doc-src/TutorialI/Inductive/document/Advanced.tex

auto-updated

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\begin{isabellebody}%
\def\isabellecontext{Advanced}%
\isanewline
\isacommand{theory}\ Advanced\ {\isacharequal}\ Even{\isacharcolon}\isanewline
\isanewline
\isanewline
\isacommand{datatype}\ {\isacharprime}f\ gterm\ {\isacharequal}\ Apply\ {\isacharprime}f\ {\isachardoublequote}{\isacharprime}f\ gterm\ list{\isachardoublequote}\isanewline
\isanewline
\isacommand{datatype}\ integer{\isacharunderscore}op\ {\isacharequal}\ Number\ int\ {\isacharbar}\ UnaryMinus\ {\isacharbar}\ Plus\isanewline
\isanewline
\isacommand{consts}\ gterms\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}f\ set\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequote}\isanewline
\isacommand{inductive}\ {\isachardoublequote}gterms\ F{\isachardoublequote}\isanewline
\isakeyword{intros}\isanewline
step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}{\isasymforall}t\ {\isasymin}\ set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ \ f\ {\isasymin}\ F{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ gterms\ F{\isachardoublequote}\isanewline
\isanewline
\isacommand{lemma}\ gterms{\isacharunderscore}mono{\isacharcolon}\ {\isachardoublequote}F{\isasymsubseteq}G\ {\isasymLongrightarrow}\ gterms\ F\ {\isasymsubseteq}\ gterms\ G{\isachardoublequote}\isanewline
\isacommand{apply}\ clarify\isanewline
\isacommand{apply}\ {\isacharparenleft}erule\ gterms{\isachardot}induct{\isacharparenright}\isanewline
\isacommand{apply}\ blast\isanewline
\isacommand{done}%
\begin{isamarkuptext}%
The situation after induction

proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{2}}\isanewline
\isanewline
goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
F\ {\isasymsubseteq}\ G\ {\isasymLongrightarrow}\ gterms\ F\ {\isasymsubseteq}\ gterms\ G\isanewline
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline
\ \ \ \ \ \ \ {\isasymlbrakk}F\ {\isasymsubseteq}\ G{\isacharsemicolon}\ {\isasymforall}t{\isasymin}set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F\ {\isasymand}\ t\ {\isasymin}\ gterms\ G{\isacharsemicolon}\ f\ {\isasymin}\ F{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ {\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ gterms\ G%
\end{isamarkuptext}%
%
\begin{isamarkuptext}%
We completely forgot about "rule inversion". 

\begin{isabelle}%
\ \ \ \ \ {\isasymlbrakk}a\ {\isasymin}\ even{\isacharsemicolon}\ a\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}n{\isachardot}\ {\isasymlbrakk}a\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharsemicolon}\ n\ {\isasymin}\ even{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
\end{isabelle}
\rulename{even.cases}

Just as a demo I include
the two forms that Markus has made available. First the one for binding the
result to a name%
\end{isamarkuptext}%
\isacommand{inductive{\isacharunderscore}cases}\ Suc{\isacharunderscore}Suc{\isacharunderscore}cases\ {\isacharbrackleft}elim{\isacharbang}{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequote}Suc{\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even{\isachardoublequote}\isanewline
\isanewline
\isacommand{thm}\ Suc{\isacharunderscore}Suc{\isacharunderscore}cases%
\begin{isamarkuptext}%
\begin{isabelle}%
\ \ \ \ \ {\isasymlbrakk}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even{\isacharsemicolon}\ n\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
\end{isabelle}
\rulename{Suc_Suc_cases}

and now the one for local usage:%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}Suc{\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P\ n{\isachardoublequote}\isanewline
\isacommand{apply}\ {\isacharparenleft}ind{\isacharunderscore}cases\ {\isachardoublequote}Suc{\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even{\isachardoublequote}{\isacharparenright}\isanewline
\isacommand{oops}\isanewline
\isanewline
\isacommand{inductive{\isacharunderscore}cases}\ gterm{\isacharunderscore}Apply{\isacharunderscore}elim\ {\isacharbrackleft}elim{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}Apply\ f\ args\ {\isasymin}\ gterms\ F{\isachardoublequote}%
\begin{isamarkuptext}%
this is what we get:

\begin{isabelle}%
\ \ \ \ \ {\isasymlbrakk}Apply\ f\ args\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ {\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ f\ {\isasymin}\ F{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
\end{isabelle}
\rulename{gterm_Apply_elim}%
\end{isamarkuptext}%
\isacommand{lemma}\ gterms{\isacharunderscore}IntI\ {\isacharbrackleft}rule{\isacharunderscore}format{\isacharbrackright}{\isacharcolon}\isanewline
\ \ \ \ \ {\isachardoublequote}t\ {\isasymin}\ gterms\ F\ {\isasymLongrightarrow}\ t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F{\isasyminter}G{\isacharparenright}{\isachardoublequote}\isanewline
\isacommand{apply}\ {\isacharparenleft}erule\ gterms{\isachardot}induct{\isacharparenright}\isanewline
\isacommand{apply}\ blast\isanewline
\isacommand{done}%
\begin{isamarkuptext}%
Subgoal after induction.  How would we cope without rule inversion?

