvspace{18pt}
\begin{verbatim};;;;;;;"A-ev" is `OK', i.e., 
;;;;;;;|- "A-ev(<FLG>, <EXP>, <VA>, <FA>, <(EV FLG EXP VA FA N)>,
;;;;;;;         [(EV-GRAPH FLG EXP VA FA N)])"
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-ev-ok-proof-proves \\ \index{a-ev-ok-proof-proves}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans})\-\index{a-ev}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{a-ev-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-ok-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-ok-proof; name this event `g0899'.\index{g0899}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  forsome-intro-proves-again \\ \index{forsome-intro-proves-again}
\=\+(\=\+{\rm{variable}}\,({\it{var\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{b\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\rm{f-implies}}\,({\it{a\/}}, {\it{b\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{forsome-intro-proof}}\,({\it{var\/}}, {\it{a\/}}, {\it{b\/}}, {\it{pf\/}})\index{forsome-intro-proof}, \\ 
{\rm{f-implies}}\,({\rm{forsome}}\,({\it{var\/}}, {\it{a\/}})\index{forsome}, {\it{b\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-good-reduc-proof} \\  
\=\+{\rm{a-ev-good-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{expr\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}}, {\it{conc\/}}, {\it{pfexpr\/}}, {\it{pf\/}})\index{a-ev-good-reduc-proof} \\ 
$=$$\;\;\;\;${\rm{chain-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{flg\/}})\index{ncode}, {\it{expr\/}}, {\rm{ncode}}\,({\it{va\/}})\index{ncode}, {\rm{ncode}}\,({\it{fa\/}})\index{ncode}))\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\it{conc\/}}, \\ 
{\rm{list}}\,({\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}), \\ 
{\rm{list}}\,(\=\+{\rm{equal-isin-imp-proof}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\it{expr\/}}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}}, \\ 
{\rm{flip-equals-proof}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\it{expr\/}}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\rm{ev-equal-z-list-proof}}\,(\=\+{\it{val1\/}}, \\ 
{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{expr\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{pfexpr\/}})\-\index{ev-equal-z-list-proof})\-\index{flip-equals-proof})\-\index{equal-isin-imp-proof}, \\ 
{\it{pf\/}})\-)\-\index{chain-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-good-reduc-proof-proves \\ \index{a-ev-good-reduc-proof-proves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, {\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{val1\/}}, {\it{trace\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pfexpr\/}}, {\rm{f-equal}}\,({\it{expr\/}}, {\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{f-equal}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\it{conc\/}})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\it{expr\/}}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\it{conc\/}})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-good-reduc-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{expr\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{conc\/}}, \\ 
{\it{pfexpr\/}}, \\ 
{\it{pf\/}})\-\index{a-ev-good-reduc-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-good-reduc-proof; name this event `g0900'.\index{g0900}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-good-step-proof} \\  
\=\+{\rm{a-ev-defn-good-step-proof}}\,({\it{cl1\/}}, {\it{cl2\/}}, {\it{dexp\/}}, {\it{rvar\/}}, {\it{rval\/}}, {\it{val1\/}}, {\it{val-bar\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{a-ev-defn-good-step-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\it{cl1\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,({\it{rvar\/}}, {\rm{ncode}}\,({\it{rval\/}})\index{ncode})\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{rvar\/}}, {\rm{ncode}}\,({\it{rval\/}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{cl2\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\it{val-bar\/}})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-and}}\,({\it{cl1\/}}, {\it{cl2\/}})\index{f-and}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\it{val-bar\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{form-equal-case-proof}}\,(\=\+{\rm{list}}\,({\it{rvar\/}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\it{rval\/}})\index{ncode}), \\ 
{\rm{f-implies}}\,(\=\+{\it{cl2\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\it{val-bar\/}})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\it{pf2\/}})\-\index{form-equal-case-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-good-step-proof-proves \\ \index{a-ev-defn-good-step-proof-prove!s}
\=\+(\=\+{\rm{variable}}\,({\it{rvar\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{cl2\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{val1\/}}, {\it{val-bar\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{f-implies}}\,({\it{cl1\/}}, {\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{rvar\/}}, {\rm{ncode}}\,({\it{rval\/}})\index{ncode})\index{f-equal})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{parallel-subst}}\,(\=\+{\rm{f-implies}}\,(\=\+{\it{cl2\/}}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\it{val-bar\/}})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,({\it{rvar\/}}), \\ 
{\rm{list}}\,({\rm{ncode}}\,({\it{rval\/}})\index{ncode}), \\ 
{\tt{0}})\-\index{parallel-subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-good-step-proof}}\,(\=\+{\it{cl1\/}}, \\ 
{\it{cl2\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{rvar\/}}, \\ 
{\it{rval\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{val-bar\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-good-step-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{cl1\/}}, {\it{cl2\/}})\index{f-and}, {\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\it{val-bar\/}})\index{f-equal})\index{f-implies})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-good-step-proof; name this event `g0901'.\index{g0901}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part-list-good-proof} \\  
\=\+{\rm{a-ev-defn-part-list-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rvar2\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{rval2\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-list-good-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{forsome}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-opair}}\,({\it{rvar1\/}}, {\it{rvar2\/}})\index{z-opair})\-\index{f-equal})\-\index{f-and})\-\index{f-and})\-\index{forsome}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\rm{cons}}\,({\it{rval1\/}}, {\it{rval2\/}}))\index{ncode})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{forsome-intro-proof}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{rvar1\/}}, \\ 
{\it{rvar2\/}})\-\index{z-opair})\-\index{f-equal})\-\index{f-and})\-\index{f-and}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{cons}}\,(\=\+{\it{rval1\/}}, \\ 
{\it{rval2\/}})\-)\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{a-ev-defn-good-step-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\it{rvar1\/}}, \\ 
{\it{rvar2\/}})\-\index{z-opair})\-\index{f-equal})\-\index{f-and}, \\ 
{\it{dexp\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{cons}}\,(\=\+{\it{rval1\/}}, \\ 
{\it{rval2\/}})\-)\index{ncode}, \\ 
{\rm{a-ev-good-reduc-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof}, \\ 
{\it{pf1\/}})\-\index{a-ev-good-reduc-proof}, \\ 
{\rm{a-ev-defn-good-step-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\it{rvar2\/}})\-\index{z-opair})\-\index{f-equal}, \\ 
{\it{dexp\/}}, \\ 
{\it{rvar2\/}}, \\ 
{\it{rval2\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{cons}}\,(\=\+{\it{rval1\/}}, \\ 
{\it{rval2\/}})\-)\index{ncode}, \\ 
{\rm{a-ev-good-reduc-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{rvar2\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\it{rval2\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{exp\/}})\index{z-cdr-ncode-proof}, \\ 
{\it{pf2\/}})\-\index{a-ev-good-reduc-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{cons}}\,(\=\+{\it{rval1\/}}, \\ 
{\it{rval2\/}})\-)\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{cons}}\,(\=\+{\it{rval1\/}}, \\ 
{\it{rval2\/}})\-)\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof})\-\index{a-ev-defn-good-step-proof})\-\index{a-ev-defn-good-step-proof})\-\index{forsome-intro-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Disable not-free-parallel-subst; name this event `g0908'.\index{g0908}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-list-good-proof-proves \\ \index{a-ev-defn-part-list-good-proof-p!roves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{rvar1\/}}, {\it{rvar2\/}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{4}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\it{rvar1\/}}, {\it{rvar2\/}}), {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{rvar1\/}}, {\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar2\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{rvar2\/}}, {\rm{ncode}}\,({\it{rval2\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-list-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{rvar1\/}}, {\it{rvar2\/}}, {\it{trace\/}})\index{a-ev-list-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\rm{cons}}\,({\it{rval1\/}}, {\it{rval2\/}}))\index{ncode})\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-list-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rvar2\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{rval2\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-list-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part-list-good-proof; name this event `g0916'.\index{g0916}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part-if-good-proof} \\  
\=\+{\rm{a-ev-defn-part-if-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf\/}})\-\index{a-ev-defn-part-if-good-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{body}}\,({\rm{a-ev-if-part}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{rvar1\/}})\-\index{a-ev-if-part}), \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies}, \\ 
{\rm{a-ev-defn-good-step-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-if}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{f-if}, \\ 
{\it{dexp\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{a-ev-good-reduc-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\rm{z-car}}\,({\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr})\index{z-car}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{z-car-cdr-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr-proof}, \\ 
{\it{pf1\/}})\-\index{a-ev-good-reduc-proof}, \\ 
{\rm{rec-form-z-if-eval-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{ncode-equal}}\,(\=\+{\it{rval1\/}}, \\ 
{\tt{0}})\-\index{ncode-equal}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{0}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,(\=\+{\it{rval1\/}}, \\ 
{\tt{0}})\-\index{ncode-nequal-proof})\-, \\ 
{\rm{a-ev-good-reduc-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{z-car-cdr-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr-proof}, \\ 
{\it{pf\/}})\-\index{a-ev-good-reduc-proof}, \\ 
{\rm{a-ev-good-reduc-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{z-car-cdr-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr-proof}, \\ 
{\it{pf\/}})\-\index{a-ev-good-reduc-proof})\-, \\ 
{\rm{z-if}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode})\-\index{f-equal}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{z-if}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{rec-form-z-if-eval-proof})\-\index{a-ev-defn-good-step-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nlistp-nnumberp-ncode \\ \index{nlistp-nnumberp-ncode}
(($\neg$ {\rm{listp}}\,({\it{x\/}})) $\wedge$ ({\it{x\/}} $\not\in\;${\bf{N}}${}$)) $\rightarrow$ ({\rm{ncode}}\,({\it{x\/}})\index{ncode} $=$ {\rm{ncode}}\,({\tt{0}})\index{ncode})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-if-good-proof-proves \\ \index{a-ev-defn-part-if-good-proof-pr!oves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{rvar1\/}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{3}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{rvar1\/}} $\not\in$ {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{rvar1\/}}, {\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\rm{ncode-equal}}\,({\it{rval1\/}}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{caddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-if-part}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{rvar1\/}})\-\index{a-ev-if-part}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-if-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf\/}})\-\index{a-ev-defn-part-if-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part-if-good-proof; name this event `g0919'.\index{g0919}

\addvspace{18pt}

\noindent{\sc Event}:   Enable nlistp-nnumberp-ncode; name this event `g0920'.\index{g0920}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part-subr-good-proof} \\  
\=\+{\rm{a-ev-defn-part-subr-good-proof}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}}, {\it{rvar1\/}}, {\it{rval1\/}}, {\it{dexp\/}}, {\it{pf\/}})\index{a-ev-defn-part-subr-good-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{body}}\,({\rm{a-ev-subr-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{rvar1\/}}, {\it{trace\/}})\index{a-ev-subr-part}), \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,({\rm{car}}\,({\it{exp\/}}), {\it{rval1\/}}))\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{a-ev-defn-good-step-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\it{rvar1\/}})\-\index{z-apply-subr})\-\index{f-equal}, \\ 
{\it{dexp\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode}, \\ 
{\rm{a-ev-good-reduc-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{exp\/}})\index{z-cdr-ncode-proof}, \\ 
{\it{pf\/}})\-\index{a-ev-good-reduc-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{z-apply-subr}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{z-apply-subr})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{z-apply-subr}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{z-apply-subr})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{compose-proof}}\,(\=\+{\rm{fn}}\,(\=\+{\tt{15}}, \\ 
{\tt{2}})\-\index{fn}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{car}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\index{ident-axiom-proof})\-, \\ 
{\rm{z-apply-subr-ncode-proof}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-\index{z-apply-subr-ncode-proof})\-\index{compose-proof}, \\ 
{\rm{trans-equal-proof3}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-apply-subr}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{z-apply-subr}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode})\-\index{trans-equal-proof3})\-)\-\index{tautconseq-proof})\-\index{a-ev-defn-good-step-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-subr-good-proof-proves \\ \index{a-ev-defn-part-subr-good-proof-p!roves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-apply-subr-all-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{z-apply-subr-all-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{rvar1\/}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{3}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{rvar1\/}} $\not\in$ {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar1\/}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{rvar1\/}}, {\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-subr-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{rvar1\/}}, {\it{trace\/}})\index{a-ev-subr-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{apply-subr}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{rval1\/}})\-)\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-subr-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{rvar1\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf\/}})\-\index{a-ev-defn-part-subr-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part-subr-good-proof; name this event `g0921'.\index{g0921}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  parallel-subst-var \\ \index{parallel-subst-var}
({\it{exp\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{parallel-subst}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}}, {\tt{0}})\index{parallel-subst} $=$ {\rm{getpf}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{getpf})
\end{tabbing}

\noindent{\sc Event}:   Enable not-free-parallel-subst; name this event `g0922'.\index{g0922}

\addvspace{18pt}

\noindent{\sc Event}:   Disable parallel-subst-nil; name this event `g0923'.\index{g0923}

\addvspace{18pt}

\noindent{\sc Event}:   Enable get-graph; name this event `g0924'.\index{g0924}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-fun-part-body} \\  
\=\+{\rm{a-ev-fun-part-body}}\,({\it{exp\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}}, {\it{rvar1\/}}, {\it{getva\/}}, {\it{tr1\/}})\index{a-ev-fun-part-body} \\ 
$=$$\;\;\;\;${\rm{f-implies}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\tt{11}})\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{getva\/}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\it{tr1\/}})\-\index{a-get}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\rm{ncode}}\,({\tt{0}})\index{ncode}, {\it{getva\/}}, {\it{rvar1\/}}, {\rm{ncode}}\,({\it{fa\/}})\index{ncode}))\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin})\-\index{f-implies}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-get-ev-good-reduc-proof} \\  
\=\+{\rm{a-get-ev-good-reduc-proof}}\,({\it{exp\/}}, {\it{fa\/}}, {\it{getval\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{tr1\/}})\index{a-get-ev-good-reduc-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{ncode}}\,({\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{tr1\/}})\-\index{isin}, \\ 
{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{tr1\/}})\-\index{isin})\-\index{f-implies}, \\ 
{\rm{a-get}}\,(\=\+{\rm{ncode}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{getval\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{tr1\/}})\-\index{a-get})\-, \\ 
{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\tt{11}})\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{getval\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{tr1\/}})\-\index{a-get}, \\ 
{\rm{list}}\,(\=\+{\rm{equal-z-list-isin-imp-proof}}\,(\=\+{\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{ncode}}\,({\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-, \\ 
{\rm{list}}\,(\=\+{\it{getval\/}}, \\ 
{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-, \\ 
{\it{tr1\/}}, \\ 
{\rm{list}}\,(\=\+{\rm{ident-axiom-proof}}\,({\it{getval\/}})\index{ident-axiom-proof}, \\ 
{\rm{flip-equals-proof}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n})\index{ncode}, \\ 
{\rm{z-sub1-n-ncode-proof}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof})\-\index{z-sub1-n-ncode-proof})\-\index{flip-equals-proof}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{fa\/}})\index{ncode})\index{ident-axiom-proof})\-)\-\index{equal-z-list-isin-imp-proof}, \\ 
{\rm{a-get-ok-proof1}}\,(\=\+{\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, \\ 
{\it{fa\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}})\-\index{a-get-ok-proof1})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-get-ev-good-reduc-proof-proves \\ \index{a-get-ev-good-reduc-proof-prove!s}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}), {\tt{3}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\it{getval\/}} $=$ {\rm{ncode}}\,({\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get})\index{ncode}) \\ 
$\wedge$$\;\;\;\;$({\it{tr1\/}} $=$ {\rm{graph-trans}}\,({\rm{get-graph}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get-graph})\index{graph-trans}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, {\tt{11}})\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{getval\/}}, \\ 
{\it{x1\/}}, \\ 
{\it{y1\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{tr1\/}})\-\index{a-get}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-get-ev-good-reduc-proof}}\,({\it{exp\/}}, {\it{fa\/}}, {\it{getval\/}}, {\it{x1\/}}, {\it{y1\/}}, {\it{val1\/}}, {\it{tr1\/}})\index{a-get-ev-good-reduc-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-get-ev-good-reduc-proof; name this event `g0925'.\index{g0925}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part-fun-good-proof} \\  
\=\+{\rm{a-ev-defn-part-fun-good-proof}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{val\/}}, {\it{trace\/}}, {\it{rval1\/}}, {\it{dexp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{a-ev-defn-part-fun-good-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\tt{11}}, \\ 
{\rm{body}}\,({\rm{a-ev-fun-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-fun-part}), \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies}, \\ 
{\rm{chain-proof}}\,(\=\+{\rm{body}}\,({\rm{a-ev-fun-part}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-fun-part}), \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-fun-part-body}}\,(\=\+{\it{exp\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-ev-fun-part-body})\-\index{f-and}), \\ 
{\rm{list}}\,(\=\+{\rm{subst-thm-b-proof}}\,(\=\+{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-fun-part-body}}\,(\=\+{\it{exp\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\tt{15}}, \\ 
{\tt{16}})\-\index{a-ev-fun-part-body})\-\index{f-and}, \\ 
{\rm{list}}\,({\tt{15}}, {\tt{16}}), \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get-graph})\index{graph-trans})\-)\-\index{subst-thm-b-proof}, \\ 
{\rm{a-ev-defn-good-step-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-fun-part-body}}\,(\=\+{\it{exp\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-ev-fun-part-body}, \\ 
{\it{dexp\/}}, \\ 
{\tt{11}}, \\ 
{\it{rval1\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{a-ev-good-reduc-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\rm{z-cdr}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-cdr}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\tt{11}}, \\ 
{\it{trace\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\rm{z-cdr-ncode-proof}}\,({\it{exp\/}})\index{z-cdr-ncode-proof}, \\ 
{\it{pf1\/}})\-\index{a-ev-good-reduc-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{a-get}}\,(\=\+{\rm{z-sub1-n}}\,(\=\+{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\tt{11}})\-\index{z-sub1-n}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-ev-fun-part-body}}\,(\=\+{\it{exp\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-ev-fun-part-body}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{a-get-ev-good-reduc-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{fa\/}}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\rm{graph-trans}}\,({\rm{get-graph}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get-graph})\index{graph-trans})\-\index{a-get-ev-good-reduc-proof}, \\ 
{\it{pf2\/}})\-)\-\index{tautconseq-proof})\-\index{a-ev-defn-good-step-proof})\-)\-\index{chain-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-fun-good-proof-proves \\ \index{a-ev-defn-part-fun-good-proof-p!roves}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{z-succ-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-succ-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-car-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-car-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-cdr-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-cdr-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-int-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-int-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-zerop-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-zerop-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{z-sub1-hyps}}\,({\it{defns\/}}, {\it{symbols\/}})\index{z-sub1-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{5}})\index{var-set} \\ 
$\wedge$$\;\;\;\;$({\tt{11}} $\not\in$ {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{11}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\tt{11}}, {\rm{ncode}}\,({\it{rval1\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{rval1\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-fun-part}}\,({\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-fun-part}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-fun-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{val\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{rval1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-fun-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part-fun-good-proof; name this event `g0926'.\index{g0926}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{sub-rvar-proof} \\  
\=\+{\rm{sub-rvar-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{trace\/}}, {\it{rvar\/}}, {\it{val1\/}}, {\it{dexp\/}}, {\it{rval\/}}, {\it{pf\/}})\index{sub-rvar-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{rval\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{val1\/}}, \\ 
{\it{rvar\/}}, \\ 
{\it{pf\/}})\-\index{subst-rule-proof}\-
\end{tabbing}

\noindent{\sc Event}:   Disable not-free-covering; name this event `g0927'.\index{g0927}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  sub-rvar-proof-proves \\ \index{sub-rvar-proof-proves}
\=\+(\=\+{\rm{var-set}}\,({\rm{list}}\,({\it{val1\/}}, {\it{trace\/}}), {\tt{2}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\it{rvar\/}}, {\tt{0}}, {\it{symbols\/}})\index{termp} \\ 
$\wedge$$\;\;\;\;$({\it{val1\/}} $\not\in$ {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{rval\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{rvar\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{rvar\/}}, {\rm{ncode}}\,({\it{rval\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{sub-rvar-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{trace\/}}, {\it{rvar\/}}, {\it{val1\/}}, {\it{dexp\/}}, {\it{rval\/}}, {\it{pf\/}})\index{sub-rvar-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable sub-rvar-proof; name this event `g0928'.\index{g0928}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{ind-form} \\  
\=\+{\rm{ind-form}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{val\/}}, {\it{trace\/}}, {\it{dexp\/}})\index{ind-form} \\ 
$=$$\;\;\;\;${\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{f-implies}}\,({\it{dexp\/}}, {\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\it{val\/}})\index{ncode})\index{f-equal})\index{f-implies})\-\index{f-implies}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  form-z-if-ev-defn-part-again \\ \index{form-z-if-ev-defn-part-again}
\=\+(\=\+{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{termp}}\,({\rm{list}}\,({\it{val1\/}}, {\it{trace\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{form-z-if}}\,({\rm{z-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{z-ev-defn-part}, {\it{symbols\/}})\index{form-z-if}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ncode-equal-ident-axiom-proof-proves \\ \index{ncode-equal-ident-axiom-proof-p!roves}
\=\+(\=\+{\rm{subset}}\,({\rm{list}}\,({\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, {\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, {\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}), {\it{symbols\/}})\index{subset} \\ 
$\wedge$$\;\;\;\;${\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{f-equal}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\rm{ncode}}\,({\it{y\/}})\index{ncode})\index{f-equal}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Disable a-ev-defn-part; name this event `g0929'.\index{g0929}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part-good-pftree} \\  
\=\+{\rm{a-ev-defn-part-good-pftree}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{val1\/}}, {\it{trace\/}}, {\it{dexp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{a-ev-defn-part-good-pftree} \\ 
$=$$\;\;\;\;${\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\it{flg\/}} $=$ {\tt{1}}, \\ 
{\rm{ident-axiom-proof}}\,({\rm{ncode}}\,({\tt{1}})\index{ncode})\index{ident-axiom-proof}, \\ 
{\rm{ncode-nequal-proof}}\,({\it{flg\/}}, {\tt{1}})\index{ncode-nequal-proof})\-, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{listp}}\,({\it{exp\/}}), \\ 
{\rm{z-listp-ncode-proof}}\,({\it{exp\/}})\index{z-listp-ncode-proof}, \\ 
{\rm{trans-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-listp}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-listp}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,({\tt{1}}, {\tt{0}})\index{ncode-nequal-proof}, \\ 
{\rm{z-listp-ncode-proof}}\,({\it{exp\/}})\index{z-listp-ncode-proof})\-\index{trans-not-equal-proof})\-, \\ 
{\rm{a-ev-defn-part-list-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\tt{11}}, \\ 
{\tt{12}}, \\ 
{\rm{ev}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\rm{sub-rvar-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\it{val1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{pf1\/}})\-\index{sub-rvar-proof}, \\ 
{\rm{sub-rvar-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{12}}, \\ 
{\it{val1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{pf2\/}})\-\index{sub-rvar-proof})\-\index{a-ev-defn-part-list-good-proof}, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\bf{nil}}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,({\it{val1\/}}, {\rm{ncode}}\,({\bf{nil}})\index{ncode})\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\bf{nil}})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\bf{nil}})\-\index{tautconseq-proof})\-, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{listp}}\,({\it{exp\/}}), \\ 
{\rm{z-listp-ncode-proof}}\,({\it{exp\/}})\index{z-listp-ncode-proof}, \\ 
{\rm{trans-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-listp}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-listp}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,({\tt{1}}, {\tt{0}})\index{ncode-nequal-proof}, \\ 
{\rm{z-listp-ncode-proof}}\,({\it{exp\/}})\index{z-listp-ncode-proof})\-\index{trans-not-equal-proof})\-, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\it{exp\/}}, \\ 
{\rm{z-zerop-ncode-proof}}\,({\rm{car}}\,({\it{exp\/}}))\index{z-zerop-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{trans-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-zerop}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car})\index{z-zerop}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,({\tt{1}}, {\tt{0}})\index{ncode-nequal-proof}, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\it{exp\/}}, \\ 
{\rm{z-zerop-ncode-proof}}\,({\rm{car}}\,({\it{exp\/}}))\index{z-zerop-ncode-proof})\-\index{compose-car-proof})\-\index{trans-not-equal-proof})\-, \\ 
{\rm{a-ev-defn-part-if-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{dexp\/}}, \\ 
{\rm{sub-rvar-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\it{val1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{pf1\/}})\-\index{sub-rvar-proof}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-if-good-proof}, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{car}}\,({\it{exp\/}}) $=$ {\tt{10}}, \\ 
{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof}, \\ 
{\rm{trans-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\tt{10}})\index{ncode}, \\ 
{\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,(\=\+{\tt{10}}, \\ 
{\rm{car}}\,({\it{exp\/}}))\-\index{ncode-nequal-proof}, \\ 
{\rm{z-car-ncode-proof}}\,({\it{exp\/}})\index{z-car-ncode-proof})\-\index{trans-not-equal-proof})\-, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}))\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode})\-\index{f-equal}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies})\-, \\ 
{\rm{f-implies}}\,(\=\+{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr})\-\index{f-equal}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,(\=\+{\rm{z-car-cdr-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{list}}\,({\tt{1}}, {\tt{2}}))\-\index{z-car-cdr-proof}, \\ 
{\rm{trans-equal-proof3}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{z-car-cdr}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{list}}\,(\=\+{\tt{1}}, \\ 
{\tt{2}})\-)\-\index{z-car-cdr}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode})\-\index{trans-equal-proof3})\-)\-\index{tautconseq-proof}, \\ 
{\rm{list}}\,(\=\+{\rm{list}}\,(\=\+{\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})), \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\it{exp\/}}, \\ 
{\rm{z-subrp-ncode-proof}}\,({\rm{car}}\,({\it{exp\/}}))\index{z-subrp-ncode-proof})\-\index{compose-car-proof}, \\ 
{\rm{trans-not-equal-proof}}\,(\=\+{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{z-subrp}}\,({\rm{z-car}}\,({\rm{ncode}}\,({\it{exp\/}})\index{ncode})\index{z-car})\index{z-subrp}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode-nequal-proof}}\,(\=\+{\tt{1}}, \\ 
{\tt{0}})\-\index{ncode-nequal-proof}, \\ 
{\rm{compose-car-proof}}\,(\=\+{\rm{fn}}\,(\=\+{\tt{16}}, \\ 
{\tt{1}})\-\index{fn}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\it{exp\/}}, \\ 
{\rm{z-subrp-ncode-proof}}\,({\rm{car}}\,({\it{exp\/}}))\index{z-subrp-ncode-proof})\-\index{compose-car-proof})\-\index{trans-not-equal-proof})\-, \\ 
{\rm{a-ev-defn-part-subr-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{dexp\/}}, \\ 
{\rm{sub-rvar-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\it{val1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{pf1\/}})\-\index{sub-rvar-proof})\-\index{a-ev-defn-part-subr-good-proof}, \\ 
{\rm{a-ev-defn-part-fun-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{dexp\/}}, \\ 
{\rm{sub-rvar-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{trace\/}}, \\ 
{\tt{11}}, \\ 
{\it{val1\/}}, \\ 
{\it{dexp\/}}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{pf1\/}})\-\index{sub-rvar-proof}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-fun-good-proof})\-)\-)\-, \\ 
{\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,({\rm{f-implies}}\,(\=\+{\rm{a-get1}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\it{val1\/}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get1}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-get1}}\,(\=\+{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\it{val1\/}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get1}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{list}}\,({\rm{a-get1-good-proof}}\,(\=\+{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{val1\/}}, \\ 
{\tt{11}}, \\ 
{\tt{13}}, \\ 
{\tt{14}}, \\ 
{\tt{16}})\-\index{a-get1-good-proof}))\-\index{tautconseq-proof})\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn-part-good-proof} \\  
\=\+{\rm{a-ev-defn-part-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{val1\/}}, {\it{trace\/}}, {\it{dexp\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{a-ev-defn-part-good-proof} \\ 
$=$$\;\;\;\;${\rm{rec-form-z-if-eval-proof}}\,(\=\+{\rm{a-ev-defn-part-good-pftree}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-pftree}, \\ 
{\rm{z-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{z-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{rec-form-z-if-eval-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-good-proof-proves-quote-case \\ \index{a-ev-defn-part-good-proof-prove!s-quote-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{6}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{dexp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-good-proof-proves-list-case \\ \index{a-ev-defn-part-good-proof-prove!s-list-case}
\=\+(\=\+({\it{flg\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{6}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\it{val1\/}}), {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\tt{0}}, {\rm{car}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-good-proof-proves-nlist-case \\ \index{a-ev-defn-part-good-proof-prove!s-nlist-case}
\=\+(\=\+({\it{flg\/}} $=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$({\it{exp\/}} $\simeq\;${{\bf{nil}}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{6}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{dexp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-good-proof-proves-if-case \\ \index{a-ev-defn-part-good-proof-prove!s-if-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{6}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\it{val1\/}}), {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\rm{cadr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }{\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\tt{0}}, {\rm{cadddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
{\bf else }{\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\tt{0}}, {\rm{caddr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}$\;${\bf  endif}\-)\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-good-proof-proves-subr-case \\ \index{a-ev-defn-part-good-proof-prove!s-subr-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;${\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{6}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\it{val1\/}}), {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-good-proof-proves-fun-case \\ \index{a-ev-defn-part-good-proof-prove!s-fun-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{exp\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\rm{car}}\,({\it{exp\/}}) $\not=$ {\tt{10}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}}))) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{6}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\it{val1\/}}), {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf1\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf2\/}}, \\ 
{\rm{ind-form}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}})\-\index{ind-form}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-defn-part-good-proof-proves-var-case \\ \index{a-ev-defn-part-good-proof-prove!s-var-case}
\=\+(\=\+({\it{flg\/}} $\not=$ {\tt{1}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{listp}}\,({\it{exp\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{formula}}\,({\it{dexp\/}}, {\it{symbols\/}})\index{formula} \\ 
$\wedge$$\;\;\;\;${\rm{var-set}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}}, {\it{val1\/}}, {\it{trace\/}}), {\tt{6}})\index{var-set} \\ 
$\wedge$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\it{val1\/}}), {\rm{collect-free}}\,({\it{dexp\/}}, {\tt{0}})\index{collect-free})\index{nil-intersect} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-implies}}\,(\=\+{\it{dexp\/}}, \\ 
{\rm{f-equal}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}}, \\ 
{\it{dexp\/}}, \\ 
{\it{pf1\/}}, \\ 
{\it{pf2\/}})\-\index{a-ev-defn-part-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part-good-proof; name this event `g0930'.\index{g0930}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  free-for-x-x \\ \index{free-for-x-x}
{\rm{variable}}\,({\it{x\/}})\index{variable} $\rightarrow$ {\rm{free-for}}\,({\it{exp\/}}, {\it{x\/}}, {\it{x\/}}, {\it{flg\/}})\index{free-for}
\end{tabbing}

\noindent{\sc Event}:   Enable free-for; name this event `g0932'.\index{g0932}

\addvspace{18pt}

\noindent{\sc Event}:   Disable free-for-list; name this event `g0933'.\index{g0933}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-reduc-proof} \\  
\=\+{\rm{a-ev-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-reduc-proof} \\ 
$=$$\;\;\;\;${\rm{f-iff-mp-proof2}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{a-ev}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-ev}, \\ 
{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev-defn-part})\-\index{f-implies}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev-defn-part1})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-defn-part})\-\index{f-implies})\-\index{f-implies}, \\ 
{\rm{good-frame1-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\it{val1\/}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev-defn-part1})\-\index{f-implies})\-\index{list-all-quantify}, \\ 
{\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{val1\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-defn-part})\-\index{good-frame1-proof}, \\ 
{\rm{subst-thm-b-proof}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,(\=\+{\tt{6}}, \\ 
{\tt{7}}, \\ 
{\tt{8}}, \\ 
{\tt{9}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev-defn-part1})\-\index{f-implies}, \\ 
{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val1\/}})\-)\-\index{subst-thm-b-proof})\-\index{f-iff-mp-proof2}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  parallel-subst-a-ev-defn-part1-again \\ \index{parallel-subst-a-ev-defn-part1-!again}
\=\+({\rm{variable}}\,({\it{trace\/}})\index{variable} $\wedge$ ({\it{trace\/}} $\not\in$ {\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}))) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{parallel-subst}}\,(\=\+{\rm{a-ev-defn-part1}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part1}, \\ 
{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\it{val-bar\/}})\-, \\ 
{\tt{0}})\-\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{val-bar\/}}, {\it{trace\/}})\index{a-ev-defn-part})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part; name this event `g0935'.\index{g0935}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  free-for-list-reduc \\ \index{free-for-list-reduc}
\=\+{\rm{free-for-list}}\,({\it{exp\/}}, {\it{vars\/}}, {\it{terms\/}})\index{free-for-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{vars\/}}) $\wedge$ {\rm{listp}}\,({\it{terms\/}}) \\ 
{\bf then }\=\+{\rm{free-for}}\,({\it{exp\/}}, {\rm{car}}\,({\it{vars\/}}), {\rm{car}}\,({\it{terms\/}}), {\tt{0}})\index{free-for} \\ 
$\wedge$$\;\;\;\;${\rm{free-for-list}}\,({\it{exp\/}}, {\rm{cdr}}\,({\it{vars\/}}), {\rm{cdr}}\,({\it{terms\/}}))\index{free-for-list}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable free-for-list; name this event `g0940'.\index{g0940}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{all-symbs} \\  
\=\+{\rm{all-symbs}}\,({\it{symbols\/}})\index{all-symbs} \\ 
$=$$\;\;\;\;${\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\it{symbols\/}})\-\index{subset}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-reduc-proof-proves \\ \index{a-ev-reduc-proof-proves}
\=\+(\=\+{\rm{all-symbs}}\,({\it{symbols\/}})\index{all-symbs} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{trace\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $\not\in$ {\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}})) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev}, \\ 
{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-reduc-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-reduc-proof; name this event `g0947'.\index{g0947}

\addvspace{18pt}

\noindent{\sc Event}:   Disable a-ev-defn-part1; name this event `g0949'.\index{g0949}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-defn} \\  
\=\+{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn} \\ 
$=$$\;\;\;\;${\rm{list-all-quantify}}\,(\=\+{\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}), \\ 
{\rm{f-implies}}\,(\=\+{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\tt{10}}, {\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}))\index{z-list}, {\it{trace\/}})\index{isin}, \\ 
{\rm{a-ev-defn-part1}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part1})\-\index{f-implies})\-\index{list-all-quantify}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-z-sub1-n \\ \index{collect-free-z-sub1-n}
\=\+({\it{var\/}} $\in$ {\rm{collect-free}}\,({\rm{z-sub1-n}}\,({\it{exp\/}}, {\it{n\/}})\index{z-sub1-n}, {\tt{0}})\index{collect-free}) \\ 
$=$$\;\;\;\;$({\it{var\/}} $\in$ {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nil-int-a-ev-defn \\ \index{nil-int-a-ev-defn}
\=\+({\rm{variable}}\,({\it{trace\/}})\index{variable} $\wedge$ ({\it{trace\/}} $\not\in$ {\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{10}}))) \\ 
$\rightarrow$$\;\;\;\;${\rm{nil-intersect}}\,({\rm{list}}\,({\tt{11}}, {\tt{12}}, {\tt{10}}), {\rm{collect-free}}\,({\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, {\tt{0}})\index{collect-free})\index{nil-intersect}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-a-ev-defn \\ \index{formula-a-ev-defn}
\=\+({\rm{all-symbs}}\,({\it{symbols\/}})\index{all-symbs} $\wedge$ {\rm{variable}}\,({\it{trace\/}})\index{variable}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part1; name this event `g0950'.\index{g0950}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  a-ev-reduc \\ \index{a-ev-reduc}
\=\+{\rm{a-ev}}\,({\it{flg-bar\/}}, {\it{exp-bar\/}}, {\it{va-bar\/}}, {\it{fa-bar\/}}, {\it{val1\/}}, {\it{trace\/}})\index{a-ev} \\ 
$=$$\;\;\;\;${\rm{f-and}}\,(\=\+{\rm{isin}}\,({\rm{z-list}}\,({\rm{list}}\,({\it{val1\/}}, {\it{flg-bar\/}}, {\it{exp-bar\/}}, {\it{va-bar\/}}, {\it{fa-bar\/}}))\index{z-list}, {\it{trace\/}})\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn})\-\index{f-and}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn; name this event `g0951'.\index{g0951}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev; name this event `g0952'.\index{g0952}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  good-frame2-proof-proves-again \\ \index{good-frame2-proof-proves-again}
\=\+(\=\+{\rm{proves}}\,({\it{pf1\/}}, {\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{c\/}})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,({\it{pf2\/}}, {\rm{f-implies}}\,({\it{c\/}}, {\rm{f-implies}}\,({\it{b\/}}, {\it{d\/}})\index{f-implies})\index{f-implies}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{good-frame2-proof}}\,({\it{a\/}}, {\it{b\/}}, {\it{c\/}}, {\it{d\/}}, {\it{pf1\/}}, {\it{pf2\/}})\index{good-frame2-proof}, \\ 
{\rm{f-implies}}\,({\rm{f-and}}\,({\it{a\/}}, {\it{b\/}})\index{f-and}, {\it{d\/}})\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable good-frame2-proof-proves; name this event `g0953'.\index{g0953}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev-good-proof} \\  
\=\+{\rm{a-ev-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{trace\/}})\index{a-ev-good-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $=$ {\tt{1}} \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-reduc-proof}, \\ 
{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{car}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{car}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof}, \\ 
{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof})\-\index{a-ev-defn-part-good-proof})\-\index{good-frame2-proof} \\ 
{\bf else }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev-defn-part}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-reduc-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev-reduc-proof}, \\ 
{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\bf{nil}}, \\ 
{\bf{nil}})\-\index{a-ev-defn-part-good-proof})\-\index{good-frame2-proof}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{exp\/}}) $\simeq {\tt{0}}$ \\ 
{\bf then }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-reduc-proof}, \\ 
{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof}, \\ 
\=\+{\bf if }{\rm{ncode-equal}}\,(\=\+{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\tt{0}})\-\index{ncode-equal} \\ 
{\bf then }{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cadddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{cadddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof} \\ 
{\bf else }{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{caddr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{caddr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof}$\;${\bf  endif}\-)\-\index{a-ev-defn-part-good-proof})\-\index{good-frame2-proof} \\ 
{\bf elseif }{\rm{car}}\,({\it{exp\/}}) $=$ {\tt{10}} \\ 
{\bf then }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-reduc-proof}, \\ 
{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\bf{nil}}, \\ 
{\bf{nil}})\-\index{a-ev-defn-part-good-proof})\-\index{good-frame2-proof} \\ 
{\bf elseif }{\rm{subrp}}\,({\rm{car}}\,({\it{exp\/}})) \\ 
{\bf then }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-reduc-proof}, \\ 
{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof}, \\ 
{\bf{nil}})\-\index{a-ev-defn-part-good-proof})\-\index{good-frame2-proof} \\ 
{\bf elseif }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{a-ev-defn-part}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev-defn-part}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-reduc-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev-reduc-proof}, \\ 
{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{1}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof}, \\ 
{\rm{good-frame3-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\tt{0}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev})\index{ncode})\-\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,(\=\+{\rm{sub1-n}}\,(\=\+{\rm{car}}\,({\it{exp\/}}), \\ 
{\tt{11}})\-\index{sub1-n}, \\ 
{\it{fa\/}})\-\index{get}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{cdr}}\,({\it{exp\/}}), \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\it{trace\/}})\-\index{a-ev-good-proof})\-\index{good-frame3-proof})\-\index{a-ev-defn-part-good-proof})\-\index{good-frame2-proof}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{good-frame2-proof}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\it{trace\/}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\rm{a-ev-defn-part}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-reduc-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-reduc-proof}, \\ 
{\rm{a-ev-defn-part-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}}, \\ 
{\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, \\ 
{\bf{nil}}, \\ 
{\bf{nil}})\-\index{a-ev-defn-part-good-proof})\-\index{good-frame2-proof}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;"A-EV" is `GOOD', i.e., 
;;;;;;; |- "(A-ev(<FLG>, <EXP>, <VA>, <FA>, val, trace) -> (val = <(EV FLG VA FA N)>))".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a-ev-good-proof-proves \\ \index{a-ev-good-proof-proves}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{trace\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $\not\in$ {\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}})) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{trace\/}})\index{a-ev-good-proof}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-ev}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal})\-\index{f-implies}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-good-proof; name this event `g0954'.\index{g0954}

\addvspace{18pt}
\begin{verbatim};;;;;;;
;;;;;;;This brings us to an entirely different section of the proof.  So far we have
;;;;;;;shown that a Lisp interpreter is representable by a Z2 formula.  In order to
;;;;;;;construct the undecidable sentence, we need to show that the metatheory of Z2
;;;;;;;is representable in Z2.  We do that by showing that the metatheoretic formalization
;;;;;;;of Z2 is computable by means of the Lisp interpreter EV.  Since EV is only aware
;;;;;;;of data-structures which are lists or numbers (num-trees), we need to first 
;;;;;;;represent the Z2 expressions and proofs in these terms.  The function GCODE
;;;;;;;translates the shell-constructors F-NOT, F-OR, FORSOME, CONS, and ADD1, into
;;;;;;;distinct num-trees.  If it is none of the above (NIL, for instance), then GCODE
;;;;;;;returns the num-tree (CONS 5 0).
;;;;;;;
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{gcode} \\  
\=\+{\rm{gcode}}\,({\it{x\/}})\index{gcode} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{cons}}\,({\tt{1}}, {\rm{cons}}\,({\rm{gcode}}\,({\rm{car}}\,({\it{x\/}}))\index{gcode}, {\rm{gcode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{gcode})) \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{x\/}})\index{f-notp}$\;\;${\bf then }{\rm{cons}}\,({\tt{2}}, {\rm{cons}}\,({\rm{gcode}}\,({\rm{arg}}\,({\it{x\/}})\index{arg})\index{gcode}, {\tt{0}})) \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{x\/}})\index{f-orp}$\;\;${\bf then }{\rm{cons}}\,({\tt{3}}, {\rm{cons}}\,({\rm{gcode}}\,({\rm{arg1}}\,({\it{x\/}})\index{arg1})\index{gcode}, {\rm{gcode}}\,({\rm{arg2}}\,({\it{x\/}})\index{arg2})\index{gcode})) \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{x\/}})\index{forsomep}$\;\;${\bf then }{\rm{cons}}\,({\tt{4}}, {\rm{cons}}\,({\rm{bind}}\,({\it{x\/}})\index{bind}, {\rm{gcode}}\,({\rm{body}}\,({\it{x\/}}))\index{gcode})) \\ 
{\bf elseif }{\it{x\/}} $\in\;${\bf{N}}${}$$\;\;${\bf then }{\it{x\/}} \\ 
{\bf else }{\rm{cons}}\,({\tt{5}}, {\tt{0}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below establishes the connection between EQL and GCODE.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  eql-gcode-equal \\ \index{eql-gcode-equal}
{\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql} $=$ ({\rm{gcode}}\,({\it{x\/}})\index{gcode} $=$ {\rm{gcode}}\,({\it{y\/}})\index{gcode})
\end{tabbing}
\begin{verbatim};;;;;;;Then we introduce functions which form e-expressions corresponding to e-function
;;;;;;;applications of the Lisp primitives.  The prefix PR stands for `primitive'.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-cons}$\;\;$
{\rm{pr-cons}}\,({\it{x\/}}, {\it{y\/}})\index{pr-cons} $=$ {\rm{list}}\,({\tt{7}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-car}$\;\;$
{\rm{pr-car}}\,({\it{x\/}})\index{pr-car} $=$ {\rm{list}}\,({\tt{8}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-cdr}$\;\;$
{\rm{pr-cdr}}\,({\it{x\/}})\index{pr-cdr} $=$ {\rm{list}}\,({\tt{9}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-equal}$\;\;$
{\rm{pr-equal}}\,({\it{x\/}}, {\it{y\/}})\index{pr-equal} $=$ {\rm{list}}\,({\tt{1}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-numberp}$\;\;$
{\rm{pr-numberp}}\,({\it{x\/}})\index{pr-numberp} $=$ {\rm{list}}\,({\tt{2}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-zerop}$\;\;$
{\rm{pr-zerop}}\,({\it{x\/}})\index{pr-zerop} $=$ {\rm{list}}\,({\tt{3}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-add1}$\;\;$
{\rm{pr-add1}}\,({\it{x\/}})\index{pr-add1} $=$ {\rm{list}}\,({\tt{4}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-sub1}$\;\;$
{\rm{pr-sub1}}\,({\it{x\/}})\index{pr-sub1} $=$ {\rm{list}}\,({\tt{5}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-listp}$\;\;$
{\rm{pr-listp}}\,({\it{x\/}})\index{pr-listp} $=$ {\rm{list}}\,({\tt{6}}, {\it{x\/}})
\end{tabbing}
\begin{verbatim};;;;;;;The functions suffixed with `N' are the Lisp analogues of the e-functions.
;;;;;;;CONS is not the Lisp analogue of PR-CONS since PR-CONS can return (BTM),
;;;;;;;whereas CONS cannot.  The function PR-CONSN returns (BTM) if one of its 
;;;;;;;arguments is (BTM), and (CONS X Y) otherwise.  Similarly, we have the 
;;;;;;;Lisp analogues for the other primitives.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-consn} \\  
\=\+{\rm{pr-consn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-consn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{cons}}\,({\it{x\/}}, {\it{y\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-carn} \\  
\=\+{\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{car}}\,({\it{x\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-cdrn} \\  
\=\+{\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{cdr}}\,({\it{x\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-equaln} \\  
\=\+{\rm{pr-equaln}}\,({\it{x\/}}, {\it{y\/}})\index{pr-equaln} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{bool-fix}}\,({\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal})\index{bool-fix}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-numberpn} \\  
\=\+{\rm{pr-numberpn}}\,({\it{x\/}})\index{pr-numberpn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{bool-fix}}\,($\neg$ {\rm{listp}}\,({\it{x\/}}))\index{bool-fix}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-zeropn} \\  
\=\+{\rm{pr-zeropn}}\,({\it{x\/}})\index{pr-zeropn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{bool-fix}}\,({\it{x\/}} $\simeq {\tt{0}}$)\index{bool-fix}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-add1n} \\  
\=\+{\rm{pr-add1n}}\,({\it{x\/}})\index{pr-add1n} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }$1\;+$ {\it{x\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-listpn} \\  
\=\+{\rm{pr-listpn}}\,({\it{x\/}})\index{pr-listpn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{bool-fix}}\,({\rm{listp}}\,({\it{x\/}}))\index{bool-fix}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable subrp; name this event `g0955'.\index{g0955}

\addvspace{18pt}

\noindent{\sc Event}:   Disable apply-subr; name this event `g0956'.\index{g0956}

\addvspace{18pt}

\noindent{\sc Event}:   Disable apply-subr1; name this event `g0957'.\index{g0957}

\addvspace{18pt}

\noindent{\sc Event}:   Disable apply-subr2; name this event `g0958'.\index{g0958}

\addvspace{18pt}
\begin{verbatim};;;;;;;The relationship between PR-CONS and PR-CONSN is established below.  The lemma
;;;;;;;asserts that the result of evaluating (PR-CONS X Y) with EV is (PR-CONSN x y),
;;;;;;;where x and y are the results of evaluating X and Y, respectively.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  pr-cons-eval \\ \index{pr-cons-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-cons}}\,({\it{x\/}}, {\it{y\/}})\index{pr-cons}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-consn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-consn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-car-eval \\ \index{pr-car-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-car}}\,({\it{x\/}})\index{pr-car}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-carn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-carn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-cdr-eval \\ \index{pr-cdr-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-cdr}}\,({\it{x\/}})\index{pr-cdr}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-cdrn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-cdrn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-equal-eval \\ \index{pr-equal-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-equal}}\,({\it{x\/}}, {\it{y\/}})\index{pr-equal}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-equaln}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-equaln}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-numberp-eval \\ \index{pr-numberp-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-numberp}}\,({\it{x\/}})\index{pr-numberp}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-numberpn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-numberpn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-zerop-eval \\ \index{pr-zerop-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-zerop}}\,({\it{x\/}})\index{pr-zerop}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-zeropn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-zeropn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-add1-eval \\ \index{pr-add1-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-add1}}\,({\it{x\/}})\index{pr-add1}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-add1n}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-add1n}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-sub1n} \\  
\=\+{\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\it{x\/}} $-\;1$$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-sub1-eval \\ \index{pr-sub1-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-sub1}}\,({\it{x\/}})\index{pr-sub1}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-sub1n}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-sub1n}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-listp-eval \\ \index{pr-listp-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-listp}}\,({\it{x\/}})\index{pr-listp}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-listpn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-listpn}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-cons; name this event `g0959'.\index{g0959}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-car; name this event `g0960'.\index{g0960}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-cdr; name this event `g0961'.\index{g0961}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-equal; name this event `g0962'.\index{g0962}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-zerop; name this event `g0963'.\index{g0963}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-numberp; name this event `g0964'.\index{g0964}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-add1; name this event `g0965'.\index{g0965}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-sub1; name this event `g0966'.\index{g0966}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-listp; name this event `g0967'.\index{g0967}

