3 \def\isabellecontext{Advanced}%
5 \isacommand{theory}\ Advanced\ {\isacharequal}\ Even{\isacharcolon}\isanewline
9 \isacommand{datatype}\ {\isacharprime}f\ gterm\ {\isacharequal}\ Apply\ {\isacharprime}f\ {\isachardoublequote}{\isacharprime}f\ gterm\ list{\isachardoublequote}\isanewline
12 \isacommand{datatype}\ integer{\isacharunderscore}op\ {\isacharequal}\ Number\ int\ {\isacharbar}\ UnaryMinus\ {\isacharbar}\ Plus\isanewline
15 \isacommand{consts}\ gterms\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}f\ set\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequote}\isanewline
17 \isacommand{inductive}\ {\isachardoublequote}gterms\ F{\isachardoublequote}\isanewline
18 \isakeyword{intros}\isanewline
19 step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}{\isasymforall}t\ {\isasymin}\ set\ args{\isachardot}\ t\ {\isasymin}\ gterms\ F{\isacharsemicolon}\ \ f\ {\isasymin}\ F{\isasymrbrakk}\isanewline
20 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ gterms\ F{\isachardoublequote}\isanewline
23 \isacommand{lemma}\ gterms{\isacharunderscore}mono{\isacharcolon}\ {\isachardoublequote}F{\isasymsubseteq}G\ {\isasymLongrightarrow}\ gterms\ F\ {\isasymsubseteq}\ gterms\ G{\isachardoublequote}\isanewline
25 \isacommand{apply}\ clarify\isanewline
27 \isacommand{apply}\ {\isacharparenleft}erule\ gterms{\isachardot}induct{\isacharparenright}\isamarkupfalse%
31 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline
32 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\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
33 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ gterms\ G%
37 \isacommand{apply}\ blast\isanewline
39 \isacommand{done}\isamarkupfalse%
41 \begin{isamarkuptext}%
43 \ \ \ \ \ {\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%
47 Just as a demo I include
48 the two forms that Markus has made available. First the one for binding the
52 \isacommand{inductive{\isacharunderscore}cases}\ Suc{\isacharunderscore}Suc{\isacharunderscore}cases\ {\isacharbrackleft}elim{\isacharbang}{\isacharbrackright}{\isacharcolon}\isanewline
53 \ \ {\isachardoublequote}Suc{\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even{\isachardoublequote}\isanewline
56 \isacommand{thm}\ Suc{\isacharunderscore}Suc{\isacharunderscore}cases\isamarkupfalse%
58 \begin{isamarkuptext}%
60 \ \ \ \ \ {\isasymlbrakk}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even{\isacharsemicolon}\ n\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P%
62 \rulename{Suc_Suc_cases}
64 and now the one for local usage:%
67 \isacommand{lemma}\ {\isachardoublequote}Suc{\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even\ {\isasymLongrightarrow}\ P\ n{\isachardoublequote}\isanewline
69 \isacommand{apply}\ {\isacharparenleft}ind{\isacharunderscore}cases\ {\isachardoublequote}Suc{\isacharparenleft}Suc\ n{\isacharparenright}\ {\isasymin}\ even{\isachardoublequote}{\isacharparenright}\isanewline
71 \isacommand{oops}\isanewline
74 \isacommand{inductive{\isacharunderscore}cases}\ gterm{\isacharunderscore}Apply{\isacharunderscore}elim\ {\isacharbrackleft}elim{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}Apply\ f\ args\ {\isasymin}\ gterms\ F{\isachardoublequote}\isamarkupfalse%
76 \begin{isamarkuptext}%
80 \ \ \ \ \ {\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%
82 \rulename{gterm_Apply_elim}%
85 \isacommand{lemma}\ gterms{\isacharunderscore}IntI\ {\isacharbrackleft}rule{\isacharunderscore}format{\isacharcomma}\ intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\isanewline
86 \ \ \ \ \ {\isachardoublequote}t\ {\isasymin}\ gterms\ F\ {\isasymLongrightarrow}\ t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F{\isasyminter}G{\isacharparenright}{\isachardoublequote}\isanewline
88 \isacommand{apply}\ {\isacharparenleft}erule\ gterms{\isachardot}induct{\isacharparenright}\isamarkupfalse%
92 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}args\ f{\isachardot}\isanewline
93 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\isanewline
94 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}\ \ \ }t\ {\isasymin}\ gterms\ F\ {\isasymand}\ {\isacharparenleft}t\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\ t\ {\isasymin}\ gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}{\isacharparenright}{\isacharsemicolon}\isanewline
95 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }f\ {\isasymin}\ F{\isasymrbrakk}\isanewline
96 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ gterms\ G\ {\isasymlongrightarrow}\isanewline
97 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymLongrightarrow}\ }Apply\ f\ args\ {\isasymin}\ gterms\ {\isacharparenleft}F\ {\isasyminter}\ G{\isacharparenright}%
101 \isacommand{apply}\ blast\isanewline