pr(latex xsymbols symbols)

proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{1}}\isanewline
\isanewline
goal\ {\isacharparenleft}lemma\ gterms{\isacharunderscore}IntI{\isacharparenright}{\isacharcolon}\isanewline
t\ {\isasymin}\ gterms\ F\ {\isasymLongrightarrow}\ t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}\isanewline
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}args\ f{\isachardot}\isanewline
\ \ \ \ \ \ \ {\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F\ {\isasymand}\ {\isacharparenleft}t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}{\isacharparenright}{\isacharsemicolon}\isanewline
\ \ \ \ \ \ \ \ \ \ f\ {\isasymin}\ F{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ {\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ Apply\ f\ args\ {\isasymin}\ gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}%
\end{isamarkuptext}%
%
\begin{isamarkuptext}%
\begin{isabelle}%
\ \ \ \ \ mono\ f\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}A\ {\isasyminter}\ B{\isacharparenright}\ {\isasymsubseteq}\ f\ A\ {\isasyminter}\ f\ B%
\end{isabelle}
\rulename{mono_Int}%
\end{isamarkuptext}%
\isacommand{lemma}\ gterms{\isacharunderscore}Int{\isacharunderscore}eq\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\isanewline
\ \ \ \ \ {\isachardoublequote}gterms\ {\isacharparenleft}F{\isasyminter}G{\isacharparenright}\ {\isacharequal}\ gterms\ F\ {\isasyminter}\ gterms\ G{\isachardoublequote}\isanewline
\isacommand{apply}\ {\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
\isacommand{apply}\ {\isacharparenleft}blast\ intro{\isacharbang}{\isacharcolon}\ mono{\isacharunderscore}Int\ monoI\ gterms{\isacharunderscore}mono{\isacharparenright}\isanewline
\isacommand{apply}\ {\isacharparenleft}blast\ intro{\isacharbang}{\isacharcolon}\ gterms{\isacharunderscore}IntI{\isacharparenright}\isanewline
\isacommand{done}\isanewline
\isanewline
\isanewline
\isacommand{consts}\ integer{\isacharunderscore}arity\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}integer{\isacharunderscore}op\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
\isacommand{primrec}\isanewline
{\isachardoublequote}integer{\isacharunderscore}arity\ {\isacharparenleft}Number\ n{\isacharparenright}\ \ \ \ \ \ \ \ {\isacharequal}\ {\isacharhash}{\isadigit{0}}{\isachardoublequote}\isanewline
{\isachardoublequote}integer{\isacharunderscore}arity\ UnaryMinus\ \ \ \ \ \ \ \ {\isacharequal}\ {\isacharhash}{\isadigit{1}}{\isachardoublequote}\isanewline
{\isachardoublequote}integer{\isacharunderscore}arity\ Plus\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharequal}\ {\isacharhash}{\isadigit{2}}{\isachardoublequote}\isanewline
\isanewline
\isacommand{consts}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequote}\isanewline
\isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequote}\isanewline
\isakeyword{intros}\isanewline
step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}{\isasymforall}t\ {\isasymin}\ set\ args{\isachardot}\ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isacharsemicolon}\ \ \isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequote}\isanewline
\isanewline
\isanewline
\isacommand{consts}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequote}\isanewline
\isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequote}\isanewline
\isakeyword{intros}\isanewline
step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}args\ {\isasymin}\ lists\ {\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharparenright}{\isacharsemicolon}\ \ \isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequote}\isanewline
\isakeyword{monos}\ lists{\isacharunderscore}mono\isanewline
\isanewline
\isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequote}\isanewline
\isacommand{apply}\ clarify\isanewline
\isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isachardot}induct{\isacharparenright}\isanewline
\isacommand{apply}\ auto\isanewline
\isacommand{done}%
\begin{isamarkuptext}%
The situation after clarify (note the induction hypothesis!)

pr(latex xsymbols symbols)

proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{2}}\isanewline
\isanewline
goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\isanewline
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline
\ \ \ \ \ \ \ {\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\isanewline
\ \ \ \ \ \ \ \ \ \ \ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymand}\ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharsemicolon}\isanewline
\ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ {\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequote}\isanewline
\isacommand{apply}\ clarify\isanewline
\isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}{\isachardot}induct{\isacharparenright}\isanewline
\isacommand{apply}\ auto\isanewline
\isacommand{done}%
\begin{isamarkuptext}%
\begin{isabelle}%
\ \ \ \ \ lists\ {\isacharparenleft}A\ {\isasyminter}\ B{\isacharparenright}\ {\isacharequal}\ lists\ A\ {\isasyminter}\ lists\ B%
\end{isabelle}
\rulename{lists_Int_eq}

The situation after clarify (note the strange induction hypothesis!)