\addvspace{18pt}
\begin{verbatim};;;;;;;Similarly we construct e-expressions for IF-expressions and QUOTE-expressions, 
;;;;;;;and show how they are evaluated.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-if}$\;\;$
{\rm{pr-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{pr-if} $=$ {\rm{list}}\,({\tt{0}}, {\it{x\/}}, {\it{y\/}}, {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-if-eval \\ \index{pr-if-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{pr-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{pr-if}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\tt{0}})\index{ncode-equal}$\;\;${\bf then }{\rm{ev}}\,({\tt{0}}, {\it{z\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
{\bf else }{\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable pr-if; name this event `g0968'.\index{g0968}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{pr-quote}$\;\;$
{\rm{pr-quote}}\,({\it{x\/}})\index{pr-quote} $=$ {\rm{list}}\,({\tt{10}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-quote-eval \\ \index{pr-quote-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-quote}}\,({\it{x\/}})\index{pr-quote}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\it{x\/}}
\end{tabbing}
\begin{verbatim};;;;;;;Having dealt with the primitives, we now get into the defined notions.
;;;;;;;There are two kinds of definitions: explicit and recursive.  The explicit
;;;;;;;definitions will be treated as abbreviations and no efunction symbol will
;;;;;;;introduced on their behalf.  E-function symbols will be introduced for
;;;;;;;recursively defined e-functions.  We first develop the basic logical connectives
;;;;;;;and analogues for the GCODEd versions of the shell-constructors/destructors/recognizers.
;;;;;;;G-CONS constructs the e-expression for the GCODE analogue of CONS, i.e., (G-CONS X Y)
;;;;;;;evaluates to (GCODE (CONS x y)).  We will adopt the convention of representing the
;;;;;;;value of X by x.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-cons}$\;\;$
{\rm{g-cons}}\,({\it{x\/}}, {\it{y\/}})\index{g-cons} $=$ {\rm{pr-cons}}\,({\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\rm{pr-cons}}\,({\it{x\/}}, {\it{y\/}})\index{pr-cons})\index{pr-cons}
\end{tabbing}
\begin{verbatim};;;;;;;The logical-and connective is defined below.  Note that PR-AND evaluates to 1 or 0.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-and} \\  
{\rm{pr-and}}\,({\it{x\/}}, {\it{y\/}})\index{pr-and} $=$ {\rm{pr-if}}\,({\it{x\/}}, {\rm{pr-if}}\,({\it{y\/}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{pr-if}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{pr-if}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-andn} \\  
\=\+{\rm{pr-andn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-andn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\it{x\/}}, {\tt{0}})\index{ncode-equal}$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }{\rm{btmp}}\,({\it{y\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\it{y\/}}, {\tt{0}})\index{ncode-equal}$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }{\tt{1}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-and-eval \\ \index{pr-and-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-and}}\,({\it{x\/}}, {\it{y\/}})\index{pr-and}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-andn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-andn}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-and; name this event `g0990'.\index{g0990}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-andn-reduc \\ \index{pr-andn-reduc}
{\rm{pr-andn}}\,({\rm{bool-fix}}\,({\it{x\/}})\index{bool-fix}, {\rm{bool-fix}}\,({\it{y\/}})\index{bool-fix})\index{pr-andn} $=$ {\rm{bool-fix}}\,({\it{x\/}} $\wedge$ {\it{y\/}})\index{bool-fix}
\end{tabbing}
\begin{verbatim};;;;;;;The logical-not connective.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-not}$\;\;$
{\rm{pr-not}}\,({\it{x\/}})\index{pr-not} $=$ {\rm{pr-if}}\,({\it{x\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{pr-if}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-notn} \\  
\=\+{\rm{pr-notn}}\,({\it{x\/}})\index{pr-notn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\it{x\/}}, {\tt{0}})\index{ncode-equal}$\;\;${\bf then }{\tt{1}} \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-not-eval \\ \index{pr-not-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-not}}\,({\it{x\/}})\index{pr-not}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-notn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-notn}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-quote; name this event `g0991'.\index{g0991}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-not; name this event `g0992'.\index{g0992}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-notn-reduc \\ \index{pr-notn-reduc}
{\rm{pr-notn}}\,({\rm{bool-fix}}\,({\it{x\/}})\index{bool-fix})\index{pr-notn} $=$ {\rm{bool-fix}}\,($\neg$ {\it{x\/}})\index{bool-fix}
\end{tabbing}
\begin{verbatim};;;;;;;The logical-or connective.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-or} \\  
{\rm{pr-or}}\,({\it{x\/}}, {\it{y\/}})\index{pr-or} $=$ {\rm{pr-if}}\,({\it{x\/}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\rm{pr-if}}\,({\it{y\/}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{pr-if})\index{pr-if}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{pr-orn} \\  
\=\+{\rm{pr-orn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-orn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\it{x\/}}, {\tt{0}})\index{ncode-equal} \\ 
{\bf then }\=\+{\bf if }{\rm{btmp}}\,({\it{y\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\it{y\/}}, {\tt{0}})\index{ncode-equal}$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }{\tt{1}}$\;${\bf  endif}\- \\ 
{\bf else }{\tt{1}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-or-eval \\ \index{pr-or-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{pr-or}}\,({\it{x\/}}, {\it{y\/}})\index{pr-or}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{pr-orn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-orn}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-or; name this event `g0993'.\index{g0993}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-orn-reduc \\ \index{pr-orn-reduc}
{\rm{pr-orn}}\,({\rm{bool-fix}}\,({\it{x\/}})\index{bool-fix}, {\rm{bool-fix}}\,({\it{y\/}})\index{bool-fix})\index{pr-orn} $=$ {\rm{bool-fix}}\,({\it{x\/}} $\vee$ {\it{y\/}})\index{bool-fix}
\end{tabbing}
\begin{verbatim};;;;;;;The next e-definition is an important one.  Given (GCODE X), we can recover
;;;;;;;the type of X by looking at (CAR X).  If it is 1, X is a list; if it is 2,
;;;;;;;X is of the form (F-NOT Y); if it is 3, X is of the form (F-OR Y Z); if it is
;;;;;;;4, X is of the form (FORSOME Y Z); otherwise it could either be a natural number
;;;;;;;for which we already have a NUMBERP primitive, or any remaining type which is
;;;;;;;coerced to NIL.  G-TYPEP checks if X is of type TYPE.  Note that TYPE is going
;;;;;;;to be a number, not an e-expression, since it appears withing PR-QUOTE in G-TYPEP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-typep} \\  
{\rm{g-typep}}\,({\it{x\/}}, {\it{type\/}})\index{g-typep} $=$ {\rm{pr-equal}}\,({\rm{pr-car}}\,({\it{x\/}})\index{pr-car}, {\rm{pr-quote}}\,({\it{type\/}})\index{pr-quote})\index{pr-equal}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-typepn}$\;\;$
{\rm{g-typepn}}\,({\it{x\/}}, {\it{type\/}})\index{g-typepn} $=$ {\rm{pr-equaln}}\,({\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, {\it{type\/}})\index{pr-equaln}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-typep-eval \\ \index{g-typep-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-typep}}\,({\it{x\/}}, {\it{type\/}})\index{g-typep}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-typepn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\it{type\/}})\index{g-typepn}
\end{tabbing}

\noindent{\sc Event}:   Enable g-typep; name this event `g0995'.\index{g0995}

\addvspace{18pt}
\begin{verbatim};;;;;;;PR-IFN is the Lisp analogue of PR-IF.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-ifn} \\  
\=\+{\rm{pr-ifn}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{pr-ifn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\rm{ncode-equal}}\,({\it{x\/}}, {\tt{0}})\index{ncode-equal}$\;\;${\bf then }{\it{z\/}} \\ 
{\bf else }{\it{y\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;PR-CAR-CDR is the iterated CAR/CDR e-expression.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{pr-car-cdr} \\  
\=\+{\rm{pr-car-cdr}}\,({\it{x\/}}, {\it{list\/}})\index{pr-car-cdr} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{list\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{list\/}}) $=$ {\tt{1}}$\;\;${\bf then }{\rm{pr-car}}\,({\rm{pr-car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{pr-car-cdr})\index{pr-car} \\ 
{\bf else }{\rm{pr-cdr}}\,({\rm{pr-car-cdr}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{pr-car-cdr})\index{pr-cdr}$\;${\bf  endif}\- \\ 
{\bf else }{\it{x\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The destructors for GCODEd objects are derived from G-ARG below.  
;;;;;;;Let us represent (GCODE X) as \X\.
;;;;;;;G-ARG selects a field from \X\ according to its type TYPE and the
;;;;;;;desired field ARGNUM.  E.g., (G-ARG Y (PR-QUOTE 1) 1) evaluates to
;;;;;;;\(CAR X)\, if Y evaluates to \X\.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-arg} \\  
\=\+{\rm{g-arg}}\,({\it{x\/}}, {\it{argnum\/}}, {\it{type\/}})\index{g-arg} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\it{x\/}}, {\it{type\/}})\index{g-typep}, \\ 
{\rm{pr-if}}\,({\it{argnum\/}}, {\rm{pr-car-cdr}}\,({\it{x\/}}, {\rm{list}}\,({\tt{2}}, {\tt{2}}))\index{pr-car-cdr}, {\rm{pr-car-cdr}}\,({\it{x\/}}, {\rm{list}}\,({\tt{1}}, {\tt{2}}))\index{pr-car-cdr})\index{pr-if}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-typepn; name this event `g0997'.\index{g0997}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-carn; name this event `g0998'.\index{g0998}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-cdrn; name this event `g0999'.\index{g0999}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-argn} \\  
\=\+{\rm{g-argn}}\,({\it{x\/}}, {\it{argnum\/}}, {\it{type\/}})\index{g-argn} \\ 
$=$$\;\;\;\;${\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{x\/}}, {\it{type\/}})\index{g-typepn}, \\ 
{\rm{pr-ifn}}\,({\it{argnum\/}}, {\rm{pr-cdrn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-cdrn}, {\rm{pr-carn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-carn})\index{pr-ifn}, \\ 
{\tt{0}})\-\index{pr-ifn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-arg-eval \\ \index{g-arg-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-arg}}\,({\it{x\/}}, {\it{argnum\/}}, {\it{type\/}})\index{g-arg}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-argn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{argnum\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\it{type\/}})\index{g-argn}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-arg; name this event `g1000'.\index{g1000}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-typepn; name this event `g1001'.\index{g1001}

\addvspace{18pt}

\noindent{\sc Event}:   Disable pr-carn; name this event `g1003'.\index{g1003}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-carn-gcode \\ \index{pr-carn-gcode}
{\rm{pr-carn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{pr-carn} $=$ {\rm{car}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Disable pr-cdrn; name this event `g1004'.\index{g1004}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-cdrn-gcode \\ \index{pr-cdrn-gcode}
{\rm{pr-cdrn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{pr-cdrn} $=$ {\rm{cdr}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})
\end{tabbing}
\begin{verbatim};;;;;;;G-TYPEPN on Lists, negations, disjunctions, and existential quantifications.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  g-typepn-gcode-list \\ \index{g-typepn-gcode-list}
{\rm{g-typepn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{1}})\index{g-typepn} $=$ {\rm{bool-fix}}\,({\rm{listp}}\,({\it{x\/}}))\index{bool-fix}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-typepn-gcode-f-notp \\ \index{g-typepn-gcode-f-notp}
{\rm{g-typepn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{2}})\index{g-typepn} $=$ {\rm{bool-fix}}\,({\rm{f-notp}}\,({\it{x\/}})\index{f-notp})\index{bool-fix}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-typepn-gcode-f-orp \\ \index{g-typepn-gcode-f-orp}
{\rm{g-typepn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{3}})\index{g-typepn} $=$ {\rm{bool-fix}}\,({\rm{f-orp}}\,({\it{x\/}})\index{f-orp})\index{bool-fix}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-typepn-gcode-forsomep \\ \index{g-typepn-gcode-forsomep}
{\rm{g-typepn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{4}})\index{g-typepn} $=$ {\rm{bool-fix}}\,({\rm{forsomep}}\,({\it{x\/}})\index{forsomep})\index{bool-fix}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  car-gcode \\ \index{car-gcode}
{\rm{g-argn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{0}}, {\tt{1}})\index{g-argn} $=$ {\rm{gcode}}\,({\rm{car}}\,({\it{x\/}}))\index{gcode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  cdr-gcode \\ \index{cdr-gcode}
{\rm{g-argn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{1}}, {\tt{1}})\index{g-argn} $=$ {\rm{gcode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{gcode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  arg-gcode \\ \index{arg-gcode}
{\rm{g-argn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{0}}, {\tt{2}})\index{g-argn} $=$ {\rm{gcode}}\,({\rm{arg}}\,({\it{x\/}})\index{arg})\index{gcode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  arg1-gcode \\ \index{arg1-gcode}
{\rm{g-argn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{0}}, {\tt{3}})\index{g-argn} $=$ {\rm{gcode}}\,({\rm{arg1}}\,({\it{x\/}})\index{arg1})\index{gcode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  arg2-gcode \\ \index{arg2-gcode}
{\rm{g-argn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{1}}, {\tt{3}})\index{g-argn} $=$ {\rm{gcode}}\,({\rm{arg2}}\,({\it{x\/}})\index{arg2})\index{gcode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  bind-gcode \\ \index{bind-gcode}
{\rm{g-argn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{0}}, {\tt{4}})\index{g-argn} $=$ {\rm{gcode}}\,({\rm{bind}}\,({\it{x\/}})\index{bind})\index{gcode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  body-gcode \\ \index{body-gcode}
{\rm{g-argn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\tt{1}}, {\tt{4}})\index{g-argn} $=$ {\rm{gcode}}\,({\rm{body}}\,({\it{x\/}}))\index{gcode}
\end{tabbing}
\begin{verbatim};;;;;;;From G-ARG, we derive the selectors (destructors) for the GCODEd objects.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-car}$\;\;$
{\rm{g-car}}\,({\it{x\/}})\index{g-car} $=$ {\rm{g-arg}}\,({\it{x\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{1}})\index{g-arg}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-cdr}$\;\;$
{\rm{g-cdr}}\,({\it{x\/}})\index{g-cdr} $=$ {\rm{g-arg}}\,({\it{x\/}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{1}})\index{g-arg}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-carn}$\;\;$
{\rm{g-carn}}\,({\it{x\/}})\index{g-carn} $=$ {\rm{g-argn}}\,({\it{x\/}}, {\tt{0}}, {\tt{1}})\index{g-argn}
\end{tabbing}
\begin{verbatim};;;;;;;G-FUNCTION is the EV analogue of the metatheoretic function FUNCTION.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-function}$\;\;$
{\rm{g-function}}\,({\it{fn\/}})\index{g-function} $=$ {\rm{pr-equal}}\,({\rm{g-car}}\,({\it{fn\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{pr-equal}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-functionn}$\;\;$
{\rm{g-functionn}}\,({\it{fn\/}})\index{g-functionn} $=$ {\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{fn\/}})\index{g-carn}, {\tt{2}})\index{pr-equaln}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-function \\ \index{eval-g-function}
{\rm{ev}}\,({\tt{0}}, {\rm{g-function}}\,({\it{fn\/}})\index{g-function}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-functionn}}\,({\rm{ev}}\,({\tt{0}}, {\it{fn\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-functionn}
\end{tabbing}

\noindent{\sc Event}:   Disable function; name this event `g0969'.\index{g0969}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-functionn-gcode \\ \index{g-functionn-gcode}
{\rm{g-functionn}}\,({\rm{gcode}}\,({\it{fn\/}})\index{gcode})\index{g-functionn} $=$ {\rm{bool-fix}}\,({\rm{function}}\,({\it{fn\/}})\index{function})\index{bool-fix}
\end{tabbing}

\noindent{\sc Event}:   Enable function; name this event `g0970'.\index{g0970}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-function; name this event `g0971'.\index{g0971}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-functionn; name this event `g0972'.\index{g0972}

\addvspace{18pt}
\begin{verbatim};;;;;;;G-VARIABLE is the EV analogue of VARIABLE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-variable}$\;\;$
{\rm{g-variable}}\,({\it{x\/}})\index{g-variable} $=$ {\rm{pr-numberp}}\,({\it{x\/}})\index{pr-numberp}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-variablen}$\;\;$
{\rm{g-variablen}}\,({\it{x\/}})\index{g-variablen} $=$ {\rm{pr-numberpn}}\,({\it{x\/}})\index{pr-numberpn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-variable \\ \index{eval-g-variable}
{\rm{ev}}\,({\tt{0}}, {\rm{g-variable}}\,({\it{x\/}})\index{g-variable}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-variablen}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-variablen}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-variablen-gcode \\ \index{g-variablen-gcode}
{\rm{g-variablen}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{g-variablen} $=$ {\rm{bool-fix}}\,({\rm{variable}}\,({\it{x\/}})\index{variable})\index{bool-fix}
\end{tabbing}

\noindent{\sc Event}:   Enable g-variable; name this event `g0973'.\index{g0973}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-variablen; name this event `g0976'.\index{g0976}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-consn; name this event `g0977'.\index{g0977}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-consn}$\;\;$
{\rm{g-consn}}\,({\it{x\/}}, {\it{y\/}})\index{g-consn} $=$ {\rm{pr-consn}}\,({\tt{1}}, {\rm{pr-consn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-consn})\index{pr-consn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-cons-eval \\ \index{g-cons-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-cons}}\,({\it{x\/}}, {\it{y\/}})\index{g-cons}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-consn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-consn}
\end{tabbing}

\noindent{\sc Event}:   Disable pr-consn; name this event `g0978'.\index{g0978}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-consn-gcode \\ \index{g-consn-gcode}
{\rm{g-consn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-consn} $=$ {\rm{gcode}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-consn; name this event `g0979'.\index{g0979}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-consn; name this event `g0980'.\index{g0980}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-cons; name this event `g0981'.\index{g0981}

\addvspace{18pt}
\begin{verbatim};;;;;;;G-FIX is the EV analogue of FIX.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-fix}$\;\;$
{\rm{g-fix}}\,({\it{x\/}})\index{g-fix} $=$ {\rm{pr-if}}\,({\rm{pr-numberp}}\,({\it{x\/}})\index{pr-numberp}, {\it{x\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{pr-if}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-fixn}$\;\;$
{\rm{g-fixn}}\,({\it{x\/}})\index{g-fixn} $=$ {\rm{pr-ifn}}\,({\rm{pr-numberpn}}\,({\it{x\/}})\index{pr-numberpn}, {\it{x\/}}, {\tt{0}})\index{pr-ifn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-fix \\ \index{eval-g-fix}
{\rm{ev}}\,({\tt{0}}, {\rm{g-fix}}\,({\it{x\/}})\index{g-fix}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-fixn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-fixn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-fix-gcode \\ \index{g-fix-gcode}
{\rm{g-fixn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{g-fixn} $=$ {\rm{gcode}}\,({\rm{fix}}\,({\it{x\/}}))\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable g-fix; name this event `g0982'.\index{g0982}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-fixn; name this event `g0983'.\index{g0983}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-f-not} \\  
{\rm{g-f-not}}\,({\it{x\/}})\index{g-f-not} $=$ {\rm{pr-cons}}\,({\rm{pr-quote}}\,({\tt{2}})\index{pr-quote}, {\rm{pr-cons}}\,({\it{x\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{pr-cons})\index{pr-cons}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-notn}$\;\;$
{\rm{g-f-notn}}\,({\it{x\/}})\index{g-f-notn} $=$ {\rm{pr-consn}}\,({\tt{2}}, {\rm{pr-consn}}\,({\it{x\/}}, {\tt{0}})\index{pr-consn})\index{pr-consn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-f-not \\ \index{eval-g-f-not}
{\rm{ev}}\,({\tt{0}}, {\rm{g-f-not}}\,({\it{x\/}})\index{g-f-not}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-f-notn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-f-notn}
\end{tabbing}

\noindent{\sc Event}:   Disable pr-consn; name this event `g0984'.\index{g0984}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-f-notn-gcode \\ \index{g-f-notn-gcode}
{\rm{g-f-notn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{g-f-notn} $=$ {\rm{gcode}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable g-f-not; name this event `g0985'.\index{g0985}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-notn; name this event `g0986'.\index{g0986}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-f-or}$\;\;$
{\rm{g-f-or}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-or} $=$ {\rm{pr-cons}}\,({\rm{pr-quote}}\,({\tt{3}})\index{pr-quote}, {\rm{pr-cons}}\,({\it{x\/}}, {\it{y\/}})\index{pr-cons})\index{pr-cons}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-orn}$\;\;$
{\rm{g-f-orn}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-orn} $=$ {\rm{pr-consn}}\,({\tt{3}}, {\rm{pr-consn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-consn})\index{pr-consn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-f-or \\ \index{eval-g-f-or}
{\rm{ev}}\,({\tt{0}}, {\rm{g-f-or}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-or}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-f-orn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-f-orn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-f-orn-gcode \\ \index{g-f-orn-gcode}
{\rm{g-f-orn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-f-orn} $=$ {\rm{gcode}}\,({\rm{f-or}}\,({\it{x\/}}, {\it{y\/}})\index{f-or})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable g-f-or; name this event `g0987'.\index{g0987}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-orn; name this event `g0988'.\index{g0988}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-forsome} \\  
{\rm{g-forsome}}\,({\it{x\/}}, {\it{y\/}})\index{g-forsome} $=$ {\rm{pr-cons}}\,({\rm{pr-quote}}\,({\tt{4}})\index{pr-quote}, {\rm{pr-cons}}\,({\rm{g-fix}}\,({\it{x\/}})\index{g-fix}, {\it{y\/}})\index{pr-cons})\index{pr-cons}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-forsomen} \\  
{\rm{g-forsomen}}\,({\it{x\/}}, {\it{y\/}})\index{g-forsomen} $=$ {\rm{pr-consn}}\,({\tt{4}}, {\rm{pr-consn}}\,({\rm{g-fixn}}\,({\it{x\/}})\index{g-fixn}, {\it{y\/}})\index{pr-consn})\index{pr-consn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-forsome \\ \index{eval-g-forsome}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-forsome}}\,({\it{x\/}}, {\it{y\/}})\index{g-forsome}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-forsomen}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-forsomen}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-forsomen-gcode \\ \index{g-forsomen-gcode}
{\rm{g-forsomen}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-forsomen} $=$ {\rm{gcode}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{y\/}})\index{forsome})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable g-forsome; name this event `g0989'.\index{g0989}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-forsomen; name this event `g0994'.\index{g0994}

\addvspace{18pt}
\begin{verbatim};;;;;;;G-PREDICATE is the EV analogue of PREDICATE.  In general G-FOO is the
;;;;;;;EV analogue of FOO.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-predicate}$\;\;$
{\rm{g-predicate}}\,({\it{x\/}})\index{g-predicate} $=$ {\rm{pr-equal}}\,({\rm{g-car}}\,({\it{x\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{3}})\index{pr-quote})\index{pr-equal}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-predicaten}$\;\;$
{\rm{g-predicaten}}\,({\it{x\/}})\index{g-predicaten} $=$ {\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{x\/}})\index{g-carn}, {\tt{3}})\index{pr-equaln}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-predicate \\ \index{eval-g-predicate}
{\rm{ev}}\,({\tt{0}}, {\rm{g-predicate}}\,({\it{x\/}})\index{g-predicate}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-predicaten}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-predicaten}
\end{tabbing}

\noindent{\sc Event}:   Disable predicate; name this event `g0996'.\index{g0996}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-predicaten-gcode \\ \index{g-predicaten-gcode}
{\rm{g-predicaten}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{g-predicaten} $=$ {\rm{bool-fix}}\,({\rm{predicate}}\,({\it{x\/}})\index{predicate})\index{bool-fix}
\end{tabbing}

\noindent{\sc Event}:   Enable predicate; name this event `g1002'.\index{g1002}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-predicate; name this event `g1005'.\index{g1005}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-predicaten; name this event `g1006'.\index{g1006}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-func-pred} \\  
{\rm{g-func-pred}}\,({\it{x\/}})\index{g-func-pred} $=$ {\rm{pr-or}}\,({\rm{g-function}}\,({\it{x\/}})\index{g-function}, {\rm{g-predicate}}\,({\it{x\/}})\index{g-predicate})\index{pr-or}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-func-predn} \\  
{\rm{g-func-predn}}\,({\it{x\/}})\index{g-func-predn} $=$ {\rm{pr-orn}}\,({\rm{g-functionn}}\,({\it{x\/}})\index{g-functionn}, {\rm{g-predicaten}}\,({\it{x\/}})\index{g-predicaten})\index{pr-orn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-func-pred \\ \index{eval-g-func-pred}
{\rm{ev}}\,({\tt{0}}, {\rm{g-func-pred}}\,({\it{x\/}})\index{g-func-pred}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-func-predn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-func-predn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-func-predn-gcode \\ \index{g-func-predn-gcode}
{\rm{g-func-predn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{g-func-predn} $=$ {\rm{bool-fix}}\,({\rm{func-pred}}\,({\it{x\/}})\index{func-pred})\index{bool-fix}
\end{tabbing}

\noindent{\sc Event}:   Enable g-func-pred; name this event `g1007'.\index{g1007}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-func-predn; name this event `g1008'.\index{g1008}

\addvspace{18pt}
\begin{verbatim};;;;;;;Now we start defining the EV analogues of the recursive metatheoretic functions.
;;;;;;;These will all be defined first, and then shown to correspond to their
;;;;;;;Boyer-Moore counterparts.  
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-append}$\;\;$
{\rm{g-append}}\,({\it{x\/}}, {\it{y\/}})\index{g-append} $=$ {\rm{list}}\,({\tt{11}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-cdrn}$\;\;$
{\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn} $=$ {\rm{g-argn}}\,({\it{x\/}}, {\tt{1}}, {\tt{1}})\index{g-argn}
\end{tabbing}

\noindent{\sc Event}:   Enable g-argn; name this event `g1009'.\index{g1009}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-typepn; name this event `g1010'.\index{g1010}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  eval-pr-if-again \\ \index{eval-pr-if-again}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{pr-if}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{pr-if}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{pr-ifn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{z\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{pr-ifn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-ifn-gcode \\ \index{pr-ifn-gcode}
\=\+{\rm{pr-ifn}}\,({\rm{bool-fix}}\,({\it{x\/}})\index{bool-fix}, {\it{y\/}}, {\it{z\/}})\index{pr-ifn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}}$\;\;${\bf then }{\it{y\/}} \\ 
{\bf else }{\it{z\/}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable pr-ifn; name this event `g1011'.\index{g1011}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  eval-var \\ \index{eval-var}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{get}}\,({\it{x\/}}, {\it{va\/}})\index{get})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-fun-zero \\ \index{eval-fun-zero}
(({\tt{10}} $<$ {\it{fn\/}}) $\wedge$ ({\it{n\/}} $\simeq {\tt{0}}$)) $\rightarrow$ ({\rm{ev}}\,({\tt{0}}, {\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\sc{btm}}\index{btm})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-fun1 \\ \index{eval-fun1}
\=\+(({\it{flg\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{listp}}\,({\it{exp\/}}) $\wedge$ ({\tt{10}} $<$ {\rm{car}}\,({\it{exp\/}}))) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{cdr}}\,({\it{exp\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev}$\;${\bf  endif}\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-fun2 \\ \index{eval-fun2}
\=\+({\tt{10}} $<$ {\it{fn\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{args\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf elseif }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\it{fn\/}}, {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\it{args\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev}$\;${\bf  endif}\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-list1 \\ \index{ev-list1}
\=\+{\rm{ev}}\,({\tt{1}}, {\rm{cons}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp} $\vee$ {\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{cons}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{1}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-list2 \\ \index{ev-list2}
{\rm{ev}}\,({\tt{1}}, {\bf{nil}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\bf{nil}}
\end{tabbing}

\noindent{\sc Event}:   Enable ev; name this event `g1012'.\index{g1012}

\addvspace{18pt}

\noindent{\sc Event}:   Disable sub1; name this event `g1013'.\index{g1013}

\addvspace{18pt}

\noindent{\sc Event}:   Disable get; name this event `g1014'.\index{g1014}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-typepn; name this event `g1015'.\index{g1015}

\addvspace{18pt}

\noindent{\sc Event}:   Enable eval-fun1; name this event `g1016'.\index{g1016}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  eval-fun3 \\ \index{eval-fun3}
\=\+(({\tt{10}} $<$ {\it{fn\/}}) $\wedge$ ({\it{n\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{ev}}\,({\tt{0}}, {\rm{cons}}\,({\it{fn\/}}, {\it{args\/}}), {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\rm{ev}}\,({\tt{1}}, {\it{args\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{ev}}\,(\=\+{\tt{0}}, \\ 
{\rm{get}}\,({\rm{sub1-n}}\,({\it{fn\/}}, {\tt{11}})\index{sub1-n}, {\it{fa\/}})\index{get}, \\ 
{\rm{ev}}\,({\tt{1}}, {\it{args\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{ev}$\;${\bf  endif}\-)\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition of G-APPEND.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-append-defn} \\  
\=\+{\sc{g-append-defn}}\index{g-append-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,({\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, {\rm{g-cons}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\rm{g-append}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}})\index{g-append})\index{g-cons}, {\tt{1}})\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-append; name this event `g1017'.\index{g1017}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-DEL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-del}$\;\;$
{\rm{g-del}}\,({\it{x\/}}, {\it{y\/}})\index{g-del} $=$ {\rm{list}}\,({\tt{12}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-del-defn} \\  
\=\+{\sc{g-del-defn}}\index{g-del-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{1}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\tt{0}}, {\rm{g-car}}\,({\tt{1}})\index{g-car})\index{pr-equal}, \\ 
{\rm{g-del}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr})\index{g-del}, \\ 
{\rm{g-cons}}\,({\rm{g-car}}\,({\tt{1}})\index{g-car}, {\rm{g-del}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr})\index{g-del})\index{g-cons})\-\index{pr-if}, \\ 
{\tt{1}})\-\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-del; name this event `g1018'.\index{g1018}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-MEMB.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-memb}$\;\;$
{\rm{g-memb}}\,({\it{x\/}}, {\it{y\/}})\index{g-memb} $=$ {\rm{list}}\,({\tt{13}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-memb-defn} \\  
\=\+{\sc{g-memb-defn}}\index{g-memb-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{1}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-if}}\,({\rm{pr-equal}}\,({\tt{0}}, {\rm{g-car}}\,({\tt{1}})\index{g-car})\index{pr-equal}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\rm{g-memb}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr})\index{g-memb})\index{pr-if}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-memb; name this event `g1019'.\index{g1019}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-COLLECT-FREE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-collect-free}$\;\;$
{\rm{g-collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{g-collect-free} $=$ {\rm{list}}\,({\tt{14}}, {\it{exp\/}}, {\it{flg\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-collect-free-defn} \\  
\=\+{\sc{g-collect-free-defn}}\index{g-collect-free-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-variable}}\,({\tt{0}})\index{g-variable}, \\ 
{\rm{g-cons}}\,({\tt{0}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\index{g-cons}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{2}})\index{g-typep}, \\ 
{\rm{g-collect-free}}\,({\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-collect-free}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{3}})\index{g-typep}, \\ 
{\rm{g-append}}\,(\=\+{\rm{g-collect-free}}\,(\=\+{\rm{g-arg}}\,(\=\+{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\tt{3}})\-\index{g-arg}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-collect-free}, \\ 
{\rm{g-collect-free}}\,(\=\+{\rm{g-arg}}\,(\=\+{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\tt{3}})\-\index{g-arg}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-collect-free})\-\index{g-append}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{4}})\index{g-typep}, \\ 
{\rm{g-del}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\rm{g-collect-free}}\,(\=\+{\rm{g-arg}}\,(\=\+{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\tt{4}})\-\index{g-arg}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-collect-free})\-\index{g-del}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{g-collect-free}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\-\index{g-collect-free}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{g-append}}\,(\=\+{\rm{g-collect-free}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-collect-free}, \\ 
{\rm{g-collect-free}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{g-collect-free})\-\index{g-append}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-collect-free; name this event `g1020'.\index{g1020}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-COVERING.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-covering}$\;\;$
{\rm{g-covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{g-covering} $=$ {\rm{list}}\,({\tt{15}}, {\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-covering-defn} \\  
\=\+{\sc{g-covering-defn}}\index{g-covering-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{2}})\index{pr-zerop}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{2}})\index{g-typep}, \\ 
{\rm{g-covering}}\,({\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, {\tt{1}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-covering}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{3}})\index{g-typep}, \\ 
{\rm{g-append}}\,(\=\+{\rm{g-covering}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-covering}, \\ 
{\rm{g-covering}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-covering})\-\index{g-append}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{4}})\index{g-typep}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-memb}}\,({\tt{1}}, {\rm{g-collect-free}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-collect-free})\index{g-memb}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\rm{g-covering}}\,(\=\+{\rm{g-arg}}\,(\=\+{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\tt{4}})\-\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-covering})\-\index{g-cons}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{g-covering}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{g-covering}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{g-append}}\,(\=\+{\rm{g-covering}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\tt{1}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-covering}, \\ 
{\rm{g-covering}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{g-covering})\-\index{g-append}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-covering; name this event `g1021'.\index{g1021}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-NIL-INTERSECT.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-nil-intersect}$\;\;$
{\rm{g-nil-intersect}}\,({\it{x\/}}, {\it{y\/}})\index{g-nil-intersect} $=$ {\rm{list}}\,({\tt{16}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1022'.\index{g1022}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-nil-intersect-defn} \\  
\=\+{\sc{g-nil-intersect-defn}}\index{g-nil-intersect-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-and}}\,({\rm{pr-not}}\,({\rm{g-memb}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\tt{1}})\index{g-memb})\index{pr-not}, {\rm{g-nil-intersect}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}})\index{g-nil-intersect})\index{pr-and}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote})\-\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-nil-intersect; name this event `g1023'.\index{g1023}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-VAR-LIST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-var-list}$\;\;$
{\rm{g-var-list}}\,({\it{list\/}}, {\it{len\/}})\index{g-var-list} $=$ {\rm{list}}\,({\tt{17}}, {\it{list\/}}, {\it{len\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-sub1n-gcode \\ \index{pr-sub1n-gcode}
{\rm{pr-sub1n}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{pr-sub1n} $=$ {\rm{gcode}}\,({\it{x\/}} $-\;1$)\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-sub1n; name this event `g1024'.\index{g1024}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-var-list-defn} \\  
\=\+{\sc{g-var-list-defn}}\index{g-var-list-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-or}}\,({\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, {\rm{pr-not}}\,({\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep})\index{pr-not})\index{pr-or}, \\ 
{\rm{pr-and}}\,({\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, {\rm{pr-not}}\,({\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep})\index{pr-not})\index{pr-and}, \\ 
{\rm{pr-and}}\,({\rm{g-variable}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car})\index{g-variable}, {\rm{g-var-list}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1})\index{g-var-list})\index{pr-and})\-\index{pr-if}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-var-list; name this event `g1025'.\index{g1025}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-TERMP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-termp} \\  
{\rm{g-termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{g-termp} $=$ {\rm{list}}\,({\tt{19}}, {\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})
\end{tabbing}

\noindent{\sc Event}:   Enable pr-equaln; name this event `g1026'.\index{g1026}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-symb} \\  
\=\+{\rm{g-symb}}\,({\it{x\/}}, {\it{symbols\/}})\index{g-symb} \\ 
$=$$\;\;\;\;${\rm{pr-or}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\it{x\/}}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode})\index{pr-quote}, \\ 
{\rm{g-cons}}\,({\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode})\index{pr-quote})\index{g-cons})\-\index{g-cons})\-\index{pr-equal}, \\ 
{\rm{g-memb}}\,({\it{x\/}}, {\it{symbols\/}})\index{g-memb})\-\index{pr-or}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-symb; name this event `g1027'.\index{g1027}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-length}$\;\;$
{\rm{g-length}}\,({\it{x\/}})\index{g-length} $=$ {\rm{list}}\,({\tt{18}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-length-defn} \\  
\=\+{\sc{g-length-defn}}\index{g-length-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,({\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, {\rm{pr-add1}}\,({\rm{g-length}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr})\index{g-length})\index{pr-add1}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote})\index{pr-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-termp-defn} \\  
\=\+{\sc{g-termp-defn}}\index{g-termp-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-function}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car})\index{g-function}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-symb}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\tt{2}})\index{g-symb}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{2}})\index{g-termp}, \\ 
{\rm{pr-equal}}\,(\=\+{\rm{g-length}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr})\index{g-length}, \\ 
{\rm{g-fix}}\,({\rm{g-cdr}}\,({\rm{g-cdr}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car})\index{g-cdr})\index{g-cdr})\index{g-fix})\-\index{pr-equal})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}, \\ 
{\rm{g-variable}}\,({\tt{0}})\index{g-variable})\-\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-termp}, \\ 
{\rm{g-termp}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{2}})\index{g-termp})\-\index{pr-and}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition of G-SUBST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-subst} \\  
{\rm{g-subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{g-subst} $=$ {\rm{list}}\,({\tt{20}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-subst-defn} \\  
\=\+{\sc{g-subst-defn}}\index{g-subst-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{3}})\index{pr-zerop}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-variable}}\,({\tt{0}})\index{g-variable}, \\ 
{\rm{pr-if}}\,({\rm{pr-equal}}\,({\tt{0}}, {\tt{1}})\index{pr-equal}, {\tt{2}}, {\tt{0}})\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{2}})\index{g-typep}, \\ 
{\rm{g-f-not}}\,({\rm{g-subst}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote})\-\index{g-subst})\index{g-f-not}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{3}})\index{g-typep}, \\ 
{\rm{g-f-or}}\,(\=\+{\rm{g-subst}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote})\-\index{g-subst}, \\ 
{\rm{g-subst}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote})\-\index{g-subst})\-\index{g-f-or}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{4}})\index{g-typep}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\tt{1}})\-\index{pr-equal}, \\ 
{\tt{0}}, \\ 
{\rm{g-forsome}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\rm{g-subst}}\,(\=\+{\rm{g-arg}}\,(\=\+{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\tt{4}})\-\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote})\-\index{g-subst})\-\index{g-forsome})\-\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-car}}\,({\tt{0}})\index{g-car}, \\ 
{\rm{g-subst}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{1}})\index{gcode})\index{pr-quote})\-\index{g-subst})\-\index{g-cons}, \\ 
{\tt{0}})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-subst}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\tt{1}}, {\tt{2}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote})\index{g-subst}, \\ 
{\rm{g-subst}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}}, {\tt{2}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{1}})\index{gcode})\index{pr-quote})\index{g-subst})\-\index{g-cons}, \\ 
{\tt{0}})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-sub}$\;\;$
{\rm{g-sub}}\,({\it{x\/}}, {\it{y\/}})\index{g-sub} $=$ {\rm{list}}\,({\tt{21}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-sub-defn} \\  
\=\+{\sc{g-sub-defn}}\index{g-sub-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-and}}\,({\rm{g-memb}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\tt{1}})\index{g-memb}, {\rm{g-sub}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}})\index{g-sub})\index{pr-and}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition of G-FORMULA.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-formula}$\;\;$
{\rm{g-formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{g-formula} $=$ {\rm{list}}\,({\tt{22}}, {\it{exp\/}}, {\it{symbols\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-atomp} \\  
\=\+{\rm{g-atomp}}\,({\it{exp\/}}, {\it{symbols\/}})\index{g-atomp} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-predicate}}\,({\rm{g-car}}\,({\it{exp\/}})\index{g-car})\index{g-predicate}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-symb}}\,({\rm{g-car}}\,({\it{exp\/}})\index{g-car}, {\it{symbols\/}})\index{g-symb}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-length}}\,({\rm{g-cdr}}\,({\it{exp\/}})\index{g-cdr})\index{g-length}, \\ 
{\rm{g-fix}}\,({\rm{g-cdr}}\,({\rm{g-cdr}}\,({\rm{g-car}}\,({\it{exp\/}})\index{g-car})\index{g-cdr})\index{g-cdr})\index{g-fix})\-\index{pr-equal}, \\ 
{\rm{g-termp}}\,({\rm{g-cdr}}\,({\it{exp\/}})\index{g-cdr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\it{symbols\/}})\index{g-termp})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-atomp; name this event `g1028'.\index{g1028}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-formula-defn} \\  
\=\+{\sc{g-formula-defn}}\index{g-formula-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{2}})\index{g-typep}, \\ 
{\rm{g-formula}}\,({\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, {\tt{1}})\index{g-formula}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{3}})\index{g-typep}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{3}})\index{g-arg}, {\tt{1}})\index{g-formula}, \\ 
{\rm{g-formula}}\,({\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{3}})\index{g-arg}, {\tt{1}})\index{g-formula})\-\index{pr-and}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{4}})\index{g-typep}, \\ 
{\rm{g-formula}}\,({\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{4}})\index{g-arg}, {\tt{1}})\index{g-formula}, \\ 
{\rm{g-atomp}}\,({\tt{0}}, {\tt{1}})\index{g-atomp})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-pairequals} \\  
{\rm{g-pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}})\index{g-pairequals} $=$ {\rm{list}}\,({\tt{23}}, {\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-and} \\  
{\rm{g-f-and}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-and} $=$ {\rm{g-f-not}}\,({\rm{g-f-or}}\,({\rm{g-f-not}}\,({\it{x\/}})\index{g-f-not}, {\rm{g-f-not}}\,({\it{y\/}})\index{g-f-not})\index{g-f-or})\index{g-f-not}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-andn} \\  
{\rm{g-f-andn}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-andn} $=$ {\rm{g-f-notn}}\,({\rm{g-f-orn}}\,({\rm{g-f-notn}}\,({\it{x\/}})\index{g-f-notn}, {\rm{g-f-notn}}\,({\it{y\/}})\index{g-f-notn})\index{g-f-orn})\index{g-f-notn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-f-and-eval \\ \index{g-f-and-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-f-and}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-and}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-f-andn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-f-andn}
\end{tabbing}

\noindent{\sc Event}:   Disable f-and; name this event `g1098'.\index{g1098}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-f-andn-gcode \\ \index{g-f-andn-gcode}
{\rm{g-f-andn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-f-andn} $=$ {\rm{gcode}}\,({\rm{f-and}}\,({\it{x\/}}, {\it{y\/}})\index{f-and})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable f-and; name this event `g1099'.\index{g1099}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-and; name this event `g1100'.\index{g1100}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-andn; name this event `g1101'.\index{g1101}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-f-implies}$\;\;$
{\rm{g-f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-implies} $=$ {\rm{g-f-or}}\,({\rm{g-f-not}}\,({\it{x\/}})\index{g-f-not}, {\it{y\/}})\index{g-f-or}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-impliesn}$\;\;$
{\rm{g-f-impliesn}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-impliesn} $=$ {\rm{g-f-orn}}\,({\rm{g-f-notn}}\,({\it{x\/}})\index{g-f-notn}, {\it{y\/}})\index{g-f-orn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-f-implies-eval \\ \index{g-f-implies-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-implies}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-f-impliesn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-f-impliesn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable f-implies; name this event `g1102'.\index{g1102}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-f-impliesn-gcode \\ \index{g-f-impliesn-gcode}
{\rm{g-f-impliesn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-f-impliesn} $=$ {\rm{gcode}}\,({\rm{f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{f-implies})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable f-implies; name this event `g1103'.\index{g1103}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-implies; name this event `g1104'.\index{g1104}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-impliesn; name this event `g1105'.\index{g1105}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-forall} \\  
{\rm{g-forall}}\,({\it{var\/}}, {\it{exp\/}})\index{g-forall} $=$ {\rm{g-f-not}}\,({\rm{g-forsome}}\,({\it{var\/}}, {\rm{g-f-not}}\,({\it{exp\/}})\index{g-f-not})\index{g-forsome})\index{g-f-not}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-foralln} \\  
{\rm{g-foralln}}\,({\it{var\/}}, {\it{exp\/}})\index{g-foralln} $=$ {\rm{g-f-notn}}\,({\rm{g-forsomen}}\,({\it{var\/}}, {\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn})\index{g-forsomen})\index{g-f-notn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-forall-eval \\ \index{g-forall-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-forall}}\,({\it{x\/}}, {\it{y\/}})\index{g-forall}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-foralln}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-foralln}
\end{tabbing}

\noindent{\sc Event}:   Disable forall; name this event `g1106'.\index{g1106}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-foralln-gcode \\ \index{g-foralln-gcode}
{\rm{g-foralln}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-foralln} $=$ {\rm{gcode}}\,($\forall\;${\it{x\/}}$\;${\it{y\/}}\index{forall})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable forall; name this event `g1107'.\index{g1107}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-forall; name this event `g1108'.\index{g1108}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-foralln; name this event `g1109'.\index{g1109}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-f-equal} \\  
\=\+{\rm{g-f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-equal} \\ 
$=$$\;\;\;\;${\rm{g-cons}}\,(\=\+{\rm{g-cons}}\,(\=\+{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode})\index{pr-quote}, \\ 
{\rm{g-cons}}\,({\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode})\index{pr-quote})\index{g-cons})\-\index{g-cons}, \\ 
{\rm{g-cons}}\,({\it{x\/}}, {\rm{g-cons}}\,({\it{y\/}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\index{g-cons})\index{g-cons})\-\index{g-cons}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-equaln} \\  
\=\+{\rm{g-f-equaln}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-equaln} \\ 
$=$$\;\;\;\;${\rm{g-consn}}\,(\=\+{\rm{g-consn}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode}, {\rm{gcode}}\,({\tt{2}})\index{gcode})\index{g-consn})\index{g-consn}, \\ 
{\rm{g-consn}}\,({\it{x\/}}, {\rm{g-consn}}\,({\it{y\/}}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn})\index{g-consn})\-\index{g-consn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-f-equal-eval \\ \index{g-f-equal-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-equal}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-f-equaln}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-f-equaln}\-
\end{tabbing}

\noindent{\sc Event}:   Disable p; name this event `g1049'.\index{g1049}

\addvspace{18pt}

\noindent{\sc Event}:   Disable f-equal; name this event `g1050'.\index{g1050}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-f-equaln-gcode \\ \index{g-f-equaln-gcode}
{\rm{g-f-equaln}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-f-equaln} $=$ {\rm{gcode}}\,({\rm{f-equal}}\,({\it{x\/}}, {\it{y\/}})\index{f-equal})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable p; name this event `g1051'.\index{g1051}

\addvspace{18pt}

\noindent{\sc Event}:   Enable f-equal; name this event `g1052'.\index{g1052}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-equal; name this event `g1053'.\index{g1053}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-equaln; name this event `g1054'.\index{g1054}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-PAIREQUALS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-pairequals-defn} \\  
\=\+{\sc{g-pairequals-defn}}\index{g-pairequals-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-and}}\,({\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, {\rm{g-typep}}\,({\tt{1}}, {\tt{1}})\index{g-typep})\index{pr-and}, \\ 
{\rm{g-f-implies}}\,(\=\+{\rm{g-f-equal}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\rm{g-car}}\,({\tt{1}})\index{g-car})\index{g-f-equal}, \\ 
{\rm{g-pairequals}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr}, {\tt{2}})\index{g-pairequals})\-\index{g-f-implies}, \\ 
{\tt{2}})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition of G-SET.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-set}$\;\;$
{\rm{g-set}}\,({\it{x\/}})\index{g-set} $=$ {\rm{list}}\,({\tt{24}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-set-defn} \\  
\=\+{\sc{g-set-defn}}\index{g-set-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-and}}\,({\rm{pr-not}}\,({\rm{g-memb}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\rm{g-cdr}}\,({\tt{0}})\index{g-cdr})\index{g-memb})\index{pr-not}, {\rm{g-set}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr})\index{g-set})\index{pr-and}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote})\-\index{pr-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-conc}$\;\;$
{\rm{g-conc}}\,({\it{pf\/}}, {\it{flg\/}})\index{g-conc} $=$ {\rm{list}}\,({\tt{25}}, {\it{pf\/}}, {\it{flg\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-cadr}$\;\;$
{\rm{g-cadr}}\,({\it{x\/}})\index{g-cadr} $=$ {\rm{g-car}}\,({\rm{g-cdr}}\,({\it{x\/}})\index{g-cdr})\index{g-car}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-caddr}$\;\;$
{\rm{g-caddr}}\,({\it{x\/}})\index{g-caddr} $=$ {\rm{g-car}}\,({\rm{g-cdr}}\,({\rm{g-cdr}}\,({\it{x\/}})\index{g-cdr})\index{g-cdr})\index{g-car}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-cadddr}$\;\;$
{\rm{g-cadddr}}\,({\it{x\/}})\index{g-cadddr} $=$ {\rm{g-car}}\,({\rm{g-cdr}}\,({\rm{g-cdr}}\,({\rm{g-cdr}}\,({\it{x\/}})\index{g-cdr})\index{g-cdr})\index{g-cdr})\index{g-car}
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition of G-CONC.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-conc-defn} \\  
\=\+{\sc{g-conc-defn}}\index{g-conc-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, \\ 
{\rm{g-caddr}}\,({\tt{0}})\index{g-caddr}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{0}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-conc}}\,({\rm{g-car}}\,({\tt{0}})\index{g-car}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-conc}, \\ 
{\rm{g-conc}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{g-conc})\-\index{g-cons}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition of G-REST-OF.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-rest-of}$\;\;$
{\rm{g-rest-of}}\,({\it{sym\/}}, {\it{symbols\/}})\index{g-rest-of} $=$ {\rm{list}}\,({\tt{26}}, {\it{sym\/}}, {\it{symbols\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-rest-of-defn} \\  
\=\+{\sc{g-rest-of-defn}}\index{g-rest-of-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{1}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-and}}\,({\rm{pr-equal}}\,({\rm{g-car}}\,({\tt{1}})\index{g-car}, {\tt{0}})\index{pr-equal}, {\rm{pr-not}}\,({\rm{g-memb}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr})\index{g-memb})\index{pr-not})\index{pr-and}, \\ 
{\rm{g-cdr}}\,({\tt{1}})\index{g-cdr}, \\ 
{\rm{g-rest-of}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr})\index{g-rest-of})\-\index{pr-if}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition of G-ASSOC.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-assoc}$\;\;$
{\rm{g-assoc}}\,({\it{x\/}}, {\it{y\/}})\index{g-assoc} $=$ {\rm{list}}\,({\tt{27}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-assoc-defn} \\  
\=\+{\sc{g-assoc-defn}}\index{g-assoc-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\tt{1}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\rm{g-car}}\,({\tt{1}})\index{g-car})\index{g-car}, {\tt{0}})\index{pr-equal}, \\ 
{\rm{g-cdr}}\,({\rm{g-car}}\,({\tt{1}})\index{g-car})\index{g-cdr}, \\ 
{\rm{g-assoc}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr})\index{g-assoc})\-\index{pr-if}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{pr-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-prf} \\  
{\rm{g-prf}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{g-prf} $=$ {\rm{list}}\,({\tt{28}}, {\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-free-for} \\  
\=\+{\rm{g-free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{g-free-for} \\ 
$=$$\;\;\;\;${\rm{g-nil-intersect}}\,(\=\+{\rm{g-covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{g-covering}, \\ 
{\rm{g-collect-free}}\,({\it{term\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-collect-free})\-\index{g-nil-intersect}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-free-for; name this event `g1029'.\index{g1029}

