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Initialisations
\xtc{
}{
\spadcommand{x:Symbol :='x \bound{x}}
}
\xtc{
}{
\spadcommand{y:Symbol :='y \bound{y}}
}
\xtc{
}{
\spadcommand{z:Symbol :='z \bound{z}}
}
\xtc{
}{
\spadcommand{word := FreeMonoid(Symbol) \bound{word}}
}
\xtc{
}{
\spadcommand{tree := FreeMagma(Symbol) \bound{tree}}
}

Let's make some trees
\xtc{
}{
\spadcommand{a:tree := x*x \free{x tree} \bound{a}}
}
\xtc{
}{
\spadcommand{b:tree := y*y \free{y tree} \bound{b}}
}
\xtc{
}{
\spadcommand{c:tree := a*b \free{a b tree} \bound{c}}
}

Query the trees
\xtc{
}{
\spadcommand{left c \free{c}}
}
\xtc{
}{
\spadcommand{right c \free{c}}
}
\xtc{
}{
\spadcommand{length c \free{c}}
}
\xtc{
Coerce to the monoid
}{
\spadcommand{c::word \free{c word}}
}

Check ordering
\xtc{
}{
\spadcommand{a < b \free{a b}}
}
\xtc{
}{
\spadcommand{a < c \free{a c}}
}
\xtc{
}{
\spadcommand{b < c \free{b c}}
}

Navigate the tree
\xtc{
}{
\spadcommand{first c \free{c}}
}
\xtc{
}{
\spadcommand{rest c \free{c}}
}
\xtc{
}{
\spadcommand{rest rest c  \free{c}}
}

Check ordering
\xtc{
}{
\spadcommand{ax:tree := a*x \free{a x tree} \bound{ax}}
}
\xtc{
}{
\spadcommand{xa:tree := x*a \free{a x tree} \bound{xa}}
}
\xtc{
}{
\spadcommand{xa < ax \free{ax xa}}
}
\xtc{
}{
\spadcommand{lexico(xa,ax) \free{ax xa}}
}