103 \isacommand{done}\isamarkupfalse%
105 \begin{isamarkuptext}%
107 \ \ \ \ \ mono\ f\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}A\ {\isasyminter}\ B{\isacharparenright}\ {\isasymsubseteq}\ f\ A\ {\isasyminter}\ f\ B%
112 \isacommand{lemma}\ gterms{\isacharunderscore}Int{\isacharunderscore}eq\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\isanewline
113 \ \ \ \ \ {\isachardoublequote}gterms\ {\isacharparenleft}F{\isasyminter}G{\isacharparenright}\ {\isacharequal}\ gterms\ F\ {\isasyminter}\ gterms\ G{\isachardoublequote}\isanewline
115 \isacommand{by}\ {\isacharparenleft}blast\ intro{\isacharbang}{\isacharcolon}\ mono{\isacharunderscore}Int\ monoI\ gterms{\isacharunderscore}mono{\isacharparenright}\isamarkupfalse%
117 \begin{isamarkuptext}%
118 the following declaration isn't actually used%
121 \isacommand{consts}\ integer{\isacharunderscore}arity\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}integer{\isacharunderscore}op\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
123 \isacommand{primrec}\isanewline
124 {\isachardoublequote}integer{\isacharunderscore}arity\ {\isacharparenleft}Number\ n{\isacharparenright}\ \ \ \ \ \ \ \ {\isacharequal}\ {\isadigit{0}}{\isachardoublequote}\isanewline
125 {\isachardoublequote}integer{\isacharunderscore}arity\ UnaryMinus\ \ \ \ \ \ \ \ {\isacharequal}\ {\isadigit{1}}{\isachardoublequote}\isanewline
126 {\isachardoublequote}integer{\isacharunderscore}arity\ Plus\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharequal}\ {\isadigit{2}}{\isachardoublequote}\isanewline
129 \isacommand{consts}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequote}\isanewline
131 \isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequote}\isanewline
132 \isakeyword{intros}\isanewline
133 step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}{\isasymforall}t\ {\isasymin}\ set\ args{\isachardot}\ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isacharsemicolon}\ \ \isanewline
134 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
135 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequote}\isanewline
139 \isacommand{consts}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}f\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}f\ gterm\ set{\isachardoublequote}\isanewline
141 \isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequote}\isanewline
142 \isakeyword{intros}\isanewline
143 step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ {\isachardoublequote}{\isasymlbrakk}args\ {\isasymin}\ lists\ {\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharparenright}{\isacharsemicolon}\ \ \isanewline
144 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
145 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ args{\isacharparenright}\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequote}\isanewline
146 \isakeyword{monos}\ lists{\isacharunderscore}mono\isanewline
149 \isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isachardoublequote}\isanewline
151 \isacommand{apply}\ clarify\isamarkupfalse%
153 \begin{isamarkuptxt}%
154 The situation after clarify
156 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymLongrightarrow}\isanewline
157 \isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ }x\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity%
161 \isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isachardot}induct{\isacharparenright}\isamarkupfalse%
163 \begin{isamarkuptxt}%
164 note the induction hypothesis!
166 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline
167 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\isanewline
168 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}\ \ \ }t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymand}\isanewline
169 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}\ \ \ }t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharsemicolon}\isanewline
170 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
171 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity%
175 \isacommand{apply}\ auto\isanewline
177 \isacommand{done}\isanewline
182 \isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasymsubseteq}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isachardoublequote}\isanewline
184 \isacommand{apply}\ clarify\isamarkupfalse%
186 \begin{isamarkuptxt}%
187 The situation after clarify
189 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasymLongrightarrow}\isanewline
190 \isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ }x\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity%
194 \isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}{\isachardot}induct{\isacharparenright}\isamarkupfalse%
196 \begin{isamarkuptxt}%
197 note the induction hypothesis!