\addvspace{18pt}
\begin{verbatim};;;;;;;The EV analogues of the FOL axiom-checkers.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-subst-axiom-proof} \\  
\=\+{\rm{g-subst-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-subst-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-variable}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\index{g-variable}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\it{symbols\/}})\index{g-termp}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-free-for}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-free-for}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-implies}}\,(\=\+{\rm{g-subst}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-subst}, \\ 
{\rm{g-forsome}}\,(\=\+{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\-\index{g-forsome})\-\index{g-f-implies})\-\index{pr-equal})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-degree}$\;\;$
{\rm{g-degree}}\,({\it{x\/}})\index{g-degree} $=$ {\rm{g-fix}}\,({\rm{g-cdr}}\,({\rm{g-cdr}}\,({\it{x\/}})\index{g-cdr})\index{g-cdr})\index{g-fix}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-fun-equal-axiom-proof} \\  
\=\+{\rm{g-fun-equal-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-fun-equal-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\it{symbols\/}})\index{g-termp}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\it{symbols\/}})\index{g-termp}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-function}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-function}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-length}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\index{g-length}, \\ 
{\rm{g-degree}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-degree})\-\index{pr-equal}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-length}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\index{g-length}, \\ 
{\rm{g-degree}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-degree})\-\index{pr-equal}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-pairequals}}\,(\=\+{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, \\ 
{\rm{g-f-equal}}\,(\=\+{\rm{g-cons}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\-\index{g-cons}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-cons})\-\index{g-f-equal})\-\index{g-pairequals})\-\index{pr-equal})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-pred-equal-axiom-proof} \\  
\=\+{\rm{g-pred-equal-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-pred-equal-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\it{symbols\/}})\index{g-termp}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\it{symbols\/}})\index{g-termp}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-predicate}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-predicate}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-length}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\index{g-length}, \\ 
{\rm{g-degree}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-degree})\-\index{pr-equal}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-length}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\index{g-length}, \\ 
{\rm{g-degree}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-degree})\-\index{pr-equal}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-pairequals}}\,(\=\+{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, \\ 
{\rm{g-f-implies}}\,(\=\+{\rm{g-cons}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\-\index{g-cons}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-cons})\-\index{g-f-implies})\-\index{g-pairequals})\-\index{pr-equal})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{verbatim};;;;;;;The EV analogue of FOL-AXIOM-PROOF.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-fol-axiom-proof} \\  
\=\+{\rm{g-fol-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-fol-axiom-proof} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-or}}\,({\rm{g-f-not}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-f-not}, {\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-f-or})\-\index{pr-equal})\-\index{pr-and}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-subst-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-subst-axiom-proof}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{3}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-termp}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\it{symbols\/}})\index{g-termp}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-equal}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\-\index{g-f-equal})\-\index{pr-equal})\-\index{pr-and}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{4}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-fun-equal-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-fun-equal-axiom-proof}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{5}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-pred-equal-axiom-proof}}\,(\=\+{\it{conc\/}}, \\ 
{\it{hints\/}}, \\ 
{\it{symbols\/}})\-\index{g-pred-equal-axiom-proof}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-quote})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The EV analogues of the various FOL rule-checkers.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-expan-rule} \\  
\=\+{\rm{g-expan-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-expan-rule} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-car}}\,({\it{sub-goals\/}})\index{g-car}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-equal}}\,({\it{conc\/}}, {\rm{g-f-or}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\rm{g-car}}\,({\it{sub-goals\/}})\index{g-car})\index{g-f-or})\index{pr-equal})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-contrac-rule} \\  
\=\+{\rm{g-contrac-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-contrac-rule} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-f-or}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-f-or}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{g-cons})\-\index{pr-equal}, \\ 
{\rm{pr-equal}}\,({\it{conc\/}}, {\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{pr-equal})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-assoc-rule} \\  
\=\+{\rm{g-assoc-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-assoc-rule} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-f-or}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-f-or}}\,(\=\+{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-f-or})\-\index{g-f-or}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{g-cons})\-\index{pr-equal}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-or}}\,(\=\+{\rm{g-f-or}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\-\index{g-f-or}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-f-or})\-\index{pr-equal})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-cut-rule} \\  
\=\+{\rm{g-cut-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-cut-rule} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-f-or}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\-\index{g-f-or}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-f-or}}\,(\=\+{\rm{g-f-not}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-f-not}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-f-or}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{g-cons})\-\index{g-cons})\-\index{pr-equal}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-or}}\,(\=\+{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-f-or})\-\index{pr-equal})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-forsome-intro-rule} \\  
\=\+{\rm{g-forsome-intro-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-forsome-intro-rule} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{g-variable}}\,({\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr})\index{g-variable}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-not}}\,({\rm{g-memb}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-collect-free}}\,(\=\+{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-collect-free})\-\index{g-memb})\index{pr-not}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-f-implies}}\,(\=\+{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-f-implies}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{g-cons})\-\index{pr-equal}, \\ 
{\rm{pr-equal}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-implies}}\,(\=\+{\rm{g-forsome}}\,(\=\+{\rm{g-cadr}}\,({\it{hints\/}})\index{g-cadr}, \\ 
{\rm{g-caddr}}\,({\it{hints\/}})\index{g-caddr})\-\index{g-forsome}, \\ 
{\rm{g-cadddr}}\,({\it{hints\/}})\index{g-cadddr})\-\index{g-f-implies})\-\index{pr-equal})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}
\begin{verbatim};;;;;;;The EV analogue of G-FOL-RULES.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-fol-rules} \\  
\=\+{\rm{g-fol-rules}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-fol-rules} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-expan-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-expan-rule}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-contrac-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-contrac-rule}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{3}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-assoc-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-assoc-rule}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{4}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-cut-rule}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-cut-rule}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{hints\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{5}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-forsome-intro-rule}}\,(\=\+{\it{conc\/}}, \\ 
{\it{hints\/}}, \\ 
{\it{sub-goals\/}}, \\ 
{\it{symbols\/}})\-\index{g-forsome-intro-rule}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-quote})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The EV analogues of the various set-theoretic operations.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-isin} \\  
\=\+{\rm{g-isin}}\,({\it{x\/}}, {\it{y\/}})\index{g-isin} \\ 
$=$$\;\;\;\;${\rm{g-cons}}\,(\=\+{\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{3}})\index{pr-quote}, {\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{g-cons})\index{g-cons}, \\ 
{\rm{g-cons}}\,({\it{x\/}}, {\rm{g-cons}}\,({\it{y\/}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\index{g-cons})\index{g-cons})\-\index{g-cons}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-phi} \\  
\=\+{\sc{g-phi}}\index{g-phi} \\ 
$=$$\;\;\;\;${\rm{g-cons}}\,(\=\+{\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{2}})\index{pr-quote}, {\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-cons})\index{g-cons}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{g-cons}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-pair} \\  
\=\+{\rm{g-z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-pair} \\ 
$=$$\;\;\;\;${\rm{g-cons}}\,(\=\+{\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{2}})\index{pr-quote}, {\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{g-cons})\index{g-cons}, \\ 
{\rm{g-cons}}\,({\it{x\/}}, {\rm{g-cons}}\,({\it{y\/}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\index{g-cons})\index{g-cons})\-\index{g-cons}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-union} \\  
\=\+{\rm{g-z-union}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-union} \\ 
$=$$\;\;\;\;${\rm{g-cons}}\,(\=\+{\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{2}})\index{pr-quote}, {\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{2}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{g-cons})\index{g-cons}, \\ 
{\rm{g-cons}}\,({\it{x\/}}, {\rm{g-cons}}\,({\it{y\/}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\index{g-cons})\index{g-cons})\-\index{g-cons}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-int} \\  
\=\+{\rm{g-z-int}}\,({\it{x\/}})\index{g-z-int} \\ 
$=$$\;\;\;\;${\rm{g-cons}}\,(\=\+{\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{3}})\index{pr-quote}, {\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{2}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{g-cons})\index{g-cons}, \\ 
{\rm{g-cons}}\,({\it{x\/}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\index{g-cons})\-\index{g-cons}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-succ} \\  
\=\+{\rm{g-z-succ}}\,({\it{x\/}})\index{g-z-succ} \\ 
$=$$\;\;\;\;${\rm{g-cons}}\,(\=\+{\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{2}})\index{pr-quote}, {\rm{g-cons}}\,({\rm{pr-quote}}\,({\tt{3}})\index{pr-quote}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{g-cons})\index{g-cons}, \\ 
{\rm{g-cons}}\,({\it{x\/}}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\index{g-cons})\-\index{g-cons}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-iff} \\  
{\rm{g-f-iff}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-iff} $=$ {\rm{g-f-and}}\,({\rm{g-f-implies}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-implies}, {\rm{g-f-implies}}\,({\it{y\/}}, {\it{x\/}})\index{g-f-implies})\index{g-f-and}
\end{tabbing}
\begin{verbatim};;;;;;;The EV analogue of Z2-AXIOMS.
\end{verbatim}%\item 
\begin{verbatim}

;;;;;;;The EV analogue of Z2-AXIOMS.
(DEFN
 G-Z2-AXIOMS
 (CONC HINTS SYMBOLS)
 (PR-IF
  (PR-EQUAL (G-CAR HINTS) (PR-QUOTE 1))
  (PR-AND
   (G-VARIABLE (G-CADDDR HINTS))
   (PR-AND
    (G-TERMP (G-CADR HINTS) (PR-QUOTE 0) SYMBOLS)
    (PR-AND
     (G-TERMP (G-CADDR HINTS) (PR-QUOTE 0) SYMBOLS)
     (PR-AND (PR-NOT (G-MEMB (G-CADDDR HINTS) (G-COLLECT-FREE (G-CADR HINTS) (PR-QUOTE 0))))
             (PR-AND (PR-NOT (G-MEMB (G-CADDDR HINTS)
                                     (G-COLLECT-FREE (G-CADDR HINTS) (PR-QUOTE 0))))
                     (PR-EQUAL CONC
                               (G-F-IFF (G-F-EQUAL (G-CADR HINTS) (G-CADDR HINTS))
                                        (G-FORALL (G-CADDDR HINTS)
                                                  (G-F-IFF (G-ISIN (G-CADDDR HINTS)
                                                                   (G-CADR HINTS))
                                                           (G-ISIN (G-CADDDR HINTS)
                                                                   (G-CADDR HINTS)))))))))))
  (PR-IF
   (PR-EQUAL (G-CAR HINTS) (PR-QUOTE 2))
   (PR-AND (G-TERMP (G-CADR HINTS) (PR-QUOTE 0) SYMBOLS)
           (PR-EQUAL CONC (G-F-NOT (G-ISIN (G-CADR HINTS) (G-PHI)))))
   (PR-IF
    (PR-EQUAL (G-CAR HINTS) (PR-QUOTE 3))
    (PR-AND (G-TERMP (G-CONS (G-CADR HINTS)
                             (G-CONS (G-CADDR HINTS)
                                     (G-CONS (G-CADDDR HINTS) (PR-QUOTE (GCODE NIL)))))
                     (PR-QUOTE 1)
                     SYMBOLS)
            (PR-EQUAL CONC
                      (G-F-IFF (G-ISIN (G-CADDDR HINTS)
                                       (G-Z-PAIR (G-CADR HINTS) (G-CADDR HINTS)))
                               (G-F-OR (G-F-EQUAL (G-CADDDR HINTS) (G-CADR HINTS))
                                       (G-F-EQUAL (G-CADDDR HINTS) (G-CADDR HINTS))))))
    (PR-IF
     (PR-EQUAL (G-CAR HINTS) (PR-QUOTE 4))
     (PR-AND (G-TERMP (G-CONS (G-CADR HINTS)
                              (G-CONS (G-CADDR HINTS)
                                      (G-CONS (G-CADDDR HINTS) (PR-QUOTE (GCODE NIL)))))
                      (PR-QUOTE 1)
                      SYMBOLS)
             (PR-EQUAL CONC
                       (G-F-IFF (G-ISIN (G-CADDDR HINTS)
                                        (G-Z-UNION (G-CADR HINTS) (G-CADDR HINTS)))
                                (G-F-OR (G-ISIN (G-CADDDR HINTS) (G-CADR HINTS))
                                        (G-ISIN (G-CADDDR HINTS) (G-CADDR HINTS))))))
     (PR-IF
      (PR-EQUAL (G-CAR HINTS) (PR-QUOTE 5))
      (PR-AND
       (G-VARIABLE (G-CADDR HINTS))
       (PR-AND
        (G-FORMULA (G-CADR HINTS) SYMBOLS)
        (PR-EQUAL
         CONC
         (G-F-IMPLIES
          (G-F-AND (G-SUBST (G-CADR HINTS) (G-CADDR HINTS) (G-PHI) (PR-QUOTE 0))
                   (G-FORALL (G-CADDR HINTS)
                             (G-F-IMPLIES (G-F-AND (G-Z-INT (G-CADDR HINTS))
                                                   (G-CADR HINTS))
                                          (G-SUBST (G-CADR HINTS)
                                                   (G-CADDR HINTS)
                                                   (G-Z-SUCC (G-CADDR HINTS))
                                                   (PR-QUOTE 0)))))
          (G-FORALL (G-CADDR HINTS)
                    (G-F-IMPLIES (G-Z-INT (G-CADDR HINTS)) (G-CADR HINTS)))))))
      (PR-QUOTE (BOOL-FIX F))))))))
\end{verbatim}

\begin{verbatim};;;;;;;The EV analogue of SYMB-DEFN-PROOF.
\end{verbatim}%\item 
\begin{verbatim}

;;;;;;;The EV analogue of SYMB-DEFN-PROOF.
(DEFN
 G-SYMB-DEFN-PROOF
 (CONC HINTS SUB-GOALS DEFN SYMBOLS)
 (PR-IF
  (G-PREDICATE (G-CAR (G-CAR HINTS)))
  (PR-AND (G-SET (G-CDR (G-CAR HINTS)))
          (PR-AND (G-VAR-LIST (G-CDR (G-CAR HINTS)) (G-DEGREE (G-CAR (G-CAR HINTS))))
                  (PR-AND (PR-EQUAL CONC (G-F-IFF (G-CAR HINTS) DEFN))
                          (PR-AND (G-SUB (G-COLLECT-FREE DEFN (PR-QUOTE 0))
                                         (G-CDR (G-CAR HINTS)))
                                  (G-FORMULA DEFN (G-REST-OF (G-CAR (G-CAR HINTS)) SYMBOLS))))))
  (PR-IF
   (G-FUNCTION (G-CAR (G-CAR HINTS)))
   (PR-AND
    (G-SET (G-CONS (G-CADDDR HINTS) (G-CONS (G-CADR HINTS) (G-CDR (G-CAR HINTS)))))
    (PR-AND
     (G-VAR-LIST (G-CDR (G-CAR HINTS)) (G-DEGREE (G-CAR (G-CAR HINTS))))
     (PR-AND
      (G-VARIABLE (G-CADR HINTS))
      (PR-AND
       (G-VARIABLE (G-CADDDR HINTS))
       (PR-AND
        (G-FORMULA (G-CADDR HINTS) (G-REST-OF (G-CAR (G-CAR HINTS)) SYMBOLS))
        (PR-AND
         (G-SUB (G-COLLECT-FREE (G-CADDR HINTS) (PR-QUOTE 0))
                (G-CONS (G-CADR HINTS) (G-CDR (G-CAR HINTS))))
         (PR-AND
          (G-FREE-FOR (G-CADDR HINTS) (G-CADR HINTS) (G-CAR HINTS) (PR-QUOTE 0))
          (PR-AND
           (G-FREE-FOR (G-CADDR HINTS) (G-CADR HINTS) (G-CADDDR HINTS) (PR-QUOTE 0))
           (PR-AND
            (PR-EQUAL CONC
                      (G-SUBST (G-CADDR HINTS)
                               (G-CADR HINTS)
                               (G-CAR HINTS)
                               (PR-QUOTE 0)))
            (PR-AND
             (PR-EQUAL CONC DEFN)
             (PR-EQUAL
              SUB-GOALS
              (G-CONS (G-FORSOME (G-CADR HINTS) (G-CADDR HINTS))
                      (G-CONS (G-F-IMPLIES (G-F-AND (G-CADDR HINTS)
                                                    (G-SUBST (G-CADDR HINTS)
                                                             (G-CADR HINTS)
                                                             (G-CADDDR HINTS)
                                                             (PR-QUOTE 0)))
                                           (G-F-EQUAL (G-CADR HINTS)
                                                      (G-CADDDR HINTS)))
                              (PR-QUOTE (GCODE NIL)))))))))))))))
   (PR-QUOTE (BOOL-FIX F)))))
\end{verbatim}


\noindent{\sc Event}:   Enable g-symb-defn-proof; name this event `g1286'.\index{g1286}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z2-axioms; name this event `g1287'.\index{g1287}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-fol-rules; name this event `g1288'.\index{g1288}

\addvspace{18pt}
\begin{verbatim};;;;;;;The e-definition of G-PRF.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-prf-defn} \\  
\=\+{\rm{g-prf-defn}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{g-prf-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\it{flg\/}})\index{pr-zerop}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\it{pf\/}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-not}}\,({\rm{g-formula}}\,({\rm{g-conc}}\,({\it{pf\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-conc}, {\it{symbols\/}})\index{g-formula})\index{pr-not}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-quote}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{pf\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-fol-axiom-proof}}\,(\=\+{\rm{g-conc}}\,({\it{pf\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-conc}, \\ 
{\rm{g-cadr}}\,({\it{pf\/}})\index{g-cadr}, \\ 
{\it{symbols\/}})\-\index{g-fol-axiom-proof}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{pf\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-fol-rules}}\,(\=\+{\rm{g-conc}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-conc}, \\ 
{\rm{g-cadr}}\,({\it{pf\/}})\index{g-cadr}, \\ 
{\rm{g-conc}}\,(\=\+{\rm{g-cadddr}}\,({\it{pf\/}})\index{g-cadddr}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\-\index{g-conc}, \\ 
{\it{symbols\/}})\-\index{g-fol-rules}, \\ 
{\rm{g-prf}}\,(\=\+{\rm{g-cadddr}}\,({\it{pf\/}})\index{g-cadddr}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\it{symbols\/}})\-\index{g-prf})\-\index{pr-and}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-car}}\,({\it{pf\/}})\index{g-car}, {\rm{pr-quote}}\,({\tt{3}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-z2-axioms}}\,(\=\+{\rm{g-conc}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-conc}, \\ 
{\rm{g-cadr}}\,({\it{pf\/}})\index{g-cadr}, \\ 
{\it{symbols\/}})\-\index{g-z2-axioms}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-car}}\,({\it{pf\/}})\index{g-car}, \\ 
{\rm{pr-quote}}\,({\tt{4}})\index{pr-quote})\-\index{pr-equal}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-symb-defn-proof}}\,(\=\+{\rm{g-conc}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-conc}, \\ 
{\rm{g-cadr}}\,({\it{pf\/}})\index{g-cadr}, \\ 
{\rm{g-conc}}\,(\=\+{\rm{g-cadddr}}\,({\it{pf\/}})\index{g-cadddr}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\-\index{g-conc}, \\ 
{\rm{g-assoc}}\,(\=\+{\rm{g-car}}\,({\rm{g-car}}\,({\rm{g-cadr}}\,({\it{pf\/}})\index{g-cadr})\index{g-car})\index{g-car}, \\ 
{\it{defns\/}})\-\index{g-assoc}, \\ 
{\it{symbols\/}})\-\index{g-symb-defn-proof}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-function}}\,({\rm{g-car}}\,({\rm{g-car}}\,({\rm{g-cadr}}\,({\it{pf\/}})\index{g-cadr})\index{g-car})\index{g-car})\index{g-function}, \\ 
{\rm{g-prf}}\,(\=\+{\rm{g-cadddr}}\,({\it{pf\/}})\index{g-cadddr}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\rm{g-rest-of}}\,(\=\+{\rm{g-car}}\,({\rm{g-car}}\,({\rm{g-cadr}}\,({\it{pf\/}})\index{g-cadr})\index{g-car})\index{g-car}, \\ 
{\it{symbols\/}})\-\index{g-rest-of})\-\index{g-prf}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote})\-\index{pr-if})\-\index{pr-and}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{g-car}}\,({\it{pf\/}})\index{g-car}, \\ 
{\rm{pr-quote}}\,({\tt{5}})\index{pr-quote})\-\index{pr-equal}, \\ 
{\rm{g-memb}}\,(\=\+{\rm{g-conc}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\-\index{g-conc}, \\ 
{\it{given\/}})\-\index{g-memb}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-quote})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-quote})\-\index{pr-if}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-typep}}\,({\it{pf\/}}, {\tt{1}})\index{g-typep}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-prf}}\,({\rm{g-car}}\,({\it{pf\/}})\index{g-car}, {\it{given\/}}, {\it{defns\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\it{symbols\/}})\index{g-prf}, \\ 
{\rm{g-prf}}\,({\rm{g-cdr}}\,({\it{pf\/}})\index{g-cdr}, {\it{given\/}}, {\it{defns\/}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\it{symbols\/}})\index{g-prf})\-\index{pr-and}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;So far we have presented the e-definitions leading to the EV analogue
;;;;;;;of the proof-checker PRF.  We need a few more e-definitions in order to
;;;;;;;construct the undecidable sentence.  
;;;;;;;Below is the e-definition for G-NUMERAL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-numeral}$\;\;$
{\rm{g-numeral}}\,({\it{x\/}})\index{g-numeral} $=$ {\rm{list}}\,({\tt{29}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-numeral-defn} \\  
\=\+{\sc{g-numeral-defn}}\index{g-numeral-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,({\rm{pr-zerop}}\,({\tt{0}})\index{pr-zerop}, {\sc{g-phi}}\index{g-phi}, {\rm{g-z-succ}}\,({\rm{g-numeral}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1})\index{g-numeral})\index{g-z-succ})\index{pr-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-sing}$\;\;$
{\rm{g-z-sing}}\,({\it{x\/}})\index{g-z-sing} $=$ {\rm{g-z-pair}}\,({\it{x\/}}, {\it{x\/}})\index{g-z-pair}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-opair} \\  
{\rm{g-z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-opair} $=$ {\rm{g-z-pair}}\,({\rm{g-z-sing}}\,({\it{x\/}})\index{g-z-sing}, {\rm{g-z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-pair})\index{g-z-pair}
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition for G-PLUS (addition).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-plus}$\;\;$
{\rm{g-plus}}\,({\it{x\/}}, {\it{y\/}})\index{g-plus} $=$ {\rm{list}}\,({\tt{30}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-plus-defn} \\  
{\sc{g-plus-defn}}\index{g-plus-defn} $=$ {\rm{pr-if}}\,({\rm{pr-zerop}}\,({\tt{0}})\index{pr-zerop}, {\rm{g-fix}}\,({\tt{1}})\index{g-fix}, {\rm{pr-add1}}\,({\rm{g-plus}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1}, {\tt{1}})\index{g-plus})\index{pr-add1})\index{pr-if}
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition for G-TIMES (multiplication).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-times}$\;\;$
{\rm{g-times}}\,({\it{x\/}}, {\it{y\/}})\index{g-times} $=$ {\rm{list}}\,({\tt{31}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-times-defn} \\  
\=\+{\sc{g-times-defn}}\index{g-times-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{0}})\index{pr-zerop}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{g-plus}}\,({\tt{1}}, {\rm{g-times}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1}, {\tt{1}})\index{g-times})\index{g-plus})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition for G-EXP (exponentiation).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-exp}$\;\;$
{\rm{g-exp}}\,({\it{x\/}}, {\it{y\/}})\index{g-exp} $=$ {\rm{list}}\,({\tt{32}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-exp-defn} \\  
\=\+{\sc{g-exp-defn}}\index{g-exp-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,({\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{1}})\index{gcode})\index{pr-quote}, {\rm{g-times}}\,({\tt{0}}, {\rm{g-exp}}\,({\tt{0}}, {\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1})\index{g-exp})\index{g-times})\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition for G-HALVE (integer division by 2).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-halve}$\;\;$
{\rm{g-halve}}\,({\it{x\/}})\index{g-halve} $=$ {\rm{list}}\,({\tt{33}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-halve-defn} \\  
\=\+{\sc{g-halve-defn}}\index{g-halve-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{0}})\index{pr-zerop}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1})\index{pr-zerop}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-add1}}\,({\rm{g-halve}}\,({\rm{pr-sub1}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1})\index{pr-sub1})\index{g-halve})\index{pr-add1})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The e-definition for G-LESSP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-lessp}$\;\;$
{\rm{g-lessp}}\,({\it{x\/}}, {\it{y\/}})\index{g-lessp} $=$ {\rm{list}}\,({\tt{34}}, {\it{x\/}}, {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-lessp-defn} \\  
\=\+{\sc{g-lessp-defn}}\index{g-lessp-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-quote}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{0}})\index{pr-zerop}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote}, \\ 
{\rm{g-lessp}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1}, {\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1})\index{g-lessp})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;With PLUS, TIMES, and EXP, we can cipher num-trees as numbers, and with these
;;;;;;;and the HALVE function, we can decipher back the num-trees from the numbers
;;;;;;;by a yet-to-be-defined function DECIPHER1.  The e-definition for G-DECIPHER1 is
;;;;;;;shown below.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-decipher1}$\;\;$
{\rm{g-decipher1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{g-decipher1} $=$ {\rm{list}}\,({\tt{35}}, {\it{x\/}}, {\it{y\/}}, {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-decipher1-defn} \\  
\=\+{\sc{g-decipher1-defn}}\index{g-decipher1-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{1}})\index{pr-zerop}, \\ 
{\rm{g-halve}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1})\index{g-halve}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-zerop}}\,({\tt{2}})\index{pr-zerop}, \\ 
{\rm{g-decipher1}}\,({\tt{0}}, {\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, {\tt{0}})\index{g-decipher1}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-and}}\,({\rm{g-lessp}}\,({\tt{1}}, {\tt{0}})\index{g-lessp}, {\rm{g-lessp}}\,({\tt{2}}, {\tt{0}})\index{g-lessp})\index{pr-and}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\tt{0}}, \\ 
{\rm{g-times}}\,(\=\+{\rm{g-exp}}\,({\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode})\index{pr-quote}, {\tt{1}})\index{g-exp}, \\ 
{\rm{g-exp}}\,({\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode})\index{pr-quote}, {\tt{2}})\index{g-exp})\-\index{g-times})\-\index{pr-equal}, \\ 
{\rm{pr-cons}}\,(\=\+{\rm{g-decipher1}}\,({\tt{1}}, {\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, {\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1})\index{g-decipher1}, \\ 
{\rm{g-decipher1}}\,({\tt{2}}, {\rm{pr-sub1}}\,({\tt{2}})\index{pr-sub1}, {\rm{pr-sub1}}\,({\tt{2}})\index{pr-sub1})\index{g-decipher1})\-\index{pr-cons}, \\ 
{\rm{g-decipher1}}\,({\tt{0}}, {\tt{1}}, {\rm{pr-sub1}}\,({\tt{2}})\index{pr-sub1})\index{g-decipher1})\-\index{pr-if}, \\ 
{\rm{g-decipher1}}\,({\tt{0}}, {\tt{1}}, {\rm{pr-sub1}}\,({\tt{2}})\index{pr-sub1})\index{g-decipher1})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-proves} \\  
\=\+{\rm{g-proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{g-proves} \\ 
$=$$\;\;\;\;${\rm{pr-and}}\,(\=\+{\rm{pr-equal}}\,({\rm{g-conc}}\,({\it{pf\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote})\index{g-conc}, {\it{exp\/}})\index{pr-equal}, \\ 
{\rm{pr-and}}\,(\=\+{\rm{g-formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{g-formula}, \\ 
{\rm{g-prf}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\it{symbols\/}})\index{g-prf})\-\index{pr-and})\-\index{pr-and}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-proves; name this event `g1035'.\index{g1035}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-decipher} \\  
{\rm{g-decipher}}\,({\it{x\/}})\index{g-decipher} $=$ {\rm{g-decipher1}}\,({\it{x\/}}, {\rm{pr-sub1}}\,({\it{x\/}})\index{pr-sub1}, {\rm{pr-sub1}}\,({\it{x\/}})\index{pr-sub1})\index{g-decipher1}
\end{tabbing}
\begin{verbatim};;;;;;;G-INV-CODE is the EV analogue of a function INV-CODE which is the inverse for GCODE.
;;;;;;;At first sight, G-INV-CODE might seem like an identity mapping but that is only
;;;;;;;true when its argument is a GCODE and as we shall see, this is not always the case.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-inv-code}$\;\;$
{\rm{g-inv-code}}\,({\it{x\/}})\index{g-inv-code} $=$ {\rm{list}}\,({\tt{36}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-inv-code-defn} \\  
\=\+{\sc{g-inv-code-defn}}\index{g-inv-code-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-listp}}\,({\tt{0}})\index{pr-listp}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{pr-car}}\,({\tt{0}})\index{pr-car}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-cons}}\,(\=\+{\rm{g-inv-code}}\,({\rm{pr-car}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-car})\index{g-inv-code}, \\ 
{\rm{g-inv-code}}\,({\rm{pr-cdr}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-cdr})\index{g-inv-code})\-\index{g-cons}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{pr-car}}\,({\tt{0}})\index{pr-car}, {\rm{pr-quote}}\,({\tt{2}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-f-not}}\,({\rm{g-inv-code}}\,({\rm{pr-car}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-car})\index{g-inv-code})\index{g-f-not}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{pr-car}}\,({\tt{0}})\index{pr-car}, {\rm{pr-quote}}\,({\tt{3}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-f-or}}\,(\=\+{\rm{g-inv-code}}\,({\rm{pr-car}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-car})\index{g-inv-code}, \\ 
{\rm{g-inv-code}}\,({\rm{pr-cdr}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-cdr})\index{g-inv-code})\-\index{g-f-or}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{pr-car}}\,({\tt{0}})\index{pr-car}, {\rm{pr-quote}}\,({\tt{4}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{g-forsome}}\,(\=\+{\rm{g-inv-code}}\,({\rm{pr-car}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-car})\index{g-inv-code}, \\ 
{\rm{g-inv-code}}\,({\rm{pr-cdr}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-cdr})\index{g-inv-code})\-\index{g-forsome}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,({\rm{pr-car}}\,({\tt{0}})\index{pr-car}, {\rm{pr-quote}}\,({\tt{5}})\index{pr-quote})\index{pr-equal}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{t}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{pr-equal}}\,(\=\+{\rm{pr-car}}\,({\tt{0}})\index{pr-car}, \\ 
{\rm{pr-quote}}\,({\tt{6}})\index{pr-quote})\-\index{pr-equal}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{f}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{pr-quote})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if})\-\index{pr-if}, \\ 
{\rm{g-fix}}\,({\tt{0}})\index{g-fix})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;G-ISTHM is the EV analogue of the `theorem-checker' ISTHM which will be described 
;;;;;;;later.  The e-definition is presented below.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-isthm} \\  
\=\+{\rm{g-isthm}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{g-isthm} \\ 
$=$$\;\;\;\;${\rm{list}}\,({\tt{37}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-isthm-defn} \\  
\=\+{\sc{g-isthm-defn}}\index{g-isthm-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{g-proves}}\,({\rm{g-inv-code}}\,({\rm{g-decipher}}\,({\tt{0}})\index{g-decipher})\index{g-inv-code}, {\tt{1}}, {\tt{2}}, {\tt{3}}, {\tt{4}})\index{g-proves}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{pr-quote}, \\ 
{\rm{pr-if}}\,(\=\+{\rm{g-proves}}\,({\rm{g-inv-code}}\,({\rm{g-decipher}}\,({\tt{0}})\index{g-decipher})\index{g-inv-code}, {\rm{g-f-not}}\,({\tt{1}})\index{g-f-not}, {\tt{2}}, {\tt{3}}, {\tt{4}})\index{g-proves}, \\ 
{\rm{pr-quote}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-quote}, \\ 
{\rm{g-isthm}}\,({\rm{pr-add1}}\,({\tt{0}})\index{pr-add1}, {\tt{1}}, {\tt{2}}, {\tt{3}}, {\tt{4}})\index{g-isthm})\-\index{pr-if})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;The last e-definition is that of GC-NCODE which is the EV analogue of NCODE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{gc-ncode}$\;\;$
{\rm{gc-ncode}}\,({\it{x\/}})\index{gc-ncode} $=$ {\rm{list}}\,({\tt{38}}, {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{gc-ncode-defn} \\  
\=\+{\sc{gc-ncode-defn}}\index{gc-ncode-defn} \\ 
$=$$\;\;\;\;${\rm{pr-if}}\,(\=\+{\rm{pr-listp}}\,({\tt{0}})\index{pr-listp}, \\ 
{\rm{g-z-opair}}\,({\rm{gc-ncode}}\,({\rm{pr-car}}\,({\tt{0}})\index{pr-car})\index{gc-ncode}, {\rm{gc-ncode}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{gc-ncode})\index{g-z-opair}, \\ 
{\rm{g-numeral}}\,({\tt{0}})\index{g-numeral})\-\index{pr-if}\-
\end{tabbing}
\begin{verbatim};;;;;;;For the time being, we disable all the e-definitions.
\end{verbatim}
\noindent{\sc Event}:   Enable g-append-defn; name this event `g1036'.\index{g1036}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-del-defn; name this event `g1037'.\index{g1037}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-memb-defn; name this event `g1038'.\index{g1038}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-collect-free-defn; name this event `g1039'.\index{g1039}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-covering-defn; name this event `g1040'.\index{g1040}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-nil-intersect-defn; name this event `g1041'.\index{g1041}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-var-list-defn; name this event `g1042'.\index{g1042}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-length-defn; name this event `g1043'.\index{g1043}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-termp-defn; name this event `g1044'.\index{g1044}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-subst-defn; name this event `g1045'.\index{g1045}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-sub-defn; name this event `g1046'.\index{g1046}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-formula-defn; name this event `g1047'.\index{g1047}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-pairequals-defn; name this event `g1048'.\index{g1048}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-set-defn; name this event `g1055'.\index{g1055}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-conc-defn; name this event `g1056'.\index{g1056}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-rest-of-defn; name this event `g1057'.\index{g1057}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-assoc-defn; name this event `g1058'.\index{g1058}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-prf-defn; name this event `g1059'.\index{g1059}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-numeral-defn; name this event `g1060'.\index{g1060}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-plus-defn; name this event `g1061'.\index{g1061}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-times-defn; name this event `g1062'.\index{g1062}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-exp-defn; name this event `g1063'.\index{g1063}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-halve-defn; name this event `g1064'.\index{g1064}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-lessp-defn; name this event `g1065'.\index{g1065}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-decipher1-defn; name this event `g1066'.\index{g1066}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-inv-code-defn; name this event `g1067'.\index{g1067}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-isthm-defn; name this event `g1068'.\index{g1068}

\addvspace{18pt}

\noindent{\sc Event}:   Enable gc-ncode-defn; name this event `g1069'.\index{g1069}

\addvspace{18pt}

\noindent{\sc Event}:   Disable sub1-n; name this event `g1070'.\index{g1070}

\addvspace{18pt}
\begin{verbatim};;;;;;;The function FA below is the big list of e-definitions.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{fa} \\  
\=\+{\sc{fa}}\index{fa} \\ 
$=$$\;\;\;\;${\rm{list}}\,(\=\+{\sc{g-append-defn}}\index{g-append-defn}, \\ 
{\sc{g-del-defn}}\index{g-del-defn}, \\ 
{\sc{g-memb-defn}}\index{g-memb-defn}, \\ 
{\sc{g-collect-free-defn}}\index{g-collect-free-defn}, \\ 
{\sc{g-covering-defn}}\index{g-covering-defn}, \\ 
{\sc{g-nil-intersect-defn}}\index{g-nil-intersect-defn}, \\ 
{\sc{g-var-list-defn}}\index{g-var-list-defn}, \\ 
{\sc{g-length-defn}}\index{g-length-defn}, \\ 
{\sc{g-termp-defn}}\index{g-termp-defn}, \\ 
{\sc{g-subst-defn}}\index{g-subst-defn}, \\ 
{\sc{g-sub-defn}}\index{g-sub-defn}, \\ 
{\sc{g-formula-defn}}\index{g-formula-defn}, \\ 
{\sc{g-pairequals-defn}}\index{g-pairequals-defn}, \\ 
{\sc{g-set-defn}}\index{g-set-defn}, \\ 
{\sc{g-conc-defn}}\index{g-conc-defn}, \\ 
{\sc{g-rest-of-defn}}\index{g-rest-of-defn}, \\ 
{\sc{g-assoc-defn}}\index{g-assoc-defn}, \\ 
{\rm{g-prf-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}}, {\tt{3}}, {\tt{4}})\index{g-prf-defn}, \\ 
{\sc{g-numeral-defn}}\index{g-numeral-defn}, \\ 
{\sc{g-plus-defn}}\index{g-plus-defn}, \\ 
{\sc{g-times-defn}}\index{g-times-defn}, \\ 
{\sc{g-exp-defn}}\index{g-exp-defn}, \\ 
{\sc{g-halve-defn}}\index{g-halve-defn}, \\ 
{\sc{g-lessp-defn}}\index{g-lessp-defn}, \\ 
{\sc{g-decipher1-defn}}\index{g-decipher1-defn}, \\ 
{\sc{g-inv-code-defn}}\index{g-inv-code-defn}, \\ 
{\sc{g-isthm-defn}}\index{g-isthm-defn}, \\ 
{\sc{gc-ncode-defn}}\index{gc-ncode-defn})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable sub1-n; name this event `g1072'.\index{g1072}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  difference-lessp \\ \index{difference-lessp}
({\it{x\/}} $\leq$ {\it{y\/}}) $\rightarrow$ (({\it{x\/}} $-$ {\it{y\/}}) $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-sub1-n \\ \index{difference-sub1-n}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{sub1-n}}\,({\it{x\/}}, {\it{y\/}})\index{sub1-n} $=$ ({\it{x\/}} $-$ {\it{y\/}}))
\end{tabbing}

\noindent{\sc Event}:   Enable sub1-n; name this event `g1073'.\index{g1073}

\addvspace{18pt}
\begin{verbatim};;;;;;;We prove that GET of the e-function symbol (minus 11) on (FA) yields the 
;;;;;;;appropriate e-definition for each of the defined e-function symbols.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  get-g-append-defn \\ \index{get-g-append-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{11}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-append-defn}}\index{g-append-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-sub1-n-nzerop \\ \index{get-sub1-n-nzerop}
{\rm{get}}\,($1\;+$ {\it{x\/}}, {\it{list\/}})\index{get} $=$ {\rm{get}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{list\/}}))\index{get}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-lessp2 \\ \index{difference-lessp2}
({\it{y\/}} $\leq$ {\it{x\/}}) $\rightarrow$ ((($1\;+$ {\it{x\/}}) $-$ {\it{y\/}}) $=$ ($1\;+$ ({\it{x\/}} $-$ {\it{y\/}})))
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  get-g-del-defn \\ \index{get-g-del-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{12}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-del-defn}}\index{g-del-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-memb-defn \\ \index{get-g-memb-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{13}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-memb-defn}}\index{g-memb-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-collect-free-defn \\ \index{get-g-collect-free-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{14}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-collect-free-defn}}\index{g-collect-free-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-covering-defn \\ \index{get-g-covering-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{15}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-covering-defn}}\index{g-covering-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-nil-intersect-defn \\ \index{get-g-nil-intersect-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{16}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-nil-intersect-defn}}\index{g-nil-intersect-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-var-list-defn \\ \index{get-g-var-list-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{17}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-var-list-defn}}\index{g-var-list-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-length-defn \\ \index{get-g-length-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{18}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-length-defn}}\index{g-length-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-termp-defn \\ \index{get-g-termp-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{19}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-termp-defn}}\index{g-termp-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-subst-defn \\ \index{get-g-subst-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{20}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-subst-defn}}\index{g-subst-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-sub-defn \\ \index{get-g-sub-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{21}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-sub-defn}}\index{g-sub-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-pairequals-defn \\ \index{get-g-pairequals-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{23}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-pairequals-defn}}\index{g-pairequals-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-set-defn \\ \index{get-g-set-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{24}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-set-defn}}\index{g-set-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-conc-defn \\ \index{get-g-conc-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{25}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-conc-defn}}\index{g-conc-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-rest-of-defn \\ \index{get-g-rest-of-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{26}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-rest-of-defn}}\index{g-rest-of-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-assoc-defn \\ \index{get-g-assoc-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{27}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-assoc-defn}}\index{g-assoc-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-prf-defn \\ \index{get-g-prf-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{28}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\rm{g-prf-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}}, {\tt{3}}, {\tt{4}})\index{g-prf-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-numeral-defn \\ \index{get-g-numeral-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{29}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-numeral-defn}}\index{g-numeral-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-plus-defn \\ \index{get-g-plus-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{30}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-plus-defn}}\index{g-plus-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-times-defn \\ \index{get-g-times-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{31}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-times-defn}}\index{g-times-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-exp-defn \\ \index{get-g-exp-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{32}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-exp-defn}}\index{g-exp-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-halve-defn \\ \index{get-g-halve-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{33}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-halve-defn}}\index{g-halve-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-lessp-defn \\ \index{get-g-lessp-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{34}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-lessp-defn}}\index{g-lessp-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-decipher1-defn \\ \index{get-g-decipher1-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{35}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-decipher1-defn}}\index{g-decipher1-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-inv-code-defn \\ \index{get-g-inv-code-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{36}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-inv-code-defn}}\index{g-inv-code-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-g-isthm-defn \\ \index{get-g-isthm-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{37}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-isthm-defn}}\index{g-isthm-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  get-gc-ncode-defn \\ \index{get-gc-ncode-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{38}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{gc-ncode-defn}}\index{gc-ncode-defn}
\end{tabbing}

\noindent{\sc Event}:   Enable fa; name this event `g1074'.\index{g1074}

\addvspace{18pt}

\noindent{\sc Event}:   Enable difference-sub1-n; name this event `g1075'.\index{g1075}

\addvspace{18pt}

\noindent{\sc Event}:   Enable difference-lessp; name this event `g1076'.\index{g1076}

\addvspace{18pt}

\noindent{\sc Event}:   Enable difference-lessp2; name this event `g1077'.\index{g1077}

\addvspace{18pt}
\begin{verbatim};;;;;;;In the next series of events, we prove that the e-functions defined correspond
;;;;;;;to their Lisp analogues.  This involves two steps, in the first step we show,
;;;;;;;for example, that G-APPEND when evaluated corresponds to G-APPENDN.  This is done
;;;;;;;in EVAL-G-APPEND.  In the
;;;;;;;second step we show that G-APPENDN when given a large enough stack bound N
;;;;;;;corresponds to APPEND.  This is done in G-APPENDN-GCODE.
;;;;;;;We do the same thing for each of the e-functions.
;;;;;;;G-APPEND-IND is the induction scheme for EVAL-G-APPEND, and is quite subtle
;;;;;;;and could not have been guessed by the theorem-prover.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-append-ind} \\  
\=\+{\rm{g-append-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-append-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-append-ind}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-append-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;G-APPENDN is the Lisp analogue of G-APPEND with N.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-appendn} \\  
\=\+{\rm{g-appendn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-appendn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $\simeq {\tt{0}}$) $\vee$ {\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{x\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-consn}}\,({\rm{g-carn}}\,({\it{x\/}})\index{g-carn}, {\rm{g-appendn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn}, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{g-appendn})\index{g-consn}, \\ 
{\it{y\/}})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-append; name this event `g1078'.\index{g1078}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-append-defn; name this event `g1079'.\index{g1079}

\addvspace{18pt}
\begin{verbatim};;;;;;;EV of G-APPEND applied to arguments is G-APPENDN of the EV of the arguments.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  eval-g-append \\ \index{eval-g-append}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-append}}\,({\it{x\/}}, {\it{y\/}})\index{g-append}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-appendn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-appendn}\-
\end{tabbing}
\begin{verbatim};;;;;;;APPENDN-IND is the induction scheme for G-APPENDN-GCODE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{appendn-ind} \\  
\=\+{\rm{appendn-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{appendn-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{appendn-ind}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{y\/}}, {\it{n\/}} $-\;1$)\index{appendn-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;For a sufficiently large N, G-APPENDN of the GCODE of the arguments is
;;;;;;;the GCODE of the APPEND of the arguments.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  g-appendn-gcode \\ \index{g-appendn-gcode}
\=\+({\rm{length}}\,({\it{x\/}})\index{length} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-appendn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-appendn} $=$ {\rm{gcode}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}))\index{gcode})\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-appendn; name this event `g1080'.\index{g1080}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-append; name this event `g1081'.\index{g1081}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-append-defn; name this event `g1082'.\index{g1082}

\addvspace{18pt}

\noindent{\sc Event}:   Enable eval-fun2; name this event `g1083'.\index{g1083}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-DEL and G-DELN, and G-DELN and DEL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-deln} \\  
\=\+{\rm{g-deln}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-deln} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $\simeq {\tt{0}}$) $\vee$ {\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{y\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\it{x\/}}, {\rm{g-carn}}\,({\it{y\/}})\index{g-carn})\index{pr-equaln}, \\ 
{\rm{g-deln}}\,({\it{x\/}}, {\rm{g-cdrn}}\,({\it{y\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-deln}, \\ 
{\rm{g-consn}}\,({\rm{g-carn}}\,({\it{y\/}})\index{g-carn}, {\rm{g-deln}}\,({\it{x\/}}, {\rm{g-cdrn}}\,({\it{y\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-deln})\index{g-consn})\-\index{pr-ifn}, \\ 
{\it{y\/}})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-del-ind} \\  
\=\+{\rm{g-del-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-del-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-del-ind}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-del-ind}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-del; name this event `g1084'.\index{g1084}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-del-defn; name this event `g1085'.\index{g1085}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  eval-g-del \\ \index{eval-g-del}
{\rm{ev}}\,({\tt{0}}, {\rm{g-del}}\,({\it{x\/}}, {\it{y\/}})\index{g-del}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-deln}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-deln}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ncode-equal-reflex \\ \index{ncode-equal-reflex}
{\rm{ncode-equal}}\,({\it{x\/}}, {\it{x\/}})\index{ncode-equal}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numberp-ncode-equal \\ \index{numberp-ncode-equal}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ ({\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal} $=$ ({\it{x\/}} $=$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ncode-gcode-equal \\ \index{ncode-gcode-equal}
{\rm{ncode-equal}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{ncode-equal} $=$ {\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql}
\end{tabbing}

\noindent{\sc Event}:   Enable numberp-ncode-equal; name this event `g1086'.\index{g1086}

\addvspace{18pt}

\noindent{\sc Event}:   Disable pr-equaln; name this event `g1087'.\index{g1087}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-deln-gcode \\ \index{g-deln-gcode}
({\rm{length}}\,({\it{y\/}})\index{length} $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-deln}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-deln} $=$ {\rm{gcode}}\,({\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-deln; name this event `g1088'.\index{g1088}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-del; name this event `g1089'.\index{g1089}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-del-defn; name this event `g1090'.\index{g1090}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-MEMB, G-MEMBN, and MEMB.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-membn} \\  
\=\+{\rm{g-membn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-membn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $\simeq {\tt{0}}$) $\vee$ {\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{y\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\it{x\/}}, {\rm{g-carn}}\,({\it{y\/}})\index{g-carn})\index{pr-equaln}, \\ 
{\tt{1}}, \\ 
{\rm{g-membn}}\,({\it{x\/}}, {\rm{g-cdrn}}\,({\it{y\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-membn})\-\index{pr-ifn}, \\ 
{\tt{0}})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-memb; name this event `g1091'.\index{g1091}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-memb-defn; name this event `g1092'.\index{g1092}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-memb-ind} \\  
\=\+{\rm{g-memb-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-memb-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-memb-ind}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-memb-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-memb-eval \\ \index{g-memb-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-memb}}\,({\it{x\/}}, {\it{y\/}})\index{g-memb}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-membn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-membn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-membn-gcode \\ \index{g-membn-gcode}
\=\+({\rm{length}}\,({\it{y\/}})\index{length} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-membn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-membn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb}$\;\;${\bf then }{\tt{1}} \\ 
{\bf else }{\tt{0}}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-memb; name this event `g1093'.\index{g1093}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-membn; name this event `g1094'.\index{g1094}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-memb-defn; name this event `g1095'.\index{g1095}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-COLLECT-FREE, G-COLLECT-FREEN, and COLLECT-FREE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-collect-freen} \\  
\=\+{\rm{g-collect-freen}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{n\/}})\index{g-collect-freen} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $\simeq {\tt{0}}$) $\vee$ {\rm{btmp}}\,({\it{exp\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{flg\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{flg\/}})\index{pr-zeropn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-variablen}}\,({\it{exp\/}})\index{g-variablen}, \\ 
{\rm{g-consn}}\,({\it{exp\/}}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{2}})\index{g-typepn}, \\ 
{\rm{g-collect-freen}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{2}})\index{g-argn}, \\ 
{\tt{0}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{3}})\index{g-typepn}, \\ 
{\rm{g-appendn}}\,(\=\+{\rm{g-collect-freen}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{0}}, \\ 
{\tt{3}})\-\index{g-argn}, \\ 
{\tt{0}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\rm{g-collect-freen}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{1}}, \\ 
{\tt{3}})\-\index{g-argn}, \\ 
{\tt{0}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-appendn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{4}})\index{g-typepn}, \\ 
{\rm{g-deln}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{4}})\index{g-argn}, \\ 
{\rm{g-collect-freen}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{1}}, \\ 
{\tt{4}})\-\index{g-argn}, \\ 
{\tt{0}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-deln}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-collect-freen}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\tt{1}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-appendn}}\,(\=\+{\rm{g-collect-freen}}\,(\=\+{\rm{g-carn}}\,({\it{exp\/}})\index{g-carn}, \\ 
{\tt{0}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\rm{g-collect-freen}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\tt{1}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-appendn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-collect-free; name this event `g1096'.\index{g1096}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-collect-free-defn; name this event `g1097'.\index{g1097}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-collect-free-ind} \\  
\=\+{\rm{g-collect-free-ind}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{va\/}}, {\it{n\/}})\index{g-collect-free-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-collect-free-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-free-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-collect-free-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-free-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-collect-free-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-free-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-collect-free-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-free-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-collect-free-ind}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-free-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-collect-free-ind}}\,(\=\+{\rm{g-car}}\,({\tt{0}})\index{g-car}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-free-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-collect-free-eval \\ \index{g-collect-free-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{g-collect-free}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-collect-freen}}\,({\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{flg\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-collect-freen}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{exp-count} \\  
\=\+{\rm{exp-count}}\,({\it{exp\/}})\index{exp-count} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }$1\;+$ ({\rm{exp-count}}\,({\rm{car}}\,({\it{exp\/}}))\index{exp-count} $+$ {\rm{exp-count}}\,({\rm{cdr}}\,({\it{exp\/}}))\index{exp-count}) \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }$1\;+$ {\rm{exp-count}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg})\index{exp-count} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }$1\;+$ ({\rm{exp-count}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1})\index{exp-count} $+$ {\rm{exp-count}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2})\index{exp-count}) \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep}$\;\;${\bf then }$1\;+$ {\rm{exp-count}}\,({\rm{body}}\,({\it{exp\/}}))\index{exp-count} \\ 
{\bf else }{\tt{1}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  exp-count-append \\ \index{exp-count-append}
{\rm{exp-count}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}))\index{exp-count} $<$ ({\rm{exp-count}}\,({\it{x\/}})\index{exp-count} $+$ {\rm{exp-count}}\,({\it{y\/}})\index{exp-count})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  exp-count-del \\ \index{exp-count-del}
{\rm{exp-count}}\,({\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del})\index{exp-count} $<$ ($1\;+$ {\rm{exp-count}}\,({\it{y\/}})\index{exp-count})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-exp-count \\ \index{length-exp-count}
{\rm{length}}\,({\it{x\/}})\index{length} $<$ ($1\;+$ {\rm{exp-count}}\,({\it{x\/}})\index{exp-count})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-append \\ \index{length-append}
{\rm{length}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}))\index{length} $=$ ({\rm{length}}\,({\it{x\/}})\index{length} $+$ {\rm{length}}\,({\it{y\/}})\index{length})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-del \\ \index{length-del}
{\rm{length}}\,({\rm{del}}\,({\it{x\/}}, {\it{y\/}})\index{del})\index{length} $<$ ($1\;+$ {\rm{length}}\,({\it{y\/}})\index{length})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-count \\ \index{collect-free-count}
{\rm{length}}\,({\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})\index{length} $<$ ($1\;+$ {\rm{exp-count}}\,({\it{exp\/}})\index{exp-count})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{collect-freen-ind} \\  
\=\+{\rm{collect-freen-ind}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{n\/}})\index{collect-freen-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{variable}}\,({\it{exp\/}})\index{variable}$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{collect-freen-ind}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{collect-freen-ind} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }\=\+{\rm{collect-freen-ind}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{collect-freen-ind} \\ 
$\wedge$$\;\;\;\;${\rm{collect-freen-ind}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{collect-freen-ind}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep}$\;\;${\bf then }{\rm{collect-freen-ind}}\,({\rm{body}}\,({\it{exp\/}}), {\tt{0}}, {\it{n\/}} $-\;1$)\index{collect-freen-ind} \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }{\rm{collect-freen-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{n\/}} $-\;1$)\index{collect-freen-ind} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\rm{collect-freen-ind}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{0}}, {\it{n\/}} $-\;1$)\index{collect-freen-ind} \\ 
$\wedge$$\;\;\;\;${\rm{collect-freen-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{n\/}} $-\;1$)\index{collect-freen-ind}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-collect-freen-gcode \\ \index{g-collect-freen-gcode}
\=\+(($\neg$ {\rm{btmp}}\,({\it{flg\/}})\index{btmp}) $\wedge$ ({\rm{exp-count}}\,({\it{exp\/}})\index{exp-count} $<$ {\it{n\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-collect-freen}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\it{flg\/}}, {\it{n\/}})\index{g-collect-freen} $=$ {\rm{gcode}}\,({\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free})\index{gcode})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numberp-gcode \\ \index{numberp-gcode}
({\rm{gcode}}\,({\it{exp\/}})\index{gcode} $\in\;${\bf{N}}${}$) $=$ ({\it{exp\/}} $\in\;${\bf{N}}${}$)
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-gcode-flg \\ \index{collect-free-gcode-flg}
{\rm{collect-free}}\,({\it{exp\/}}, {\rm{gcode}}\,({\it{flg\/}})\index{gcode})\index{collect-free} $=$ {\rm{collect-free}}\,({\it{exp\/}}, {\it{flg\/}})\index{collect-free}
\end{tabbing}

