doc-src/TutorialI/Datatype/document/Nested.tex

 \noindent
 Note that we need to quote \isa{term} on the left to avoid confusion with
 the command \isacommand{term}.
 Parameter \isa{{\isacharprime}a} is the type of variables and \isa{{\isacharprime}b} the type of
 function symbols.
-A mathematical term like $f(x,g(y))$ becomes \isa{App\ f\ {\isacharbrackleft}term{\isachardot}Var\ x{\isacharcomma}\ App\ g\ {\isacharbrackleft}term{\isachardot}Var\ y{\isacharbrackright}{\isacharbrackright}}, where \isa{f}, \isa{g}, \isa{x}, \isa{y} are
+A mathematical term like $f(x,g(y))$ becomes \isa{App\ f\ {\isacharbrackleft}Var\ x{\isacharcomma}\ App\ g\ {\isacharbrackleft}Var\ y{\isacharbrackright}{\isacharbrackright}}, where \isa{f}, \isa{g}, \isa{x}, \isa{y} are
 suitable values, e.g.\ numbers or strings.
 
 What complicates the definition of \isa{term} is the nested occurrence of
 \isa{term} inside \isa{list} on the right-hand side. In principle,
 nested recursion can be eliminated in favour of mutual recursion by unfolding

 \isanewline
 \ \ {\isachardoublequote}substs\ s\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}substs\ s\ {\isacharparenleft}t\ {\isacharhash}\ ts{\isacharparenright}\ {\isacharequal}\ subst\ s\ t\ {\isacharhash}\ substs\ s\ ts{\isachardoublequote}%
 \begin{isamarkuptext}%
 \noindent
-You should ignore the label \isa{subst{\isacharunderscore}App} for the moment.
+(Please ignore the label \isa{subst{\isacharunderscore}App} for the moment.)
 
 Similarly, when proving a statement about terms inductively, we need
 to prove a related statement about term lists simultaneously. For example,
 the fact that the identity substitution does not change a term needs to be
 strengthened and proved as follows:%

 \isacommand{lemma}\ {\isachardoublequote}subst\ \ Var\ t\ \ {\isacharequal}\ {\isacharparenleft}t\ {\isacharcolon}{\isacharcolon}{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}b{\isacharparenright}term{\isacharparenright}\ \ {\isasymand}\isanewline
 \ \ \ \ \ \ \ \ substs\ Var\ ts\ {\isacharequal}\ {\isacharparenleft}ts{\isacharcolon}{\isacharcolon}{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}b{\isacharparenright}term\ list{\isacharparenright}{\isachardoublequote}\isanewline
 \isacommand{by}{\isacharparenleft}induct{\isacharunderscore}tac\ t\ \isakeyword{and}\ ts{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}%
 \begin{isamarkuptext}%
 \noindent
-Note that \isa{term{\isachardot}Var} is the identity substitution because by definition it
-leaves variables unchanged: \isa{subst\ term{\isachardot}Var\ {\isacharparenleft}term{\isachardot}Var\ x{\isacharparenright}\ {\isacharequal}\ term{\isachardot}Var\ x}. Note also
+Note that \isa{Var} is the identity substitution because by definition it
+leaves variables unchanged: \isa{subst\ Var\ {\isacharparenleft}Var\ x{\isacharparenright}\ {\isacharequal}\ Var\ x}. Note also
 that the type annotations are necessary because otherwise there is nothing in
 the goal to enforce that both halves of the goal talk about the same type
 parameters \isa{{\isacharparenleft}{\isacharprime}a{\isacharcomma}{\isacharprime}b{\isacharparenright}}. As a result, induction would fail
 because the two halves of the goal would be unrelated.
 

 \begin{isamarkuptext}%
 \noindent
 What is more, we can now disable the old defining equation as a
 simplification rule:%
 \end{isamarkuptext}%
-\isacommand{lemmas}\ {\isacharbrackleft}simp\ del{\isacharbrackright}\ {\isacharequal}\ subst{\isacharunderscore}App%
+\isacommand{declare}\ subst{\isacharunderscore}App\ {\isacharbrackleft}simp\ del{\isacharbrackright}%
 \begin{isamarkuptext}%
 \noindent
 The advantage is that now we have replaced \isa{substs} by
 \isa{map}, we can profit from the large number of pre-proved lemmas
 about \isa{map}.  Unfortunately inductive proofs about type