199 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline
200 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymlbrakk}args\isanewline
201 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}}{\isasymin}\ lists\isanewline
202 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}{\isasymin}\ \ }{\isacharparenleft}well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity\ {\isasyminter}\isanewline
203 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ {\isasymlbrakk}{\isasymin}\ \ {\isacharparenleft}}{\isacharbraceleft}x{\isachardot}\ x\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity{\isacharbraceright}{\isacharparenright}{\isacharsemicolon}\isanewline
204 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ \ }length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
205 \isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }{\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity%
209 \isacommand{apply}\ auto\isanewline
211 \isacommand{done}\isamarkupfalse%
213 \begin{isamarkuptext}%
215 \ \ \ \ \ lists\ {\isacharparenleft}A\ {\isasyminter}\ B{\isacharparenright}\ {\isacharequal}\ lists\ A\ {\isasyminter}\ lists\ B%
220 \begin{isamarkuptext}%
221 the rest isn't used: too complicated. OK for an exercise though.%
224 \isacommand{consts}\ integer{\isacharunderscore}signature\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}integer{\isacharunderscore}op\ {\isacharasterisk}\ {\isacharparenleft}unit\ list\ {\isacharasterisk}\ unit{\isacharparenright}{\isacharparenright}\ set{\isachardoublequote}\isanewline
226 \isacommand{inductive}\ {\isachardoublequote}integer{\isacharunderscore}signature{\isachardoublequote}\isanewline
227 \isakeyword{intros}\isanewline
228 Number{\isacharcolon}\ \ \ \ \ {\isachardoublequote}{\isacharparenleft}Number\ n{\isacharcomma}\ \ \ {\isacharparenleft}{\isacharbrackleft}{\isacharbrackright}{\isacharcomma}\ {\isacharparenleft}{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymin}\ integer{\isacharunderscore}signature{\isachardoublequote}\isanewline
229 UnaryMinus{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}UnaryMinus{\isacharcomma}\ {\isacharparenleft}{\isacharbrackleft}{\isacharparenleft}{\isacharparenright}{\isacharbrackright}{\isacharcomma}\ {\isacharparenleft}{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isasymin}\ integer{\isacharunderscore}signature{\isachardoublequote}\isanewline
230 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
234 \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
236 \isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isachardoublequote}\isanewline
237 \isakeyword{intros}\isanewline
238 step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ \isanewline
239 \ \ \ \ {\isachardoublequote}{\isasymlbrakk}{\isasymforall}pair\ {\isasymin}\ set\ args{\isachardot}\ pair\ {\isasymin}\ well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isacharsemicolon}\ \isanewline
240 \ \ \ \ \ \ sig\ f\ {\isacharequal}\ {\isacharparenleft}map\ snd\ args{\isacharcomma}\ rtype{\isacharparenright}{\isasymrbrakk}\isanewline
241 \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ {\isacharparenleft}map\ fst\ args{\isacharparenright}{\isacharcomma}\ rtype{\isacharparenright}\ \isanewline
242 \ \ \ \ \ \ \ \ \ {\isasymin}\ well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isachardoublequote}\isanewline
245 \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
247 \isacommand{inductive}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isachardoublequote}\isanewline
248 \isakeyword{intros}\isanewline
249 step{\isacharbrackleft}intro{\isacharbang}{\isacharbrackright}{\isacharcolon}\ \isanewline
250 \ \ \ \ {\isachardoublequote}{\isasymlbrakk}args\ {\isasymin}\ lists{\isacharparenleft}well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isacharparenright}{\isacharsemicolon}\ \isanewline
251 \ \ \ \ \ \ sig\ f\ {\isacharequal}\ {\isacharparenleft}map\ snd\ args{\isacharcomma}\ rtype{\isacharparenright}{\isasymrbrakk}\isanewline
252 \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}Apply\ f\ {\isacharparenleft}map\ fst\ args{\isacharparenright}{\isacharcomma}\ rtype{\isacharparenright}\ \isanewline
253 \ \ \ \ \ \ \ \ \ {\isasymin}\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isachardoublequote}\isanewline
254 \isakeyword{monos}\ lists{\isacharunderscore}mono\isanewline
258 \isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig\ {\isasymsubseteq}\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig{\isachardoublequote}\isanewline
260 \isacommand{apply}\ clarify\isanewline
262 \isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isachardot}induct{\isacharparenright}\isanewline
264 \isacommand{apply}\ auto\isanewline
266 \isacommand{done}\isanewline
269 \isacommand{lemma}\ {\isachardoublequote}well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}\ sig\ {\isasymsubseteq}\ well{\isacharunderscore}typed{\isacharunderscore}gterm\ sig{\isachardoublequote}\isanewline
271 \isacommand{apply}\ clarify\isanewline
273 \isacommand{apply}\ {\isacharparenleft}erule\ well{\isacharunderscore}typed{\isacharunderscore}gterm{\isacharprime}{\isachardot}induct{\isacharparenright}\isanewline
275 \isacommand{apply}\ auto\isanewline
277 \isacommand{done}\isanewline
281 \isacommand{end}\isanewline
287 %%% TeX-master: "root"