\noindent{\sc Event}:   Enable g-collect-free; name this event `g1110'.\index{g1110}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-collect-freen; name this event `g1111'.\index{g1111}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-collect-free-defn; name this event `g1112'.\index{g1112}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-andn; name this event `g1113'.\index{g1113}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-COVERING, G-COVERINGN, and COVERING.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-coveringn} \\  
\=\+{\rm{g-coveringn}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}, {\it{n\/}})\index{g-coveringn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $\simeq {\tt{0}}$) $\vee$ {\rm{btmp}}\,({\it{exp\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{var\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{flg\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{flg\/}})\index{pr-zeropn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{2}})\index{g-typepn}, \\ 
{\rm{g-coveringn}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{2}})\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-coveringn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{3}})\index{g-typepn}, \\ 
{\rm{g-appendn}}\,(\=\+{\rm{g-coveringn}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{3}})\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-coveringn}, \\ 
{\rm{g-coveringn}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{1}}, {\tt{3}})\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-coveringn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-appendn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{4}})\index{g-typepn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-membn}}\,(\=\+{\it{var\/}}, \\ 
{\rm{g-collect-freen}}\,(\=\+{\it{exp\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-collect-freen}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-membn}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{4}})\index{g-argn}, \\ 
{\rm{g-coveringn}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{1}}, \\ 
{\tt{4}})\-\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-coveringn})\-\index{g-consn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-coveringn}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\it{var\/}}, \\ 
{\rm{gcode}}\,({\tt{1}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-coveringn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-appendn}}\,(\=\+{\rm{g-coveringn}}\,(\=\+{\rm{g-carn}}\,({\it{exp\/}})\index{g-carn}, \\ 
{\it{var\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-coveringn}, \\ 
{\rm{g-coveringn}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\it{var\/}}, \\ 
{\rm{gcode}}\,({\tt{1}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-coveringn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-appendn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-covering; name this event `g1114'.\index{g1114}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-covering-defn; name this event `g1115'.\index{g1115}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-covering-ind} \\  
\=\+{\rm{g-covering-ind}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}, {\it{va\/}}, {\it{n\/}})\index{g-covering-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-covering-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-covering-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-covering-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-covering-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-covering-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-covering-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-covering-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-covering-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-covering-ind}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-covering-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-covering-ind}}\,(\=\+{\rm{g-car}}\,({\tt{0}})\index{g-car}, \\ 
{\tt{1}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-covering-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-covering-eval \\ \index{g-covering-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{g-covering}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-coveringn}}\,({\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{var\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{flg\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-coveringn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  covering-count \\ \index{covering-count}
{\rm{length}}\,({\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering})\index{length} $<$ {\rm{exp-count}}\,({\it{exp\/}})\index{exp-count}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-zeropn-gcode \\ \index{pr-zeropn-gcode}
{\rm{pr-zeropn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{pr-zeropn} $=$ {\rm{bool-fix}}\,({\it{x\/}} $\simeq {\tt{0}}$)\index{bool-fix}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-zeropn; name this event `g1116'.\index{g1116}

\addvspace{18pt}

\noindent{\sc Event}:   Disable bool-fix; name this event `g1117'.\index{g1117}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-membn-gcode-again \\ \index{g-membn-gcode-again}
\=\+({\rm{length}}\,({\it{y\/}})\index{length} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-membn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-membn} $=$ {\rm{bool-fix}}\,({\rm{memb}}\,({\it{x\/}}, {\it{y\/}})\index{memb})\index{bool-fix})\-
\end{tabbing}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1118'.\index{g1118}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-membn-gcode; name this event `g1119'.\index{g1119}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{coveringn-ind} \\  
\=\+{\rm{coveringn-ind}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}, {\it{n\/}})\index{coveringn-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{coveringn-ind}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{var\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{coveringn-ind} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }\=\+{\rm{coveringn-ind}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{var\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{coveringn-ind} \\ 
$\wedge$$\;\;\;\;${\rm{coveringn-ind}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{var\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{coveringn-ind}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep} \\ 
{\bf then }\=\+{\bf if }{\rm{memb}}\,({\it{var\/}}, {\rm{collect-free}}\,({\it{exp\/}}, {\tt{0}})\index{collect-free})\index{memb} \\ 
{\bf then }{\rm{coveringn-ind}}\,({\rm{body}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{coveringn-ind} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }{\rm{coveringn-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{1}}, {\it{n\/}} $-\;1$)\index{coveringn-ind} \\ 
{\bf elseif }{\it{exp\/}} $\in\;${\bf{N}}${}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\rm{coveringn-ind}}\,({\rm{car}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{coveringn-ind} \\ 
$\wedge$$\;\;\;\;${\rm{coveringn-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\tt{1}}, {\it{n\/}} $-\;1$)\index{coveringn-ind}\- \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-coveringn-gcode \\ \index{g-coveringn-gcode}
\=\+(($1\;+$ {\rm{exp-count}}\,({\it{exp\/}})\index{exp-count}) $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-coveringn}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\rm{gcode}}\,({\it{var\/}})\index{gcode}, {\rm{gcode}}\,({\it{flg\/}})\index{gcode}, {\it{n\/}})\index{g-coveringn} \\ 
$=$$\;\;\;\;${\rm{gcode}}\,({\rm{covering}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}})\index{covering})\index{gcode})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-coveringn; name this event `g1120'.\index{g1120}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-covering; name this event `g1121'.\index{g1121}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-covering-defn; name this event `g1122'.\index{g1122}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-NIL-INTERSECT, G-NIL-INTERSECTN, and NIL-INTERSECT.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-nil-intersectn} \\  
\=\+{\rm{g-nil-intersectn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-nil-intersectn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{x\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-notn}}\,({\rm{g-membn}}\,({\rm{g-carn}}\,({\it{x\/}})\index{g-carn}, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{g-membn})\index{pr-notn}, \\ 
{\rm{g-nil-intersectn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn}, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{g-nil-intersectn})\-\index{pr-andn}, \\ 
{\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-nil-intersect-defn; name this event `g1123'.\index{g1123}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-nil-intersect; name this event `g1124'.\index{g1124}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-nil-intersect-eval \\ \index{g-nil-intersect-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-nil-intersect}}\,({\it{x\/}}, {\it{y\/}})\index{g-nil-intersect}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-nil-intersectn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-nil-intersectn}\-
\end{tabbing}

\noindent{\sc Event}:   Enable pr-notn; name this event `g1125'.\index{g1125}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{nil-intersect-ind} \\  
\=\+{\rm{nil-intersect-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{nil-intersect-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{nil-intersect-ind}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{y\/}}, {\it{n\/}} $-\;1$)\index{nil-intersect-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-nil-intersectn-gcode \\ \index{g-nil-intersectn-gcode}
\=\+(({\rm{length}}\,({\it{x\/}})\index{length} $+$ {\rm{length}}\,({\it{y\/}})\index{length}) $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-nil-intersectn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-nil-intersectn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{nil-intersect}}\,({\it{x\/}}, {\it{y\/}})\index{nil-intersect})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-nil-intersect; name this event `g1126'.\index{g1126}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-nil-intersectn; name this event `g1127'.\index{g1127}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-nil-intersect-defn; name this event `g1128'.\index{g1128}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-free-for; name this event `g1129'.\index{g1129}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-FREE-FOR, G-FREE-FORN, and FREE-FOR.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-free-forn} \\  
\=\+{\rm{g-free-forn}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{n\/}})\index{g-free-forn} \\ 
$=$$\;\;\;\;${\rm{g-nil-intersectn}}\,(\=\+{\rm{g-coveringn}}\,({\it{exp\/}}, {\it{var\/}}, {\it{flg\/}}, {\it{n\/}})\index{g-coveringn}, \\ 
{\rm{g-collect-freen}}\,({\it{term\/}}, {\rm{gcode}}\,({\tt{0}})\index{gcode}, {\it{n\/}})\index{g-collect-freen}, \\ 
{\it{n\/}})\-\index{g-nil-intersectn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-free-for \\ \index{eval-g-free-for}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{g-free-for}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-free-forn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{var\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{term\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{flg\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-free-forn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable free-for; name this event `g1130'.\index{g1130}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-free-forn-gcode \\ \index{g-free-forn-gcode}
\=\+(($1\;+$ ({\rm{exp-count}}\,({\it{exp\/}})\index{exp-count} $+$ {\rm{exp-count}}\,({\it{term\/}})\index{exp-count})) $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-free-forn}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\rm{gcode}}\,({\it{var\/}})\index{gcode}, {\rm{gcode}}\,({\it{term\/}})\index{gcode}, {\rm{gcode}}\,({\it{flg\/}})\index{gcode}, {\it{n\/}})\index{g-free-forn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{free-for}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{free-for})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-free-for; name this event `g1131'.\index{g1131}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-free-forn; name this event `g1132'.\index{g1132}

\addvspace{18pt}

\noindent{\sc Event}:   Enable free-for; name this event `g1133'.\index{g1133}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-VAR-LIST, G-VAR-LISTN, and VAR-LIST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-var-listn} \\  
\=\+{\rm{g-var-listn}}\,({\it{list\/}}, {\it{len\/}}, {\it{n\/}})\index{g-var-listn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $\simeq {\tt{0}}$) $\vee$ {\rm{btmp}}\,({\it{list\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{len\/}})\index{btmp}$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-orn}}\,({\rm{pr-zeropn}}\,({\it{len\/}})\index{pr-zeropn}, {\rm{pr-notn}}\,({\rm{g-typepn}}\,({\it{list\/}}, {\tt{1}})\index{g-typepn})\index{pr-notn})\index{pr-orn}, \\ 
{\rm{pr-andn}}\,({\rm{pr-zeropn}}\,({\it{len\/}})\index{pr-zeropn}, {\rm{pr-notn}}\,({\rm{g-typepn}}\,({\it{list\/}}, {\tt{1}})\index{g-typepn})\index{pr-notn})\index{pr-andn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-variablen}}\,({\rm{g-carn}}\,({\it{list\/}})\index{g-carn})\index{g-variablen}, \\ 
{\rm{g-var-listn}}\,(\=\+{\rm{g-cdrn}}\,({\it{list\/}})\index{g-cdrn}, \\ 
{\rm{pr-sub1n}}\,({\it{len\/}})\index{pr-sub1n}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-var-listn})\-\index{pr-andn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-var-list; name this event `g1139'.\index{g1139}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-var-list-defn; name this event `g1140'.\index{g1140}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-var-list-ind} \\  
\=\+{\rm{g-var-list-ind}}\,({\it{list\/}}, {\it{len\/}}, {\it{va\/}}, {\it{n\/}})\index{g-var-list-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-var-list-ind}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{list\/}}, {\it{len\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-var-list-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-var-list-eval \\ \index{g-var-list-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-var-list}}\,({\it{list\/}}, {\it{len\/}})\index{g-var-list}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-var-listn}}\,({\rm{ev}}\,({\tt{0}}, {\it{list\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{len\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-var-listn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{var-listn-ind} \\  
\=\+{\rm{var-listn-ind}}\,({\it{list\/}}, {\it{len\/}}, {\it{n\/}})\index{var-listn-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }({\it{len\/}} $\simeq {\tt{0}}$) $\vee$ ({\it{list\/}} $\simeq\;${{\bf{nil}}}${}$)$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{var-listn-ind}}\,({\rm{cdr}}\,({\it{list\/}}), {\it{len\/}} $-\;1$, {\it{n\/}} $-\;1$)\index{var-listn-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-var-listn-gcode \\ \index{g-var-listn-gcode}
\=\+({\it{len\/}} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-var-listn}}\,({\rm{gcode}}\,({\it{list\/}})\index{gcode}, {\rm{gcode}}\,({\it{len\/}})\index{gcode}, {\it{n\/}})\index{g-var-listn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{var-list}}\,({\it{list\/}}, {\it{len\/}})\index{var-list})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-var-listn; name this event `g1141'.\index{g1141}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-var-list; name this event `g1142'.\index{g1142}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-var-list-defn; name this event `g1143'.\index{g1143}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-equaln-reduc \\ \index{pr-equaln-reduc}
{\rm{pr-equaln}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{pr-equaln} $=$ {\rm{bool-fix}}\,({\rm{eql}}\,({\it{x\/}}, {\it{y\/}})\index{eql})\index{bool-fix}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-equaln; name this event `g1147'.\index{g1147}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-orn; name this event `g1150'.\index{g1150}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-SYMB, G-SYMBN, and SYMB.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-symbn} \\  
\=\+{\rm{g-symbn}}\,({\it{x\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-symbn} \\ 
$=$$\;\;\;\;${\rm{pr-orn}}\,(\=\+{\rm{pr-equaln}}\,({\it{x\/}}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode}, {\rm{gcode}}\,({\tt{2}})\index{gcode})\index{g-consn})\index{g-consn})\index{pr-equaln}, \\ 
{\rm{g-membn}}\,({\it{x\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-membn})\-\index{pr-orn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-symb; name this event `g1151'.\index{g1151}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-symb-eval \\ \index{g-symb-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-symb}}\,({\it{x\/}}, {\it{symbols\/}})\index{g-symb}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-symbn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-symbn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable p; name this event `g1152'.\index{g1152}

\addvspace{18pt}

\noindent{\sc Event}:   Disable bool-fix; name this event `g1153'.\index{g1153}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  gcode-p \\ \index{gcode-p}
\=\+{\rm{bool-fix}}\,({\it{x\/}} $=$ {\rm{cons}}\,({\tt{3}}, {\rm{cons}}\,({\tt{0}}, {\tt{2}})))\index{bool-fix} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode} $=$ {\rm{gcode}}\,({\rm{cons}}\,({\tt{3}}, {\rm{cons}}\,({\tt{0}}, {\tt{2}})))\index{gcode})\index{bool-fix}\-
\end{tabbing}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1154'.\index{g1154}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-symbn-gcode \\ \index{g-symbn-gcode}
\=\+({\rm{length}}\,({\it{symbols\/}})\index{length} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-symbn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-symbn} $=$ {\rm{bool-fix}}\,({\rm{symb}}\,({\it{x\/}}, {\it{symbols\/}})\index{symb})\index{bool-fix})\-
\end{tabbing}

\noindent{\sc Event}:   Enable p; name this event `g1155'.\index{g1155}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-symb; name this event `g1156'.\index{g1156}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-symbn; name this event `g1157'.\index{g1157}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-LENGTH, G-LENGTHN, and LENGTH.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-lengthn} \\  
\=\+{\rm{g-lengthn}}\,({\it{x\/}}, {\it{n\/}})\index{g-lengthn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{x\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-add1n}}\,({\rm{g-lengthn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-lengthn})\index{pr-add1n}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-length-defn; name this event `g1158'.\index{g1158}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-length; name this event `g1159'.\index{g1159}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-length-ind} \\  
\=\+{\rm{g-length-ind}}\,({\it{x\/}}, {\it{va\/}}, {\it{n\/}})\index{g-length-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-length-ind}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-length-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  eval-g-length \\ \index{eval-g-length}
{\rm{ev}}\,({\tt{0}}, {\rm{g-length}}\,({\it{x\/}})\index{g-length}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-lengthn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-lengthn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  add1-gcode \\ \index{add1-gcode}
($1\;+$ {\rm{gcode}}\,({\it{x\/}})\index{gcode}) $=$ {\rm{gcode}}\,($1\;+$ {\it{x\/}})\index{gcode}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-lengthn-gcode \\ \index{g-lengthn-gcode}
({\rm{length}}\,({\it{x\/}})\index{length} $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-lengthn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\it{n\/}})\index{g-lengthn} $=$ {\rm{gcode}}\,({\rm{length}}\,({\it{x\/}})\index{length})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-length; name this event `g1160'.\index{g1160}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-lengthn; name this event `g1161'.\index{g1161}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-length-defn; name this event `g1162'.\index{g1162}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-TERMP, G-TERMPN, and TERMP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-termpn} \\  
\=\+{\rm{g-termpn}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-termpn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{exp\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{flg\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{symbols\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{flg\/}})\index{pr-zeropn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-functionn}}\,({\rm{g-carn}}\,({\it{exp\/}})\index{g-carn})\index{g-functionn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-symbn}}\,(\=\+{\rm{g-carn}}\,({\it{exp\/}})\index{g-carn}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-symbn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\rm{gcode}}\,({\tt{1}})\index{gcode}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-termpn}, \\ 
{\rm{pr-equaln}}\,(\=\+{\rm{g-lengthn}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-lengthn}, \\ 
{\rm{g-fixn}}\,({\rm{g-cdrn}}\,({\rm{g-cdrn}}\,({\rm{g-carn}}\,({\it{exp\/}})\index{g-carn})\index{g-cdrn})\index{g-cdrn})\index{g-fixn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}, \\ 
{\rm{g-variablen}}\,({\it{exp\/}})\index{g-variablen})\-\index{pr-ifn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,(\=\+{\rm{g-carn}}\,({\it{exp\/}})\index{g-carn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-termpn}, \\ 
{\rm{g-termpn}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\rm{gcode}}\,({\tt{1}})\index{gcode}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-termpn})\-\index{pr-andn}, \\ 
{\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-termp; name this event `g1163'.\index{g1163}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-termp-defn; name this event `g1164'.\index{g1164}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-termp-ind} \\  
\=\+{\rm{g-termp-ind}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}}, {\it{va\/}}, {\it{n\/}})\index{g-termp-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-termp-ind}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\tt{2}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-termp-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-termp-ind}}\,(\=\+{\rm{g-car}}\,({\tt{0}})\index{g-car}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\tt{2}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-termp-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-termp-eval \\ \index{g-termp-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{g-termp}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-termpn}}\,({\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{flg\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-termpn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{termp-ind} \\  
\=\+{\rm{termp-ind}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}}, {\it{n\/}})\index{termp-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }{\rm{termp-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{termp-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\rm{termp-ind}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{0}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{termp-ind} \\ 
$\wedge$$\;\;\;\;${\rm{termp-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{termp-ind}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-listp-cdr \\ \index{length-listp-cdr}
{\rm{listp}}\,({\it{x\/}}) $\rightarrow$ ({\rm{length}}\,({\rm{cdr}}\,({\it{x\/}}))\index{length} $<$ {\rm{length}}\,({\it{x\/}})\index{length})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-length-cdr \\ \index{not-length-cdr}
{\rm{length}}\,({\it{x\/}})\index{length} $\not<$ {\rm{length}}\,({\rm{cdr}}\,({\it{x\/}}))\index{length}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{termp-cnt} \\  
\=\+{\rm{termp-cnt}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }$1\;+$ {\rm{termp-cnt}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp-cnt} \\ 
{\bf else }$1\;+$ ($1\;+$ {\rm{length}}\,({\it{symbols\/}})\index{length})$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }$1\;+$ (\=\+{\rm{termp-cnt}}\,({\rm{car}}\,({\it{exp\/}}), {\tt{0}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp-cnt})\- \\ 
{\bf else }$1\;+$ ($1\;+$ {\rm{length}}\,({\it{symbols\/}})\index{length})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-symbols-termp-cnt \\ \index{lessp-symbols-termp-cnt}
{\rm{length}}\,({\it{symbols\/}})\index{length} $<$ ({\rm{termp-cnt}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp-cnt} $-\;1$)
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-length-termp-cnt \\ \index{lessp-length-termp-cnt}
{\rm{length}}\,({\it{exp\/}})\index{length} $<$ {\rm{termp-cnt}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp-cnt}
\end{tabbing}

\noindent{\sc Event}:   Disable degree; name this event `g1165'.\index{g1165}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  numberp-gcode-equal \\ \index{numberp-gcode-equal}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\vee$ ({\it{y\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ (({\rm{gcode}}\,({\it{x\/}})\index{gcode} $=$ {\rm{gcode}}\,({\it{y\/}})\index{gcode}) $=$ ({\it{x\/}} $=$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-termpn-gcode \\ \index{g-termpn-gcode}
\=\+({\rm{termp-cnt}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-termpn}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\rm{gcode}}\,({\it{flg\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-termpn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{termp}}\,({\it{exp\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{termp})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-termp; name this event `g1166'.\index{g1166}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-termpn; name this event `g1167'.\index{g1167}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-termp-defn; name this event `g1168'.\index{g1168}

\addvspace{18pt}

\noindent{\sc Event}:   Enable termp-cnt; name this event `g1169'.\index{g1169}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-ATOMP, G-ATOMPN, and ATOMP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-atompn} \\  
\=\+{\rm{g-atompn}}\,({\it{exp\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-atompn} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-predicaten}}\,({\rm{g-carn}}\,({\it{exp\/}})\index{g-carn})\index{g-predicaten}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-symbn}}\,({\rm{g-carn}}\,({\it{exp\/}})\index{g-carn}, {\it{symbols\/}}, {\it{n\/}})\index{g-symbn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{g-lengthn}}\,({\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, {\it{n\/}})\index{g-lengthn}, \\ 
{\rm{g-fixn}}\,({\rm{g-cdrn}}\,({\rm{g-cdrn}}\,({\rm{g-carn}}\,({\it{exp\/}})\index{g-carn})\index{g-cdrn})\index{g-cdrn})\index{g-fixn})\-\index{pr-equaln}, \\ 
{\rm{g-termpn}}\,({\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, {\rm{gcode}}\,({\tt{1}})\index{gcode}, {\it{symbols\/}}, {\it{n\/}})\index{g-termpn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-atomp; name this event `g1170'.\index{g1170}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-atomp-eval \\ \index{g-atomp-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-atomp}}\,({\it{exp\/}}, {\it{symbols\/}})\index{g-atomp}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-atompn}}\,({\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-atompn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable pr-andn; name this event `g1171'.\index{g1171}

\addvspace{18pt}

\noindent{\sc Event}:   Disable bool-fix; name this event `g1172'.\index{g1172}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  lazy-pr-andn-reduc \\ \index{lazy-pr-andn-reduc}
{\rm{pr-andn}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}, {\it{y\/}})\index{pr-andn} $=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lazy-pr-andn-reduc2 \\ \index{lazy-pr-andn-reduc2}
{\rm{pr-andn}}\,({\rm{bool-fix}}\,({\bf{t}})\index{bool-fix}, {\rm{bool-fix}}\,({\it{y\/}})\index{bool-fix})\index{pr-andn} $=$ {\rm{bool-fix}}\,({\it{y\/}})\index{bool-fix}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-andn; name this event `g1173'.\index{g1173}

\addvspace{18pt}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1174'.\index{g1174}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-atompn-gcode \\ \index{g-atompn-gcode}
\=\+({\rm{termp-cnt}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-atompn}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-atompn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{atomp}}\,({\it{exp\/}}, {\it{symbols\/}})\index{atomp})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-atomp; name this event `g1175'.\index{g1175}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-atompn; name this event `g1176'.\index{g1176}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-SUBST, G-SUBSTN, and SUBST.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-substn} \\  
\=\+{\rm{g-substn}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{n\/}})\index{g-substn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{exp\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{var\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{term\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{flg\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$) \\ 
{\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{flg\/}})\index{pr-zeropn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-variablen}}\,({\it{exp\/}})\index{g-variablen}, \\ 
{\rm{pr-ifn}}\,({\rm{pr-equaln}}\,({\it{exp\/}}, {\it{var\/}})\index{pr-equaln}, {\it{term\/}}, {\it{exp\/}})\index{pr-ifn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{2}})\index{g-typepn}, \\ 
{\rm{g-f-notn}}\,({\rm{g-substn}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{2}})\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-substn})\index{g-f-notn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{3}})\index{g-typepn}, \\ 
{\rm{g-f-orn}}\,(\=\+{\rm{g-substn}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{0}}, \\ 
{\tt{3}})\-\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-substn}, \\ 
{\rm{g-substn}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{1}}, \\ 
{\tt{3}})\-\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-substn})\-\index{g-f-orn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{4}})\index{g-typepn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{0}}, \\ 
{\tt{4}})\-\index{g-argn}, \\ 
{\it{var\/}})\-\index{pr-equaln}, \\ 
{\it{exp\/}}, \\ 
{\rm{g-forsomen}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{0}}, \\ 
{\tt{4}})\-\index{g-argn}, \\ 
{\rm{g-substn}}\,(\=\+{\rm{g-argn}}\,(\=\+{\it{exp\/}}, \\ 
{\tt{1}}, \\ 
{\tt{4}})\-\index{g-argn}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-substn})\-\index{g-forsomen})\-\index{pr-ifn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-carn}}\,({\it{exp\/}})\index{g-carn}, \\ 
{\rm{g-substn}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\rm{gcode}}\,({\tt{1}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-substn})\-\index{g-consn}, \\ 
{\it{exp\/}})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-substn}}\,(\=\+{\rm{g-carn}}\,({\it{exp\/}})\index{g-carn}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-substn}, \\ 
{\rm{g-substn}}\,(\=\+{\rm{g-cdrn}}\,({\it{exp\/}})\index{g-cdrn}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\rm{gcode}}\,({\tt{1}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-substn})\-\index{g-consn}, \\ 
{\it{exp\/}})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-subst; name this event `g1177'.\index{g1177}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-subst-defn; name this event `g1178'.\index{g1178}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-subst-ind} \\  
\=\+{\rm{g-subst-ind}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{va\/}}, {\it{n\/}})\index{g-subst-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-subst-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-subst-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-subst-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-subst-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-subst-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-subst-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-subst-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-subst-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-subst-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{1}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{1}})\index{gcode})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-subst-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-subst-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{1}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-subst-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-subst-eval \\ \index{g-subst-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{g-subst}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-substn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{var\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{term\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{flg\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-substn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{subst-ind} \\  
\=\+{\rm{subst-ind}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{n\/}})\index{subst-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{variable}}\,({\it{exp\/}})\index{variable}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{subst-ind}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{var\/}}, {\it{term\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{subst-ind} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }\=\+{\rm{subst-ind}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{var\/}}, {\it{term\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{subst-ind} \\ 
$\wedge$$\;\;\;\;${\rm{subst-ind}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{var\/}}, {\it{term\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{subst-ind}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep}$\;\;${\bf then }{\rm{subst-ind}}\,({\rm{body}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{subst-ind} \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}})$\;\;${\bf then }{\rm{subst-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{1}}, {\it{n\/}} $-\;1$)\index{subst-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{exp\/}}) \\ 
{\bf then }\=\+{\rm{subst-ind}}\,({\rm{car}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{0}}, {\it{n\/}} $-\;1$)\index{subst-ind} \\ 
$\wedge$$\;\;\;\;${\rm{subst-ind}}\,({\rm{cdr}}\,({\it{exp\/}}), {\it{var\/}}, {\it{term\/}}, {\tt{1}}, {\it{n\/}} $-\;1$)\index{subst-ind}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-substn-gcode \\ \index{g-substn-gcode}
\=\+({\rm{exp-count}}\,({\it{exp\/}})\index{exp-count} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-substn}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\rm{gcode}}\,({\it{var\/}})\index{gcode}, {\rm{gcode}}\,({\it{term\/}})\index{gcode}, {\rm{gcode}}\,({\it{flg\/}})\index{gcode}, {\it{n\/}})\index{g-substn} \\ 
$=$$\;\;\;\;${\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst})\index{gcode})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-subst; name this event `g1179'.\index{g1179}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-substn; name this event `g1180'.\index{g1180}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-subst-defn; name this event `g1181'.\index{g1181}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-SUB, G-SUBN, and SUB.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-subn} \\  
\=\+{\rm{g-subn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-subn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{x\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-membn}}\,({\rm{g-carn}}\,({\it{x\/}})\index{g-carn}, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{g-membn}, \\ 
{\rm{g-subn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn}, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{g-subn})\-\index{pr-andn}, \\ 
{\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-sub; name this event `g1182'.\index{g1182}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-sub-defn; name this event `g1183'.\index{g1183}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-sub-ind} \\  
\=\+{\rm{g-sub-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-sub-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-sub-ind}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\tt{1}}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-sub-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-sub-eval \\ \index{g-sub-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-sub}}\,({\it{x\/}}, {\it{y\/}})\index{g-sub}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-subn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-subn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-subn-gcode \\ \index{g-subn-gcode}
\=\+(({\rm{length}}\,({\it{x\/}})\index{length} $+$ {\rm{length}}\,({\it{y\/}})\index{length}) $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-subn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-subn} $=$ {\rm{bool-fix}}\,({\rm{sub}}\,({\it{x\/}}, {\it{y\/}})\index{sub})\index{bool-fix})\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-sub; name this event `g1184'.\index{g1184}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-subn; name this event `g1185'.\index{g1185}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-sub-defn; name this event `g1186'.\index{g1186}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-FORMULA, G-FORMULAN, and FORMULA.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-formulan} \\  
\=\+{\rm{g-formulan}}\,({\it{exp\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{exp\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{symbols\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{2}})\index{g-typepn}, \\ 
{\rm{g-formulan}}\,({\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{2}})\index{g-argn}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{g-formulan}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{3}})\index{g-typepn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{0}}, {\tt{3}})\index{g-argn}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-formulan}, \\ 
{\rm{g-formulan}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{1}}, {\tt{3}})\index{g-argn}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-formulan})\-\index{pr-andn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{exp\/}}, {\tt{4}})\index{g-typepn}, \\ 
{\rm{g-formulan}}\,(\=\+{\rm{g-argn}}\,({\it{exp\/}}, {\tt{1}}, {\tt{4}})\index{g-argn}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-formulan}, \\ 
{\rm{g-atompn}}\,({\it{exp\/}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{g-atompn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-formula; name this event `g1187'.\index{g1187}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-formula-defn; name this event `g1188'.\index{g1188}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-formula-ind} \\  
\=\+{\rm{g-formula-ind}}\,({\it{exp\/}}, {\it{symbols\/}}, {\it{va\/}}, {\it{n\/}})\index{g-formula-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-formula-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{2}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-formula-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-formula-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-formula-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-formula-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{3}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-formula-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-formula-ind}}\,(\=\+{\rm{g-arg}}\,({\tt{0}}, {\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, {\tt{4}})\index{g-arg}, \\ 
{\tt{1}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{exp\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-formula-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable difference-sub1-n; name this event `g1189'.\index{g1189}

\addvspace{18pt}

\noindent{\sc Event}:   Disable difference-lessp2; name this event `g1190'.\index{g1190}

\addvspace{18pt}

\noindent{\sc Event}:   Disable fa; name this event `g1191'.\index{g1191}

\addvspace{18pt}

\noindent{\sc Event}:   Disable get-sub1-n-nzerop; name this event `g1192'.\index{g1192}

\addvspace{18pt}

\noindent{\sc Event}:   Disable difference-lessp; name this event `g1193'.\index{g1193}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  get-g-formula-defn \\ \index{get-g-formula-defn}
{\rm{get}}\,({\rm{sub1-n}}\,({\tt{22}}, {\tt{11}})\index{sub1-n}, {\sc{fa}}\index{fa})\index{get} $=$ {\sc{g-formula-defn}}\index{g-formula-defn}
\end{tabbing}

\noindent{\sc Event}:   Enable difference-sub1-n; name this event `g1194'.\index{g1194}

\addvspace{18pt}

\noindent{\sc Event}:   Enable difference-lessp2; name this event `g1195'.\index{g1195}

\addvspace{18pt}

\noindent{\sc Event}:   Enable fa; name this event `g1196'.\index{g1196}

\addvspace{18pt}

\noindent{\sc Event}:   Enable get-sub1-n-nzerop; name this event `g1197'.\index{g1197}

\addvspace{18pt}

\noindent{\sc Event}:   Enable difference-lessp; name this event `g1198'.\index{g1198}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-formula-eval \\ \index{g-formula-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{g-formula}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-formulan}}\,({\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-formulan}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{formula-ind} \\  
\=\+{\rm{formula-ind}}\,({\it{exp\/}}, {\it{symbols\/}}, {\it{n\/}})\index{formula-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }{\rm{formula-ind}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{formula-ind} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }\=\+{\rm{formula-ind}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{formula-ind} \\ 
$\wedge$$\;\;\;\;${\rm{formula-ind}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{formula-ind}\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep}$\;\;${\bf then }{\rm{formula-ind}}\,({\rm{body}}\,({\it{exp\/}}), {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{formula-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{formula-cnt} \\  
\=\+{\rm{formula-cnt}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula-cnt} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{f-notp}}\,({\it{exp\/}})\index{f-notp}$\;\;${\bf then }$1\;+$ {\rm{formula-cnt}}\,({\rm{arg}}\,({\it{exp\/}})\index{arg}, {\it{symbols\/}})\index{formula-cnt} \\ 
{\bf elseif }{\rm{f-orp}}\,({\it{exp\/}})\index{f-orp} \\ 
{\bf then }$1\;+$ (\=\+{\rm{formula-cnt}}\,({\rm{arg1}}\,({\it{exp\/}})\index{arg1}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{arg2}}\,({\it{exp\/}})\index{arg2}, {\it{symbols\/}})\index{formula-cnt})\- \\ 
{\bf elseif }{\rm{forsomep}}\,({\it{exp\/}})\index{forsomep}$\;\;${\bf then }$1\;+$ {\rm{formula-cnt}}\,({\rm{body}}\,({\it{exp\/}}), {\it{symbols\/}})\index{formula-cnt} \\ 
{\bf else }$1\;+$ {\rm{termp-cnt}}\,({\rm{cdr}}\,({\it{exp\/}}), {\tt{1}}, {\it{symbols\/}})\index{termp-cnt}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-formulan-gcode \\ \index{g-formulan-gcode}
\=\+({\rm{formula-cnt}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-formulan}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-formulan} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{formula}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-formula; name this event `g1199'.\index{g1199}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-formulan; name this event `g1200'.\index{g1200}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-formula-defn; name this event `g1201'.\index{g1201}

\addvspace{18pt}

\noindent{\sc Event}:   Enable formula-cnt; name this event `g1202'.\index{g1202}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence between G-PAIREQUALS, G-PAIREQUALSN, and PAIREQUALS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-pairequalsn} \\  
\=\+{\rm{g-pairequalsn}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}}, {\it{n\/}})\index{g-pairequalsn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{vars1\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{vars2\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{exp\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-andn}}\,({\rm{g-typepn}}\,({\it{vars1\/}}, {\tt{1}})\index{g-typepn}, {\rm{g-typepn}}\,({\it{vars2\/}}, {\tt{1}})\index{g-typepn})\index{pr-andn}, \\ 
{\rm{g-f-impliesn}}\,(\=\+{\rm{g-f-equaln}}\,({\rm{g-carn}}\,({\it{vars1\/}})\index{g-carn}, {\rm{g-carn}}\,({\it{vars2\/}})\index{g-carn})\index{g-f-equaln}, \\ 
{\rm{g-pairequalsn}}\,(\=\+{\rm{g-cdrn}}\,({\it{vars1\/}})\index{g-cdrn}, \\ 
{\rm{g-cdrn}}\,({\it{vars2\/}})\index{g-cdrn}, \\ 
{\it{exp\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-pairequalsn})\-\index{g-f-impliesn}, \\ 
{\it{exp\/}})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-pairequals; name this event `g1203'.\index{g1203}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-pairequals-defn; name this event `g1204'.\index{g1204}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-pairequals-ind} \\  
\=\+{\rm{g-pairequals-ind}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{n\/}})\index{g-pairequals-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-pairequals-ind}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\rm{g-cdr}}\,({\tt{1}})\index{g-cdr}, \\ 
{\tt{2}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-pairequals-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-pairequals-eval \\ \index{g-pairequals-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}})\index{g-pairequals}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-pairequalsn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{vars1\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{vars2\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-pairequalsn}\-
\end{tabbing}

\noindent{\sc Event}:   Enable pairequals-reduc; name this event `g1205'.\index{g1205}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{pairequals-ind} \\  
\=\+{\rm{pairequals-ind}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}}, {\it{n\/}})\index{pairequals-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{listp}}\,({\it{vars1\/}}) $\wedge$ {\rm{listp}}\,({\it{vars2\/}}) \\ 
{\bf then }{\rm{pairequals-ind}}\,({\rm{cdr}}\,({\it{vars1\/}}), {\rm{cdr}}\,({\it{vars2\/}}), {\it{exp\/}}, {\it{n\/}} $-\;1$)\index{pairequals-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-pairequalsn-gcode \\ \index{g-pairequalsn-gcode}
\=\+({\rm{length}}\,({\it{vars1\/}})\index{length} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-pairequalsn}}\,({\rm{gcode}}\,({\it{vars1\/}})\index{gcode}, {\rm{gcode}}\,({\it{vars2\/}})\index{gcode}, {\rm{gcode}}\,({\it{exp\/}})\index{gcode}, {\it{n\/}})\index{g-pairequalsn} \\ 
$=$$\;\;\;\;${\rm{gcode}}\,({\rm{pairequals}}\,({\it{vars1\/}}, {\it{vars2\/}}, {\it{exp\/}})\index{pairequals})\index{gcode})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-pairequals; name this event `g1206'.\index{g1206}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-pairequalsn; name this event `g1207'.\index{g1207}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-pairequals-defn; name this event `g1208'.\index{g1208}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-cadrn}$\;\;$
{\rm{g-cadrn}}\,({\it{x\/}})\index{g-cadrn} $=$ {\rm{g-carn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn})\index{g-carn}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-caddrn}$\;\;$
{\rm{g-caddrn}}\,({\it{x\/}})\index{g-caddrn} $=$ {\rm{g-carn}}\,({\rm{g-cdrn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn})\index{g-cdrn})\index{g-carn}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-cadddrn}$\;\;$
{\rm{g-cadddrn}}\,({\it{x\/}})\index{g-cadddrn} $=$ {\rm{g-carn}}\,({\rm{g-cdrn}}\,({\rm{g-cdrn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn})\index{g-cdrn})\index{g-cdrn})\index{g-carn}
\end{tabbing}
\begin{verbatim};;;;;;;SUBST-AXIOM correspondence.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-subst-axiom-proofn} \\  
\=\+{\rm{g-subst-axiom-proofn}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-subst-axiom-proofn} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-variablen}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn})\index{g-variablen}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,({\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, {\rm{gcode}}\,({\tt{0}})\index{gcode}, {\it{symbols\/}}, {\it{n\/}})\index{g-termpn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-free-forn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-free-forn}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-impliesn}}\,(\=\+{\rm{g-substn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-substn}, \\ 
{\rm{g-forsomen}}\,(\=\+{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\-\index{g-forsomen})\-\index{g-f-impliesn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-degreen}$\;\;$
{\rm{g-degreen}}\,({\it{x\/}})\index{g-degreen} $=$ {\rm{g-fixn}}\,({\rm{g-cdrn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn})\index{g-cdrn})\index{g-fixn}
\end{tabbing}
\begin{verbatim};;;;;;;Function equality axiom correspondence.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-fun-equal-axiom-proofn} \\  
\=\+{\rm{g-fun-equal-axiom-proofn}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-fun-equal-axiom-proofn} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, {\rm{gcode}}\,({\tt{1}})\index{gcode}, {\it{symbols\/}}, {\it{n\/}})\index{g-termpn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,({\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, {\rm{gcode}}\,({\tt{1}})\index{gcode}, {\it{symbols\/}}, {\it{n\/}})\index{g-termpn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-functionn}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-functionn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{g-lengthn}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, {\it{n\/}})\index{g-lengthn}, \\ 
{\rm{g-degreen}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-degreen})\-\index{pr-equaln}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{g-lengthn}}\,(\=\+{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, \\ 
{\it{n\/}})\-\index{g-lengthn}, \\ 
{\rm{g-degreen}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-degreen})\-\index{pr-equaln}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-pairequalsn}}\,(\=\+{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, \\ 
{\rm{g-f-equaln}}\,(\=\+{\rm{g-consn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn})\-\index{g-consn}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-consn})\-\index{g-f-equaln}, \\ 
{\it{n\/}})\-\index{g-pairequalsn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{verbatim};;;;;;;Predicate equality axiom correspondence.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-pred-equal-axiom-proofn} \\  
\=\+{\rm{g-pred-equal-axiom-proofn}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-pred-equal-axiom-proofn} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, {\rm{gcode}}\,({\tt{1}})\index{gcode}, {\it{symbols\/}}, {\it{n\/}})\index{g-termpn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,({\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, {\rm{gcode}}\,({\tt{1}})\index{gcode}, {\it{symbols\/}}, {\it{n\/}})\index{g-termpn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-predicaten}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-predicaten}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{g-lengthn}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, {\it{n\/}})\index{g-lengthn}, \\ 
{\rm{g-degreen}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-degreen})\-\index{pr-equaln}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{g-lengthn}}\,(\=\+{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, \\ 
{\it{n\/}})\-\index{g-lengthn}, \\ 
{\rm{g-degreen}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-degreen})\-\index{pr-equaln}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-pairequalsn}}\,(\=\+{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, \\ 
{\rm{g-f-impliesn}}\,(\=\+{\rm{g-consn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn})\-\index{g-consn}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-consn})\-\index{g-f-impliesn}, \\ 
{\it{n\/}})\-\index{g-pairequalsn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{verbatim};;;;;;;Correspondence for FOL-AXIOM-PROOF.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-fol-axiom-proofn} \\  
\=\+{\rm{g-fol-axiom-proofn}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-fol-axiom-proofn} \\ 
$=$$\;\;\;\;${\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{1}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-orn}}\,(\=\+{\rm{g-f-notn}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-f-notn}, \\ 
{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\-\index{g-f-orn})\-\index{pr-equaln})\-\index{pr-andn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{2}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-subst-axiom-proofn}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-subst-axiom-proofn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{3}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-termpn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{g-termpn}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-equaln}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\-\index{g-f-equaln})\-\index{pr-equaln})\-\index{pr-andn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{4}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-fun-equal-axiom-proofn}}\,(\=\+{\it{conc\/}}, \\ 
{\it{hints\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{g-fun-equal-axiom-proofn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{5}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-pred-equal-axiom-proofn}}\,(\=\+{\it{conc\/}}, \\ 
{\it{hints\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{g-pred-equal-axiom-proofn}, \\ 
{\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-fol-axiom-proof-eval \\ \index{g-fol-axiom-proof-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-fol-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-fol-axiom-proof}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-fol-axiom-proofn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{conc\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{hints\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-fol-axiom-proofn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{fol-axiom-cnt} \\  
\=\+{\rm{fol-axiom-cnt}}\,({\it{hints\/}}, {\it{symbols\/}})\index{fol-axiom-cnt} \\ 
$=$$\;\;\;\;$($1\;+$ (\=\+{\rm{formula-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{0}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\tt{0}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{1}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\tt{1}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{length}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{length} \\ 
$+$$\;\;\;\;${\rm{length}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{length})\-)\-
\end{tabbing}

\noindent{\sc Event}:   Disable fol-axiom-proof; name this event `g1213'.\index{g1213}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-fol-axiom-proofn-gcode \\ \index{g-fol-axiom-proofn-gcode}
\=\+({\rm{fol-axiom-cnt}}\,({\it{hints\/}}, {\it{symbols\/}})\index{fol-axiom-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-fol-axiom-proofn}}\,({\rm{gcode}}\,({\it{conc\/}})\index{gcode}, {\rm{gcode}}\,({\it{hints\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-fol-axiom-proofn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{fol-axiom-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{fol-axiom-proof})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-fol-axiom-proof; name this event `g1214'.\index{g1214}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-fol-axiom-proofn; name this event `g1215'.\index{g1215}

\addvspace{18pt}

\noindent{\sc Event}:   Enable fol-axiom-proof; name this event `g1216'.\index{g1216}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-expan-rulen} \\  
\=\+{\rm{g-expan-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-expan-rulen} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-carn}}\,({\it{sub-goals\/}})\index{g-carn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-orn}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, {\rm{g-carn}}\,({\it{sub-goals\/}})\index{g-carn})\index{g-f-orn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-contrac-rulen} \\  
\=\+{\rm{g-contrac-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-contrac-rulen} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-f-orn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\-\index{g-f-orn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{g-consn})\-\index{pr-equaln}, \\ 
{\rm{pr-equaln}}\,({\it{conc\/}}, {\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-assoc-rulen} \\  
\=\+{\rm{g-assoc-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-assoc-rulen} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-f-orn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-f-orn}}\,(\=\+{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-f-orn})\-\index{g-f-orn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{g-consn})\-\index{pr-equaln}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-orn}}\,(\=\+{\rm{g-f-orn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn})\-\index{g-f-orn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-f-orn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-cut-rulen} \\  
\=\+{\rm{g-cut-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-cut-rulen} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-f-orn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn})\-\index{g-f-orn}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-f-orn}}\,(\=\+{\rm{g-f-notn}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-f-notn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-f-orn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{g-consn})\-\index{g-consn})\-\index{pr-equaln}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-orn}}\,(\=\+{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-f-orn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-forsome-intro-rulen} \\  
\=\+{\rm{g-forsome-intro-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-forsome-intro-rulen} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{g-variablen}}\,({\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn})\index{g-variablen}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-notn}}\,({\rm{g-membn}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-collect-freen}}\,(\=\+{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-collect-freen}, \\ 
{\it{n\/}})\-\index{g-membn})\index{pr-notn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\it{sub-goals\/}}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-f-impliesn}}\,(\=\+{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-f-impliesn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{g-consn})\-\index{pr-equaln}, \\ 
{\rm{pr-equaln}}\,(\=\+{\it{conc\/}}, \\ 
{\rm{g-f-impliesn}}\,(\=\+{\rm{g-forsomen}}\,(\=\+{\rm{g-cadrn}}\,({\it{hints\/}})\index{g-cadrn}, \\ 
{\rm{g-caddrn}}\,({\it{hints\/}})\index{g-caddrn})\-\index{g-forsomen}, \\ 
{\rm{g-cadddrn}}\,({\it{hints\/}})\index{g-cadddrn})\-\index{g-f-impliesn})\-\index{pr-equaln})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}
\begin{verbatim};;;;;;;Correspondence for FOL-RULES.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-fol-rulesn} \\  
\=\+{\rm{g-fol-rulesn}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-fol-rulesn} \\ 
$=$$\;\;\;\;${\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{1}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-expan-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-expan-rulen}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{2}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-contrac-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-contrac-rulen}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{3}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-assoc-rulen}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-assoc-rulen}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{4}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-cut-rulen}}\,(\=\+{\it{conc\/}}, \\ 
{\it{hints\/}}, \\ 
{\it{sub-goals\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{g-cut-rulen}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{hints\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{5}})\index{gcode})\index{pr-equaln}, \\ 
{\rm{g-forsome-intro-rulen}}\,(\=\+{\it{conc\/}}, \\ 
{\it{hints\/}}, \\ 
{\it{sub-goals\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{g-forsome-intro-rulen}, \\ 
{\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-fol-rules; name this event `g1217'.\index{g1217}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-fol-rules-eval \\ \index{g-fol-rules-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-fol-rules}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{g-fol-rules}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-fol-rulesn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{conc\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{hints\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{sub-goals\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-fol-rulesn}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-fol-rules; name this event `g1218'.\index{g1218}

\addvspace{18pt}

\noindent{\sc Event}:   Disable fol-rules; name this event `g1219'.\index{g1219}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-fol-rulesn-gcode \\ \index{g-fol-rulesn-gcode}
\=\+(\=\+(\=\+{\rm{formula-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{car}}\,({\it{sub-goals\/}}), {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{exp-count})\- \\ 
$<$$\;\;\;\;${\it{n\/}})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-fol-rulesn}}\,(\=\+{\rm{gcode}}\,({\it{conc\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{hints\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{sub-goals\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-fol-rulesn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{fol-rules}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{fol-rules})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable fol-rules; name this event `g1220'.\index{g1220}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-fol-rules; name this event `g1221'.\index{g1221}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-fol-rulesn; name this event `g1222'.\index{g1222}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-isinn} \\  
\=\+{\rm{g-isinn}}\,({\it{x\/}}, {\it{y\/}})\index{g-isinn} \\ 
$=$$\;\;\;\;${\rm{g-consn}}\,(\=\+{\rm{g-consn}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{1}})\index{gcode}, {\rm{gcode}}\,({\tt{2}})\index{gcode})\index{g-consn})\index{g-consn}, \\ 
{\rm{g-consn}}\,({\it{x\/}}, {\rm{g-consn}}\,({\it{y\/}}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn})\index{g-consn})\-\index{g-consn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-phin} \\  
\=\+{\sc{g-phin}}\index{g-phin} \\ 
$=$$\;\;\;\;${\rm{g-consn}}\,({\rm{g-consn}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode}, {\rm{gcode}}\,({\tt{0}})\index{gcode})\index{g-consn})\index{g-consn}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-pairn} \\  
\=\+{\rm{g-z-pairn}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-pairn} \\ 
$=$$\;\;\;\;${\rm{g-consn}}\,(\=\+{\rm{g-consn}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{1}})\index{gcode}, {\rm{gcode}}\,({\tt{2}})\index{gcode})\index{g-consn})\index{g-consn}, \\ 
{\rm{g-consn}}\,({\it{x\/}}, {\rm{g-consn}}\,({\it{y\/}}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn})\index{g-consn})\-\index{g-consn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-unionn} \\  
\=\+{\rm{g-z-unionn}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-unionn} \\ 
$=$$\;\;\;\;${\rm{g-consn}}\,(\=\+{\rm{g-consn}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode}, {\rm{gcode}}\,({\tt{2}})\index{gcode})\index{g-consn})\index{g-consn}, \\ 
{\rm{g-consn}}\,({\it{x\/}}, {\rm{g-consn}}\,({\it{y\/}}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn})\index{g-consn})\-\index{g-consn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-intn} \\  
\=\+{\rm{g-z-intn}}\,({\it{x\/}})\index{g-z-intn} \\ 
$=$$\;\;\;\;${\rm{g-consn}}\,(\=\+{\rm{g-consn}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode}, {\rm{gcode}}\,({\tt{1}})\index{gcode})\index{g-consn})\index{g-consn}, \\ 
{\rm{g-consn}}\,({\it{x\/}}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn})\-\index{g-consn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-succn} \\  
\=\+{\rm{g-z-succn}}\,({\it{x\/}})\index{g-z-succn} \\ 
$=$$\;\;\;\;${\rm{g-consn}}\,(\=\+{\rm{g-consn}}\,({\rm{gcode}}\,({\tt{2}})\index{gcode}, {\rm{g-consn}}\,({\rm{gcode}}\,({\tt{3}})\index{gcode}, {\rm{gcode}}\,({\tt{1}})\index{gcode})\index{g-consn})\index{g-consn}, \\ 
{\rm{g-consn}}\,({\it{x\/}}, {\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{g-consn})\-\index{g-consn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-f-iffn} \\  
{\rm{g-f-iffn}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-iffn} $=$ {\rm{g-f-andn}}\,({\rm{g-f-impliesn}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-impliesn}, {\rm{g-f-impliesn}}\,({\it{y\/}}, {\it{x\/}})\index{g-f-impliesn})\index{g-f-andn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-f-iff-eval \\ \index{g-f-iff-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-f-iff}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-iff}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev} $=$ {\rm{g-f-iffn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{g-f-iffn}
\end{tabbing}

