<s>
In	O
mathematical	O
logic	O
,	O
the	O
De	B-Application
Bruijn	I-Application
notation	I-Application
is	O
a	O
syntax	O
for	O
terms	O
in	O
the	O
λ	B-Language
calculus	I-Language
invented	O
by	O
the	O
Dutch	O
mathematician	O
Nicolaas	O
Govert	O
de	O
Bruijn	O
.	O
</s>
<s>
It	O
can	O
be	O
seen	O
as	O
a	O
reversal	O
of	O
the	O
usual	O
syntax	O
for	O
the	O
λ	B-Language
calculus	I-Language
where	O
the	O
argument	O
in	O
an	O
application	O
is	O
placed	O
next	O
to	O
its	O
corresponding	O
binder	O
in	O
the	O
function	O
instead	O
of	O
after	O
the	O
latter	O
's	O
body	O
.	O
</s>
<s>
Terms	O
(	O
)	O
in	O
the	O
De	B-Application
Bruijn	I-Application
notation	I-Application
are	O
either	O
variables	O
(	O
)	O
,	O
or	O
have	O
one	O
of	O
two	O
wagon	O
prefixes	O
.	O
</s>
<s>
The	O
abstractor	O
wagon	O
,	O
written	O
,	O
corresponds	O
to	O
the	O
usual	O
λ-binder	O
of	O
the	O
λ	B-Language
calculus	I-Language
,	O
and	O
the	O
applicator	O
wagon	O
,	O
written	O
,	O
corresponds	O
to	O
the	O
argument	O
in	O
an	O
application	O
in	O
the	O
λ	B-Language
calculus	I-Language
.	O
</s>
<s>
Terms	O
in	O
the	O
traditional	O
syntax	O
can	O
be	O
converted	O
to	O
the	O
De	B-Application
Bruijn	I-Application
notation	I-Application
by	O
defining	O
an	O
inductive	O
function	O
for	O
which	O
:	O
</s>
<s>
in	O
the	O
De	B-Application
Bruijn	I-Application
notation	I-Application
is	O
,	O
predictably	O
,	O
</s>
<s>
One	O
thus	O
obtains	O
a	O
generalised	O
conversion	O
primitive	O
for	O
λ-terms	O
in	O
the	O
De	B-Application
Bruijn	I-Application
notation	I-Application
.	O
</s>
<s>
Several	O
properties	O
of	O
λ-terms	O
that	O
are	O
difficult	O
to	O
state	O
and	O
prove	O
using	O
the	O
traditional	O
notation	O
are	O
easily	O
expressed	O
in	O
the	O
De	B-Application
Bruijn	I-Application
notation	I-Application
.	O
</s>
<s>
De	B-Application
Bruijn	I-Application
notation	I-Application
has	O
also	O
been	O
shown	O
to	O
be	O
useful	O
in	O
calculi	O
for	O
explicit	O
substitution	O
in	O
pure	O
type	O
systems	O
.	O
</s>