pr(latex xsymbols symbols)

proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{2}}\isanewline
\isanewline
goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\isanewline
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline
\ \ \ \ \ \ \ {\isasymlbrakk}args\isanewline
\ \ \ \ \ \ \ \ {\isasymin}\ lists\isanewline
\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasyminter}\ {\isacharbraceleft}u{\isachardot}\ u\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isacharbraceright}{\isacharparenright}{\isacharsemicolon}\isanewline
\ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
\ \ \ \ \ \ \ {\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity%
\end{isamarkuptext}%
%
\begin{isamarkuptext}%
the rest isn't used: too complicated.  OK for an exercise though.%
\end{isamarkuptext}%
\isacommand{consts}\ integer{\isacharunderscore}signature\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}integer{\isacharunderscore}op\ {\isacharasterisk}\ {\isacharparenleft}unit\ list\ {\isacharasterisk}\ unit{\isacharparenright}{\isacharparenright}\ set{\isachardoublequote}\isanewline
\isacommand{inductive}\ {\isachardoublequote}integer{\isacharunderscore}signature{\isachardoublequote}\isanewline
\isakeyword{intros}\isanewline
Number{\isacharcolon}\ \ \ \ \ {\isachardoublequote}{\isacharparenleft}Number\ n{\isacharcomma}\ \ \ {\isacharparenleft}{\isacharbrackleft}{\isacharbrackright}{\isacharcomma}\ {\isacharparenleft}{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymin}\ integer{\isacharunderscore}signature{\isachardoublequote}\isanewline
UnaryMinus{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}UnaryMinus{\isacharcomma}\ {\isacharparenleft}{\isacharbrackleft}{\isacharparenleft}{\isacharparenright}{\isacharbrackright}{\isacharcomma}\ {\isacharparenleft}{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymin}\ integer{\isacharunderscore}signature{\isachardoublequote}\isanewline
Plus{\isacharcolon}\ \ \ \ \ \ \ {\isachardoublequote}{\isacharparenleft}Plus{\isacharcomma}\ \ \ \ \ \ \ {\isacharparenleft}{\isacharbrackleft}{\isacharparenleft}{\isacharparenright}{\isacharcomma}{\isacharparenleft}{\isacharparenright}{\isacharbrackright}{\isacharcomma}\ {\isacharparenleft}{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymin}\ integer{\isacharunderscore}signature{\isachardoublequote}\isanewline
\isanewline
\isanewline
\isacommand{consts}\ well{\isacharunderscore}typed{\isacharunderscore}gterm\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ {\isacharprime}t\ list\ {\isacharasterisk}\ {\isacharprime}t{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}f\ gterm\ {\isacharasterisk}\ {\isacharprime}t{\isacharparenright}set{\isachardoublequote}\isanewline
\isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isachardoublequote}\isanewline
\isakeyword{intros}\isanewline
step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ \isanewline
\ \ \ \ {\isachardoublequote}{\isasymlbrakk}{\isasymforall}pair\ {\isasymin}\ set\ args{\isachardot}\ pair\ {\isasymin}\ well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isacharsemicolon}\ \isanewline
\ \ \ \ \ \ sig\ f\ {\isacharequal}\ {\isacharparenleft}map\ snd\ args{\isacharcomma}\ rtype{\isacharparenright}{\isasymrbrakk}\isanewline
\ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ {\isacharparenleft}map\ fst\ args{\isacharparenright}{\isacharcomma}\ rtype{\isacharparenright}\ \isanewline
\ \ \ \ \ \ \ \ \ {\isasymin}\ well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isachardoublequote}\isanewline
\isanewline
\isacommand{consts}\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ {\isacharprime}t\ list\ {\isacharasterisk}\ {\isacharprime}t{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}f\ gterm\ {\isacharasterisk}\ {\isacharprime}t{\isacharparenright}set{\isachardoublequote}\isanewline
\isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isachardoublequote}\isanewline
\isakeyword{intros}\isanewline
step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ \isanewline
\ \ \ \ {\isachardoublequote}{\isasymlbrakk}args\ {\isasymin}\ lists{\isacharparenleft}well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isacharparenright}{\isacharsemicolon}\ \isanewline
\ \ \ \ \ \ sig\ f\ {\isacharequal}\ {\isacharparenleft}map\ snd\ args{\isacharcomma}\ rtype{\isacharparenright}{\isasymrbrakk}\isanewline
\ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ {\isacharparenleft}map\ fst\ args{\isacharparenright}{\isacharcomma}\ rtype{\isacharparenright}\ \isanewline
\ \ \ \ \ \ \ \ \ {\isasymin}\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isachardoublequote}\isanewline
\isakeyword{monos}\ lists{\isacharunderscore}mono\isanewline
\isanewline
\isanewline
\isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig\ {\isasymsubseteq}\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isachardoublequote}\isanewline
\isacommand{apply}\ clarify\isanewline
\isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isachardot}induct{\isacharparenright}\isanewline
\isacommand{apply}\ auto\isanewline
\isacommand{done}\isanewline
\isanewline
\isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig\ {\isasymsubseteq}\ well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isachardoublequote}\isanewline
\isacommand{apply}\ clarify\isanewline
\isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}{\isachardot}induct{\isacharparenright}\isanewline
\isacommand{apply}\ auto\isanewline
\isacommand{done}\isanewline
\isanewline
\isanewline
\isacommand{end}\isanewline
\isanewline
\end{isabellebody}%
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