\noindent{\sc Event}:   Disable f-iff; name this event `g1223'.\index{g1223}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-f-iffn-gcode \\ \index{g-f-iffn-gcode}
{\rm{g-f-iffn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-f-iffn} $=$ {\rm{gcode}}\,({\rm{f-iff}}\,({\it{x\/}}, {\it{y\/}})\index{f-iff})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable f-iff; name this event `g1224'.\index{g1224}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-iff; name this event `g1225'.\index{g1225}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-iffn; name this event `g1226'.\index{g1226}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for Z2-AXIOMS.
\end{verbatim}%\item 
\begin{verbatim}

;;;;;;;Correspondence for Z2-AXIOMS.
(DEFN
 G-Z2-AXIOMSN
 (CONC HINTS SYMBOLS N)
 (PR-IFN
  (PR-EQUALN (G-CARN HINTS) (GCODE 1))
  (PR-ANDN
   (G-VARIABLEN (G-CADDDRN HINTS))
   (PR-ANDN
    (G-TERMPN (G-CADRN HINTS) (GCODE 0) SYMBOLS N)
    (PR-ANDN
     (G-TERMPN (G-CADDRN HINTS) (GCODE 0) SYMBOLS N)
     (PR-ANDN
      (PR-NOTN (G-MEMBN (G-CADDDRN HINTS) (G-COLLECT-FREEN (G-CADRN HINTS) (GCODE 0) N) N))
      (PR-ANDN (PR-NOTN (G-MEMBN (G-CADDDRN HINTS)
                                 (G-COLLECT-FREEN (G-CADDRN HINTS) (GCODE 0) N)
                                 N))
               (PR-EQUALN CONC
                          (G-F-IFFN (G-F-EQUALN (G-CADRN HINTS) (G-CADDRN HINTS))
                                    (G-FORALLN (G-CADDDRN HINTS)
                                               (G-F-IFFN (G-ISINN (G-CADDDRN HINTS)
                                                                  (G-CADRN HINTS))
                                                         (G-ISINN (G-CADDDRN HINTS)
                                                                  (G-CADDRN HINTS)))))))))))
  (PR-IFN
   (PR-EQUALN (G-CARN HINTS) (GCODE 2))
   (PR-ANDN (G-TERMPN (G-CADRN HINTS) (GCODE 0) SYMBOLS N)
            (PR-EQUALN CONC (G-F-NOTN (G-ISINN (G-CADRN HINTS) (G-PHIN)))))
   (PR-IFN
    (PR-EQUALN (G-CARN HINTS) (GCODE 3))
    (PR-ANDN (G-TERMPN (G-CONSN (G-CADRN HINTS)
                                (G-CONSN (G-CADDRN HINTS)
                                         (G-CONSN (G-CADDDRN HINTS) (GCODE NIL))))
                       (GCODE 1)
                       SYMBOLS
                       N)
             (PR-EQUALN CONC
                        (G-F-IFFN (G-ISINN (G-CADDDRN HINTS)
                                           (G-Z-PAIRN (G-CADRN HINTS) (G-CADDRN HINTS)))
                                  (G-F-ORN (G-F-EQUALN (G-CADDDRN HINTS) (G-CADRN HINTS))
                                           (G-F-EQUALN (G-CADDDRN HINTS) (G-CADDRN HINTS))))))
    (PR-IFN
     (PR-EQUALN (G-CARN HINTS) (GCODE 4))
     (PR-ANDN (G-TERMPN (G-CONSN (G-CADRN HINTS)
                                 (G-CONSN (G-CADDRN HINTS)
                                          (G-CONSN (G-CADDDRN HINTS) (GCODE NIL))))
                        (GCODE 1)
                        SYMBOLS
                        N)
              (PR-EQUALN CONC
                         (G-F-IFFN (G-ISINN (G-CADDDRN HINTS)
                                            (G-Z-UNIONN (G-CADRN HINTS) (G-CADDRN HINTS)))
                                   (G-F-ORN (G-ISINN (G-CADDDRN HINTS) (G-CADRN HINTS))
                                            (G-ISINN (G-CADDDRN HINTS) (G-CADDRN HINTS))))))
     (PR-IFN
      (PR-EQUALN (G-CARN HINTS) (GCODE 5))
      (PR-ANDN
       (G-VARIABLEN (G-CADDRN HINTS))
       (PR-ANDN
        (G-FORMULAN (G-CADRN HINTS) SYMBOLS N)
        (PR-EQUALN
         CONC
         (G-F-IMPLIESN
          (G-F-ANDN (G-SUBSTN (G-CADRN HINTS) (G-CADDRN HINTS) (G-PHIN) (GCODE 0) N)
                    (G-FORALLN (G-CADDRN HINTS)
                               (G-F-IMPLIESN (G-F-ANDN (G-Z-INTN (G-CADDRN HINTS))
                                                       (G-CADRN HINTS))
                                             (G-SUBSTN (G-CADRN HINTS)
                                                       (G-CADDRN HINTS)
                                                       (G-Z-SUCCN (G-CADDRN HINTS))
                                                       (GCODE 0)
                                                       N))))
          (G-FORALLN (G-CADDRN HINTS)
                     (G-F-IMPLIESN (G-Z-INTN (G-CADDRN HINTS)) (G-CADRN HINTS)))))))
      (BOOL-FIX F)))))))
\end{verbatim}


\noindent{\sc Event}:   Disable g-z2-axioms; name this event `g1238'.\index{g1238}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-z2-axioms-eval \\ \index{g-z2-axioms-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-z2-axioms}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{g-z2-axioms}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-z2-axiomsn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{conc\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{hints\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-z2-axiomsn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable p; name this event `g1239'.\index{g1239}

\addvspace{18pt}

\noindent{\sc Event}:   Disable fn; name this event `g1240'.\index{g1240}

\addvspace{18pt}

\noindent{\sc Event}:   Disable isin; name this event `g1241'.\index{g1241}

\addvspace{18pt}

\noindent{\sc Event}:   Disable phi; name this event `g1242'.\index{g1242}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-pair; name this event `g1243'.\index{g1243}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-union; name this event `g1244'.\index{g1244}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-int; name this event `g1245'.\index{g1245}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-succ; name this event `g1246'.\index{g1246}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z2-axioms; name this event `g1247'.\index{g1247}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{z2-axioms-cnt} \\  
\=\+{\rm{z2-axioms-cnt}}\,({\it{hints\/}}, {\it{symbols\/}})\index{z2-axioms-cnt} \\ 
$=$$\;\;\;\;$(\=\+{\rm{termp-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\tt{0}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{0}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,(\=\+{\rm{list}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}), \\ 
{\tt{1}}, \\ 
{\it{symbols\/}})\-\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula-cnt})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-z2-axiomsn-gcode \\ \index{g-z2-axiomsn-gcode}
\=\+(\=\+(\=\+{\rm{termp-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\tt{0}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\tt{0}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{termp-cnt}}\,({\rm{list}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{third}}\,({\it{hints\/}})\index{third}, {\rm{fourth}}\,({\it{hints\/}})\index{fourth}), {\tt{1}}, {\it{symbols\/}})\index{termp-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula-cnt})\- \\ 
$<$$\;\;\;\;${\it{n\/}})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-z2-axiomsn}}\,({\rm{gcode}}\,({\it{conc\/}})\index{gcode}, {\rm{gcode}}\,({\it{hints\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-z2-axiomsn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{z2-axioms}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{z2-axioms})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable fn; name this event `g1248'.\index{g1248}

\addvspace{18pt}

\noindent{\sc Event}:   Enable isin; name this event `g1249'.\index{g1249}

\addvspace{18pt}

\noindent{\sc Event}:   Enable phi; name this event `g1250'.\index{g1250}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-pair; name this event `g1251'.\index{g1251}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-union; name this event `g1252'.\index{g1252}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-int; name this event `g1253'.\index{g1253}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-succ; name this event `g1254'.\index{g1254}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z2-axioms; name this event `g1255'.\index{g1255}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z2-axiomsn; name this event `g1256'.\index{g1256}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z2-axioms; name this event `g1257'.\index{g1257}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for SET.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-setn} \\  
\=\+{\rm{g-setn}}\,({\it{x\/}}, {\it{n\/}})\index{g-setn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{x\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{pr-notn}}\,({\rm{g-membn}}\,({\rm{g-carn}}\,({\it{x\/}})\index{g-carn}, {\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-membn})\index{pr-notn}, \\ 
{\rm{g-setn}}\,({\rm{g-cdrn}}\,({\it{x\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-setn})\-\index{pr-andn}, \\ 
{\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-set; name this event `g1258'.\index{g1258}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-set-defn; name this event `g1259'.\index{g1259}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-set-ind} \\  
\=\+{\rm{g-set-ind}}\,({\it{x\/}}, {\it{va\/}}, {\it{n\/}})\index{g-set-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-set-ind}}\,({\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-set-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-set-eval \\ \index{g-set-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-set}}\,({\it{x\/}})\index{g-set}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-setn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-setn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-setn-gcode \\ \index{g-setn-gcode}
({\rm{length}}\,({\it{x\/}})\index{length} $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-setn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\it{n\/}})\index{g-setn} $=$ {\rm{bool-fix}}\,({\rm{set}}\,({\it{x\/}})\index{set})\index{bool-fix})
\end{tabbing}

\noindent{\sc Event}:   Enable g-set; name this event `g1260'.\index{g1260}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-setn; name this event `g1261'.\index{g1261}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-set-defn; name this event `g1262'.\index{g1262}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for ASSOC.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-assocn} \\  
\=\+{\rm{g-assocn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-assocn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{y\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\rm{g-carn}}\,({\it{y\/}})\index{g-carn})\index{g-carn}, {\it{x\/}})\index{pr-equaln}, \\ 
{\rm{g-cdrn}}\,({\rm{g-carn}}\,({\it{y\/}})\index{g-carn})\index{g-cdrn}, \\ 
{\rm{g-assocn}}\,({\it{x\/}}, {\rm{g-cdrn}}\,({\it{y\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-assocn})\-\index{pr-ifn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-assoc-defn; name this event `g1265'.\index{g1265}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-assoc-ind} \\  
\=\+{\rm{g-assoc-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-assoc-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-assoc-ind}}\,({\tt{0}}, {\rm{g-cdr}}\,({\tt{1}})\index{g-cdr}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-assoc-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-assoc-eval \\ \index{g-assoc-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-assoc}}\,({\it{x\/}}, {\it{y\/}})\index{g-assoc}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-assocn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-assocn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-assocn-gcode \\ \index{g-assocn-gcode}
({\rm{length}}\,({\it{y\/}})\index{length} $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-assocn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-assocn} $=$ {\rm{gcode}}\,({\rm{assoc}}\,({\it{x\/}}, {\it{y\/}}))\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-assoc; name this event `g1266'.\index{g1266}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-assocn; name this event `g1267'.\index{g1267}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-assoc-defn; name this event `g1268'.\index{g1268}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for CONC.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-concn} \\  
\=\+{\rm{g-concn}}\,({\it{pf\/}}, {\it{flg\/}}, {\it{n\/}})\index{g-concn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{pf\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{flg\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{flg\/}})\index{pr-zeropn}, \\ 
{\rm{g-caddrn}}\,({\it{pf\/}})\index{g-caddrn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{pf\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-concn}}\,({\rm{g-carn}}\,({\it{pf\/}})\index{g-carn}, {\rm{gcode}}\,({\tt{0}})\index{gcode}, {\it{n\/}} $-\;1$)\index{g-concn}, \\ 
{\rm{g-concn}}\,({\rm{g-cdrn}}\,({\it{pf\/}})\index{g-cdrn}, {\rm{gcode}}\,({\tt{1}})\index{gcode}, {\it{n\/}} $-\;1$)\index{g-concn})\-\index{g-consn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-conc; name this event `g1269'.\index{g1269}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-conc-defn; name this event `g1270'.\index{g1270}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-conc-ind} \\  
\=\+{\rm{g-conc-ind}}\,({\it{pf\/}}, {\it{flg\/}}, {\it{va\/}}, {\it{n\/}})\index{g-conc-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-conc-ind}}\,(\=\+{\rm{g-car}}\,({\tt{0}})\index{g-car}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{pf\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-conc-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-conc-ind}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{pf\/}}, {\it{flg\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-conc-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-conc-eval \\ \index{g-conc-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-conc}}\,({\it{pf\/}}, {\it{flg\/}})\index{g-conc}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-concn}}\,({\rm{ev}}\,({\tt{0}}, {\it{pf\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{flg\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-concn}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{conc-ind} \\  
\=\+{\rm{conc-ind}}\,({\it{pf\/}}, {\it{flg\/}}, {\it{n\/}})\index{conc-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{flg\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{conc-ind}}\,({\rm{car}}\,({\it{pf\/}}), {\tt{0}}, {\it{n\/}} $-\;1$)\index{conc-ind} $\wedge$ {\rm{conc-ind}}\,({\rm{cdr}}\,({\it{pf\/}}), {\tt{1}}, {\it{n\/}} $-\;1$)\index{conc-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-concn-gcode \\ \index{g-concn-gcode}
\=\+({\rm{exp-count}}\,({\it{pf\/}})\index{exp-count} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-concn}}\,({\rm{gcode}}\,({\it{pf\/}})\index{gcode}, {\rm{gcode}}\,({\it{flg\/}})\index{gcode}, {\it{n\/}})\index{g-concn} $=$ {\rm{gcode}}\,({\rm{conc}}\,({\it{pf\/}}, {\it{flg\/}})\index{conc})\index{gcode})\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-conc; name this event `g1271'.\index{g1271}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-concn; name this event `g1272'.\index{g1272}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-conc-defn; name this event `g1273'.\index{g1273}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for REST-OF.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-rest-ofn} \\  
\=\+{\rm{g-rest-ofn}}\,({\it{sym\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-rest-ofn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{sym\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{symbols\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-typepn}}\,({\it{symbols\/}}, {\tt{1}})\index{g-typepn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-carn}}\,({\it{symbols\/}})\index{g-carn}, {\it{sym\/}})\index{pr-equaln}, \\ 
{\rm{pr-notn}}\,({\rm{g-membn}}\,(\=\+{\it{sym\/}}, \\ 
{\rm{g-cdrn}}\,({\it{symbols\/}})\index{g-cdrn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-membn})\index{pr-notn})\-\index{pr-andn}, \\ 
{\rm{g-cdrn}}\,({\it{symbols\/}})\index{g-cdrn}, \\ 
{\rm{g-rest-ofn}}\,({\it{sym\/}}, {\rm{g-cdrn}}\,({\it{symbols\/}})\index{g-cdrn}, {\it{n\/}} $-\;1$)\index{g-rest-ofn})\-\index{pr-ifn}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-rest-of; name this event `g1274'.\index{g1274}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-rest-of-defn; name this event `g1275'.\index{g1275}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-rest-of-ind} \\  
\=\+{\rm{g-rest-of-ind}}\,({\it{sym\/}}, {\it{symbols\/}}, {\it{va\/}}, {\it{n\/}})\index{g-rest-of-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-rest-of-ind}}\,(\=\+{\tt{0}}, \\ 
{\rm{g-cdr}}\,({\tt{1}})\index{g-cdr}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{sym\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-rest-of-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-rest-of-eval \\ \index{g-rest-of-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-rest-of}}\,({\it{sym\/}}, {\it{symbols\/}})\index{g-rest-of}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-rest-ofn}}\,({\rm{ev}}\,({\tt{0}}, {\it{sym\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-rest-ofn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable rest-of; name this event `g1276'.\index{g1276}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-rest-ofn-gcode \\ \index{g-rest-ofn-gcode}
\=\+(($1\;+$ {\rm{length}}\,({\it{symbols\/}})\index{length}) $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-rest-ofn}}\,({\rm{gcode}}\,({\it{sym\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-rest-ofn} \\ 
$=$$\;\;\;\;${\rm{gcode}}\,({\rm{rest-of}}\,({\it{sym\/}}, {\it{symbols\/}})\index{rest-of})\index{gcode})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable rest-of; name this event `g1277'.\index{g1277}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-rest-of; name this event `g1278'.\index{g1278}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-rest-ofn; name this event `g1279'.\index{g1279}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-rest-of-defn; name this event `g1280'.\index{g1280}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-symb-defn-proof; name this event `g1281'.\index{g1281}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for SYMB-DEFN-PROOF.
\end{verbatim}%\item 
\begin{verbatim}

;;;;;;;Correspondence for SYMB-DEFN-PROOF.
(DEFN
 G-SYMB-DEFN-PROOFN
 (CONC HINTS SUB-GOALS DEFN SYMBOLS N)
 (PR-IFN
  (G-PREDICATEN (G-CARN (G-CARN HINTS)))
  (PR-ANDN (G-SETN (G-CDRN (G-CARN HINTS)) N)
           (PR-ANDN (G-VAR-LISTN (G-CDRN (G-CARN HINTS)) (G-DEGREEN (G-CARN (G-CARN HINTS))) N)
                    (PR-ANDN (PR-EQUALN CONC (G-F-IFFN (G-CARN HINTS) DEFN))
                             (PR-ANDN (G-SUBN (G-COLLECT-FREEN DEFN (GCODE 0) N)
                                              (G-CDRN (G-CARN HINTS))
                                              N)
                                      (G-FORMULAN DEFN
                                                  (G-REST-OFN (G-CARN (G-CARN HINTS))
                                                              SYMBOLS
                                                              N)
                                                  N)))))
  (PR-IFN
   (G-FUNCTIONN (G-CARN (G-CARN HINTS)))
   (PR-ANDN
    (G-SETN (G-CONSN (G-CADDDRN HINTS) (G-CONSN (G-CADRN HINTS) (G-CDRN (G-CARN HINTS)))) N)
    (PR-ANDN
     (G-VAR-LISTN (G-CDRN (G-CARN HINTS)) (G-DEGREEN (G-CARN (G-CARN HINTS))) N)
     (PR-ANDN
      (G-VARIABLEN (G-CADRN HINTS))
      (PR-ANDN
       (G-VARIABLEN (G-CADDDRN HINTS))
       (PR-ANDN
        (G-FORMULAN (G-CADDRN HINTS) (G-REST-OFN (G-CARN (G-CARN HINTS)) SYMBOLS N) N)
        (PR-ANDN
         (G-SUBN (G-COLLECT-FREEN (G-CADDRN HINTS) (GCODE 0) N)
                 (G-CONSN (G-CADRN HINTS) (G-CDRN (G-CARN HINTS)))
                 N)
         (PR-ANDN
          (G-FREE-FORN (G-CADDRN HINTS) (G-CADRN HINTS) (G-CARN HINTS) (GCODE 0) N)
          (PR-ANDN
           (G-FREE-FORN (G-CADDRN HINTS) (G-CADRN HINTS) (G-CADDDRN HINTS) (GCODE 0) N)
           (PR-ANDN
            (PR-EQUALN CONC
                       (G-SUBSTN (G-CADDRN HINTS)
                                 (G-CADRN HINTS)
                                 (G-CARN HINTS)
                                 (GCODE 0)
                                 N))
            (PR-ANDN
             (PR-EQUALN CONC DEFN)
             (PR-EQUALN
              SUB-GOALS
              (G-CONSN (G-FORSOMEN (G-CADRN HINTS) (G-CADDRN HINTS))
                       (G-CONSN (G-F-IMPLIESN (G-F-ANDN (G-CADDRN HINTS)
                                                        (G-SUBSTN (G-CADDRN HINTS)
                                                                  (G-CADRN HINTS)
                                                                  (G-CADDDRN HINTS)
                                                                  (GCODE 0)
                                                                  N))
                                              (G-F-EQUALN (G-CADRN HINTS)
                                                          (G-CADDDRN HINTS)))
                                (GCODE NIL))))))))))))))
   (BOOL-FIX F))))
\end{verbatim}

\begin{tabbing}{\sc Theorem}:  g-symb-defn-proof-eval \\ \index{g-symb-defn-proof-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-symb-defn-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{defn\/}}, {\it{symbols\/}})\index{g-symb-defn-proof}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-symb-defn-proofn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{conc\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{hints\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{sub-goals\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{defn\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-symb-defn-proofn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lazy-pr-andn-reduc3 \\ \index{lazy-pr-andn-reduc3}
{\rm{pr-andn}}\,({\rm{bool-fix}}\,({\it{x\/}})\index{bool-fix}, {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\index{pr-andn} $=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-symb-count} \\  
\=\+{\rm{g-symb-count}}\,({\it{hints\/}}, {\it{defn\/}}, {\it{symbols\/}})\index{g-symb-count} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{predicate}}\,({\rm{caar}}\,({\it{hints\/}}))\index{predicate} \\ 
{\bf then }\=\+{\rm{length}}\,({\rm{cdar}}\,({\it{hints\/}}))\index{length} \\ 
$+$$\;\;\;\;${\rm{cddaar}}\,({\it{hints\/}}) \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\it{defn\/}})\index{exp-count} \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{length}}\,({\it{symbols\/}})\index{length}) \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\it{defn\/}}, {\rm{rest-of}}\,({\rm{caar}}\,({\it{hints\/}}), {\it{symbols\/}})\index{rest-of})\index{formula-cnt}\- \\ 
{\bf else }\=\+{\rm{length}}\,({\rm{cons}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\rm{cons}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\rm{cdar}}\,({\it{hints\/}}))))\index{length} \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{length}}\,({\it{symbols\/}})\index{length}) \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,(\=\+{\rm{third}}\,({\it{hints\/}})\index{third}, \\ 
{\rm{rest-of}}\,({\rm{caar}}\,({\it{hints\/}}), {\it{symbols\/}})\index{rest-of})\-\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{cddaar}}\,({\it{hints\/}}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{exp-count}}\,({\rm{third}}\,({\it{hints\/}})\index{third})\index{exp-count}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{exp-count}}\,({\rm{second}}\,({\it{hints\/}})\index{second})\index{exp-count}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{exp-count}}\,({\rm{car}}\,({\it{hints\/}}))\index{exp-count}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{exp-count}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{exp-count})\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-symb-defn-proofn-gcode \\ \index{g-symb-defn-proofn-gcode}
\=\+({\rm{g-symb-count}}\,({\it{hints\/}}, {\it{defn\/}}, {\it{symbols\/}})\index{g-symb-count} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-symb-defn-proofn}}\,(\=\+{\rm{gcode}}\,({\it{conc\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{hints\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{sub-goals\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-symb-defn-proofn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{symb-defn-proof}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{defn\/}}, {\it{symbols\/}})\index{symb-defn-proof})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable symb-defn-proof; name this event `g1290'.\index{g1290}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-symb-defn-proof; name this event `g1291'.\index{g1291}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-symb-defn-proofn; name this event `g1292'.\index{g1292}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for PRF.
\end{verbatim}%\item 
\begin{verbatim}

;;;;;;;Correspondence for PRF.
(DEFN
 G-PRFN
 (PF GIVEN DEFNS FLG SYMBOLS N)
 (IF
  (OR (ZEROP N)
      (BTMP PF)
      (BTMP GIVEN)
      (BTMP DEFNS)
      (BTMP FLG)
      (BTMP SYMBOLS))
  (BTM)
  (PR-IFN
   (PR-ZEROPN FLG)
   (PR-IFN
    (G-TYPEPN PF 1)
    (PR-IFN
     (PR-NOTN (G-FORMULAN (G-CONCN PF (GCODE 0) (SUB1 N)) SYMBOLS (SUB1 N)))
     (BOOL-FIX F)
     (PR-IFN
      (PR-EQUALN (G-CARN PF) (GCODE 1))
      (G-FOL-AXIOM-PROOFN (G-CONCN PF (GCODE 0) (SUB1 N)) (G-CADRN PF) SYMBOLS (SUB1 N))
      (PR-IFN
       (PR-EQUALN (G-CARN PF) (GCODE 2))
       (PR-ANDN (G-FOL-RULESN (G-CONCN PF (GCODE 0) (SUB1 N))
                              (G-CADRN PF)
                              (G-CONCN (G-CADDDRN PF) (GCODE 1) (SUB1 N))
                              SYMBOLS
                              (SUB1 N))
                (G-PRFN (G-CADDDRN PF) GIVEN DEFNS (GCODE 1) SYMBOLS (SUB1 N)))
       (PR-IFN (PR-EQUALN (G-CARN PF) (GCODE 3))
               (G-Z2-AXIOMSN (G-CONCN PF (GCODE 0) (SUB1 N)) (G-CADRN PF) SYMBOLS (SUB1 N))
               (PR-IFN (PR-EQUALN (G-CARN PF) (GCODE 4))
                       (PR-ANDN (G-SYMB-DEFN-PROOFN (G-CONCN PF (GCODE 0) (SUB1 N))
                                                    (G-CADRN PF)
                                                    (G-CONCN (G-CADDDRN PF)
                                                             (GCODE 1)
                                                             (SUB1 N))
                                                    (G-ASSOCN (G-CARN (G-CARN (G-CADRN PF)))
                                                              DEFNS
                                                              (SUB1 N))
                                                    SYMBOLS
                                                    (SUB1 N))
                                (PR-IFN (G-FUNCTIONN (G-CARN (G-CARN (G-CADRN PF))))
                                        (G-PRFN (G-CADDDRN PF)
                                                GIVEN
                                                DEFNS
                                                (GCODE 1)
                                                (G-REST-OFN (G-CARN (G-CARN (G-CADRN PF)))
                                                            SYMBOLS
                                                            (SUB1 N))
                                                (SUB1 N))
                                        (BOOL-FIX T)))
                       (PR-IFN (PR-EQUALN (G-CARN PF) (GCODE 5))
                               (G-MEMBN (G-CONCN PF (GCODE 0) (SUB1 N)) GIVEN (SUB1 N))
                               (BOOL-FIX F)))))))
    (BOOL-FIX F))
   (PR-IFN (G-TYPEPN PF 1)
           (PR-ANDN (G-PRFN (G-CARN PF) GIVEN DEFNS (GCODE 0) SYMBOLS (SUB1 N))
                    (G-PRFN (G-CDRN PF) GIVEN DEFNS (GCODE 1) SYMBOLS (SUB1 N)))
           (BOOL-FIX T)))))
\end{verbatim}


\noindent{\sc Event}:   Disable g-prf-defn; name this event `g1293'.\index{g1293}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-prf-ind} \\  
\=\+{\rm{g-prf-ind}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}}, {\it{va\/}}, {\it{n\/}})\index{g-prf-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-prf-ind}}\,(\=\+{\rm{g-cadddr}}\,({\tt{0}})\index{g-cadddr}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\tt{4}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-prf-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-prf-ind}}\,(\=\+{\rm{g-cadddr}}\,({\tt{0}})\index{g-cadddr}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\rm{g-rest-of}}\,({\rm{g-car}}\,({\rm{g-car}}\,({\rm{g-cadr}}\,({\tt{0}})\index{g-cadr})\index{g-car})\index{g-car}, {\tt{4}})\index{g-rest-of}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{list}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}}), \\ 
{\it{va\/}}, \\ 
{\sc{fa}}\index{fa}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-prf-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-prf-ind}}\,(\=\+{\rm{g-car}}\,({\tt{0}})\index{g-car}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\tt{0}})\index{pr-quote}, \\ 
{\tt{4}}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{list}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}}), \\ 
{\it{va\/}}, \\ 
{\sc{fa}}\index{fa}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-prf-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-prf-ind}}\,(\=\+{\rm{g-cdr}}\,({\tt{0}})\index{g-cdr}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\rm{pr-quote}}\,({\tt{1}})\index{pr-quote}, \\ 
{\tt{4}}, \\ 
{\rm{ev}}\,(\=\+{\tt{1}}, \\ 
{\rm{list}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}}), \\ 
{\it{va\/}}, \\ 
{\sc{fa}}\index{fa}, \\ 
{\it{n\/}})\-\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-prf-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-prf-eval \\ \index{g-prf-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-prf}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{g-prf}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-prfn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{pf\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{given\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{defns\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{flg\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-prfn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable prf; name this event `g1294'.\index{g1294}

\addvspace{18pt}

\noindent{\sc Event}:   Enable fol-axiom-cnt; name this event `g1296'.\index{g1296}

\addvspace{18pt}

\noindent{\sc Event}:   Enable fol-axiom-cnt; name this event `g1297'.\index{g1297}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{fol-rules-cnt} \\  
\=\+{\rm{fol-rules-cnt}}\,({\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{fol-rules-cnt} \\ 
$=$$\;\;\;\;$(\=\+{\rm{formula-cnt}}\,({\rm{second}}\,({\it{hints\/}})\index{second}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{third}}\,({\it{hints\/}})\index{third}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{car}}\,({\it{sub-goals\/}}), {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{fourth}}\,({\it{hints\/}})\index{fourth})\index{exp-count})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-fol-rulesn-gcode1 \\ \index{g-fol-rulesn-gcode1}
\=\+({\rm{fol-rules-cnt}}\,({\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{fol-rules-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-fol-rulesn}}\,(\=\+{\rm{gcode}}\,({\it{conc\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{hints\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{sub-goals\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-fol-rulesn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{fol-rules}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{sub-goals\/}}, {\it{symbols\/}})\index{fol-rules})\index{bool-fix})\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prf-cnt-part} \\  
\=\+{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part} \\ 
$=$$\;\;\;\;$(($1\;+$ {\rm{formula-cnt}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\it{symbols\/}})\index{formula-cnt}) $+$ ($1\;+$ {\rm{exp-count}}\,({\it{pf\/}})\index{exp-count}))\-
\end{tabbing}

\noindent{\sc Event}:   Enable fol-rules-cnt; name this event `g1298'.\index{g1298}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{prf-ind} \\  
\=\+{\rm{prf-ind}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}}, {\it{n\/}})\index{prf-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{pf\/}}) \\ 
{\bf then }\=\+{\bf if }$\neg$ {\rm{formula}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\it{symbols\/}})\index{formula}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{1}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{2}} \\ 
{\bf then }{\rm{prf-ind}}\,({\rm{cadddr}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{1}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{prf-ind} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{3}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{4}} \\ 
{\bf then }{\rm{prf-ind}}\,(\=\+{\rm{cadddr}}\,({\it{pf\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\tt{1}}, \\ 
{\rm{rest-of}}\,({\rm{caar}}\,({\rm{cadr}}\,({\it{pf\/}})), {\it{symbols\/}})\index{rest-of}, \\ 
{\it{n\/}} $-\;1$)\-\index{prf-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{pf\/}}) \\ 
{\bf then }\=\+{\rm{prf-ind}}\,({\rm{car}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{prf-ind} \\ 
$\wedge$$\;\;\;\;${\rm{prf-ind}}\,({\rm{cdr}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{1}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{prf-ind}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-symb-count; name this event `g1299'.\index{g1299}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-z2-axiomsn-gcode1 \\ \index{g-z2-axiomsn-gcode1}
\=\+({\rm{z2-axioms-cnt}}\,({\it{hints\/}}, {\it{symbols\/}})\index{z2-axioms-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-z2-axiomsn}}\,({\rm{gcode}}\,({\it{conc\/}})\index{gcode}, {\rm{gcode}}\,({\it{hints\/}})\index{gcode}, {\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, {\it{n\/}})\index{g-z2-axiomsn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{z2-axioms}}\,({\it{conc\/}}, {\it{hints\/}}, {\it{symbols\/}})\index{z2-axioms})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable z2-axioms-cnt; name this event `g1300'.\index{g1300}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z2-axiomsn-gcode; name this event `g1301'.\index{g1301}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-fol-rulesn-gcode; name this event `g1302'.\index{g1302}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{prf-cnt} \\  
\=\+{\rm{prf-cnt}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{prf-cnt} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{flg\/}} $\simeq {\tt{0}}$ \\ 
{\bf then }\=\+{\bf if }{\rm{listp}}\,({\it{pf\/}}) \\ 
{\bf then }\=\+{\bf if }$\neg$ {\rm{formula}}\,({\rm{conc}}\,({\it{pf\/}}, {\tt{0}})\index{conc}, {\it{symbols\/}})\index{formula} \\ 
{\bf then }{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part} \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{1}} \\ 
{\bf then }\=\+($1\;+$ {\rm{fol-axiom-cnt}}\,({\rm{second}}\,({\it{pf\/}})\index{second}, {\it{symbols\/}})\index{fol-axiom-cnt}) \\ 
$+$$\;\;\;\;${\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{2}} \\ 
{\bf then }\=\+{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part} \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{fol-rules-cnt}}\,(\=\+{\rm{second}}\,({\it{pf\/}})\index{second}, \\ 
{\rm{conc}}\,({\rm{cadddr}}\,({\it{pf\/}}), {\tt{1}})\index{conc}, \\ 
{\it{symbols\/}})\-\index{fol-rules-cnt}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{prf-cnt}}\,({\rm{cadddr}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{1}}, {\it{symbols\/}})\index{prf-cnt})\- \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{3}} \\ 
{\bf then }\=\+{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part} \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{z2-axioms-cnt}}\,({\rm{second}}\,({\it{pf\/}})\index{second}, {\it{symbols\/}})\index{z2-axioms-cnt})\- \\ 
{\bf elseif }{\rm{car}}\,({\it{pf\/}}) $=$ {\tt{4}} \\ 
{\bf then }\=\+{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part} \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{g-symb-count}}\,(\=\+{\rm{second}}\,({\it{pf\/}})\index{second}, \\ 
{\rm{assoc}}\,({\rm{caar}}\,({\rm{cadr}}\,({\it{pf\/}})), {\it{defns\/}}), \\ 
{\it{symbols\/}})\-\index{g-symb-count}) \\ 
$+$$\;\;\;\;$($1\;+$ ($1\;+$ {\rm{length}}\,({\it{symbols\/}})\index{length})) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{length}}\,({\it{defns\/}})\index{length}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{exp-count}}\,({\rm{fourth}}\,({\it{pf\/}})\index{fourth})\index{exp-count}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{prf-cnt}}\,(\=\+{\rm{cadddr}}\,({\it{pf\/}}), \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\tt{1}}, \\ 
{\rm{rest-of}}\,({\rm{caar}}\,({\rm{cadr}}\,({\it{pf\/}})), {\it{symbols\/}})\index{rest-of})\-\index{prf-cnt})\- \\ 
{\bf else }\=\+{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part} \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{length}}\,({\it{given\/}})\index{length})\-$\;${\bf  endif}\- \\ 
{\bf else }{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part}$\;${\bf  endif}\- \\ 
{\bf elseif }{\rm{listp}}\,({\it{pf\/}}) \\ 
{\bf then }$1\;+$ (\=\+($1\;+$ {\rm{prf-cnt}}\,({\rm{car}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf-cnt}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{prf-cnt}}\,({\rm{cdr}}\,({\it{pf\/}}), {\it{given\/}}, {\it{defns\/}}, {\tt{1}}, {\it{symbols\/}})\index{prf-cnt}))\- \\ 
{\bf else }{\rm{prf-cnt-part}}\,({\it{pf\/}}, {\it{symbols\/}})\index{prf-cnt-part}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-prfn-gcode \\ \index{g-prfn-gcode}
\=\+({\rm{prf-cnt}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{prf-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-prfn}}\,(\=\+{\rm{gcode}}\,({\it{pf\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{flg\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-prfn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{prf}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{flg\/}}, {\it{symbols\/}})\index{prf})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-prf; name this event `g1303'.\index{g1303}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-prfn; name this event `g1304'.\index{g1304}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-prf-defn; name this event `g1305'.\index{g1305}

\addvspace{18pt}

\noindent{\sc Event}:   Enable prf-cnt; name this event `g1306'.\index{g1306}

\addvspace{18pt}

\noindent{\sc Event}:   Enable prf; name this event `g1307'.\index{g1307}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for PROVES.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-provesn} \\  
\=\+{\rm{g-provesn}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-provesn} \\ 
$=$$\;\;\;\;${\rm{pr-andn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{g-concn}}\,({\it{pf\/}}, {\rm{gcode}}\,({\tt{0}})\index{gcode}, {\it{n\/}})\index{g-concn}, {\it{exp\/}})\index{pr-equaln}, \\ 
{\rm{pr-andn}}\,(\=\+{\rm{g-formulan}}\,({\it{exp\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-formulan}, \\ 
{\rm{g-prfn}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\rm{gcode}}\,({\tt{0}})\index{gcode}, {\it{symbols\/}}, {\it{n\/}})\index{g-prfn})\-\index{pr-andn})\-\index{pr-andn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-proves; name this event `g1308'.\index{g1308}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-proves-eval \\ \index{g-proves-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{g-proves}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-provesn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{pf\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{given\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{defns\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-provesn}\-
\end{tabbing}

\noindent{\sc Event}:   Disable proves; name this event `g1309'.\index{g1309}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-provesn-gcode \\ \index{g-provesn-gcode}
\=\+(\=\+(\=\+{\rm{exp-count}}\,({\it{pf\/}})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{prf-cnt}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf-cnt})\- \\ 
$<$$\;\;\;\;${\it{n\/}})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-provesn}}\,(\=\+{\rm{gcode}}\,({\it{pf\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{exp\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-provesn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\index{bool-fix})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable proves; name this event `g1310'.\index{g1310}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-proves; name this event `g1311'.\index{g1311}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-provesn; name this event `g1312'.\index{g1312}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence for NUMERAL.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-numeraln} \\  
\=\+{\rm{g-numeraln}}\,({\it{x\/}}, {\it{n\/}})\index{g-numeraln} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{x\/}})\index{pr-zeropn}, \\ 
{\sc{g-phin}}\index{g-phin}, \\ 
{\rm{g-z-succn}}\,({\rm{g-numeraln}}\,({\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n}, {\it{n\/}} $-\;1$)\index{g-numeraln})\index{g-z-succn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-numeral-defn; name this event `g1313'.\index{g1313}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-numeral-ind} \\  
\=\+{\rm{g-numeral-ind}}\,({\it{x\/}}, {\it{va\/}}, {\it{n\/}})\index{g-numeral-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-numeral-ind}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-numeral-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-numeral-eval \\ \index{g-numeral-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-numeral}}\,({\it{x\/}})\index{g-numeral}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-numeraln}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-numeraln}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{numeral-ind} \\  
\=\+{\rm{numeral-ind}}\,({\it{x\/}}, {\it{n\/}})\index{numeral-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{numeral-ind}}\,({\it{x\/}} $-\;1$, {\it{n\/}} $-\;1$)\index{numeral-ind}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable phi; name this event `g1314'.\index{g1314}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-succ; name this event `g1315'.\index{g1315}

\addvspace{18pt}

\noindent{\sc Event}:   Disable fn; name this event `g1316'.\index{g1316}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-numeraln-gcode \\ \index{g-numeraln-gcode}
({\it{x\/}} $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-numeraln}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\it{n\/}})\index{g-numeraln} $=$ {\rm{gcode}}\,({\rm{numeral}}\,({\it{x\/}})\index{numeral})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable phi; name this event `g1317'.\index{g1317}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-succ; name this event `g1318'.\index{g1318}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-numeral; name this event `g1319'.\index{g1319}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-numeraln; name this event `g1320'.\index{g1320}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-numeral-defn; name this event `g1321'.\index{g1321}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-z-singn}$\;\;$
{\rm{g-z-singn}}\,({\it{x\/}})\index{g-z-singn} $=$ {\rm{g-z-pairn}}\,({\it{x\/}}, {\it{x\/}})\index{g-z-pairn}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-z-opairn} \\  
{\rm{g-z-opairn}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-opairn} $=$ {\rm{g-z-pairn}}\,({\rm{g-z-singn}}\,({\it{x\/}})\index{g-z-singn}, {\rm{g-z-pairn}}\,({\it{x\/}}, {\it{y\/}})\index{g-z-pairn})\index{g-z-pairn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev-good-proof-proves-again \\ \index{a-ev-good-proof-proves-again}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{variable}}\,({\it{trace\/}})\index{variable} \\ 
$\wedge$$\;\;\;\;$({\it{trace\/}} $\not\in$ {\rm{list}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\tt{11}}, {\tt{12}}, {\tt{15}}, {\tt{16}})) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{10}}, \\ 
{\it{trace\/}})\-\index{a-ev}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{trace\/}})\index{a-ev-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev1} \\  
\=\+{\rm{a-ev1}}\,({\it{flg-bar\/}}, {\it{exp-bar\/}}, {\it{va-bar\/}}, {\it{fa-bar\/}}, {\it{val-bar\/}})\index{a-ev1} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,({\tt{1}}, {\rm{a-ev}}\,({\it{flg-bar\/}}, {\it{exp-bar\/}}, {\it{va-bar\/}}, {\it{fa-bar\/}}, {\it{val-bar\/}}, {\tt{1}})\index{a-ev})\index{forsome}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev; name this event `g1323'.\index{g1323}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev1-ok-proof} \\  
\=\+{\rm{a-ev1-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev1-ok-proof} \\ 
$=$$\;\;\;\;${\rm{eg-proof}}\,(\=\+{\rm{a-ev}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode}, \\ 
{\tt{1}})\-\index{a-ev}, \\ 
{\rm{list}}\,({\tt{1}}), \\ 
{\rm{list}}\,({\rm{graph-trans}}\,({\rm{ev-graph}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev-graph})\index{graph-trans}), \\ 
{\rm{a-ev-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev-ok-proof})\-\index{eg-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  formula-a-ev-defn-again \\ \index{formula-a-ev-defn-again}
\=\+({\rm{variable}}\,({\it{trace\/}})\index{variable} $\wedge$ {\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps}) \\ 
$\rightarrow$$\;\;\;\;${\rm{formula}}\,({\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}, {\it{symbols\/}})\index{formula}\-
\end{tabbing}

\noindent{\sc Event}:   Disable a-ev-defn-part1; name this event `g1324'.\index{g1324}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  parallel-subst-trace-a-ev-defn-part1 \\ \index{parallel-subst-trace-a-ev-defn-p!art1}
\=\+{\rm{parallel-subst}}\,({\rm{a-ev-defn-part1}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\tt{1}})\index{a-ev-defn-part1}, {\rm{list}}\,({\tt{1}}), {\rm{list}}\,({\it{trace\/}}), {\tt{0}})\index{parallel-subst} \\ 
$=$$\;\;\;\;${\rm{a-ev-defn-part1}}\,({\tt{6}}, {\tt{7}}, {\tt{8}}, {\tt{9}}, {\tt{10}}, {\it{trace\/}})\index{a-ev-defn-part1}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part1; name this event `g1325'.\index{g1325}

\addvspace{18pt}

\noindent{\sc Event}:   Disable a-ev-defn; name this event `g1326'.\index{g1326}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  parallel-subst-trace-a-ev-defn \\ \index{parallel-subst-trace-a-ev-defn}
{\rm{parallel-subst}}\,({\rm{a-ev-defn}}\,({\tt{1}})\index{a-ev-defn}, {\rm{list}}\,({\tt{1}}), {\rm{list}}\,({\it{trace\/}}), {\tt{0}})\index{parallel-subst} $=$ {\rm{a-ev-defn}}\,({\it{trace\/}})\index{a-ev-defn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev1-ok-proof-proves \\ \index{a-ev1-ok-proof-proves}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{a-ev1}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{a-ev1-ok-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev1-ok-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev1-good-proof} \\  
\=\+{\rm{a-ev1-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev1-good-proof} \\ 
$=$$\;\;\;\;${\rm{forsome-intro-proof}}\,(\=\+{\tt{1}}, \\ 
{\rm{a-ev}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{10}}, \\ 
{\tt{1}})\-\index{a-ev}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal}, \\ 
{\rm{a-ev-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\tt{1}})\index{a-ev-good-proof})\-\index{forsome-intro-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev1-good-proof-proves \\ \index{a-ev1-good-proof-proves}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-implies}}\,(\=\+{\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{10}})\-\index{a-ev1}, \\ 
{\rm{f-equal}}\,({\tt{10}}, {\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\index{f-equal})\-\index{f-implies}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,({\rm{a-ev1-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{a-ev1-good-proof}, {\it{concl\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev1-ok-proof; name this event `g1327'.\index{g1327}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev1-good-proof; name this event `g1328'.\index{g1328}

\addvspace{18pt}

\noindent{\sc Event}:   Enable free-for; name this event `g1329'.\index{g1329}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev1-not-good-proof} \\  
\=\+{\rm{a-ev1-not-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{val\/}})\index{a-ev1-not-good-proof} \\ 
$=$$\;\;\;\;${\rm{tautconseq-proof}}\,(\=\+{\rm{list}}\,(\=\+{\rm{f-not}}\,({\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\index{f-not}, \\ 
{\rm{f-implies}}\,(\=\+{\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}, \\ 
{\rm{f-equal}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies})\-, \\ 
{\rm{f-not}}\,({\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1})\index{f-not}, \\ 
{\rm{list}}\,(\=\+{\rm{ncode-nequal-proof}}\,({\it{val\/}}, {\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode-nequal-proof}, \\ 
{\rm{subst-rule-proof}}\,(\=\+{\rm{f-implies}}\,(\=\+{\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\tt{10}})\-\index{a-ev1}, \\ 
{\rm{f-equal}}\,(\=\+{\tt{10}}, \\ 
{\rm{ncode}}\,({\rm{ev}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{ev})\index{ncode})\-\index{f-equal})\-\index{f-implies}, \\ 
{\tt{10}}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{a-ev1-good-proof}}\,(\=\+{\it{flg\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{va\/}}, \\ 
{\it{fa\/}}, \\ 
{\it{n\/}})\-\index{a-ev1-good-proof})\-\index{subst-rule-proof})\-)\-\index{tautconseq-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev1-not-good-proof-proves \\ \index{a-ev1-not-good-proof-proves}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{btmp}}\,({\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{ncode-equal}}\,({\it{val\/}}, {\rm{ev}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}})\index{ev})\index{ncode-equal}) \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-not}}\,({\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{va\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1})\index{f-not}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev1-not-good-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{va\/}}, {\it{fa\/}}, {\it{n\/}}, {\it{val\/}})\index{a-ev1-not-good-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{a-ev2} \\  
\=\+{\rm{a-ev2}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}})\index{a-ev2} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\tt{6}}, \\ 
{\rm{z-opair}}\,({\tt{7}}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\-\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\tt{1}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\tt{1}})\index{a-ev-defn})\-\index{f-and})\-\index{forsome}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  replace-va-a-ev2 \\ \index{replace-va-a-ev2}
\=\+{\rm{subst}}\,({\rm{subst}}\,({\rm{a-ev2}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}})\index{a-ev2}, {\tt{7}}, {\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\tt{0}})\index{subst}, {\tt{6}}, {\rm{ncode}}\,({\it{va1\/}})\index{ncode}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\rm{ncode}}\,({\it{va1\/}})\index{ncode}, {\rm{z-opair}}\,({\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  replace-va-forall-a-ev2 \\ \index{replace-va-forall-a-ev2}
\=\+{\rm{subst}}\,($\forall\;${\tt{6}}$\;${\rm{a-ev2}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}})\index{a-ev2}\index{forall}, {\tt{7}}, {\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;$$\forall\;${\tt{6}}$\;${\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\tt{6}}, {\rm{z-opair}}\,({\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}\index{forall}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  replace-va1-a-ev2 \\ \index{replace-va1-a-ev2}
\=\+{\rm{subst}}\,(\=\+{\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\tt{6}}, {\rm{z-opair}}\,({\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\it{va1\/}})\index{ncode}, \\ 
{\tt{0}})\-\index{subst} \\ 
$=$$\;\;\;\;${\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\rm{ncode}}\,({\it{va1\/}})\index{ncode}, {\rm{z-opair}}\,({\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev1; name this event `g1330'.\index{g1330}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev-defn; name this event `g1331'.\index{g1331}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev; name this event `g1332'.\index{g1332}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{a-ev2-inst-proof} \\  
\=\+{\rm{a-ev2-inst-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{va1\/}}, {\it{va2\/}}, {\it{pf\/}})\index{a-ev2-inst-proof} \\ 
$=$$\;\;\;\;${\rm{subst-rule-proof}}\,(\=\+{\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\tt{6}}, \\ 
{\rm{z-opair}}\,({\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\-\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\it{va1\/}})\index{ncode}, \\ 
{\rm{detach-proof}}\,(\=\+$\forall\;${\tt{6}}$\;${\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\tt{6}}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{va2\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\-\index{z-opair})\-\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}\index{forall}, \\ 
{\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\tt{6}}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{va2\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\-\index{z-opair})\-\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}, \\ 
{\it{pf\/}}, \\ 
{\rm{all-elim-proof}}\,(\=\+{\tt{6}}, \\ 
{\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\tt{6}}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{va2\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\-\index{z-opair})\-\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1})\-\index{all-elim-proof})\-\index{detach-proof})\-\index{subst-rule-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  a-ev2-inst-proof-proves \\ \index{a-ev2-inst-proof-proves}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
$\forall\;${\tt{6}}$\;${\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\tt{6}}, {\rm{z-opair}}\,({\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}\index{forall}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,(\=\+{\rm{ncode}}\,({\it{va1\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\rm{ncode}}\,({\it{va2\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair})\-\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{a-ev2-inst-proof}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}}, {\it{va1\/}}, {\it{va2\/}}, {\it{pf\/}})\index{a-ev2-inst-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Disable bool-fix; name this event `g1333'.\index{g1333}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  ncode-nequal-bool-fix-t-f \\ \index{ncode-nequal-bool-fix-t-f}
$\neg$ {\rm{ncode-equal}}\,({\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}, {\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\index{ncode-equal}
\end{tabbing}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1334'.\index{g1334}

\addvspace{18pt}
\begin{verbatim};;;;;;;Exponentiation.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{exp} \\  
\=\+{\rm{exp}}\,({\it{x\/}}, {\it{y\/}})\index{exp} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\tt{1}} \\ 
{\bf else }{\rm{exp}}\,({\it{x\/}}, {\it{y\/}} $-\;1$)\index{exp} $*$ {\it{x\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;Remainder.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{rem} \\  
\=\+{\rm{rem}}\,({\it{x\/}}, {\it{y\/}})\index{rem} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{fix}}\,({\it{x\/}}) \\ 
{\bf elseif }{\it{x\/}} $<$ {\it{y\/}}$\;\;${\bf then }{\rm{fix}}\,({\it{x\/}}) \\ 
{\bf else }{\rm{rem}}\,({\it{x\/}} $-$ {\it{y\/}}, {\it{y\/}})\index{rem}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;Development of number theory leading to Unique Prime factorization theorem.
;;;;;;;Taken from Boyer-Moore's "proveall".
\end{verbatim}\begin{tabbing}{\sc Theorem}:  plus-right-id2 \\ \index{plus-right-id2}
({\it{y\/}} $\not\in\;${\bf{N}}${}$) $\rightarrow$ (({\it{x\/}} $+$ {\it{y\/}}) $=$ {\rm{fix}}\,({\it{x\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  plus-add1 \\ \index{plus-add1}
\=\+({\it{x\/}} $+$ ($1\;+$ {\it{y\/}})) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\in\;${\bf{N}}${}$$\;\;${\bf then }$1\;+$ ({\it{x\/}} $+$ {\it{y\/}}) \\ 
{\bf else }$1\;+$ {\it{x\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  commutativity2-of-plus \\ \index{commutativity2-of-plus}
({\it{x\/}} $+$ ({\it{y\/}} $+$ {\it{z\/}})) $=$ ({\it{y\/}} $+$ ({\it{x\/}} $+$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  commutativity-of-plus \\ \index{commutativity-of-plus}
({\it{x\/}} $+$ {\it{y\/}}) $=$ ({\it{y\/}} $+$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  associativity-of-plus \\ \index{associativity-of-plus}
(({\it{x\/}} $+$ {\it{y\/}}) $+$ {\it{z\/}}) $=$ ({\it{x\/}} $+$ ({\it{y\/}} $+$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  plus-equal-0 \\ \index{plus-equal-0}
(({\it{a\/}} $+$ {\it{b\/}}) $=$ {\tt{0}}) $=$ (({\it{a\/}} $\simeq {\tt{0}}$) $\wedge$ ({\it{b\/}} $\simeq {\tt{0}}$))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-x-x \\ \index{difference-x-x}
({\it{x\/}} $-$ {\it{x\/}}) $=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus \\ \index{difference-plus}
((({\it{x\/}} $+$ {\it{y\/}}) $-$ {\it{x\/}}) $=$ {\rm{fix}}\,({\it{y\/}})) $\wedge$ ((({\it{y\/}} $+$ {\it{x\/}}) $-$ {\it{x\/}}) $=$ {\rm{fix}}\,({\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  plus-cancellation \\ \index{plus-cancellation}
(({\it{a\/}} $+$ {\it{b\/}}) $=$ ({\it{a\/}} $+$ {\it{c\/}})) $=$ ({\rm{fix}}\,({\it{b\/}}) $=$ {\rm{fix}}\,({\it{c\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-0 \\ \index{difference-0}
({\it{y\/}} $\not<$ {\it{x\/}}) $\rightarrow$ (({\it{x\/}} $-$ {\it{y\/}}) $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-difference-0 \\ \index{equal-difference-0}
({\tt{0}} $=$ ({\it{x\/}} $-$ {\it{y\/}})) $=$ ({\it{y\/}} $\not<$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-cancellation-0 \\ \index{difference-cancellation-0}
({\it{x\/}} $=$ ({\it{x\/}} $-$ {\it{y\/}})) $=$ (({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ (({\it{x\/}} $=$ {\tt{0}}) $\vee$ ({\it{y\/}} $\simeq {\tt{0}}$)))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-cancellation-1 \\ \index{difference-cancellation-1}
\=\+(({\it{x\/}} $-$ {\it{y\/}}) $=$ ({\it{z\/}} $-$ {\it{y\/}})) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $<$ {\it{y\/}}$\;\;${\bf then }{\it{y\/}} $\not<$ {\it{z\/}} \\ 
{\bf elseif }{\it{z\/}} $<$ {\it{y\/}}$\;\;${\bf then }{\it{y\/}} $\not<$ {\it{x\/}} \\ 
{\bf else }{\rm{fix}}\,({\it{x\/}}) $=$ {\rm{fix}}\,({\it{z\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-zero2 \\ \index{times-zero2}
({\it{y\/}} $\not\in\;${\bf{N}}${}$) $\rightarrow$ (({\it{x\/}} $*$ {\it{y\/}}) $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  distributivity-of-times-over-plus \\ \index{distributivity-of-times-over-pl!us}
({\it{x\/}} $*$ ({\it{y\/}} $+$ {\it{z\/}})) $=$ (({\it{x\/}} $*$ {\it{y\/}}) $+$ ({\it{x\/}} $*$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-add1 \\ \index{times-add1}
\=\+({\it{x\/}} $*$ ($1\;+$ {\it{y\/}})) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\in\;${\bf{N}}${}$$\;\;${\bf then }{\it{x\/}} $+$ ({\it{x\/}} $*$ {\it{y\/}}) \\ 
{\bf else }{\rm{fix}}\,({\it{x\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  commutativity-of-times \\ \index{commutativity-of-times}
({\it{x\/}} $*$ {\it{y\/}}) $=$ ({\it{y\/}} $*$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  commutativity2-of-times \\ \index{commutativity2-of-times}
({\it{x\/}} $*$ ({\it{y\/}} $*$ {\it{z\/}})) $=$ ({\it{y\/}} $*$ ({\it{x\/}} $*$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  associativity-of-times \\ \index{associativity-of-times}
(({\it{x\/}} $*$ {\it{y\/}}) $*$ {\it{z\/}}) $=$ ({\it{x\/}} $*$ ({\it{y\/}} $*$ {\it{z\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-times-0 \\ \index{equal-times-0}
(({\it{x\/}} $*$ {\it{y\/}}) $=$ {\tt{0}}) $=$ (({\it{x\/}} $\simeq {\tt{0}}$) $\vee$ ({\it{y\/}} $\simeq {\tt{0}}$))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  equal-lessp \\ \index{equal-lessp}
\=\+(({\it{x\/}} $<$ {\it{y\/}}) $=$ {\it{z\/}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $<$ {\it{y\/}}$\;\;${\bf then }{\bf{t}} $=$ {\it{z\/}} \\ 
{\bf else }{\bf{f}} $=$ {\it{z\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-elim \\ \index{difference-elim}
(({\it{y\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $\not<$ {\it{x\/}})) $\rightarrow$ (({\it{x\/}} $+$ ({\it{y\/}} $-$ {\it{x\/}})) $=$ {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-quotient \\ \index{rem-quotient}
({\rm{rem}}\,({\it{x\/}}, {\it{y\/}})\index{rem} $+$ ({\it{y\/}} $*$ ({\it{x\/}} $\div$ {\it{y\/}}))) $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-wrt-1 \\ \index{rem-wrt-1}
{\rm{rem}}\,({\it{y\/}}, {\tt{1}})\index{rem} $=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-wrt-12 \\ \index{rem-wrt-12}
({\it{x\/}} $\not\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{rem}}\,({\it{y\/}}, {\it{x\/}})\index{rem} $=$ {\rm{fix}}\,({\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-rem2 \\ \index{lessp-rem2}
({\rm{rem}}\,({\it{x\/}}, {\it{y\/}})\index{rem} $<$ {\it{y\/}}) $=$ ({\it{y\/}} $\not\simeq {\tt{0}}$)
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-x-x \\ \index{rem-x-x}
{\rm{rem}}\,({\it{x\/}}, {\it{x\/}})\index{rem} $=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-quotient-elim \\ \index{rem-quotient-elim}
(({\it{y\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ (({\rm{rem}}\,({\it{x\/}}, {\it{y\/}})\index{rem} $+$ ({\it{y\/}} $*$ ({\it{x\/}} $\div$ {\it{y\/}}))) $=$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-times-1 \\ \index{lessp-times-1}
({\it{i\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ (({\it{i\/}} $*$ {\it{j\/}}) $\not<$ {\it{j\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-times-2 \\ \index{lessp-times-2}
({\it{i\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ (({\it{j\/}} $*$ {\it{i\/}}) $\not<$ {\it{j\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-quotient1 \\ \index{lessp-quotient1}
(({\it{i\/}} $\div$ {\it{j\/}}) $<$ {\it{i\/}}) $=$ (({\it{i\/}} $\not\simeq {\tt{0}}$) $\wedge$ (({\it{j\/}} $\simeq {\tt{0}}$) $\vee$ ({\it{j\/}} $\not=$ {\tt{1}})))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-rem1 \\ \index{lessp-rem1}
({\rm{rem}}\,({\it{x\/}}, {\it{y\/}})\index{rem} $<$ {\it{x\/}}) $=$ (({\it{y\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $\not<$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{gcd} \\  
\=\+{\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{fix}}\,({\it{y\/}}) \\ 
{\bf elseif }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{x\/}} \\ 
{\bf elseif }{\it{x\/}} $<$ {\it{y\/}}$\;\;${\bf then }{\rm{gcd}}\,({\it{x\/}}, {\it{y\/}} $-$ {\it{x\/}})\index{gcd} \\ 
{\bf else }{\rm{gcd}}\,({\it{x\/}} $-$ {\it{y\/}}, {\it{y\/}})\index{gcd}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numberp-gcd \\ \index{numberp-gcd}
{\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd} $\in\;${\bf{N}}${}$
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gcd-equal-0 \\ \index{gcd-equal-0}
({\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd} $=$ {\tt{0}}) $=$ (({\it{x\/}} $\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\simeq {\tt{0}}$))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gcd-0 \\ \index{gcd-0}
{\rm{gcd}}\,({\tt{0}}, {\it{y\/}})\index{gcd} $=$ {\rm{fix}}\,({\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  commutativity-of-gcd \\ \index{commutativity-of-gcd}
{\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd} $=$ {\rm{gcd}}\,({\it{y\/}}, {\it{x\/}})\index{gcd}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus1 \\ \index{difference-plus1}
(({\it{x\/}} $+$ {\it{y\/}}) $-$ {\it{x\/}}) $=$ {\rm{fix}}\,({\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus2 \\ \index{difference-plus2}
(({\it{y\/}} $+$ {\it{x\/}}) $-$ {\it{x\/}}) $=$ {\rm{fix}}\,({\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus-cancelation \\ \index{difference-plus-cancelation}
(({\it{x\/}} $+$ {\it{y\/}}) $-$ ({\it{x\/}} $+$ {\it{z\/}})) $=$ ({\it{y\/}} $-$ {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-difference \\ \index{times-difference}
({\it{x\/}} $*$ ({\it{c\/}} $-$ {\it{w\/}})) $=$ (({\it{c\/}} $*$ {\it{x\/}}) $-$ ({\it{w\/}} $*$ {\it{x\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{divides}$\;\;$
{\rm{divides}}\,({\it{x\/}}, {\it{y\/}})\index{divides} $=$ ({\rm{rem}}\,({\it{y\/}}, {\it{x\/}})\index{rem} $\simeq {\tt{0}}$)
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-times \\ \index{divides-times}
{\rm{rem}}\,({\it{x\/}} $*$ {\it{z\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-plus3 \\ \index{difference-plus3}
(({\it{b\/}} $+$ ({\it{a\/}} $+$ {\it{c\/}})) $-$ {\it{a\/}}) $=$ ({\it{b\/}} $+$ {\it{c\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  difference-add1-cancellation \\ \index{difference-add1-cancellation}
(($1\;+$ ({\it{y\/}} $+$ {\it{z\/}})) $-$ {\it{z\/}}) $=$ ($1\;+$ {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-add1 \\ \index{rem-add1}
(({\it{y\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\not=$ {\tt{1}})) $\rightarrow$ ({\rm{rem}}\,($1\;+$ ({\it{x\/}} $*$ {\it{y\/}}), {\it{y\/}})\index{rem} $\not=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-plus-rewrite1 \\ \index{divides-plus-rewrite1}
(({\rm{rem}}\,({\it{x\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) $\wedge$ ({\rm{rem}}\,({\it{y\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}})) $\rightarrow$ ({\rm{rem}}\,({\it{x\/}} $+$ {\it{y\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-plus-rewrite2 \\ \index{divides-plus-rewrite2}
(({\rm{rem}}\,({\it{x\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) $\wedge$ ({\rm{rem}}\,({\it{y\/}}, {\it{z\/}})\index{rem} $\not=$ {\tt{0}})) $\rightarrow$ ({\rm{rem}}\,({\it{x\/}} $+$ {\it{y\/}}, {\it{z\/}})\index{rem} $\not=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-plus-rewrite \\ \index{divides-plus-rewrite}
({\rm{rem}}\,({\it{x\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) $\rightarrow$ (({\rm{rem}}\,({\it{x\/}} $+$ {\it{y\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) $=$ ({\rm{rem}}\,({\it{y\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-plus-cancelation \\ \index{lessp-plus-cancelation}
(({\it{x\/}} $+$ {\it{y\/}}) $<$ ({\it{x\/}} $+$ {\it{z\/}})) $=$ ({\it{y\/}} $<$ {\it{z\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-plus-rewrite-commuted \\ \index{divides-plus-rewrite-commuted}
({\rm{rem}}\,({\it{x\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) $\rightarrow$ (({\rm{rem}}\,({\it{y\/}} $+$ {\it{x\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) $=$ ({\rm{rem}}\,({\it{y\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  euclid \\ \index{euclid}
\=\+({\rm{rem}}\,({\it{x\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{rem}}\,({\it{y\/}} $-$ {\it{x\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $<$ {\it{y\/}}$\;\;${\bf then }{\rm{rem}}\,({\it{y\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-times-cancellation \\ \index{lessp-times-cancellation}
(({\it{x\/}} $*$ {\it{z\/}}) $<$ ({\it{y\/}} $*$ {\it{z\/}})) $=$ (({\it{z\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $<$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-plus-cancellation3 \\ \index{lessp-plus-cancellation3}
({\it{y\/}} $<$ ({\it{x\/}} $+$ {\it{y\/}})) $=$ ({\it{x\/}} $\not\simeq {\tt{0}}$)
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  distributivity-of-times-over-gcd \\ \index{distributivity-of-times-over-gc!d}
{\rm{gcd}}\,({\it{x\/}} $*$ {\it{z\/}}, {\it{y\/}} $*$ {\it{z\/}})\index{gcd} $=$ ({\it{z\/}} $*$ {\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gcd-divides-both \\ \index{gcd-divides-both}
({\rm{rem}}\,({\it{x\/}}, {\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd})\index{rem} $=$ {\tt{0}}) $\wedge$ ({\rm{rem}}\,({\it{y\/}}, {\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd})\index{rem} $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gcd-is-the-greatest \\ \index{gcd-is-the-greatest}
\=\+(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$) $\wedge$ {\rm{divides}}\,({\it{z\/}}, {\it{x\/}})\index{divides} $\wedge$ {\rm{divides}}\,({\it{z\/}}, {\it{y\/}})\index{divides}) \\ 
$\rightarrow$$\;\;\;\;$({\it{z\/}} $\leq$ {\rm{gcd}}\,({\it{x\/}}, {\it{y\/}})\index{gcd})\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prime1} \\  
\=\+{\rm{prime1}}\,({\it{x\/}}, {\it{y\/}})\index{prime1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\it{y\/}} $=$ {\tt{1}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }($\neg$ {\rm{divides}}\,({\it{y\/}}, {\it{x\/}})\index{divides}) $\wedge$ {\rm{prime1}}\,({\it{x\/}}, {\it{y\/}} $-\;1$)\index{prime1}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prime} \\  
{\rm{prime}}\,({\it{x\/}})\index{prime} $=$ (({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{prime1}}\,({\it{x\/}}, {\it{x\/}} $-\;1$)\index{prime1})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{greatest-factor} \\  
\=\+{\rm{greatest-factor}}\,({\it{x\/}}, {\it{y\/}})\index{greatest-factor} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{y\/}} $\simeq {\tt{0}}$) $\vee$ ({\it{y\/}} $=$ {\tt{1}})$\;\;${\bf then }{\it{x\/}} \\ 
{\bf elseif }{\rm{divides}}\,({\it{y\/}}, {\it{x\/}})\index{divides}$\;\;${\bf then }{\it{y\/}} \\ 
{\bf else }{\rm{greatest-factor}}\,({\it{x\/}}, {\it{y\/}} $-\;1$)\index{greatest-factor}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  greatest-factor-lessp \\ \index{greatest-factor-lessp}
\=\+(\=\+({\it{y\/}} $<$ {\it{x\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{prime1}}\,({\it{x\/}}, {\it{y\/}})\index{prime1}) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$(({\it{x\/}} $-\;1$) $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $\not\simeq {\tt{0}}$))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{greatest-factor}}\,({\it{x\/}}, {\it{y\/}})\index{greatest-factor} $<$ {\it{x\/}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  greatest-factor-divides \\ \index{greatest-factor-divides}
\=\+(({\it{y\/}} $<$ {\it{x\/}}) $\wedge$ ($\neg$ {\rm{prime1}}\,({\it{x\/}}, {\it{y\/}})\index{prime1}) $\wedge$ ({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $\not=$ {\tt{1}}) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$({\rm{rem}}\,({\it{x\/}}, {\rm{greatest-factor}}\,({\it{x\/}}, {\it{y\/}})\index{greatest-factor})\index{rem} $=$ {\tt{0}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  greatest-factor-0 \\ \index{greatest-factor-0}
({\rm{greatest-factor}}\,({\it{x\/}}, {\it{y\/}})\index{greatest-factor} $=$ {\tt{0}}) $=$ ((({\it{y\/}} $\simeq {\tt{0}}$) $\vee$ ({\it{y\/}} $=$ {\tt{1}})) $\wedge$ ({\it{x\/}} $=$ {\tt{0}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-0-crock \\ \index{rem-0-crock}
{\rm{rem}}\,({\tt{0}}, {\it{y\/}})\index{rem} $=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  greatest-factor-1 \\ \index{greatest-factor-1}
({\rm{greatest-factor}}\,({\it{x\/}}, {\it{y\/}})\index{greatest-factor} $=$ {\tt{1}}) $=$ ({\it{x\/}} $=$ {\tt{1}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numberp-greatest-factor \\ \index{numberp-greatest-factor}
({\rm{greatest-factor}}\,({\it{x\/}}, {\it{y\/}})\index{greatest-factor} $\in\;${\bf{N}}${}$) $=$ ($\neg$ ((({\it{y\/}} $\simeq {\tt{0}}$) $\vee$ ({\it{y\/}} $=$ {\tt{1}})) $\wedge$ ({\it{x\/}} $\not\in\;${\bf{N}}${}$)))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prime-factors} \\  
\=\+{\rm{prime-factors}}\,({\it{x\/}})\index{prime-factors} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{x\/}} $\simeq {\tt{0}}$) $\vee$ (({\it{x\/}} $-\;1$) $=$ {\tt{0}})$\;\;${\bf then }{\bf{nil}} \\ 
{\bf elseif }{\rm{prime1}}\,({\it{x\/}}, {\it{x\/}} $-\;1$)\index{prime1}$\;\;${\bf then }{\rm{cons}}\,({\it{x\/}}, {\bf{nil}}) \\ 
{\bf else }{\rm{append}}\,(\=\+{\rm{prime-factors}}\,({\rm{greatest-factor}}\,({\it{x\/}}, {\it{x\/}} $-\;1$)\index{greatest-factor})\index{prime-factors}, \\ 
{\rm{prime-factors}}\,({\it{x\/}} $\div$ {\rm{greatest-factor}}\,({\it{x\/}}, {\it{x\/}} $-\;1$)\index{greatest-factor})\index{prime-factors})\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{prime-list} \\  
\=\+{\rm{prime-list}}\,({\it{l\/}})\index{prime-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{l\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{prime}}\,({\rm{car}}\,({\it{l\/}}))\index{prime} $\wedge$ {\rm{prime-list}}\,({\rm{cdr}}\,({\it{l\/}}))\index{prime-list}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{times-list} \\  
\=\+{\rm{times-list}}\,({\it{l\/}})\index{times-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{l\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\tt{1}} \\ 
{\bf else }{\rm{car}}\,({\it{l\/}}) $*$ {\rm{times-list}}\,({\rm{cdr}}\,({\it{l\/}}))\index{times-list}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-list-append \\ \index{times-list-append}
{\rm{times-list}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}))\index{times-list} $=$ ({\rm{times-list}}\,({\it{x\/}})\index{times-list} $*$ {\rm{times-list}}\,({\it{y\/}})\index{times-list})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-list-append \\ \index{prime-list-append}
{\rm{prime-list}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}))\index{prime-list} $=$ ({\rm{prime-list}}\,({\it{x\/}})\index{prime-list} $\wedge$ {\rm{prime-list}}\,({\it{y\/}})\index{prime-list})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-list-prime-factors \\ \index{prime-list-prime-factors}
{\rm{prime-list}}\,({\rm{prime-factors}}\,({\it{x\/}})\index{prime-factors})\index{prime-list}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  quotient-times1 \\ \index{quotient-times1}
\=\+(({\it{y\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{x\/}} $\not=$ {\tt{0}}) $\wedge$ {\rm{divides}}\,({\it{x\/}}, {\it{y\/}})\index{divides}) \\ 
$\rightarrow$$\;\;\;\;$(({\it{x\/}} $*$ ({\it{y\/}} $\div$ {\it{x\/}})) $=$ {\it{y\/}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  quotient-lessp \\ \index{quotient-lessp}
(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $<$ {\it{y\/}})) $\rightarrow$ (({\it{y\/}} $\div$ {\it{x\/}}) $\not=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  enough-factors \\ \index{enough-factors}
({\it{x\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ ({\rm{times-list}}\,({\rm{prime-factors}}\,({\it{x\/}})\index{prime-factors})\index{times-list} $=$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-factorization-existence \\ \index{prime-factorization-existence}
\=\+({\it{x\/}} $\not\simeq {\tt{0}}$) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{times-list}}\,({\rm{prime-factors}}\,({\it{x\/}})\index{prime-factors})\index{times-list} $=$ {\it{x\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{prime-list}}\,({\rm{prime-factors}}\,({\it{x\/}})\index{prime-factors})\index{prime-list})\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{greatereqpr} \\  
\=\+{\rm{greatereqpr}}\,({\it{w\/}}, {\it{z\/}})\index{greatereqpr} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{w\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\it{z\/}} $\simeq {\tt{0}}$ \\ 
{\bf elseif }{\it{w\/}} $=$ {\it{z\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{greatereqpr}}\,({\it{w\/}} $-\;1$, {\it{z\/}})\index{greatereqpr}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-id-iff-1 \\ \index{times-id-iff-1}
({\it{z\/}} $=$ ({\it{w\/}} $*$ {\it{z\/}})) $=$ (({\it{z\/}} $\in\;${\bf{N}}${}$) $\wedge$ (({\it{z\/}} $=$ {\tt{0}}) $\vee$ ({\it{w\/}} $=$ {\tt{1}})))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime1-basic \\ \index{prime1-basic}
\=\+(({\it{z\/}} $\not=$ {\tt{1}}) $\wedge$ ({\it{z\/}} $\not=$ {\tt{0}}) $\wedge$ ({\it{z\/}} $\in\;${\bf{N}}${}$) $\wedge$ {\rm{greatereqpr}}\,({\it{u\/}}, {\it{z\/}})\index{greatereqpr}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{prime1}}\,({\it{w\/}} $*$ {\it{z\/}}, {\it{u\/}})\index{prime1})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  greatereqpr-lessp \\ \index{greatereqpr-lessp}
{\rm{greatereqpr}}\,({\it{x\/}}, {\it{y\/}})\index{greatereqpr} $=$ ({\it{x\/}} $\not<$ {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  greatereqpr-rem \\ \index{greatereqpr-rem}
(({\it{z\/}} $\not=$ ($1\;+$ {\it{v\/}})) $\wedge$ {\rm{divides}}\,({\it{z\/}}, $1\;+$ {\it{v\/}})\index{divides}) $\rightarrow$ {\rm{greatereqpr}}\,({\it{v\/}}, {\it{z\/}})\index{greatereqpr}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-basic \\ \index{prime-basic}
\=\+(({\it{z\/}} $\not=$ {\tt{1}}) $\wedge$ ({\it{z\/}} $\not=$ {\it{x\/}}) $\wedge$ ({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $\not=$ {\tt{1}}) $\wedge$ {\rm{divides}}\,({\it{z\/}}, {\it{x\/}})\index{divides}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{prime1}}\,({\it{x\/}}, {\it{x\/}} $-\;1$)\index{prime1})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-gcd \\ \index{rem-gcd}
({\rm{gcd}}\,({\it{b\/}}, {\it{x\/}})\index{gcd} $=$ {\it{y\/}}) $\rightarrow$ ({\rm{rem}}\,({\it{b\/}}, {\it{y\/}})\index{rem} $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-gcd-1 \\ \index{rem-gcd-1}
({\rm{rem}}\,({\it{b\/}}, {\it{x\/}})\index{rem} $\not=$ {\tt{0}}) $\rightarrow$ ({\rm{gcd}}\,({\it{b\/}}, {\it{x\/}})\index{gcd} $\not=$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-times1 \\ \index{divides-times1}
({\it{a\/}} $=$ ({\it{z\/}} $*$ {\it{y\/}})) $\rightarrow$ ({\rm{rem}}\,({\it{a\/}}, {\it{z\/}})\index{rem} $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-identity1 \\ \index{times-identity1}
(({\it{y\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $\not=$ {\tt{1}}) $\wedge$ ({\it{y\/}} $\not=$ {\tt{0}}) $\wedge$ ({\it{x\/}} $\not=$ {\tt{0}})) $\rightarrow$ ({\it{x\/}} $\not=$ ({\it{x\/}} $*$ {\it{y\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-identity \\ \index{times-identity}
({\it{x\/}} $=$ ({\it{x\/}} $*$ {\it{y\/}})) $=$ (({\it{x\/}} $=$ {\tt{0}}) $\vee$ (({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $=$ {\tt{1}})))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  kludge-bridge \\ \index{kludge-bridge}
({\it{y\/}} $=$ ({\it{k\/}} $*$ {\it{x\/}})) $\rightarrow$ ({\rm{gcd}}\,({\it{y\/}}, {\it{a\/}} $*$ {\it{x\/}})\index{gcd} $=$ ({\it{x\/}} $*$ {\rm{gcd}}\,({\it{a\/}}, {\it{k\/}})\index{gcd}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  hack1 \\ \index{hack1}
\=\+(($\neg$ {\rm{divides}}\,({\it{x\/}}, {\it{a\/}})\index{divides}) $\wedge$ ({\it{a\/}} $=$ {\rm{gcd}}\,({\it{x\/}} $*$ {\it{a\/}}, {\it{b\/}} $*$ {\it{a\/}})\index{gcd})) \\ 
$\rightarrow$$\;\;\;\;$(({\it{k\/}} $*$ {\it{x\/}}) $\not=$ ({\it{b\/}} $*$ {\it{a\/}}))\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-gcd \\ \index{prime-gcd}
\=\+(($\neg$ {\rm{divides}}\,({\it{x\/}}, {\it{b\/}})\index{divides}) $\wedge$ ({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ (({\it{x\/}} $-\;1$) $\not=$ {\tt{0}}) $\wedge$ {\rm{prime1}}\,({\it{x\/}}, {\it{x\/}} $-\;1$)\index{prime1}) \\ 
$\rightarrow$$\;\;\;\;$(({\rm{gcd}}\,({\it{b\/}}, {\it{x\/}})\index{gcd} $=$ {\tt{1}}) $=$ {\bf{t}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gcd-distributes-over-an-opened-up-times \\ \index{gcd-distributes-over-an-opened-!up-times}
\=\+(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{x\/}} $\not=$ {\tt{0}}) $\wedge$ ({\it{free\/}} $=$ ({\it{x\/}} $*$ {\it{z\/}}))) \\ 
$\rightarrow$$\;\;\;\;$({\rm{gcd}}\,({\it{b\/}} $*$ {\it{z\/}}, {\it{free\/}})\index{gcd} $=$ ({\it{z\/}} $*$ {\rm{gcd}}\,({\it{b\/}}, {\it{x\/}})\index{gcd}))\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-key \\ \index{prime-key}
\=\+(({\it{z\/}} $\in\;${\bf{N}}${}$) $\wedge$ {\rm{prime}}\,({\it{x\/}})\index{prime} $\wedge$ ($\neg$ {\rm{divides}}\,({\it{x\/}}, {\it{z\/}})\index{divides}) $\wedge$ ($\neg$ {\rm{divides}}\,({\it{x\/}}, {\it{b\/}})\index{divides})) \\ 
$\rightarrow$$\;\;\;\;$(({\it{x\/}} $*$ {\it{k\/}}) $\not=$ ({\it{b\/}} $*$ {\it{z\/}}))\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  quotient-divides \\ \index{quotient-divides}
(({\it{y\/}} $\in\;${\bf{N}}${}$) $\wedge$ (({\it{x\/}} $*$ ({\it{y\/}} $\div$ {\it{x\/}})) $\not=$ {\it{y\/}})) $\rightarrow$ ({\rm{rem}}\,({\it{y\/}}, {\it{x\/}})\index{rem} $\not=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  little-step \\ \index{little-step}
({\rm{prime}}\,({\it{x\/}})\index{prime} $\wedge$ ({\it{y\/}} $\not=$ {\tt{1}}) $\wedge$ ({\it{x\/}} $\not=$ {\it{y\/}})) $\rightarrow$ ({\rm{rem}}\,({\it{x\/}}, {\it{y\/}})\index{rem} $\not=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{delete} \\  
\=\+{\rm{delete}}\,({\it{x\/}}, {\it{y\/}})\index{delete} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{y\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $=$ {\rm{car}}\,({\it{y\/}})$\;\;${\bf then }{\rm{cdr}}\,({\it{y\/}}) \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{y\/}}), {\rm{delete}}\,({\it{x\/}}, {\rm{cdr}}\,({\it{y\/}}))\index{delete})$\;${\bf  endif}\- \\ 
{\bf else }{\it{y\/}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  delete-non-member \\ \index{delete-non-member}
({\it{x\/}} $\not\in$ {\it{y\/}}) $\rightarrow$ ({\rm{delete}}\,({\it{x\/}}, {\it{y\/}})\index{delete} $=$ {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-delete \\ \index{member-delete}
({\it{x\/}} $\in$ {\rm{delete}}\,({\it{u\/}}, {\it{v\/}})\index{delete}) $\rightarrow$ ({\it{x\/}} $\in$ {\it{v\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-count-delete \\ \index{lessp-count-delete}
({\it{n\/}} $\in$ {\it{l\/}}) $\rightarrow$ ({\rm{count}}\,({\rm{delete}}\,({\it{n\/}}, {\it{l\/}})\index{delete}) $<$ {\rm{count}}\,({\it{l\/}}))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{perm} \\  
\=\+{\rm{perm}}\,({\it{a\/}}, {\it{b\/}})\index{perm} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{a\/}} $\simeq\;${{\bf{nil}}}${}$$\;\;${\bf then }{\it{b\/}} $\simeq\;${{\bf{nil}}}${}$ \\ 
{\bf elseif }{\rm{car}}\,({\it{a\/}}) $\in$ {\it{b\/}}$\;\;${\bf then }{\rm{perm}}\,({\rm{cdr}}\,({\it{a\/}}), {\rm{delete}}\,({\rm{car}}\,({\it{a\/}}), {\it{b\/}})\index{delete})\index{perm} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-times \\ \index{rem-times}
{\rm{rem}}\,({\it{y\/}} $*$ {\it{x\/}}, {\it{y\/}})\index{rem} $=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-list-delete \\ \index{prime-list-delete}
{\rm{prime-list}}\,({\it{l2\/}})\index{prime-list} $\rightarrow$ {\rm{prime-list}}\,({\rm{delete}}\,({\it{x\/}}, {\it{l2\/}})\index{delete})\index{prime-list}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-times-list \\ \index{divides-times-list}
(({\it{c\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{c\/}} $\in$ {\it{l\/}})) $\rightarrow$ ({\rm{rem}}\,({\rm{times-list}}\,({\it{l\/}})\index{times-list}, {\it{c\/}})\index{rem} $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  quotient-times \\ \index{quotient-times}
\=\+(({\it{y\/}} $*$ {\it{x\/}}) $\div$ {\it{y\/}}) \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }{\rm{fix}}\,({\it{x\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  distributivity-of-divides \\ \index{distributivity-of-divides}
(({\it{a\/}} $\not\simeq {\tt{0}}$) $\wedge$ {\rm{divides}}\,({\it{a\/}}, {\it{w\/}})\index{divides}) $\rightarrow$ (({\it{c\/}} $*$ ({\it{w\/}} $\div$ {\it{a\/}})) $=$ (({\it{c\/}} $*$ {\it{w\/}}) $\div$ {\it{a\/}}))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-list-delete \\ \index{times-list-delete}
\=\+(({\it{c\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{c\/}} $\in$ {\it{l\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{times-list}}\,({\rm{delete}}\,({\it{c\/}}, {\it{l\/}})\index{delete})\index{times-list} $=$ ({\rm{times-list}}\,({\it{l\/}})\index{times-list} $\div$ {\it{c\/}}))\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-list-times-list \\ \index{prime-list-times-list}
\=\+({\rm{prime}}\,({\it{c\/}})\index{prime} $\wedge$ {\rm{prime-list}}\,({\it{l2\/}})\index{prime-list} $\wedge$ ({\it{c\/}} $\not\in$ {\it{l2\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{rem}}\,({\rm{times-list}}\,({\it{l2\/}})\index{times-list}, {\it{c\/}})\index{rem} $\not=$ {\tt{0}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  if-times-then-divides \\ \index{if-times-then-divides}
(({\it{c\/}} $\not\simeq {\tt{0}}$) $\wedge$ ($\neg$ {\rm{divides}}\,({\it{c\/}}, {\it{x\/}})\index{divides})) $\rightarrow$ (({\it{c\/}} $*$ {\it{y\/}}) $\not=$ {\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-equal-1 \\ \index{times-equal-1}
\=\+(({\it{a\/}} $*$ {\it{b\/}}) $=$ {\tt{1}}) \\ 
$=$$\;\;\;\;$(\=\+({\it{a\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{b\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{a\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{b\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$(({\it{a\/}} $-\;1$) $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$(({\it{b\/}} $-\;1$) $=$ {\tt{0}}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-member \\ \index{prime-member}
\=\+((({\it{c\/}} $*$ {\rm{times-list}}\,({\it{l1\/}})\index{times-list}) $=$ {\rm{times-list}}\,({\it{l2\/}})\index{times-list}) $\wedge$ {\rm{prime}}\,({\it{c\/}})\index{prime} $\wedge$ {\rm{prime-list}}\,({\it{l2\/}})\index{prime-list}) \\ 
$\rightarrow$$\;\;\;\;$({\it{c\/}} $\in$ {\it{l2\/}})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  divides-implies-times \\ \index{divides-implies-times}
(({\it{a\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{c\/}} $\in\;${\bf{N}}${}$) $\wedge$ (({\it{a\/}} $*$ {\it{c\/}}) $=$ {\it{b\/}})) $\rightarrow$ (({\it{c\/}} $=$ ({\it{b\/}} $\div$ {\it{a\/}})) $=$ {\bf{t}})
\end{tabbing}
\begin{verbatim};;;;;;;Two prime factorizations with the same product are permutations of one another.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  prime-factorization-uniqueness \\ \index{prime-factorization-uniqueness}
\=\+({\rm{prime-list}}\,({\it{l1\/}})\index{prime-list} $\wedge$ {\rm{prime-list}}\,({\it{l2\/}})\index{prime-list} $\wedge$ ({\rm{times-list}}\,({\it{l1\/}})\index{times-list} $=$ {\rm{times-list}}\,({\it{l2\/}})\index{times-list})) \\ 
$\rightarrow$$\;\;\;\;${\rm{perm}}\,({\it{l1\/}}, {\it{l2\/}})\index{perm}\-
\end{tabbing}
\begin{verbatim};;;;;;;NUM-TREEP checks if X is constructed from numbers using CONS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{num-treep} \\  
\=\+{\rm{num-treep}}\,({\it{x\/}})\index{num-treep} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{num-treep}}\,({\rm{car}}\,({\it{x\/}}))\index{num-treep} $\wedge$ {\rm{num-treep}}\,({\rm{cdr}}\,({\it{x\/}}))\index{num-treep} \\ 
{\bf else }{\it{x\/}} $\in\;${\bf{N}}${}$$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;DOUBLE multiplies X by 2.  The recursive definition is there to provide an induction
;;;;;;;scheme.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{double} \\  
\=\+{\rm{double}}\,({\it{x\/}})\index{double} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }$1\;+$ ($1\;+$ {\rm{double}}\,({\it{x\/}} $-\;1$)\index{double})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;HALVE integer-divides X by 2.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{halve} \\  
\=\+{\rm{halve}}\,({\it{x\/}})\index{halve} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf elseif }({\it{x\/}} $-\;1$) $\simeq {\tt{0}}$$\;\;${\bf then }{\tt{0}} \\ 
{\bf else }$1\;+$ {\rm{halve}}\,(({\it{x\/}} $-\;1$) $-\;1$)\index{halve}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  odd-rem-two \\ \index{odd-rem-two}
{\rm{rem}}\,($1\;+$ {\rm{double}}\,({\it{x\/}})\index{double}, {\tt{2}})\index{rem} $=$ {\tt{1}}
\end{tabbing}
\begin{verbatim};;;;;;;NUMCODE is a one-to-one mapping from num-trees to numbers.  A number N is mapped
;;;;;;;to the odd number (2N + 1).  The list (CONS X Y) is mapped to ((x**2)(y**3)), 
;;;;;;;if X is mapped to x, and Y to y.  The unique prime factorization theorem is
;;;;;;;needed to decipher the num-tree back from a NUMCODE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{numcode} \\  
\=\+{\rm{numcode}}\,({\it{x\/}})\index{numcode} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{exp}}\,({\tt{2}}, {\rm{numcode}}\,({\rm{car}}\,({\it{x\/}}))\index{numcode})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\rm{numcode}}\,({\rm{cdr}}\,({\it{x\/}}))\index{numcode})\index{exp} \\ 
{\bf else }$1\;+$ {\rm{double}}\,({\it{x\/}})\index{double}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{y-x-list} \\  
\=\+{\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,({\it{x\/}}, {\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}} $-\;1$)\index{y-x-list})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  exp-times-list-y-x-list \\ \index{exp-times-list-y-x-list}
{\rm{times-list}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list})\index{times-list} $=$ {\rm{exp}}\,({\it{x\/}}, {\it{y\/}})\index{exp}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{perm-y-x-ind} \\  
\=\+{\rm{perm-y-x-ind}}\,({\it{y\/}}, {\it{z\/}})\index{perm-y-x-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{z\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{perm-y-x-ind}}\,({\it{y\/}} $-\;1$, {\it{z\/}} $-\;1$)\index{perm-y-x-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  perm-y-x-list-append \\ \index{perm-y-x-list-append}
\=\+{\rm{perm}}\,({\rm{append}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list}, {\it{list1\/}}), {\rm{append}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{z\/}})\index{y-x-list}, {\it{list2\/}}))\index{perm} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $<$ {\it{z\/}}$\;\;${\bf then }{\rm{perm}}\,({\it{list1\/}}, {\rm{append}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{z\/}} $-$ {\it{y\/}})\index{y-x-list}, {\it{list2\/}}))\index{perm} \\ 
{\bf else }{\rm{perm}}\,({\rm{append}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}} $-$ {\it{z\/}})\index{y-x-list}, {\it{list1\/}}), {\it{list2\/}})\index{perm}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-member-y-x-list \\ \index{not-member-y-x-list}
({\it{x\/}} $\not=$ {\it{x1\/}}) $\rightarrow$ ({\it{x\/}} $\not\in$ {\rm{y-x-list}}\,({\it{x1\/}}, {\it{y1\/}})\index{y-x-list})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nequal-perm-y-x-list \\ \index{nequal-perm-y-x-list}
\=\+(({\it{x\/}} $\not=$ {\it{x1\/}}) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{perm}}\,({\rm{append}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list}, {\it{list1\/}}), {\rm{y-x-list}}\,({\it{x1\/}}, {\it{y1\/}})\index{y-x-list})\index{perm})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nequal-perm-y-x-step \\ \index{nequal-perm-y-x-step}
({\rm{listp}}\,({\it{y\/}}) $\wedge$ ({\rm{car}}\,({\it{y\/}}) $\not\in$ {\it{x\/}})) $\rightarrow$ ($\neg$ {\rm{perm}}\,({\it{x\/}}, {\it{y\/}})\index{perm})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  nequal-perm-y-x-list-another \\ \index{nequal-perm-y-x-list-another}
\=\+(({\it{x\/}} $\not=$ {\it{x1\/}}) $\wedge$ ({\it{y1\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{perm}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list}, {\rm{append}}\,({\rm{y-x-list}}\,({\it{x1\/}}, {\it{y1\/}})\index{y-x-list}, {\it{list1\/}}))\index{perm})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-perm-numcode \\ \index{not-perm-numcode}
\=\+(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{x\/}} $\not=$ {\it{x1\/}})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{perm}}\,(\=\+{\rm{append}}\,({\rm{y-x-list}}\,({\tt{2}}, {\it{x\/}})\index{y-x-list}, {\rm{y-x-list}}\,({\tt{3}}, {\it{y\/}})\index{y-x-list}), \\ 
{\rm{append}}\,({\rm{y-x-list}}\,({\tt{2}}, {\it{x1\/}})\index{y-x-list}, {\rm{y-x-list}}\,({\tt{3}}, {\it{y1\/}})\index{y-x-list}))\-\index{perm})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-list-y-x-list \\ \index{prime-list-y-x-list}
{\rm{prime}}\,({\it{x\/}})\index{prime} $\rightarrow$ {\rm{prime-list}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list})\index{prime-list}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem-rec-reduc \\ \index{rem-rec-reduc}
(({\it{y\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $<$ {\it{x\/}})) $\rightarrow$ ({\rm{rem}}\,({\it{x\/}}, {\it{y\/}})\index{rem} $=$ {\rm{rem}}\,({\it{x\/}} $-$ {\it{y\/}}, {\it{y\/}})\index{rem})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  rem32 \\ \index{rem32}
{\rm{rem}}\,({\tt{3}}, {\tt{2}})\index{rem} $=$ {\tt{1}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime1-3-2 \\ \index{prime1-3-2}
{\rm{prime1}}\,({\tt{3}}, {\tt{2}})\index{prime1} $=$ ($\neg$ {\rm{divides}}\,({\tt{2}}, {\tt{3}})\index{divides})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-3 \\ \index{prime-3}
{\rm{prime}}\,({\tt{3}})\index{prime}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  prime-2 \\ \index{prime-2}
{\rm{prime}}\,({\tt{2}})\index{prime}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-list-append23-numcode \\ \index{times-list-append23-numcode}
\=\+({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}) \\ 
$=$$\;\;\;\;${\rm{times-list}}\,({\rm{append}}\,({\rm{y-x-list}}\,({\tt{2}}, {\it{x\/}})\index{y-x-list}, {\rm{y-x-list}}\,({\tt{3}}, {\it{y\/}})\index{y-x-list}))\index{times-list}\-
\end{tabbing}
\begin{verbatim};;;;;;;The NUMCODEs of two lists are distinct, if the NUMCODEs of the their first elements are.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  numcode-uniqueness1 \\ \index{numcode-uniqueness1}
\=\+(({\it{x\/}} $\not=$ {\it{x1\/}}) $\wedge$ ({\it{x\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$(({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}) $\not=$ ({\rm{exp}}\,({\tt{2}}, {\it{x1\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y1\/}})\index{exp}))\-
\end{tabbing}

\noindent{\sc Event}:   Enable times-list-append23-numcode; name this event `g1335'.\index{g1335}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  perm-append-x-x \\ \index{perm-append-x-x}
{\rm{perm}}\,({\rm{append}}\,({\it{x\/}}, {\it{y\/}}), {\rm{append}}\,({\it{x\/}}, {\it{y1\/}}))\index{perm} $=$ {\rm{perm}}\,({\it{y\/}}, {\it{y1\/}})\index{perm}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-delete \\ \index{length-delete}
\=\+{\rm{length}}\,({\rm{delete}}\,({\it{x\/}}, {\it{y\/}})\index{delete})\index{length} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\in$ {\it{y\/}}$\;\;${\bf then }{\rm{length}}\,({\it{y\/}})\index{length} $-\;1$ \\ 
{\bf else }{\rm{length}}\,({\it{y\/}})\index{length}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  member-length \\ \index{member-length}
({\rm{length}}\,({\it{y\/}})\index{length} $\simeq {\tt{0}}$) $\rightarrow$ ({\it{x\/}} $\not\in$ {\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  perm-length \\ \index{perm-length}
({\rm{length}}\,({\it{x\/}})\index{length} $\not=$ {\rm{length}}\,({\it{y\/}})\index{length}) $\rightarrow$ ($\neg$ {\rm{perm}}\,({\it{x\/}}, {\it{y\/}})\index{perm})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  length-y-x-list \\ \index{length-y-x-list}
{\rm{length}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list})\index{length} $=$ {\rm{fix}}\,({\it{y\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  not-perm-nequal \\ \index{not-perm-nequal}
(({\it{y\/}} $\not=$ {\it{y1\/}}) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) $\rightarrow$ ($\neg$ {\rm{perm}}\,({\rm{y-x-list}}\,({\it{x\/}}, {\it{y\/}})\index{y-x-list}, {\rm{y-x-list}}\,({\it{x\/}}, {\it{y1\/}})\index{y-x-list})\index{perm})
\end{tabbing}
\begin{verbatim};;;;;;;The NUMCODEs of two lists are distinct, if the NUMCODEs of their second elements are.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  numcode-uniqueness2 \\ \index{numcode-uniqueness2}
\=\+(({\it{x\/}} $=$ {\it{x1\/}}) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\not=$ {\it{y1\/}})) \\ 
$\rightarrow$$\;\;\;\;$(({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}) $\not=$ ({\rm{exp}}\,({\tt{2}}, {\it{x1\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y1\/}})\index{exp}))\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{double-ind} \\  
\=\+{\rm{double-ind}}\,({\it{x\/}}, {\it{y\/}})\index{double-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{double-ind}}\,({\it{x\/}} $-\;1$, {\it{y\/}} $-\;1$)\index{double-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  double-nequal-add1-double \\ \index{double-nequal-add1-double}
{\rm{double}}\,({\it{x\/}})\index{double} $\not=$ ($1\;+$ {\rm{double}}\,({\it{y\/}})\index{double})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-2-double \\ \index{times-2-double}
({\tt{2}} $*$ {\it{x\/}}) $=$ {\rm{double}}\,({\it{x\/}})\index{double}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  plus-double \\ \index{plus-double}
({\rm{double}}\,({\it{x\/}})\index{double} $+$ {\rm{double}}\,({\it{y\/}})\index{double}) $=$ {\rm{double}}\,({\it{x\/}} $+$ {\it{y\/}})\index{double}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  times-double \\ \index{times-double}
({\it{x\/}} $*$ {\rm{double}}\,({\it{y\/}})\index{double}) $=$ {\rm{double}}\,({\it{x\/}} $*$ {\it{y\/}})\index{double}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numcode-reduc \\ \index{numcode-reduc}
({\it{y\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ (({\rm{exp}}\,({\tt{2}}, {\it{y\/}})\index{exp} $*$ {\it{z\/}}) $=$ {\rm{double}}\,({\rm{exp}}\,({\tt{2}}, {\it{y\/}} $-\;1$)\index{exp} $*$ {\it{z\/}})\index{double})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  numcode-nequal \\ \index{numcode-nequal}
({\it{x1\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ (($1\;+$ {\rm{double}}\,({\it{x\/}})\index{double}) $\not=$ ({\rm{exp}}\,({\tt{2}}, {\it{x1\/}})\index{exp} $*$ {\it{y\/}}))
\end{tabbing}

\noindent{\sc Event}:   Enable numcode-reduc; name this event `g1336'.\index{g1336}

\addvspace{18pt}

\noindent{\sc Event}:   Enable times-2-double; name this event `g1337'.\index{g1337}

\addvspace{18pt}

\noindent{\sc Event}:   Enable times-double; name this event `g1338'.\index{g1338}

\addvspace{18pt}
\begin{verbatim};;;;;;;DECIPHER1 is the function that deciphers the num-tree back from a NUMCODE.
;;;;;;;It essentially starts from Y and Z, and works its way down lexicographically
;;;;;;;until it has decomposed X into a pair of NUMCODEs, which it recursively deciphers.
;;;;;;;In the base case, when X is not the NUMCODE of any pair, it halves (SUB1 X).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{decipher1} \\  
\=\+{\rm{decipher1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{decipher1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{halve}}\,({\it{x\/}} $-\;1$)\index{halve} \\ 
{\bf elseif }{\it{z\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{decipher1}}\,({\it{x\/}}, {\it{y\/}} $-\;1$, {\it{x\/}})\index{decipher1} \\ 
{\bf elseif }({\it{y\/}} $<$ {\it{x\/}}) $\wedge$ ({\it{z\/}} $<$ {\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $=$ ({\rm{exp}}\,({\tt{2}}, {\it{y\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{z\/}})\index{exp}) \\ 
{\bf then }{\rm{cons}}\,({\rm{decipher1}}\,({\it{y\/}}, {\it{y\/}} $-\;1$, {\it{y\/}} $-\;1$)\index{decipher1}, {\rm{decipher1}}\,({\it{z\/}}, {\it{z\/}} $-\;1$, {\it{z\/}} $-\;1$)\index{decipher1}) \\ 
{\bf else }{\rm{decipher1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}} $-\;1$)\index{decipher1}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{decipher1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}} $-\;1$)\index{decipher1}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-exp \\ \index{lessp-exp}
(({\it{y\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\tt{1}} $<$ {\it{x\/}})) $\rightarrow$ ({\it{y\/}} $<$ {\rm{exp}}\,({\it{x\/}}, {\it{y\/}})\index{exp})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-times \\ \index{lessp-times}
(({\tt{1}} $<$ {\it{x\/}}) $\wedge$ ({\tt{1}} $<$ {\it{y\/}})) $\rightarrow$ (({\it{x\/}} $<$ ({\it{x\/}} $*$ {\it{y\/}})) $\wedge$ ({\it{y\/}} $<$ ({\it{x\/}} $*$ {\it{y\/}})))
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-numcode1 \\ \index{lessp-numcode1}
\=\+(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\it{x\/}} $<$ ({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp})) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $<$ ({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp})))\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable times; name this event `g1339'.\index{g1339}

\addvspace{18pt}

\noindent{\sc Event}:   Enable exp; name this event `g1340'.\index{g1340}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  decipher-listp \\ \index{decipher-listp}
\=\+(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{decipher1}}\,({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}, {\it{x\/}}, {\it{y\/}})\index{decipher1} \\ 
$=$$\;\;\;\;${\rm{cons}}\,({\rm{decipher1}}\,({\it{x\/}}, {\it{x\/}} $-\;1$, {\it{x\/}} $-\;1$)\index{decipher1}, {\rm{decipher1}}\,({\it{y\/}}, {\it{y\/}} $-\;1$, {\it{y\/}} $-\;1$)\index{decipher1}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{decipher-ind} \\  
\=\+{\rm{decipher-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}})\index{decipher-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{x1\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{y1\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{decipher-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}} $-\;1$, {\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp})\index{decipher-ind} \\ 
{\bf elseif }{\it{x\/}} $<$ {\it{x1\/}}$\;\;${\bf then }{\rm{decipher-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}} $-\;1$)\index{decipher-ind} \\ 
{\bf elseif }{\it{y\/}} $=$ {\it{y1\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{decipher-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{x1\/}}, {\it{y1\/}} $-\;1$)\index{decipher-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  decipher1-listp-reduc \\ \index{decipher1-listp-reduc}
\=\+(\=\+({\it{x\/}} $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\it{y\/}} $\not\simeq {\tt{0}}$) \\ 
$\wedge$$\;\;\;\;$({\it{x\/}} $\leq$ {\it{x1\/}}) \\ 
$\wedge$$\;\;\;\;$\=\+{\bf if }{\it{x\/}} $=$ {\it{x1\/}}$\;\;${\bf then }{\it{y\/}} $\leq$ {\it{y1\/}} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{decipher1}}\,({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}, {\it{x1\/}}, {\it{y1\/}})\index{decipher1} \\ 
$=$$\;\;\;\;${\rm{decipher1}}\,({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}, {\it{x\/}}, {\it{y\/}})\index{decipher1})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  halve-double \\ \index{halve-double}
{\rm{halve}}\,({\rm{double}}\,({\it{x\/}})\index{double})\index{halve} $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-numcode \\ \index{lessp-numcode}
{\tt{0}} $<$ {\rm{numcode}}\,({\it{x\/}})\index{numcode}
\end{tabbing}
\begin{verbatim};;;;;;;NUMTREEP is the same as NUM-TREEP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{numtreep} \\  
\=\+{\rm{numtreep}}\,({\it{x\/}})\index{numtreep} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }{\rm{numtreep}}\,({\rm{car}}\,({\it{x\/}}))\index{numtreep} $\wedge$ {\rm{numtreep}}\,({\rm{cdr}}\,({\it{x\/}}))\index{numtreep} \\ 
{\bf else }{\it{x\/}} $\in\;${\bf{N}}${}$$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-sub1-numcode \\ \index{lessp-sub1-numcode}
\=\+(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) \\ 
$\rightarrow$$\;\;\;\;$(\=\+((({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}) $-\;1$) $\not<$ {\it{x\/}}) \\ 
$\wedge$$\;\;\;\;$((({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}) $-\;1$) $\not<$ {\it{y\/}}))\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-numcode-again \\ \index{lessp-numcode-again}
{\rm{numcode}}\,({\it{x\/}})\index{numcode} $\not=$ {\tt{0}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-numcode-case \\ \index{lessp-numcode-case}
(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ ({\it{y\/}} $\not\simeq {\tt{0}}$)) $\rightarrow$ ({\it{x\/}} $\not=$ (({\rm{exp}}\,({\tt{2}}, {\it{x\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{y\/}})\index{exp}) $-\;1$))
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{decipher-ind2} \\  
\=\+{\rm{decipher-ind2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{decipher-ind2} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{z\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{decipher-ind2}}\,({\it{x\/}}, {\it{y\/}} $-\;1$, $1\;+$ {\rm{double}}\,({\it{x\/}})\index{double})\index{decipher-ind2} \\ 
{\bf else }{\rm{decipher-ind2}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}} $-\;1$)\index{decipher-ind2}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  decipher1-nlistp \\ \index{decipher1-nlistp}
{\rm{decipher1}}\,($1\;+$ {\rm{double}}\,({\it{x\/}})\index{double}, {\it{y\/}}, {\it{z\/}})\index{decipher1} $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}
\begin{verbatim};;;;;;;This is the main lemma relating DECIPHER1 and NUMCODE.  It asserts that 
;;;;;;;DECIPHER1 applied to (NUMCODE X) (with appropriate values for the Y, Z, arguments)
;;;;;;;returns X.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  decipher1-numcode \\ \index{decipher1-numcode}
{\rm{numtreep}}\,({\it{x\/}})\index{numtreep} $\rightarrow$ ({\rm{decipher1}}\,({\rm{numcode}}\,({\it{x\/}})\index{numcode}, {\rm{numcode}}\,({\it{x\/}})\index{numcode} $-\;1$, {\rm{numcode}}\,({\it{x\/}})\index{numcode} $-\;1$)\index{decipher1} $=$ {\it{x\/}})
\end{tabbing}
\begin{verbatim};;;;;;;Correspondence proofs for PLUS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-plusn} \\  
\=\+{\rm{g-plusn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-plusn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{x\/}})\index{pr-zeropn}, \\ 
{\rm{g-fixn}}\,({\it{y\/}})\index{g-fixn}, \\ 
{\rm{pr-add1n}}\,({\rm{g-plusn}}\,({\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n}, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{g-plusn})\index{pr-add1n})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-plus; name this event `g1341'.\index{g1341}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-plus-defn; name this event `g1342'.\index{g1342}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-plus-ind} \\  
\=\+{\rm{g-plus-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-plus-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-plus-ind}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1}, {\tt{1}}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-plus-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-plus-eval \\ \index{g-plus-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-plus}}\,({\it{x\/}}, {\it{y\/}})\index{g-plus}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-plusn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-plusn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-add1n-gcode \\ \index{pr-add1n-gcode}
{\rm{pr-add1n}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{pr-add1n} $=$ {\rm{gcode}}\,($1\;+$ {\it{x\/}})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable pr-add1n; name this event `g1343'.\index{g1343}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{plusn-ind} \\  
\=\+{\rm{plusn-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{plusn-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{plusn-ind}}\,({\it{x\/}} $-\;1$, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{plusn-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-plusn-gcode \\ \index{g-plusn-gcode}
({\it{x\/}} $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-plusn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-plusn} $=$ {\rm{gcode}}\,({\it{x\/}} $+$ {\it{y\/}})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-plus; name this event `g1344'.\index{g1344}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-plusn; name this event `g1345'.\index{g1345}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-plus-ind; name this event `g1346'.\index{g1346}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence proofs for TIMES.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-timesn} \\  
\=\+{\rm{g-timesn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-timesn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{x\/}})\index{pr-zeropn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\rm{g-plusn}}\,({\it{y\/}}, {\rm{g-timesn}}\,({\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n}, {\it{y\/}}, {\it{n\/}} $-\;1$)\index{g-timesn}, {\it{n\/}} $-\;1$)\index{g-plusn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-times; name this event `g1347'.\index{g1347}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-times-defn; name this event `g1348'.\index{g1348}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-times-eval \\ \index{g-times-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-times}}\,({\it{x\/}}, {\it{y\/}})\index{g-times}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-timesn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-timesn}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-timesn-gcode \\ \index{g-timesn-gcode}
\=\+(($1\;+$ ({\it{x\/}} $+$ {\it{y\/}})) $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-timesn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-timesn} $=$ {\rm{gcode}}\,({\it{x\/}} $*$ {\it{y\/}})\index{gcode})\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-times; name this event `g1349'.\index{g1349}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-timesn; name this event `g1350'.\index{g1350}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-times-defn; name this event `g1351'.\index{g1351}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence proofs for EXP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-expn} \\  
\=\+{\rm{g-expn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-expn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{y\/}})\index{pr-zeropn}, \\ 
{\rm{gcode}}\,({\tt{1}})\index{gcode}, \\ 
{\rm{g-timesn}}\,({\it{x\/}}, {\rm{g-expn}}\,({\it{x\/}}, {\rm{pr-sub1n}}\,({\it{y\/}})\index{pr-sub1n}, {\it{n\/}} $-\;1$)\index{g-expn}, {\it{n\/}} $-\;1$)\index{g-timesn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-exp; name this event `g1373'.\index{g1373}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-exp-defn; name this event `g1374'.\index{g1374}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-exp-ind} \\  
\=\+{\rm{g-exp-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-exp-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-exp-ind}}\,({\tt{0}}, {\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{g-exp-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-exp-eval \\ \index{g-exp-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-exp}}\,({\it{x\/}}, {\it{y\/}})\index{g-exp}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-expn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-expn}
\end{tabbing}

\noindent{\sc Event}:   Disable exp; name this event `g1375'.\index{g1375}

\addvspace{18pt}

\noindent{\sc Event}:   Disable times; name this event `g1376'.\index{g1376}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-expn-gcode \\ \index{g-expn-gcode}
\=\+(({\it{x\/}} $\not\simeq {\tt{0}}$) $\wedge$ (($1\;+$ ({\it{x\/}} $+$ {\it{y\/}} $+$ {\rm{exp}}\,({\it{x\/}}, {\it{y\/}})\index{exp})) $<$ {\it{n\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-expn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-expn} $=$ {\rm{gcode}}\,({\rm{exp}}\,({\it{x\/}}, {\it{y\/}})\index{exp})\index{gcode})\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-exp; name this event `g1377'.\index{g1377}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-expn; name this event `g1378'.\index{g1378}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-exp-defn; name this event `g1379'.\index{g1379}

\addvspace{18pt}

\noindent{\sc Event}:   Enable times; name this event `g1380'.\index{g1380}

\addvspace{18pt}

\noindent{\sc Event}:   Enable exp; name this event `g1381'.\index{g1381}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence proofs for HALVE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-halven} \\  
\=\+{\rm{g-halven}}\,({\it{x\/}}, {\it{n\/}})\index{g-halven} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{x\/}})\index{pr-zeropn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n})\index{pr-zeropn}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\rm{pr-add1n}}\,({\rm{g-halven}}\,(\=\+{\rm{pr-sub1n}}\,({\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n})\index{pr-sub1n}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-halven})\index{pr-add1n})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-halve; name this event `g1382'.\index{g1382}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-halve-defn; name this event `g1383'.\index{g1383}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-halve-ind} \\  
\=\+{\rm{g-halve-ind}}\,({\it{x\/}}, {\it{va\/}}, {\it{n\/}})\index{g-halve-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-halve-ind}}\,(\=\+{\rm{pr-sub1}}\,({\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1})\index{pr-sub1}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-halve-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-halve-eval \\ \index{g-halve-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-halve}}\,({\it{x\/}})\index{g-halve}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-halven}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-halven}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{halven-ind} \\  
\=\+{\rm{halven-ind}}\,({\it{x\/}}, {\it{n\/}})\index{halven-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{halven-ind}}\,(({\it{x\/}} $-\;1$) $-\;1$, {\it{n\/}} $-\;1$)\index{halven-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-halven-gcode \\ \index{g-halven-gcode}
({\it{x\/}} $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-halven}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\it{n\/}})\index{g-halven} $=$ {\rm{gcode}}\,({\rm{halve}}\,({\it{x\/}})\index{halve})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-halve; name this event `g1384'.\index{g1384}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-halven; name this event `g1385'.\index{g1385}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-halve-defn; name this event `g1386'.\index{g1386}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence proofs for LESSP.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-lesspn} \\  
\=\+{\rm{g-lesspn}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{g-lesspn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{y\/}})\index{pr-zeropn}, \\ 
{\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{x\/}})\index{pr-zeropn}, \\ 
{\rm{bool-fix}}\,({\bf{t}})\index{bool-fix}, \\ 
{\rm{g-lesspn}}\,({\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n}, {\rm{pr-sub1n}}\,({\it{y\/}})\index{pr-sub1n}, {\it{n\/}} $-\;1$)\index{g-lesspn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-lessp; name this event `g1387'.\index{g1387}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-lessp-defn; name this event `g1388'.\index{g1388}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-lessp-ind} \\  
\=\+{\rm{g-lessp-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{va\/}}, {\it{n\/}})\index{g-lessp-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-lessp-ind}}\,(\=\+{\rm{pr-sub1}}\,({\tt{0}})\index{pr-sub1}, \\ 
{\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-lessp-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-lessp-eval \\ \index{g-lessp-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-lessp}}\,({\it{x\/}}, {\it{y\/}})\index{g-lessp}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-lesspn}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-lesspn}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{lesspn-ind} \\  
\=\+{\rm{lesspn-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{n\/}})\index{lesspn-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{x\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{lesspn-ind}}\,({\it{x\/}} $-\;1$, {\it{y\/}} $-\;1$, {\it{n\/}} $-\;1$)\index{lesspn-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-lesspn-gcode \\ \index{g-lesspn-gcode}
(({\it{x\/}} $+$ {\it{y\/}}) $<$ {\it{n\/}}) $\rightarrow$ ({\rm{g-lesspn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\it{n\/}})\index{g-lesspn} $=$ {\rm{bool-fix}}\,({\it{x\/}} $<$ {\it{y\/}})\index{bool-fix})
\end{tabbing}

\noindent{\sc Event}:   Enable g-lessp; name this event `g1389'.\index{g1389}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-lesspn; name this event `g1390'.\index{g1390}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-lessp-defn; name this event `g1391'.\index{g1391}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence proofs for DECIPHER1.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-decipher1n} \\  
\=\+{\rm{g-decipher1n}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{n\/}})\index{g-decipher1n} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{y\/}})\index{btmp} $\vee$ {\rm{btmp}}\,({\it{z\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{y\/}})\index{pr-zeropn}, \\ 
{\rm{g-halven}}\,({\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n}, {\it{n\/}} $-\;1$)\index{g-halven}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-zeropn}}\,({\it{z\/}})\index{pr-zeropn}, \\ 
{\rm{g-decipher1n}}\,({\it{x\/}}, {\rm{pr-sub1n}}\,({\it{y\/}})\index{pr-sub1n}, {\it{x\/}}, {\it{n\/}} $-\;1$)\index{g-decipher1n}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-andn}}\,(\=\+{\rm{g-lesspn}}\,({\it{y\/}}, {\it{x\/}}, {\it{n\/}} $-\;1$)\index{g-lesspn}, \\ 
{\rm{g-lesspn}}\,({\it{z\/}}, {\it{x\/}}, {\it{n\/}} $-\;1$)\index{g-lesspn})\-\index{pr-andn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\it{x\/}}, \\ 
{\rm{g-timesn}}\,(\=\+{\rm{g-expn}}\,(\=\+{\rm{gcode}}\,({\tt{2}})\index{gcode}, \\ 
{\it{y\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-expn}, \\ 
{\rm{g-expn}}\,(\=\+{\rm{gcode}}\,({\tt{3}})\index{gcode}, \\ 
{\it{z\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-expn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-timesn})\-\index{pr-equaln}, \\ 
{\rm{pr-consn}}\,(\=\+{\rm{g-decipher1n}}\,(\=\+{\it{y\/}}, \\ 
{\rm{pr-sub1n}}\,({\it{y\/}})\index{pr-sub1n}, \\ 
{\rm{pr-sub1n}}\,({\it{y\/}})\index{pr-sub1n}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-decipher1n}, \\ 
{\rm{g-decipher1n}}\,(\=\+{\it{z\/}}, \\ 
{\rm{pr-sub1n}}\,({\it{z\/}})\index{pr-sub1n}, \\ 
{\rm{pr-sub1n}}\,({\it{z\/}})\index{pr-sub1n}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-decipher1n})\-\index{pr-consn}, \\ 
{\rm{g-decipher1n}}\,(\=\+{\it{x\/}}, \\ 
{\it{y\/}}, \\ 
{\rm{pr-sub1n}}\,({\it{z\/}})\index{pr-sub1n}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-decipher1n})\-\index{pr-ifn}, \\ 
{\rm{g-decipher1n}}\,({\it{x\/}}, {\it{y\/}}, {\rm{pr-sub1n}}\,({\it{z\/}})\index{pr-sub1n}, {\it{n\/}} $-\;1$)\index{g-decipher1n})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-decipher1; name this event `g1392'.\index{g1392}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-decipher1-defn; name this event `g1393'.\index{g1393}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-decipher1-ind} \\  
\=\+{\rm{g-decipher1-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{va\/}}, {\it{n\/}})\index{g-decipher1-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-decipher1-ind}}\,(\=\+{\tt{0}}, \\ 
{\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, \\ 
{\tt{0}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-decipher1-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-decipher1-ind}}\,(\=\+{\tt{0}}, \\ 
{\tt{1}}, \\ 
{\rm{pr-sub1}}\,({\tt{2}})\index{pr-sub1}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-decipher1-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-decipher1-ind}}\,(\=\+{\tt{1}}, \\ 
{\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, \\ 
{\rm{pr-sub1}}\,({\tt{1}})\index{pr-sub1}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-decipher1-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-decipher1-ind}}\,(\=\+{\tt{2}}, \\ 
{\rm{pr-sub1}}\,({\tt{2}})\index{pr-sub1}, \\ 
{\rm{pr-sub1}}\,({\tt{2}})\index{pr-sub1}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-decipher1-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-decipher1-eval \\ \index{g-decipher1-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-decipher1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{g-decipher1}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-decipher1n}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{y\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\rm{ev}}\,({\tt{0}}, {\it{z\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-decipher1n}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{dec-count} \\  
\=\+{\rm{dec-count}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{dec-count} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }$1\;+$ {\it{x\/}} \\ 
{\bf elseif }{\it{z\/}} $\simeq {\tt{0}}$$\;\;${\bf then }$1\;+$ {\rm{dec-count}}\,({\it{x\/}}, {\it{y\/}} $-\;1$, {\it{x\/}})\index{dec-count} \\ 
{\bf elseif }({\it{y\/}} $<$ {\it{x\/}}) $\wedge$ ({\it{z\/}} $<$ {\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $=$ ({\rm{exp}}\,({\tt{2}}, {\it{y\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{z\/}})\index{exp}) \\ 
{\bf then }$1\;+$ ($1\;+$ (\=\+{\it{x\/}} \\ 
$+$$\;\;\;\;${\it{y\/}} \\ 
$+$$\;\;\;\;${\it{z\/}} \\ 
$+$$\;\;\;\;${\rm{exp}}\,({\tt{2}}, {\it{y\/}})\index{exp} \\ 
$+$$\;\;\;\;${\rm{exp}}\,({\tt{3}}, {\it{z\/}})\index{exp} \\ 
$+$$\;\;\;\;${\rm{dec-count}}\,({\it{y\/}}, {\it{y\/}} $-\;1$, {\it{y\/}} $-\;1$)\index{dec-count} \\ 
$+$$\;\;\;\;${\rm{dec-count}}\,({\it{z\/}}, {\it{z\/}} $-\;1$, {\it{z\/}} $-\;1$)\index{dec-count})\-) \\ 
{\bf else }$1\;+$ ($1\;+$ (\=\+{\it{x\/}} \\ 
$+$$\;\;\;\;${\it{y\/}} \\ 
$+$$\;\;\;\;${\it{z\/}} \\ 
$+$$\;\;\;\;${\rm{exp}}\,({\tt{2}}, {\it{y\/}})\index{exp} \\ 
$+$$\;\;\;\;${\rm{exp}}\,({\tt{3}}, {\it{z\/}})\index{exp} \\ 
$+$$\;\;\;\;${\rm{dec-count}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}} $-\;1$)\index{dec-count})\-)$\;${\bf  endif}\- \\ 
{\bf else }$1\;+$ ($1\;+$ (\=\+{\it{x\/}} \\ 
$+$$\;\;\;\;${\it{y\/}} \\ 
$+$$\;\;\;\;${\it{z\/}} \\ 
$+$$\;\;\;\;${\rm{exp}}\,({\tt{2}}, {\it{y\/}})\index{exp} \\ 
$+$$\;\;\;\;${\rm{exp}}\,({\tt{3}}, {\it{z\/}})\index{exp} \\ 
$+$$\;\;\;\;${\rm{dec-count}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}} $-\;1$)\index{dec-count})\-)$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gcode-halve \\ \index{gcode-halve}
{\rm{gcode}}\,({\rm{halve}}\,({\it{x\/}})\index{halve})\index{gcode} $=$ {\rm{halve}}\,({\it{x\/}})\index{halve}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{decipher1n-ind} \\  
\=\+{\rm{decipher1n-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}}, {\it{n\/}})\index{decipher1n-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{y\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{z\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\rm{decipher1n-ind}}\,({\it{x\/}}, {\it{y\/}} $-\;1$, {\it{x\/}}, {\it{n\/}} $-\;1$)\index{decipher1n-ind} \\ 
{\bf elseif }({\it{y\/}} $<$ {\it{x\/}}) $\wedge$ ({\it{z\/}} $<$ {\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{x\/}} $=$ ({\rm{exp}}\,({\tt{2}}, {\it{y\/}})\index{exp} $*$ {\rm{exp}}\,({\tt{3}}, {\it{z\/}})\index{exp}) \\ 
{\bf then }\=\+{\rm{decipher1n-ind}}\,({\it{y\/}}, {\it{y\/}} $-\;1$, {\it{y\/}} $-\;1$, {\it{n\/}} $-\;1$)\index{decipher1n-ind} \\ 
$\wedge$$\;\;\;\;${\rm{decipher1n-ind}}\,({\it{z\/}}, {\it{z\/}} $-\;1$, {\it{z\/}} $-\;1$, {\it{n\/}} $-\;1$)\index{decipher1n-ind}\- \\ 
{\bf else }{\rm{decipher1n-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}} $-\;1$, {\it{n\/}} $-\;1$)\index{decipher1n-ind}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{decipher1n-ind}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}} $-\;1$, {\it{n\/}} $-\;1$)\index{decipher1n-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-decipher1n-gcode \\ \index{g-decipher1n-gcode}
\=\+({\rm{dec-count}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{dec-count} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-decipher1n}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode}, {\rm{gcode}}\,({\it{z\/}})\index{gcode}, {\it{n\/}})\index{g-decipher1n} $=$ {\rm{decipher1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{decipher1})\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-decipher1; name this event `g1394'.\index{g1394}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-decipher1n; name this event `g1395'.\index{g1395}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-decipher1-defn; name this event `g1396'.\index{g1396}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  num-treep-gcode \\ \index{num-treep-gcode}
{\rm{num-treep}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{num-treep}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-deciphern} \\  
{\rm{g-deciphern}}\,({\it{x\/}}, {\it{n\/}})\index{g-deciphern} $=$ {\rm{g-decipher1n}}\,({\it{x\/}}, {\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n}, {\rm{pr-sub1n}}\,({\it{x\/}})\index{pr-sub1n}, {\it{n\/}})\index{g-decipher1n}
\end{tabbing}
\begin{verbatim};;;;;;;Correspondence proofs for INV-CODE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-inv-coden} \\  
\=\+{\rm{g-inv-coden}}\,({\it{x\/}}, {\it{n\/}})\index{g-inv-coden} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-listpn}}\,({\it{x\/}})\index{pr-listpn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, {\tt{1}})\index{pr-equaln}, \\ 
{\rm{g-consn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{pr-carn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-carn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-inv-coden}}\,(\=\+{\rm{pr-cdrn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-cdrn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden})\-\index{g-consn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, {\tt{2}})\index{pr-equaln}, \\ 
{\rm{g-f-notn}}\,({\rm{g-inv-coden}}\,(\=\+{\rm{pr-carn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-carn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden})\index{g-f-notn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, {\tt{3}})\index{pr-equaln}, \\ 
{\rm{g-f-orn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{pr-carn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-carn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-inv-coden}}\,(\=\+{\rm{pr-cdrn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-cdrn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden})\-\index{g-f-orn}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,({\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, {\tt{4}})\index{pr-equaln}, \\ 
{\rm{g-forsomen}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{pr-carn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-carn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-inv-coden}}\,(\=\+{\rm{pr-cdrn}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn})\index{pr-cdrn}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden})\-\index{g-forsomen}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, \\ 
{\tt{5}})\-\index{pr-equaln}, \\ 
{\rm{gcode}}\,({\bf{t}})\index{gcode}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{pr-equaln}}\,(\=\+{\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, \\ 
{\tt{6}})\-\index{pr-equaln}, \\ 
{\rm{gcode}}\,({\bf{f}})\index{gcode}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn})\-\index{pr-ifn}, \\ 
{\rm{g-fixn}}\,({\it{x\/}})\index{g-fixn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-inv-code; name this event `g1398'.\index{g1398}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-inv-code-defn; name this event `g1399'.\index{g1399}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-inv-code-ind} \\  
\=\+{\rm{g-inv-code-ind}}\,({\it{x\/}}, {\it{va\/}}, {\it{n\/}})\index{g-inv-code-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{g-inv-code-ind}}\,(\=\+{\rm{pr-car}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-car}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-code-ind} \\ 
$\wedge$$\;\;\;\;${\rm{g-inv-code-ind}}\,(\=\+{\rm{pr-cdr}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr})\index{pr-cdr}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-code-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-inv-code-eval \\ \index{g-inv-code-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{g-inv-code}}\,({\it{x\/}})\index{g-inv-code}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{g-inv-coden}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{g-inv-coden}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-carn-reduc1 \\ \index{pr-carn-reduc1}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn} $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-carn-reduc2 \\ \index{pr-carn-reduc2}
{\rm{pr-carn}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{pr-carn} $=$ {\it{x\/}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-cdrn-reduc1 \\ \index{pr-cdrn-reduc1}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn} $=$ {\tt{0}})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-cdrn-reduc \\ \index{pr-cdrn-reduc}
{\rm{pr-cdrn}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{pr-cdrn} $=$ {\it{y\/}}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-listp-reduc1 \\ \index{pr-listp-reduc1}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{pr-listpn}}\,({\it{x\/}})\index{pr-listpn} $=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-listp-reduc2 \\ \index{pr-listp-reduc2}
{\rm{pr-listpn}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{pr-listpn} $=$ {\rm{bool-fix}}\,({\bf{t}})\index{bool-fix}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{inv-code} \\  
\=\+{\rm{inv-code}}\,({\it{x\/}})\index{inv-code} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{1}}$\;\;${\bf then }{\rm{cons}}\,({\rm{inv-code}}\,({\rm{cadr}}\,({\it{x\/}}))\index{inv-code}, {\rm{inv-code}}\,({\rm{cddr}}\,({\it{x\/}}))\index{inv-code}) \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{2}}$\;\;${\bf then }{\rm{f-not}}\,({\rm{inv-code}}\,({\rm{cadr}}\,({\it{x\/}}))\index{inv-code})\index{f-not} \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{3}} \\ 
{\bf then }{\rm{f-or}}\,({\rm{inv-code}}\,({\rm{cadr}}\,({\it{x\/}}))\index{inv-code}, {\rm{inv-code}}\,({\rm{cddr}}\,({\it{x\/}}))\index{inv-code})\index{f-or} \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{4}} \\ 
{\bf then }{\rm{forsome}}\,({\rm{inv-code}}\,({\rm{cadr}}\,({\it{x\/}}))\index{inv-code}, {\rm{inv-code}}\,({\rm{cddr}}\,({\it{x\/}}))\index{inv-code})\index{forsome} \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{5}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{6}}$\;\;${\bf then }{\bf{f}} \\ 
{\bf else }{\bf{nil}}$\;${\bf  endif}\- \\ 
{\bf else }{\rm{fix}}\,({\it{x\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{inv-coden-ind} \\  
\=\+{\rm{inv-coden-ind}}\,({\it{x\/}}, {\it{n\/}})\index{inv-coden-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{listp}}\,({\it{x\/}}) \\ 
{\bf then }\=\+{\bf if }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{1}} \\ 
{\bf then }\=\+{\rm{inv-coden-ind}}\,({\rm{cadr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{inv-coden-ind} \\ 
$\wedge$$\;\;\;\;${\rm{inv-coden-ind}}\,({\rm{cddr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{inv-coden-ind}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{2}}$\;\;${\bf then }{\rm{inv-coden-ind}}\,({\rm{cadr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{inv-coden-ind} \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{3}} \\ 
{\bf then }\=\+{\rm{inv-coden-ind}}\,({\rm{cadr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{inv-coden-ind} \\ 
$\wedge$$\;\;\;\;${\rm{inv-coden-ind}}\,({\rm{cddr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{inv-coden-ind}\- \\ 
{\bf elseif }{\rm{car}}\,({\it{x\/}}) $=$ {\tt{4}} \\ 
{\bf then }\=\+{\rm{inv-coden-ind}}\,({\rm{cadr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{inv-coden-ind} \\ 
$\wedge$$\;\;\;\;${\rm{inv-coden-ind}}\,({\rm{cddr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{inv-coden-ind}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-fixn; name this event `g1400'.\index{g1400}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-fixn-reduc \\ \index{g-fixn-reduc}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{g-fixn}}\,({\it{x\/}})\index{g-fixn} $=$ {\rm{gcode}}\,({\it{x\/}})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-fixn; name this event `g1401'.\index{g1401}

\addvspace{18pt}

\noindent{\sc Event}:   Disable pr-equaln; name this event `g1402'.\index{g1402}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-equaln-reduc2 \\ \index{pr-equaln-reduc2}
\=\+(($\neg$ {\rm{btmp}}\,({\it{x\/}})\index{btmp}) $\wedge$ ($\neg$ {\rm{btmp}}\,({\it{y\/}})\index{btmp})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{pr-equaln}}\,({\it{x\/}}, {\it{y\/}})\index{pr-equaln} $=$ {\rm{bool-fix}}\,({\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal})\index{bool-fix})\-
\end{tabbing}

\noindent{\sc Event}:   Enable pr-equaln; name this event `g1403'.\index{g1403}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-consn; name this event `g1404'.\index{g1404}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-f-notn; name this event `g1405'.\index{g1405}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-f-orn; name this event `g1406'.\index{g1406}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-forsomen; name this event `g1407'.\index{g1407}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-consn-numberp \\ \index{g-consn-numberp}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ ({\rm{g-consn}}\,({\it{x\/}}, {\it{y\/}})\index{g-consn} $=$ {\rm{gcode}}\,({\rm{cons}}\,({\it{x\/}}, {\it{y\/}}))\index{gcode})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-f-notn-numberp \\ \index{g-f-notn-numberp}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{g-f-notn}}\,({\it{x\/}})\index{g-f-notn} $=$ {\rm{gcode}}\,({\rm{f-not}}\,({\it{x\/}})\index{f-not})\index{gcode})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-f-orn-numberp \\ \index{g-f-orn-numberp}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ ({\rm{g-f-orn}}\,({\it{x\/}}, {\it{y\/}})\index{g-f-orn} $=$ {\rm{gcode}}\,({\rm{f-or}}\,({\it{x\/}}, {\it{y\/}})\index{f-or})\index{gcode})
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-forsomen-numberp \\ \index{g-forsomen-numberp}
(({\it{x\/}} $\in\;${\bf{N}}${}$) $\wedge$ ({\it{y\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ ({\rm{g-forsomen}}\,({\it{x\/}}, {\it{y\/}})\index{g-forsomen} $=$ {\rm{gcode}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{y\/}})\index{forsome})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-consn; name this event `g1408'.\index{g1408}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-notn; name this event `g1409'.\index{g1409}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-f-orn; name this event `g1410'.\index{g1410}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-equaln-reduc3 \\ \index{pr-equaln-reduc3}
\=\+({\rm{num-treep}}\,({\it{x\/}})\index{num-treep} $\wedge$ {\rm{num-treep}}\,({\it{y\/}})\index{num-treep}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{pr-equaln}}\,({\it{x\/}}, {\it{y\/}})\index{pr-equaln} $=$ {\rm{bool-fix}}\,({\rm{ncode-equal}}\,({\it{x\/}}, {\it{y\/}})\index{ncode-equal})\index{bool-fix})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-forsomen-numberp2 \\ \index{g-forsomen-numberp2}
({\it{x\/}} $\in\;${\bf{N}}${}$) $\rightarrow$ ({\rm{g-forsomen}}\,({\it{x\/}}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-forsomen} $=$ {\rm{gcode}}\,({\rm{forsome}}\,({\it{x\/}}, {\it{y\/}})\index{forsome})\index{gcode})
\end{tabbing}

\noindent{\sc Event}:   Enable g-forsomen; name this event `g1411'.\index{g1411}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-inv-coden-gcode \\ \index{g-inv-coden-gcode}
\=\+({\rm{num-treep}}\,({\it{x\/}})\index{num-treep} $\wedge$ ({\rm{exp-count}}\,({\it{x\/}})\index{exp-count} $<$ {\it{n\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-inv-coden}}\,({\it{x\/}}, {\it{n\/}})\index{g-inv-coden} $=$ {\rm{gcode}}\,({\rm{inv-code}}\,({\it{x\/}})\index{inv-code})\index{gcode})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  inv-code-gcode \\ \index{inv-code-gcode}
{\rm{gcode}}\,({\rm{inv-code}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{inv-code})\index{gcode} $=$ {\rm{gcode}}\,({\it{x\/}})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable g-inv-code; name this event `g1412'.\index{g1412}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-inv-coden; name this event `g1413'.\index{g1413}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-inv-code-defn; name this event `g1414'.\index{g1414}

\addvspace{18pt}
\begin{verbatim};;;;;;;Correspondence proofs for the "theorem-checker".  There is no Lisp function
;;;;;;;ISTHM (pronounced "is theorem") because it would have to search upwards
;;;;;;;along the enumeration of proofs given by (NUMCODE (GCODE PF)), until it found a 
;;;;;;;a proof or a disproof of EXP.  This search might not terminate.  
;;;;;;;The function G-ISTHMN however has a stack-counter N which is decremented with
;;;;;;;each recursive call, and hence does terminate.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-isthmn} \\  
\=\+{\rm{g-isthmn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-isthmn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }\=\+{\rm{btmp}}\,({\it{pfn\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{exp\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{given\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{defns\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{symbols\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;$({\it{n\/}} $\simeq {\tt{0}}$)\-$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,({\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, {\it{n\/}} $-\;1$)\index{g-inv-coden}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\rm{bool-fix}}\,({\bf{t}})\index{bool-fix}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}, \\ 
{\rm{g-isthmn}}\,(\=\+{\rm{pr-add1n}}\,({\it{pfn\/}})\index{pr-add1n}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-isthmn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-isthm; name this event `g1415'.\index{g1415}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-isthm-defn; name this event `g1416'.\index{g1416}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{g-thm-ind} \\  
\=\+{\rm{g-thm-ind}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{va\/}}, {\it{n\/}})\index{g-thm-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{g-thm-ind}}\,(\=\+{\rm{pr-add1}}\,({\tt{0}})\index{pr-add1}, \\ 
{\tt{1}}, \\ 
{\tt{2}}, \\ 
{\tt{3}}, \\ 
{\tt{4}}, \\ 
{\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-thm-ind}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-isthm-eval \\ \index{g-isthm-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-isthm}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{g-isthm}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-isthmn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{pfn\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{given\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{defns\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-isthmn}\-
\end{tabbing}
\begin{verbatim};;;;;;;Analogous to G-ISTHMN, we can define G-FIND-PROOFN which locates the number of the
;;;;;;;proof/disproof at which G-ISTHMN terminated.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-find-proofn} \\  
\=\+{\rm{g-find-proofn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proofn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }\=\+{\rm{btmp}}\,({\it{pfn\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{exp\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{given\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{defns\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{symbols\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;$({\it{n\/}} $\simeq {\tt{0}}$)\-$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,({\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, {\it{n\/}} $-\;1$)\index{g-inv-coden}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{pfn\/}}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{pfn\/}}, \\ 
{\rm{g-find-proofn}}\,(\=\+{\rm{pr-add1n}}\,({\it{pfn\/}})\index{pr-add1n}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-find-proofn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;DECIPHER is defined in terms of DECIPHER1.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{decipher}$\;\;$
{\rm{decipher}}\,({\it{x\/}})\index{decipher} $=$ {\rm{decipher1}}\,({\it{x\/}}, {\it{x\/}} $-\;1$, {\it{x\/}} $-\;1$)\index{decipher1}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{find-proofn} \\  
\=\+{\rm{find-proofn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{find-proofn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }\=\+{\rm{btmp}}\,({\it{exp\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{given\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{defns\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{symbols\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;$({\it{n\/}} $\simeq {\tt{0}}$)\-$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,({\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, {\it{n\/}} $-\;1$)\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{pfn\/}}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{pfn\/}}, \\ 
{\rm{find-proofn}}\,(\=\+$1\;+$ {\it{pfn\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{find-proofn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{find-proof-ind} \\  
\=\+{\rm{find-proof-ind}}\,({\it{pfn\/}}, {\it{n\/}})\index{find-proof-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{find-proof-ind}}\,($1\;+$ {\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{find-proof-ind}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable pr-ifn; name this event `g1417'.\index{g1417}

\addvspace{18pt}

\noindent{\sc Event}:   Disable bool-fix; name this event `g1418'.\index{g1418}

\addvspace{18pt}

\noindent{\sc Event}:   Disable pr-andn; name this event `g1419'.\index{g1419}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  pr-andn-t-or-f1 \\ \index{pr-andn-t-or-f1}
\=\+(($\neg$ {\rm{btmp}}\,({\rm{pr-andn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-andn})\index{btmp}) $\wedge$ ({\rm{pr-andn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-andn} $\not=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{pr-andn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-andn} $=$ {\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  pr-andn-t-or-f2 \\ \index{pr-andn-t-or-f2}
\=\+(($\neg$ {\rm{btmp}}\,({\rm{pr-andn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-andn})\index{btmp}) $\wedge$ ({\rm{pr-andn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-andn} $\not=$ {\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{pr-andn}}\,({\it{x\/}}, {\it{y\/}})\index{pr-andn} $=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-
\end{tabbing}

\noindent{\sc Event}:   Enable pr-andn; name this event `g1420'.\index{g1420}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-provesn; name this event `g1421'.\index{g1421}

\addvspace{18pt}
\begin{verbatim};;;;;;;If G-PROVESN terminates, it returns either (BOOL-FIX T) or (BOOL-FIX F).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  g-provesn-t-or-f1 \\ \index{g-provesn-t-or-f1}
\=\+(\=\+($\neg$ {\rm{btmp}}\,({\rm{g-provesn}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-provesn})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\rm{g-provesn}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-provesn} $\not=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-provesn}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-provesn} $=$ {\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-provesn-t-or-f2 \\ \index{g-provesn-t-or-f2}
\=\+(\=\+($\neg$ {\rm{btmp}}\,({\rm{g-provesn}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-provesn})\index{btmp}) \\ 
$\wedge$$\;\;\;\;$({\rm{g-provesn}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-provesn} $\not=$ {\rm{bool-fix}}\,({\bf{t}})\index{bool-fix}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{g-provesn}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-provesn} $=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-provesn; name this event `g1422'.\index{g1422}

\addvspace{18pt}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1423'.\index{g1423}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-find-proofn-gcode \\ \index{g-find-proofn-gcode}
\=\+($\neg$ {\rm{btmp}}\,({\rm{g-isthmn}}\,({\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-isthmn})\index{btmp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-find-proofn}}\,({\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proofn} \\ 
$=$$\;\;\;\;${\rm{gcode}}\,({\rm{find-proofn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{find-proofn})\index{gcode})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable pr-ifn; name this event `g1424'.\index{g1424}

\addvspace{18pt}
\begin{verbatim};;;;;;;DECIPHER1 always returns a num-tree.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  num-treep-decipher1 \\ \index{num-treep-decipher1}
{\rm{num-treep}}\,({\rm{decipher1}}\,({\it{x\/}}, {\it{y\/}}, {\it{z\/}})\index{decipher1})\index{num-treep}
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below essentially says that G-INV-CODEN returns the GCODE of whatever
;;;;;;;INV-CODE returns.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  inv-code-decipher \\ \index{inv-code-decipher}
\=\+(({\rm{dec-count}}\,({\it{x\/}}, {\it{x\/}} $-\;1$, {\it{x\/}} $-\;1$)\index{dec-count} $+$ {\rm{exp-count}}\,({\rm{decipher}}\,({\it{x\/}})\index{decipher})\index{exp-count}) $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-inv-coden}}\,({\rm{g-deciphern}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\it{n\/}})\index{g-deciphern}, {\it{n\/}})\index{g-inv-coden} \\ 
$=$$\;\;\;\;${\rm{gcode}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{x\/}})\index{decipher})\index{inv-code})\index{gcode})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-inv-coden; name this event `g1425'.\index{g1425}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-deciphern; name this event `g1426'.\index{g1426}

\addvspace{18pt}

\noindent{\sc Event}:   Enable inv-code; name this event `g1427'.\index{g1427}

\addvspace{18pt}

\noindent{\sc Event}:   Enable decipher; name this event `g1428'.\index{g1428}

\addvspace{18pt}
\begin{verbatim};;;;;;;G-FINDN returns the value of stack counter N at the point at which G-ISTHMN located
;;;;;;;a proof/disproof.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-findn} \\  
\=\+{\rm{g-findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }\=\+{\rm{btmp}}\,({\it{pfn\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{exp\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{given\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{defns\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{symbols\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;$({\it{n\/}} $\simeq {\tt{0}}$)\-$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,({\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, {\it{n\/}} $-\;1$)\index{g-inv-coden}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\rm{g-findn}}\,(\=\+{\rm{pr-add1n}}\,({\it{pfn\/}})\index{pr-add1n}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-findn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{verbatim};;;;;;;G-FIND-PROOF returns the GCODE of the proof/disproof.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-find-proof} \\  
\=\+{\rm{g-find-proof}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proof} \\ 
$=$$\;\;\;\;$\=\+{\bf if }\=\+{\rm{btmp}}\,({\it{pfn\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{exp\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{given\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{defns\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{symbols\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;$({\it{n\/}} $\simeq {\tt{0}}$)\-$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,({\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, {\it{n\/}} $-\;1$)\index{g-inv-coden}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\rm{g-inv-coden}}\,({\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, {\it{n\/}} $-\;1$)\index{g-inv-coden}, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\rm{g-inv-coden}}\,({\rm{g-deciphern}}\,({\it{pfn\/}}, {\it{n\/}} $-\;1$)\index{g-deciphern}, {\it{n\/}} $-\;1$)\index{g-inv-coden}, \\ 
{\rm{g-find-proof}}\,(\=\+{\rm{pr-add1n}}\,({\it{pfn\/}})\index{pr-add1n}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-find-proof})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable pr-ifn; name this event `g1429'.\index{g1429}

\addvspace{18pt}

\noindent{\sc Event}:   Disable bool-fix; name this event `g1430'.\index{g1430}

\addvspace{18pt}
\begin{verbatim};;;;;;;If G-ISTHMN returns (BOOL-FIX T), then G-FIND-PROOF returns the GCODE of a proof.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  find-proof-proves-t-case \\ \index{find-proof-proves-t-case}
\=\+({\rm{g-isthmn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-isthmn} $=$ {\rm{bool-fix}}\,({\bf{t}})\index{bool-fix}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-find-proof}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proof}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\rm{g-findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn})\-\index{g-provesn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;If G-ISTHMN returns (BOOL-FIX F), then G-FIND-PROOF returns the GCODE of a disproof.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  find-proof-proves-f-case \\ \index{find-proof-proves-f-case}
\=\+({\rm{g-isthmn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-isthmn} $=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-find-proof}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proof}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\rm{g-findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn})\-\index{g-provesn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-find-proof}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proof}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\rm{g-findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn})\-\index{g-provesn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-)\-\-
\end{tabbing}
\begin{verbatim};;;;;;;This lemma relates G-FIND-PROOF and G-FIND-PROOFN (which returns the proof number).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  g-find-proof-gcode \\ \index{g-find-proof-gcode}
\=\+($\neg$ {\rm{btmp}}\,({\rm{g-isthmn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-isthmn})\index{btmp}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-find-proof}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proof} \\ 
$=$$\;\;\;\;${\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,(\=\+{\rm{g-find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{g-find-proofn}, \\ 
{\rm{g-findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn})\-\index{g-deciphern}, \\ 
{\rm{g-findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn})\-\index{g-inv-coden})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-find-proof; name this event `g1431'.\index{g1431}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-find-proofn; name this event `g1432'.\index{g1432}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-findn; name this event `g1433'.\index{g1433}

\addvspace{18pt}

\noindent{\sc Event}:   Enable find-proofn; name this event `g1434'.\index{g1434}

\addvspace{18pt}

\noindent{\sc Event}:   Enable dec-count; name this event `g1435'.\index{g1435}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{dec-countn}$\;\;$
{\rm{dec-countn}}\,({\it{x\/}})\index{dec-countn} $=$ {\rm{dec-count}}\,({\it{x\/}}, {\it{x\/}} $-\;1$, {\it{x\/}} $-\;1$)\index{dec-count}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{thm-count} \\  
\=\+{\rm{thm-count}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count} \\ 
$=$$\;\;\;\;$(\=\+({\rm{dec-countn}}\,({\it{pfn\/}})\index{dec-countn} $+$ {\rm{exp-count}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{exp-count}) \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{prf-cnt}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf-cnt})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The same lemmas as above with PROVES, INV-CODE, and DECIPHER, instead of
;;;;;;;G-PROVESN, G-INV-CODEN, and G-DECIPHERN.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  find-proof-proves-t-case-gcode \\ \index{find-proof-proves-t-case-gcode}
\=\+(\=\+(\=\+($1\;+$ {\rm{thm-count}}\,(\=\+{\rm{find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn}, \\ 
{\it{exp\/}}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\it{flg\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{thm-count}) \\ 
$<$$\;\;\;\;${\rm{g-findn}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-findn})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{inv-code}}\,({\rm{decipher}}\,({\rm{find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn})\index{decipher})\index{inv-code}, \\ 
{\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  find-proof-proves-f-case-case \\ \index{find-proof-proves-f-case-case}
\=\+({\rm{g-isthmn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-isthmn} $=$ {\rm{bool-fix}}\,({\bf{f}})\index{bool-fix}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-find-proof}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-find-proof}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\rm{g-findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn})\-\index{g-provesn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-thm-count1 \\ \index{lessp-thm-count1}
\=\+(\=\+{\rm{exp-count}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{prf-cnt}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf-cnt})\- \\ 
$<$$\;\;\;\;$($1\;+$ {\rm{thm-count}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count})\-
\end{tabbing}

\noindent{\sc Event}:   Enable find-proof-proves-f-case; name this event `g1436'.\index{g1436}

\addvspace{18pt}

\noindent{\sc Event}:   Enable subst-ident; name this event `g1437'.\index{g1437}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  lessp-thm-count2 \\ \index{lessp-thm-count2}
\=\+({\rm{dec-countn}}\,({\it{pfn\/}})\index{dec-countn} $+$ {\rm{exp-count}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{exp-count}) \\ 
$<$$\;\;\;\;$($1\;+$ {\rm{thm-count}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  find-proof-proves-f-case-gcode \\ \index{find-proof-proves-f-case-gcode}
\=\+(\=\+(\=\+($1\;+$ {\rm{thm-count}}\,(\=\+{\rm{find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn}, \\ 
{\rm{f-not}}\,({\it{exp\/}})\index{f-not}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\tt{0}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{thm-count}) \\ 
$<$$\;\;\;\;${\rm{g-findn}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-findn})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{inv-code}}\,({\rm{decipher}}\,({\rm{find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn})\index{decipher})\index{inv-code}, \\ 
{\rm{f-not}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable thm-count; name this event `g1438'.\index{g1438}

\addvspace{18pt}
\begin{verbatim};;;;;;;Now we start constructing the undecidable sentence.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-undec1} \\  
\=\+{\rm{a-undec1}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}})\index{a-undec1} \\ 
$=$$\;\;\;\;${\rm{forsome}}\,(\=\+{\tt{1}}, \\ 
{\rm{f-and}}\,(\=\+{\rm{isin}}\,(\=\+{\rm{z-list}}\,({\rm{list}}\,(\=\+{\rm{ncode}}\,({\it{val\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\tt{6}}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode})\-)\index{z-list}, \\ 
{\tt{1}})\-\index{isin}, \\ 
{\rm{a-ev-defn}}\,({\tt{1}})\index{a-ev-defn})\-\index{f-and})\-\index{forsome}\-
\end{tabbing}

\noindent{\sc Event}:   Disable a-ev1; name this event `g1439'.\index{g1439}

\addvspace{18pt}

\noindent{\sc Event}:   Disable a-ev-defn; name this event `g1440'.\index{g1440}

\addvspace{18pt}

\noindent{\sc Event}:   Disable a-ev; name this event `g1441'.\index{g1441}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  replace-va-a-undec1 \\ \index{replace-va-a-undec1}
\=\+{\rm{subst}}\,({\rm{a-undec1}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}})\index{a-undec1}, {\tt{6}}, {\rm{ncode}}\,({\it{va1\/}})\index{ncode}, {\tt{0}})\index{subst} \\ 
$=$$\;\;\;\;${\rm{a-ev1}}\,(\=\+{\rm{ncode}}\,({\it{flg\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{exp\/}})\index{ncode}, \\ 
{\rm{z-opair}}\,({\rm{ncode}}\,({\it{va1\/}})\index{ncode}, {\rm{ncode}}\,({\rm{gcode}}\,({\bf{nil}})\index{gcode})\index{ncode})\index{z-opair}, \\ 
{\rm{ncode}}\,({\it{fa\/}})\index{ncode}, \\ 
{\rm{ncode}}\,({\it{val\/}})\index{ncode})\-\index{a-ev1}\-
\end{tabbing}

\noindent{\sc Event}:   Disable a-ev-defn-part1; name this event `g1442'.\index{g1442}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  collect-free-a-undec1 \\ \index{collect-free-a-undec1}
\=\+({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\it{term\/}}, {\tt{0}})\index{collect-free}) \\ 
$\rightarrow$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{subst}}\,({\rm{a-undec1}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}})\index{a-undec1}, {\tt{6}}, {\it{term\/}}, {\tt{0}})\index{subst}, {\tt{0}})\index{collect-free})\-
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev-defn-part1; name this event `g1443'.\index{g1443}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev-defn; name this event `g1444'.\index{g1444}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-a-undec1 \\ \index{formula-a-undec1}
{\rm{all-symbs}}\,({\it{symbols\/}})\index{all-symbs} $\rightarrow$ {\rm{formula}}\,({\rm{a-undec1}}\,({\it{flg\/}}, {\it{exp\/}}, {\it{fa\/}}, {\it{val\/}})\index{a-undec1}, {\it{symbols\/}})\index{formula}
\end{tabbing}

\noindent{\sc Event}:   Enable a-ev1; name this event `g1445'.\index{g1445}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev; name this event `g1446'.\index{g1446}

\addvspace{18pt}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1447'.\index{g1447}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-isthm; name this event `g1448'.\index{g1448}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-isthmn; name this event `g1449'.\index{g1449}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  numeral-gcode \\ \index{numeral-gcode}
{\rm{numeral}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{numeral} $=$ {\rm{numeral}}\,({\it{x\/}})\index{numeral}
\end{tabbing}
\begin{verbatim};;;;;;;Correspondence for GC-NCODE.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{gc-ncoden} \\  
\=\+{\rm{gc-ncoden}}\,({\it{x\/}}, {\it{n\/}})\index{gc-ncoden} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{btmp}}\,({\it{x\/}})\index{btmp} $\vee$ ({\it{n\/}} $\simeq {\tt{0}}$)$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{pr-listpn}}\,({\it{x\/}})\index{pr-listpn}, \\ 
{\rm{g-z-opairn}}\,(\=\+{\rm{gc-ncoden}}\,({\rm{pr-carn}}\,({\it{x\/}})\index{pr-carn}, {\it{n\/}} $-\;1$)\index{gc-ncoden}, \\ 
{\rm{gc-ncoden}}\,({\rm{pr-cdrn}}\,({\it{x\/}})\index{pr-cdrn}, {\it{n\/}} $-\;1$)\index{gc-ncoden})\-\index{g-z-opairn}, \\ 
{\rm{g-numeraln}}\,({\it{x\/}}, {\it{n\/}} $-\;1$)\index{g-numeraln})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable gc-ncode; name this event `g1450'.\index{g1450}

\addvspace{18pt}

\noindent{\sc Event}:   Disable gc-ncode-defn; name this event `g1451'.\index{g1451}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{gc-ncode-ind} \\  
\=\+{\rm{gc-ncode-ind}}\,({\it{x\/}}, {\it{va\/}}, {\it{n\/}})\index{gc-ncode-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }\=\+{\rm{gc-ncode-ind}}\,({\rm{pr-car}}\,({\tt{0}})\index{pr-car}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{gc-ncode-ind} \\ 
$\wedge$$\;\;\;\;${\rm{gc-ncode-ind}}\,({\rm{pr-cdr}}\,({\tt{0}})\index{pr-cdr}, {\rm{ev}}\,({\tt{1}}, {\rm{list}}\,({\it{x\/}}), {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}} $-\;1$)\index{gc-ncode-ind}\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gc-ncode-eval \\ \index{gc-ncode-eval}
{\rm{ev}}\,({\tt{0}}, {\rm{gc-ncode}}\,({\it{x\/}})\index{gc-ncode}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} $=$ {\rm{gc-ncoden}}\,({\rm{ev}}\,({\tt{0}}, {\it{x\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, {\it{n\/}})\index{gc-ncoden}
\end{tabbing}

\noindent{\sc Event}:   Disable z-opair; name this event `g1452'.\index{g1452}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-sing; name this event `g1453'.\index{g1453}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-pair; name this event `g1454'.\index{g1454}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-pairn-gcode \\ \index{z-pairn-gcode}
{\rm{g-z-pairn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-z-pairn} $=$ {\rm{gcode}}\,({\rm{z-pair}}\,({\it{x\/}}, {\it{y\/}})\index{z-pair})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable z-pair; name this event `g1455'.\index{g1455}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-pair; name this event `g1456'.\index{g1456}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z-pairn; name this event `g1457'.\index{g1457}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-opairn-gcode \\ \index{z-opairn-gcode}
{\rm{g-z-opairn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\rm{gcode}}\,({\it{y\/}})\index{gcode})\index{g-z-opairn} $=$ {\rm{gcode}}\,({\rm{z-opair}}\,({\it{x\/}}, {\it{y\/}})\index{z-opair})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable z-sing; name this event `g1458'.\index{g1458}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z-singn; name this event `g1459'.\index{g1459}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-opair; name this event `g1460'.\index{g1460}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z-opairn; name this event `g1461'.\index{g1461}

\addvspace{18pt}

\noindent{\sc Event}:   Disable g-numeraln; name this event `g1462'.\index{g1462}

\addvspace{18pt}

\noindent{\sc Event}:   Disable ncode; name this event `g1463'.\index{g1463}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-succ; name this event `g1464'.\index{g1464}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-z-succ-gcode \\ \index{g-z-succ-gcode}
{\rm{g-z-succn}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{g-z-succn} $=$ {\rm{gcode}}\,({\rm{z-succ}}\,({\it{x\/}})\index{z-succ})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable z-succ; name this event `g1465'.\index{g1465}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-z-succn; name this event `g1466'.\index{g1466}

\addvspace{18pt}

\noindent{\sc Event}:   Disable pr-zeropn; name this event `g1467'.\index{g1467}

\addvspace{18pt}

\noindent{\sc Event}:   Disable pr-sub1n; name this event `g1468'.\index{g1468}

\addvspace{18pt}

\noindent{\sc Event}:   Disable pr-add1n; name this event `g1469'.\index{g1469}

\addvspace{18pt}

\noindent{\sc Event}:   Disable phi; name this event `g1470'.\index{g1470}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-phin-gcode \\ \index{g-phin-gcode}
{\sc{g-phin}}\index{g-phin} $=$ {\rm{gcode}}\,({\sc{phi}}\index{phi})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable phi; name this event `g1471'.\index{g1471}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-phin; name this event `g1472'.\index{g1472}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  g-numeraln-ncode \\ \index{g-numeraln-ncode}
(({\it{x\/}} $<$ {\it{n\/}}) $\wedge$ ({\it{x\/}} $\in\;${\bf{N}}${}$)) $\rightarrow$ ({\rm{g-numeraln}}\,({\it{x\/}}, {\it{n\/}})\index{g-numeraln} $=$ {\rm{gcode}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{gcode})
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ncode-cnt} \\  
\=\+{\rm{ncode-cnt}}\,({\it{x\/}})\index{ncode-cnt} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{x\/}})$\;\;${\bf then }$1\;+$ ({\rm{ncode-cnt}}\,({\rm{car}}\,({\it{x\/}}))\index{ncode-cnt} $+$ {\rm{ncode-cnt}}\,({\rm{cdr}}\,({\it{x\/}}))\index{ncode-cnt}) \\ 
{\bf else }$1\;+$ {\rm{fix}}\,({\it{x\/}})$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{ncode-ind} \\  
\=\+{\rm{ncode-ind}}\,({\it{x\/}}, {\it{n\/}})\index{ncode-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{listp}}\,({\it{x\/}}) \\ 
{\bf then }{\rm{ncode-ind}}\,({\rm{car}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{ncode-ind} $\wedge$ {\rm{ncode-ind}}\,({\rm{cdr}}\,({\it{x\/}}), {\it{n\/}} $-\;1$)\index{ncode-ind} \\ 
{\bf else }{\bf{t}}$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  gc-ncoden-gcode \\ \index{gc-ncoden-gcode}
\=\+(({\rm{ncode-cnt}}\,({\it{x\/}})\index{ncode-cnt} $<$ {\it{n\/}}) $\wedge$ {\rm{num-treep}}\,({\it{x\/}})\index{num-treep}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{gc-ncoden}}\,({\it{x\/}}, {\it{n\/}})\index{gc-ncoden} $=$ {\rm{gcode}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode})\index{gcode})\-
\end{tabbing}

\noindent{\sc Event}:   Enable gc-ncode; name this event `g1473'.\index{g1473}

\addvspace{18pt}

\noindent{\sc Event}:   Enable gc-ncoden; name this event `g1474'.\index{g1474}

\addvspace{18pt}

\noindent{\sc Event}:   Enable ncode; name this event `g1475'.\index{g1475}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  gc-ncoden-gcode-again \\ \index{gc-ncoden-gcode-again}
\=\+({\rm{ncode-cnt}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{ncode-cnt} $<$ {\it{n\/}}) \\ 
$\rightarrow$$\;\;\;\;$({\rm{gc-ncoden}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode}, {\it{n\/}})\index{gc-ncoden} $=$ {\rm{gcode}}\,({\rm{ncode}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{ncode})\index{gcode})\-
\end{tabbing}

\noindent{\sc Event}:   Enable gc-ncode-defn; name this event `g1476'.\index{g1476}

\addvspace{18pt}
\begin{verbatim};;;;;;;G-UNDEC (when PFN is 0) is the EV analogue of a the metatheoretic statement, 
;;;;;;;"There exists a proof of the result of substituting (NCODE (GCODE EXP)) for the 
;;;;;;;variable VAR in EXP (from GIVEN, DEFNS, and SYMBOLS)".

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{g-undec} \\  
\=\+{\rm{g-undec}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{g-undec} \\ 
$=$$\;\;\;\;${\rm{g-isthm}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{g-subst}}\,({\it{exp\/}}, {\it{var\/}}, {\rm{gc-ncode}}\,({\it{exp\/}})\index{gc-ncode}, {\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote})\index{g-subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{g-isthm}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{g-undecn} \\  
\=\+{\rm{g-undecn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-undecn} \\ 
$=$$\;\;\;\;${\rm{g-isthmn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{g-substn}}\,({\it{exp\/}}, {\it{var\/}}, {\rm{gc-ncoden}}\,({\it{exp\/}}, {\it{n\/}})\index{gc-ncoden}, {\rm{gcode}}\,({\tt{0}})\index{gcode}, {\it{n\/}})\index{g-substn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{g-isthmn}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  g-undec-eval \\ \index{g-undec-eval}
\=\+{\rm{ev}}\,({\tt{0}}, {\rm{g-undec}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{g-undec}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev} \\ 
$=$$\;\;\;\;${\rm{g-undecn}}\,(\=\+{\rm{ev}}\,({\tt{0}}, {\it{pfn\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{exp\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{var\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{given\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{defns\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\rm{ev}}\,({\tt{0}}, {\it{symbols\/}}, {\it{va\/}}, {\sc{fa}}\index{fa}, {\it{n\/}})\index{ev}, \\ 
{\it{n\/}})\-\index{g-undecn}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-undec; name this event `g1477'.\index{g1477}

\addvspace{18pt}
\begin{verbatim};;;;;;;A-UNDEC is the formula from which the undecidable sentence is constructed.
;;;;;;;It is got from A-EV and asserts that the value of G-UNDEC on a particular
;;;;;;;EXP (represented by a free variable, say "y", in A-UNDEC) with GIVEN, DEFNS,
;;;;;;;and SYMBOLS, is (BOOL-FIX F).  This means that it is asserting, "there is
;;;;;;;no proof of the result of substituting (NCODE (GCODE y)) for the variable "y"
;;;;;;;in y, from GIVEN, DEFNS, and SYMBOLS."   The undecidable sentence will be got
;;;;;;;by substituting the formula the NCODE of the GCODE of A-UNDEC for the free "y"
;;;;;;;in A-UNDEC itself.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{a-undec} \\  
\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec} \\ 
$=$$\;\;\;\;${\rm{a-undec1}}\,(\=\+{\tt{0}}, \\ 
{\rm{g-undec}}\,(\=\+{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{6}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{given\/}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{defns\/}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{symbols\/}})\index{gcode})\index{pr-quote})\-\index{g-undec}, \\ 
{\sc{fa}}\index{fa}, \\ 
{\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-\index{a-undec1}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{undec-count} \\  
\=\+{\rm{undec-count}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{undec-count} \\ 
$=$$\;\;\;\;$(\=\+($1\;+$ {\rm{thm-count}}\,(\=\+{\rm{find-proofn}}\,(\=\+{\tt{0}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn}, \\ 
{\rm{f-not}}\,({\it{exp\/}})\index{f-not}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\tt{0}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{thm-count}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{thm-count}}\,(\=\+{\rm{find-proofn}}\,(\=\+{\tt{0}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn}, \\ 
{\it{exp\/}}, \\ 
{\it{var\/}}, \\ 
{\it{term\/}}, \\ 
{\tt{0}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{thm-count}) \\ 
$+$$\;\;\;\;${\rm{ncode-cnt}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode})\index{ncode-cnt})\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{isthm-a-proves-a-proof} \\  
\=\+{\rm{isthm-a-proves-a-proof}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{isthm-a-proves-a-proof} \\ 
$=$$\;\;\;\;${\rm{inv-code}}\,({\rm{decipher}}\,({\rm{find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn})\index{decipher})\index{inv-code}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{findn-count} \\  
\=\+{\rm{findn-count}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{findn-count} \\ 
$=$$\;\;\;\;${\rm{g-findn}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-findn}\-
\end{tabbing}

\noindent{\sc Event}:   Enable gcode; name this event `g1478'.\index{g1478}

\addvspace{18pt}

\noindent{\sc Event}:   Enable exp-count; name this event `g1479'.\index{g1479}

\addvspace{18pt}
\begin{verbatim};;;;;;;The lemma below begins the first of four important lemmas.
;;;;;;;The undecidable sentence is of the form;
;;;;;;;(SUBST (A-UNDEC GIVEN DEFNS SYMBOLS)
;;;;;;;       6
;;;;;;;       (NCODE (GCODE (A-UNDEC GIVEN DEFNS SYMBOLS))) 0),
;;;;;;;where DEFNS and SYMBOLS will be instantiated by a specific list of
;;;;;;;definitions and symbols, respectively.  The lemma asserts that if
;;;;;;;G-ISTHMN returns (BOOL-FIX T) on the GCODE of the undecidable sentence,
;;;;;;;then the undecidable sentence is provable.  This makes sense because
;;;;;;;G-ISTHMN returns (BOOL-FIX T) when it has found a proof in the enumeration
;;;;;;;of proofs.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  isthm-a-proves-a \\ \index{isthm-a-proves-a}
\=\+(\=\+(\=\+{\rm{undec-count}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{undec-count} \\ 
$<$$\;\;\;\;${\rm{findn-count}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{findn-count})\- \\ 
$\wedge$$\;\;\;\;$({\rm{exp-count}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{exp-count} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\rm{g-substn}}\,(\=\+{\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode}, \\ 
{\rm{gcode}}\,({\tt{6}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode})\index{gcode}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-substn}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isthm-a-proves-a-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{isthm-a-proves-a-proof}, \\ 
{\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;;Similarly, if G-ISTHMN returns (BOOL-FIX F), it means that it found a disproof of
;;;;;;;the undecidable sentence, and its negation is therefore provable.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  not-isthm-a-proves-not-a \\ \index{not-isthm-a-proves-not-a}
\=\+(\=\+(\=\+{\rm{undec-count}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{undec-count} \\ 
$<$$\;\;\;\;${\rm{findn-count}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{findn-count})\- \\ 
$\wedge$$\;\;\;\;$({\rm{exp-count}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{exp-count} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\rm{g-substn}}\,(\=\+{\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode}, \\ 
{\rm{gcode}}\,({\tt{6}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode})\index{gcode}, \\ 
{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-substn}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-)\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isthm-a-proves-a-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}})\-\index{isthm-a-proves-a-proof}, \\ 
{\rm{f-not}}\,({\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isthm-a-proves-a-proof; name this event `g1480'.\index{g1480}

\addvspace{18pt}

\noindent{\sc Event}:   Enable a-ev1-not-good-proof; name this event `g1481'.\index{g1481}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{isthm-a-proves-not-a-proof} \\  
\=\+{\rm{isthm-a-proves-not-a-proof}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{isthm-a-proves-not-a-proof} \\ 
$=$$\;\;\;\;${\rm{a-ev1-not-good-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{g-undec}}\,(\=\+{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{6}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{given\/}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{defns\/}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{symbols\/}})\index{gcode})\index{pr-quote})\-\index{g-undec}, \\ 
{\rm{cons}}\,(\=\+{\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode}, \\ 
{\rm{gcode}}\,({\bf{nil}})\index{gcode})\-, \\ 
{\sc{fa}}\index{fa}, \\ 
{\it{n\/}}, \\ 
{\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-\index{a-ev1-not-good-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The more surprising pair of facts are that if G-ISTHMN returns (BOOL-FIX T)
;;;;;;;the GCODE of the undecidable sentence, then the negation of the undecidable 
;;;;;;;sentence is provable.  This is because the undecidable sentence through the 
;;;;;;;fact that it represents some instance of the Lisp interpreter asserts that G-ISTHMN
;;;;;;;returns (BOOL-FIX F) on the undecidable sentence itself, which in this case is false.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  isthm-a-proves-not-a-proof-proves \\ \index{isthm-a-proves-not-a-proof-prove!s}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$({\rm{ncode-cnt}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode-cnt} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{exp-count}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{exp-count} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{f-not}}\,({\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst})\index{f-not}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isthm-a-proves-not-a-proof}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{isthm-a-proves-not-a-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable isthm-a-proves-not-a-proof; name this event `g1482'.\index{g1482}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{not-isthm-a-proves-a-proof} \\  
\=\+{\rm{not-isthm-a-proves-a-proof}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{not-isthm-a-proves-a-proof} \\ 
$=$$\;\;\;\;${\rm{a-ev1-ok-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{g-undec}}\,(\=\+{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{0}})\index{gcode})\index{pr-quote}, \\ 
{\tt{0}}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\tt{6}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{given\/}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{defns\/}})\index{gcode})\index{pr-quote}, \\ 
{\rm{pr-quote}}\,({\rm{gcode}}\,({\it{symbols\/}})\index{gcode})\index{pr-quote})\-\index{g-undec}, \\ 
{\rm{cons}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode}, {\rm{gcode}}\,({\bf{nil}})\index{gcode}), \\ 
{\sc{fa}}\index{fa}, \\ 
{\it{n\/}})\-\index{a-ev1-ok-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;If G-ISTHMN returns (BOOL-FIX F) on the the GCODE of the undecidable sentence, 
;;;;;;;then the undecidable sentence is provable.  This is because the undecidable 
;;;;;;;sentence through the fact that it represents some instance of the Lisp interpreter 
;;;;;;;asserts that G-ISTHMN does return (BOOL-FIX F) on the undecidable sentence itself.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  not-isthm-a-proves-a-proof-proves \\ \index{not-isthm-a-proves-a-proof-prove!s}
\=\+(\=\+{\rm{ev-hyps}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{ev-hyps} \\ 
$\wedge$$\;\;\;\;$({\rm{ncode-cnt}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode-cnt} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{exp-count}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{exp-count} $<$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\tt{0}})\index{gcode}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\- \\ 
$\wedge$$\;\;\;\;$({\it{concl\/}} $=$ {\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{not-isthm-a-proves-a-proof}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{not-isthm-a-proves-a-proof}, \\ 
{\it{concl\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves}\-
\end{tabbing}
\begin{verbatim};;;;;;;Now we need to prove that if Z2 is complete, then G-ISTHMN returns either (BOOL-FIX T)
;;;;;;;or (BOOL-FIX F).  Then by the above four lemmas, we have that the undecidable sentence
;;;;;;;is both provable and disprovable.  Therefore, we would have established that Z2 is
;;;;;;;either incomplete or inconsistent.
\end{verbatim}
\noindent{\sc Event}:   Enable isthm-a-proves-not-a-proof; name this event `g1483'.\index{g1483}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{thm-count1} \\  
\=\+{\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1} \\ 
$=$$\;\;\;\;$(\=\+({\rm{dec-countn}}\,({\it{pfn\/}})\index{dec-countn} $+$ {\rm{exp-count}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{exp-count}) \\ 
$+$$\;\;\;\;${\rm{exp-count}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{prf-cnt}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf-cnt})\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  thm-count1-lessp \\ \index{thm-count1-lessp}
\=\+({\rm{dec-count}}\,({\it{pfn\/}}, {\it{pfn\/}} $-\;1$, {\it{pfn\/}} $-\;1$)\index{dec-count} $+$ {\rm{exp-count}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{exp-count}) \\ 
$<$$\;\;\;\;$($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1})\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  thm-count1-lessp1 \\ \index{thm-count1-lessp1}
\=\+(\=\+{\rm{exp-count}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code})\index{exp-count} \\ 
$+$$\;\;\;\;${\rm{formula-cnt}}\,({\it{exp\/}}, {\it{symbols\/}})\index{formula-cnt} \\ 
$+$$\;\;\;\;${\rm{prf-cnt}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\it{given\/}}, {\it{defns\/}}, {\tt{0}}, {\it{symbols\/}})\index{prf-cnt})\- \\ 
$<$$\;\;\;\;$($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1})\-
\end{tabbing}

\noindent{\sc Event}:   Enable thm-count1; name this event `g1484'.\index{g1484}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{comp-count} \\  
\=\+{\rm{comp-count}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{extras\/}})\index{comp-count} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{pf\/}} $\leq$ {\it{pfn\/}} \\ 
{\bf then }\=\+($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\it{extras\/}})\- \\ 
{\bf elseif }{\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
{\bf then }\=\+($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\it{extras\/}})\- \\ 
{\bf elseif }{\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
{\bf then }\=\+($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\it{extras\/}})\- \\ 
{\bf else }\=\+($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{thm-count1}}\,({\it{pfn\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\it{extras\/}}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{comp-count}}\,(\=\+{\it{pf\/}}, \\ 
$1\;+$ {\it{pfn\/}}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{extras\/}})\-\index{comp-count})\-$\;${\bf  endif}\-\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{proves-isthmn-ind} \\  
\=\+{\rm{proves-isthmn-ind}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{proves-isthmn-ind} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\it{pf\/}} $\leq$ {\it{pfn\/}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}$\;\;${\bf then }{\bf{t}} \\ 
{\bf elseif }{\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pfn\/}})\index{decipher})\index{inv-code}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
{\bf then }{\bf{t}} \\ 
{\bf else }{\rm{proves-isthmn-ind}}\,({\it{pf\/}}, $1\;+$ {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{proves-isthmn-ind}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-isthmn; name this event `g1485'.\index{g1485}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  lessp-comp-count1 \\ \index{lessp-comp-count1}
\=\+{\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1} \\ 
$<$$\;\;\;\;${\rm{comp-count}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{extras\/}})\index{comp-count}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-comp-count2 \\ \index{lessp-comp-count2}
\=\+{\rm{thm-count1}}\,({\it{pfn\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1} \\ 
$<$$\;\;\;\;${\rm{comp-count}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{extras\/}})\index{comp-count}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-extras-comp-count \\ \index{lessp-extras-comp-count}
{\it{extras\/}} $<$ {\rm{comp-count}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{extras\/}})\index{comp-count}
\end{tabbing}

\noindent{\sc Event}:   Enable find-proof-proves-f-case-gcode; name this event `g1486'.\index{g1486}

\addvspace{18pt}

\noindent{\sc Event}:   Enable find-proof-proves-t-case-gcode; name this event `g1487'.\index{g1487}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-provesn-t-or-f1; name this event `g1488'.\index{g1488}

\addvspace{18pt}

\noindent{\sc Event}:   Enable g-provesn-t-or-f2; name this event `g1489'.\index{g1489}

\addvspace{18pt}

\noindent{\sc Event}:   Enable pr-ifn; name this event `g1490'.\index{g1490}

\addvspace{18pt}
\begin{verbatim};;;;;;;The lemma below is the crucial one in the final proof of the incompleteness theorem.
;;;;;;;It asserts that if there is a number PF which corresponds to a proof or disproof of EXP,
;;;;;;;then G-ISTHMN (with a large enough stack-counter) when started on a proof-number PFN
;;;;;;;less than or equal to PF, will either find PF or some proof/disproof before it, and
;;;;;;;will therefore either return (BOOL-FIX T) or (BOOL-FIX F).  This with the four lemmas
;;;;;;;above provides enough material to prove the incompleteness theorem, but we still need
;;;;;;;to build up a large enough stack-counter to cover all these five lemmas.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  proves-t-not-btmp-isthmn \\ \index{proves-t-not-btmp-isthmn}
\=\+(\=\+({\it{pfn\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{pf\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pf\/}})\index{decipher})\index{inv-code}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\vee$$\;\;\;\;${\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pf\/}})\index{decipher})\index{inv-code}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\wedge$$\;\;\;\;$({\it{pfn\/}} $\leq$ {\it{pf\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{comp-count}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{extras\/}})\index{comp-count} $<$ {\it{n\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{exp\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{t}})\index{bool-fix})\- \\ 
$\vee$$\;\;\;\;$(\=\+{\rm{g-isthmn}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{exp\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{g-isthmn} \\ 
$=$$\;\;\;\;${\rm{bool-fix}}\,({\bf{f}})\index{bool-fix})\-)\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable proves-t-not-btmp-isthmn; name this event `g1491'.\index{g1491}

\addvspace{18pt}

\noindent{\sc Event}:   Disable find-proofn; name this event `g1492'.\index{g1492}

\addvspace{18pt}
\begin{tabbing}{\sc Definition}:\index{findn} \\  
\=\+{\rm{findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{findn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }\=\+{\rm{btmp}}\,({\it{exp\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{given\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{defns\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;${\rm{btmp}}\,({\it{symbols\/}})\index{btmp} \\ 
$\vee$$\;\;\;\;$({\it{n\/}} $\simeq {\tt{0}}$)\-$\;\;${\bf then }{\sc{btm}}\index{btm} \\ 
{\bf else }{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,({\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, {\it{n\/}} $-\;1$)\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\rm{pr-ifn}}\,(\=\+{\rm{g-provesn}}\,(\=\+{\rm{g-inv-coden}}\,(\=\+{\rm{g-deciphern}}\,(\=\+{\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-deciphern}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-inv-coden}, \\ 
{\rm{g-f-notn}}\,({\it{exp\/}})\index{g-f-notn}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\it{n\/}} $-\;1$)\-\index{g-provesn}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\rm{findn}}\,($1\;+$ {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}} $-\;1$)\index{findn})\-\index{pr-ifn})\-\index{pr-ifn}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable pr-ifn; name this event `g1493'.\index{g1493}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  lessp-comp-count3 \\ \index{lessp-comp-count3}
\=\+(\=\+{\rm{thm-count1}}\,({\it{pfn\/}}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1} \\ 
$+$$\;\;\;\;${\rm{thm-count1}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1} \\ 
$+$$\;\;\;\;${\it{extras\/}})\- \\ 
$<$$\;\;\;\;${\rm{comp-count}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{extras\/}})\index{comp-count}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  findn-thm-count-lessp \\ \index{findn-thm-count-lessp}
\=\+(\=\+({\it{pfn\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{pf\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pf\/}})\index{decipher})\index{inv-code}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$\vee$$\;\;\;\;${\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\it{pf\/}})\index{decipher})\index{inv-code}, {\rm{f-not}}\,({\it{exp\/}})\index{f-not}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves})\- \\ 
$\wedge$$\;\;\;\;$({\it{pfn\/}} $\leq$ {\it{pf\/}}) \\ 
$\wedge$$\;\;\;\;$(($1\;+$ ($1\;+$ {\rm{comp-count}}\,({\it{pf\/}}, {\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{extras\/}})\index{comp-count})) $<$ {\it{n\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+($1\;+$ {\rm{thm-count1}}\,(\=\+{\rm{find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\it{exp\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn}, \\ 
{\rm{f-not}}\,({\it{exp\/}})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{thm-count1}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{thm-count1}}\,(\=\+{\rm{find-proofn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\it{exp\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{find-proofn}, \\ 
{\it{exp\/}}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{thm-count1}) \\ 
$+$$\;\;\;\;${\it{extras\/}})\- \\ 
$<$$\;\;\;\;${\rm{findn}}\,(\=\+{\it{pfn\/}}, \\ 
{\rm{gcode}}\,({\it{exp\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{findn})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable g-findn; name this event `g1494'.\index{g1494}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  findn-g-findn \\ \index{findn-g-findn}
\=\+{\rm{g-findn}}\,({\rm{gcode}}\,({\it{pfn\/}})\index{gcode}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{g-findn} \\ 
$=$$\;\;\;\;${\rm{findn}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{findn}\-
\end{tabbing}

\noindent{\sc Event}:   Enable g-findn; name this event `g1495'.\index{g1495}

\addvspace{18pt}

\noindent{\sc Event}:   Disable thm-count; name this event `g1496'.\index{g1496}

\addvspace{18pt}

\noindent{\sc Event}:   Disable thm-count1; name this event `g1497'.\index{g1497}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  thm-count-thm-count1 \\ \index{thm-count-thm-count1}
\=\+{\rm{thm-count}}\,({\it{pfn\/}}, {\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count} \\ 
$=$$\;\;\;\;${\rm{thm-count1}}\,({\it{pfn\/}}, {\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{flg\/}})\index{subst}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{thm-count1}\-
\end{tabbing}

\noindent{\sc Event}:   Enable thm-count; name this event `g1498'.\index{g1498}

\addvspace{18pt}

\noindent{\sc Event}:   Enable thm-count1; name this event `g1499'.\index{g1499}

\addvspace{18pt}

\noindent{\sc Event}:   Disable decipher; name this event `g1500'.\index{g1500}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  num-treep-num-treep \\ \index{num-treep-num-treep}
{\rm{numtreep}}\,({\it{x\/}})\index{numtreep} $=$ {\rm{num-treep}}\,({\it{x\/}})\index{num-treep}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  inv-code-decipher-numcode \\ \index{inv-code-decipher-numcode}
{\rm{gcode}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\rm{numcode}}\,({\rm{gcode}}\,({\it{x\/}})\index{gcode})\index{numcode})\index{decipher})\index{inv-code})\index{gcode} $=$ {\rm{gcode}}\,({\it{x\/}})\index{gcode}
\end{tabbing}

\noindent{\sc Event}:   Enable decipher; name this event `g1501'.\index{g1501}

\addvspace{18pt}

\noindent{\sc Event}:   Disable bool-fix; name this event `g1502'.\index{g1502}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  proves-inv-code-decipher \\ \index{proves-inv-code-decipher}
\=\+{\rm{proves}}\,({\rm{inv-code}}\,({\rm{decipher}}\,({\rm{numcode}}\,({\rm{gcode}}\,({\it{pf\/}})\index{gcode})\index{numcode})\index{decipher})\index{inv-code}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves} \\ 
$=$$\;\;\;\;${\rm{proves}}\,({\it{pf\/}}, {\it{exp\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{proves}\-
\end{tabbing}

\noindent{\sc Event}:   Enable findn; name this event `g1503'.\index{g1503}

\addvspace{18pt}

\noindent{\sc Event}:   Enable bool-fix; name this event `g1504'.\index{g1504}

\addvspace{18pt}
\begin{verbatim};;;;;;;The undecidable sentence - DEFNS and SYMBOLS will have to be instantiated, but
;;;;;;;GIVEN will be left as a parameter so that for any specific GIVEN, we get a specific
;;;;;;;undecidable sentence.  This would show that the addition of the undecidable sentence
;;;;;;;or any finite list of new axioms will still leave the theory incomplete or inconsistent.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{undec-sent} \\  
\=\+{\rm{undec-sent}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{undec-sent} \\ 
$=$$\;\;\;\;${\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{final-count} \\  
\=\+{\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{final-count} \\ 
$=$$\;\;\;\;$($1\;+$ (\=\+($1\;+$ ($1\;+$ {\rm{comp-count}}\,(\=\+{\rm{numcode}}\,({\rm{gcode}}\,({\it{pf\/}})\index{gcode})\index{numcode}, \\ 
{\tt{0}}, \\ 
{\rm{undec-sent}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{undec-sent}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\rm{ncode-cnt}}\,({\rm{gcode}}\,({\rm{a-undec}}\,(\=\+{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{a-undec})\index{gcode})\index{ncode-cnt})\-\index{comp-count})) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{ncode-cnt}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode-cnt}) \\ 
$+$$\;\;\;\;$($1\;+$ {\rm{exp-count}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{exp-count}))\-)\-
\end{tabbing}

\noindent{\sc Event}:   Enable ncode-cnt; name this event `g1505'.\index{g1505}

\addvspace{18pt}

\noindent{\sc Event}:   Enable proves-is-formula-again; name this event `g1506'.\index{g1506}

\addvspace{18pt}

\noindent{\sc Event}:   Enable comp-count; name this event `g1507'.\index{g1507}

\addvspace{18pt}

\noindent{\sc Event}:   Enable numcode; name this event `g1508'.\index{g1508}

\addvspace{18pt}

\noindent{\sc Event}:   Enable exp-count; name this event `g1509'.\index{g1509}

\addvspace{18pt}

\noindent{\sc Event}:   Enable findn-thm-count-lessp; name this event `g1510'.\index{g1510}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  undec-count-lessp \\ \index{undec-count-lessp}
\=\+(\=\+({\it{pf\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{proves}}\,(\=\+{\rm{inv-code}}\,({\rm{decipher}}\,({\it{pf\/}})\index{decipher})\index{inv-code}, \\ 
{\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves} \\ 
$\vee$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{inv-code}}\,({\rm{decipher}}\,({\it{pf\/}})\index{decipher})\index{inv-code}, \\ 
{\rm{f-not}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}})\-\index{proves})\- \\ 
$\wedge$$\;\;\;\;$(\=\+($1\;+$ ($1\;+$ {\rm{comp-count}}\,(\=\+{\it{pf\/}}, \\ 
{\tt{0}}, \\ 
{\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\rm{ncode-cnt}}\,({\rm{gcode}}\,({\it{exp\/}})\index{gcode})\index{ncode-cnt})\-\index{comp-count})) \\ 
$<$$\;\;\;\;${\it{n\/}})\-)\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{undec-count}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}}, {\it{n\/}})\index{undec-count} \\ 
$<$$\;\;\;\;${\rm{findn}}\,(\=\+{\tt{0}}, \\ 
{\rm{gcode}}\,({\rm{subst}}\,({\it{exp\/}}, {\it{var\/}}, {\it{term\/}}, {\tt{0}})\index{subst})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{given\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{defns\/}})\index{gcode}, \\ 
{\rm{gcode}}\,({\it{symbols\/}})\index{gcode}, \\ 
{\it{n\/}})\-\index{findn})\-\-
\end{tabbing}

\noindent{\sc Event}:   Enable undec-count; name this event `g1511'.\index{g1511}

\addvspace{18pt}

\noindent{\sc Event}:   Enable not-isthm-a-proves-a-proof; name this event `g1512'.\index{g1512}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  comp-count-final-count-lessp \\ \index{comp-count-final-count-lessp}
\=\+($1\;+$ ($1\;+$ {\rm{comp-count}}\,(\=\+{\rm{numcode}}\,({\rm{gcode}}\,({\it{pf\/}})\index{gcode})\index{numcode}, \\ 
{\tt{0}}, \\ 
{\rm{subst}}\,(\=\+{\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\tt{0}})\-\index{subst}, \\ 
{\it{given\/}}, \\ 
{\it{defns\/}}, \\ 
{\it{symbols\/}}, \\ 
{\rm{ncode-cnt}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode-cnt})\-\index{comp-count})) \\ 
$<$$\;\;\;\;${\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{final-count}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-ncode-cnt-final-count \\ \index{lessp-ncode-cnt-final-count}
\=\+{\rm{ncode-cnt}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{gcode})\index{ncode-cnt} \\ 
$<$$\;\;\;\;${\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{final-count}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-exp-count-final-count \\ \index{lessp-exp-count-final-count}
\=\+{\rm{exp-count}}\,({\rm{a-undec}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{a-undec})\index{exp-count} \\ 
$<$$\;\;\;\;${\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{final-count}\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  lessp-num-final-count \\ \index{lessp-num-final-count}
{\tt{4}} $<$ {\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\it{defns\/}}, {\it{symbols\/}})\index{final-count}
\end{tabbing}
\begin{verbatim};;;;;;;The final list of symbols, THM-SYMBS.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{thm-symbs} \\  
\=\+{\sc{thm-symbs}}\index{thm-symbs} \\ 
$=$$\;\;\;\;${\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn})\-\-
\end{tabbing}
\begin{verbatim};;;;;;;The final list of function/predicate definitions.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{thm-defns} \\  
\=\+{\sc{thm-defns}}\index{thm-defns} \\ 
$=$$\;\;\;\;${\rm{list}}\,(\=\+{\rm{cons}}\,({\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, {\rm{z-int-defn}}\,({\tt{0}}, {\tt{1}}, {\tt{2}})\index{z-int-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, {\rm{f-equal}}\,({\rm{z-succ}}\,({\tt{0}})\index{z-succ}, {\rm{z-union}}\,({\rm{z-sing}}\,({\tt{0}})\index{z-sing}, {\tt{0}})\index{z-union})\index{f-equal}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, {\rm{z-car-defn}}\,({\tt{0}}, {\rm{z-car}}\,({\tt{0}})\index{z-car}, {\tt{1}}, {\tt{2}})\index{z-car-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, {\rm{z-cdr-defn}}\,({\tt{0}}, {\rm{z-cdr}}\,({\tt{0}})\index{z-cdr}, {\tt{1}}, {\tt{2}})\index{z-cdr-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, {\rm{z-listp-defn}}\,({\tt{0}}, {\rm{z-listp}}\,({\tt{0}})\index{z-listp}, {\tt{1}}, {\tt{2}})\index{z-listp-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, {\rm{z-add1-defn}}\,({\tt{0}}, {\rm{z-add1}}\,({\tt{0}})\index{z-add1}, {\tt{1}}, {\tt{2}})\index{z-add1-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, {\rm{z-numberp-defn}}\,({\tt{0}}, {\rm{z-numberp}}\,({\tt{0}})\index{z-numberp})\index{z-numberp-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, {\rm{z-zerop-defn}}\,({\tt{0}}, {\rm{z-zerop}}\,({\tt{0}})\index{z-zerop})\index{z-zerop-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, {\rm{z-sub1-defn}}\,({\tt{0}}, {\rm{z-sub1}}\,({\tt{0}})\index{z-sub1}, {\tt{3}})\index{z-sub1-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, {\rm{z-equal-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-equal}}\,({\tt{0}}, {\tt{1}})\index{z-equal})\index{z-equal-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, {\rm{z-cons-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-cons}}\,({\tt{0}}, {\tt{1}})\index{z-cons})\index{z-cons-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, {\rm{z-apply-subr1-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr1}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr1})\index{z-apply-subr1-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, {\rm{z-apply-subr2-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr2}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr2})\index{z-apply-subr2-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, {\rm{z-apply-subr-defn}}\,({\tt{0}}, {\tt{1}}, {\rm{z-apply-subr}}\,({\tt{0}}, {\tt{1}})\index{z-apply-subr})\index{z-apply-subr-defn}), \\ 
{\rm{cons}}\,({\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, {\rm{z-subrp-defn}}\,({\tt{0}}, {\rm{z-subrp}}\,({\tt{0}})\index{z-subrp})\index{z-subrp-defn}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable z-subrp-hyps; name this event `g1549'.\index{g1549}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-apply-subr-hyps; name this event `g1550'.\index{g1550}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-apply-subr1-hyps; name this event `g1551'.\index{g1551}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-apply-subr2-hyps; name this event `g1552'.\index{g1552}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-succ-hyps; name this event `g1553'.\index{g1553}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-int-hyps; name this event `g1554'.\index{g1554}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-car-hyps; name this event `g1555'.\index{g1555}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-cdr-hyps; name this event `g1556'.\index{g1556}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-listp-hyps; name this event `g1557'.\index{g1557}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-cons-hyps; name this event `g1558'.\index{g1558}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-add1-hyps; name this event `g1559'.\index{g1559}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-sub1-hyps; name this event `g1560'.\index{g1560}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-equal-hyps; name this event `g1561'.\index{g1561}

\addvspace{18pt}

\noindent{\sc Event}:   Disable z-numberp-hyps; name this event `g1562'.\index{g1562}

\addvspace{18pt}

\noindent{\sc Event}:   Disable rest-of; name this event `g1563'.\index{g1563}

\addvspace{18pt}

\noindent{\sc Event}:   Enable eql-gcode-equal; name this event `g1564'.\index{g1564}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  ev-hyps-z-int \\ \index{ev-hyps-z-int}
{\rm{z-int-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-int-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-hyps-z-succ \\ \index{ev-hyps-z-succ}
{\rm{z-succ-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-succ-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-hyps-z-car \\ \index{ev-hyps-z-car}
{\rm{z-car-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-car-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  ev-hyps-z-cdr \\ \index{ev-hyps-z-cdr}
{\rm{z-cdr-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-cdr-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-listp-ev-hyps \\ \index{z-listp-ev-hyps}
{\rm{z-listp-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-listp-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-cons-ev-hyps \\ \index{z-cons-ev-hyps}
{\rm{z-cons-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-cons-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-numberp-ev-hyps \\ \index{z-numberp-ev-hyps}
{\rm{z-numberp-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-numberp-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-equal-ev-hyps \\ \index{z-equal-ev-hyps}
{\rm{z-equal-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-equal-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-add1-ev-hyps \\ \index{z-add1-ev-hyps}
{\rm{z-add1-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-add1-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-sub1-ev-hyps \\ \index{z-sub1-ev-hyps}
{\rm{z-sub1-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-sub1-hyps}
\end{tabbing}

\noindent{\sc Event}:   Enable subset-cons; name this event `g1565'.\index{g1565}

\addvspace{18pt}

\noindent{\sc Event}:   Enable prime-member; name this event `g1566'.\index{g1566}

\addvspace{18pt}

\noindent{\sc Event}:   Enable member-delete; name this event `g1567'.\index{g1567}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-apply-subr1-ev-hyps \\ \index{z-apply-subr1-ev-hyps}
{\rm{z-apply-subr1-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-apply-subr1-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr2-ev-hyps \\ \index{z-apply-subr2-ev-hyps}
{\rm{z-apply-subr2-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-apply-subr2-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-apply-subr-ev-hyps \\ \index{z-apply-subr-ev-hyps}
{\rm{z-apply-subr-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-apply-subr-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  z-subrp-ev-hyps \\ \index{z-subrp-ev-hyps}
{\rm{z-subrp-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-subrp-hyps}
\end{tabbing}

\noindent{\sc Event}:   Enable z-subrp-hyps; name this event `g1568'.\index{g1568}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-apply-subr-hyps; name this event `g1569'.\index{g1569}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-apply-subr1-hyps; name this event `g1570'.\index{g1570}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-apply-subr2-hyps; name this event `g1571'.\index{g1571}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-succ-hyps; name this event `g1572'.\index{g1572}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-int-hyps; name this event `g1573'.\index{g1573}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-car-hyps; name this event `g1574'.\index{g1574}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-cdr-hyps; name this event `g1575'.\index{g1575}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-listp-hyps; name this event `g1576'.\index{g1576}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-cons-hyps; name this event `g1577'.\index{g1577}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-add1-hyps; name this event `g1578'.\index{g1578}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-sub1-hyps; name this event `g1579'.\index{g1579}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-equal-hyps; name this event `g1580'.\index{g1580}

\addvspace{18pt}

\noindent{\sc Event}:   Enable z-numberp-hyps; name this event `g1581'.\index{g1581}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  subset-thm-symbs \\ \index{subset-thm-symbs}
{\rm{subset}}\,(\=\+{\rm{list}}\,(\=\+{\rm{fn}}\,({\tt{16}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{15}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{14}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{13}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{12}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{0}}, {\tt{0}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{3}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{4}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{5}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{6}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{7}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{8}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{9}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{10}}, {\tt{1}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{11}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{1}}, {\tt{2}})\index{fn}, \\ 
{\rm{fn}}\,({\tt{2}}, {\tt{2}})\index{fn}, \\ 
{\rm{p}}\,({\tt{2}}, {\tt{1}})\index{p}, \\ 
{\rm{p}}\,({\tt{1}}, {\tt{2}})\index{p})\-, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{subset}
\end{tabbing}

\noindent{\sc Event}:   Disable z-zerop-hyps; name this event `g1585'.\index{g1585}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  z-zerop-ev-hyps \\ \index{z-zerop-ev-hyps}
{\rm{z-zerop-hyps}}\,({\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{z-zerop-hyps}
\end{tabbing}

\noindent{\sc Event}:   Enable rest-of; name this event `g1586'.\index{g1586}

\addvspace{18pt}

\noindent{\sc Event}:   Enable thm-defns; name this event `g1587'.\index{g1587}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  all-symbs-thm-symbs \\ \index{all-symbs-thm-symbs}
{\rm{all-symbs}}\,({\sc{thm-symbs}}\index{thm-symbs})\index{all-symbs}
\end{tabbing}

\noindent{\sc Event}:   Enable thm-symbs; name this event `g1589'.\index{g1589}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  ev-hyps-thm-defns-symbols \\ \index{ev-hyps-thm-defns-symbols}
{\rm{ev-hyps}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{ev-hyps}
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  all-symbs-termp-ncode \\ \index{all-symbs-termp-ncode}
{\rm{all-symbs}}\,({\it{symbs\/}})\index{all-symbs} $\rightarrow$ {\rm{termp}}\,({\rm{ncode}}\,({\it{x\/}})\index{ncode}, {\tt{0}}, {\it{symbs\/}})\index{termp}
\end{tabbing}

\noindent{\sc Event}:   Enable all-symbs; name this event `g1591'.\index{g1591}

\addvspace{18pt}
\begin{tabbing}{\sc Theorem}:  formula-undec-sent \\ \index{formula-undec-sent}
{\rm{all-symbs}}\,({\it{symbs\/}})\index{all-symbs} $\rightarrow$ {\rm{formula}}\,({\rm{undec-sent}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbs\/}})\index{undec-sent}, {\it{symbs\/}})\index{formula}
\end{tabbing}
\begin{verbatim};;;;;;;The lemma below is very nearly the final one.  It asserts that if PF is either
;;;;;;;a proof or disproof of the undecidable sentence, then in one of two possible ways 
;;;;;;;(as given by the four lemmas above), the undecidable sentence is both provable and
;;;;;;;disprovable.  The observation is that if Z2 was complete, the undecidable sentence
;;;;;;;would either be a provable or disprovable.  PF below could be instantiated with
;;;;;;;the given proof/disproof, yielding the conclusion of the lemma that Z2 is then
;;;;;;;inconsistent.  We still need to show that the undecidable sentence below is a 
;;;;;;;Z2 formula and that it contains no free variables.  This is done next.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  incompleteness-main-part \\ \index{incompleteness-main-part}
\=\+(\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{undec-sent}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{undec-sent}, {\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{proves} \\ 
$\vee$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-not}}\,({\rm{undec-sent}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{undec-sent})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{proves}}\,(\=\+{\rm{isthm-a-proves-a-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,(\=\+{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,(\=\+{\it{pf\/}}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{final-count})\-\index{isthm-a-proves-a-proof}, \\ 
{\rm{undec-sent}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{undec-sent}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isthm-a-proves-not-a-proof}}\,(\=\+{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,(\=\+{\it{pf\/}}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{final-count})\-\index{isthm-a-proves-not-a-proof}, \\ 
{\rm{f-not}}\,({\rm{undec-sent}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{undec-sent})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves})\- \\ 
$\vee$$\;\;\;\;$(\=\+{\rm{proves}}\,(\=\+{\rm{not-isthm-a-proves-a-proof}}\,(\=\+{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,(\=\+{\it{pf\/}}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{final-count})\-\index{not-isthm-a-proves-a-proof}, \\ 
{\rm{undec-sent}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{undec-sent}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{isthm-a-proves-a-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,(\=\+{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,(\=\+{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,(\=\+{\it{pf\/}}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{final-count})\-\index{isthm-a-proves-a-proof}, \\ 
{\rm{f-not}}\,({\rm{undec-sent}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{undec-sent})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves})\-)\-\-
\end{tabbing}
\begin{tabbing}{\sc Theorem}:  collect-free-undec-sent \\ \index{collect-free-undec-sent}
{\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{undec-sent}}\,({\it{given\/}}, {\it{defns\/}}, {\it{symbs\/}})\index{undec-sent}, {\tt{0}})\index{collect-free}
\end{tabbing}

\noindent{\sc Event}:   Enable undec-sent; name this event `g1593'.\index{g1593}

\addvspace{18pt}
\begin{verbatim};;;;;;;The undecidable sentence: UNDEC-SENT with DEFNS and SYMBOLS instantiated.o
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{undecidable-sentence} \\  
{\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence} $=$ {\rm{undec-sent}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{undec-sent}
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{some-proof1} \\  
\=\+{\rm{some-proof1}}\,({\it{given\/}}, {\it{pf\/}})\index{some-proof1} \\ 
$=$$\;\;\;\;${\rm{isthm-a-proves-a-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{final-count})\-\index{isthm-a-proves-a-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{some-disproof1} \\  
\=\+{\rm{some-disproof1}}\,({\it{given\/}}, {\it{pf\/}})\index{some-disproof1} \\ 
$=$$\;\;\;\;${\rm{isthm-a-proves-not-a-proof}}\,(\=\+{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{final-count})\-\index{isthm-a-proves-not-a-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{some-proof2} \\  
\=\+{\rm{some-proof2}}\,({\it{given\/}}, {\it{pf\/}})\index{some-proof2} \\ 
$=$$\;\;\;\;${\rm{not-isthm-a-proves-a-proof}}\,(\=\+{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{final-count})\-\index{not-isthm-a-proves-a-proof}\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{some-disproof2} \\  
\=\+{\rm{some-disproof2}}\,({\it{given\/}}, {\it{pf\/}})\index{some-disproof2} \\ 
$=$$\;\;\;\;${\rm{isthm-a-proves-a-proof}}\,(\=\+{\tt{0}}, \\ 
{\rm{a-undec}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{a-undec}, \\ 
{\tt{6}}, \\ 
{\rm{ncode}}\,({\rm{gcode}}\,({\rm{a-undec}}\,({\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{a-undec})\index{gcode})\index{ncode}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs}, \\ 
{\rm{final-count}}\,({\it{pf\/}}, {\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{final-count})\-\index{isthm-a-proves-a-proof}\-
\end{tabbing}
\begin{verbatim};;;;;;;The final statement of the incompleteness theorem: [1] The undecidable sentence
;;;;;;;is a Z2 formula in a theory whose symbols are from (THM-SYMBS), 
;;;;;;;[2] the undecidable sentence is a sentence, i.e., contains no variables VAR free,
;;;;;;;[3] if the undecidable sentence is either provable or disprovable, then it is
;;;;;;;both provable and disprovable. 
\end{verbatim}\begin{tabbing}{\sc Theorem}:  incompleteness-theorem \\ \index{incompleteness-theorem}
\=\+{\rm{formula}}\,({\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence}, {\sc{thm-symbs}}\index{thm-symbs})\index{formula} \\ 
$\wedge$$\;\;\;\;$({\it{var\/}} $\not\in$ {\rm{collect-free}}\,({\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence}, {\tt{0}})\index{collect-free}) \\ 
$\wedge$$\;\;\;\;$(\=\+(\=\+{\rm{proves}}\,({\it{pf\/}}, {\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence}, {\it{given\/}}, {\sc{thm-defns}}\index{thm-defns}, {\sc{thm-symbs}}\index{thm-symbs})\index{proves} \\ 
$\vee$$\;\;\;\;${\rm{proves}}\,(\=\+{\it{pf\/}}, \\ 
{\rm{f-not}}\,({\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+(\=\+{\rm{proves}}\,(\=\+{\rm{some-proof1}}\,({\it{given\/}}, {\it{pf\/}})\index{some-proof1}, \\ 
{\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{some-disproof1}}\,({\it{given\/}}, {\it{pf\/}})\index{some-disproof1}, \\ 
{\rm{f-not}}\,({\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves})\- \\ 
$\vee$$\;\;\;\;$(\=\+{\rm{proves}}\,(\=\+{\rm{some-proof2}}\,({\it{given\/}}, {\it{pf\/}})\index{some-proof2}, \\ 
{\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves} \\ 
$\wedge$$\;\;\;\;${\rm{proves}}\,(\=\+{\rm{some-disproof2}}\,({\it{given\/}}, {\it{pf\/}})\index{some-disproof2}, \\ 
{\rm{f-not}}\,({\rm{undecidable-sentence}}\,({\it{given\/}})\index{undecidable-sentence})\index{f-not}, \\ 
{\it{given\/}}, \\ 
{\sc{thm-defns}}\index{thm-defns}, \\ 
{\sc{thm-symbs}}\index{thm-symbs})\-\index{proves})\-)\-)\-\-
\end{tabbing}





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\end{document}
