<s>
In	O
the	O
lambda	B-Language
calculus	I-Language
,	O
a	O
term	O
is	O
in	O
beta	B-Application
normal	I-Application
form	I-Application
if	O
no	O
beta	O
reduction	O
is	O
possible	O
.	O
</s>
<s>
A	O
term	O
is	O
in	O
beta-eta	B-Application
normal	I-Application
form	I-Application
if	O
neither	O
a	O
beta	O
reduction	O
nor	O
an	O
eta	O
reduction	O
is	O
possible	O
.	O
</s>
<s>
A	O
term	O
is	O
in	O
head	B-Application
normal	I-Application
form	I-Application
if	O
there	O
is	O
no	O
beta-redex	O
in	O
head	O
position	O
.	O
</s>
<s>
However	O
,	O
a	O
term	O
may	O
have	O
more	O
than	O
one	O
head	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
In	O
the	O
lambda	B-Language
calculus	I-Language
,	O
a	O
beta	B-Application
redex	I-Application
is	O
a	O
term	O
of	O
the	O
form	O
:	O
</s>
<s>
A	O
beta	O
reduction	O
is	O
an	O
application	O
of	O
the	O
following	O
rewrite	O
rule	O
to	O
a	O
beta	B-Application
redex	I-Application
contained	O
in	O
a	O
term	O
:	O
</s>
<s>
A	O
normal	O
form	O
is	O
a	O
term	O
that	O
does	O
not	O
contain	O
any	O
beta	B-Application
redex	I-Application
,	O
i.e.	O
</s>
<s>
A	O
head	B-Application
normal	I-Application
form	I-Application
is	O
a	O
term	O
that	O
does	O
not	O
contain	O
a	O
beta	B-Application
redex	I-Application
in	O
head	O
position	O
,	O
i.e.	O
</s>
<s>
When	O
considering	O
the	O
simple	O
lambda	B-Language
calculus	I-Language
(	O
viz	O
.	O
</s>
<s>
without	O
the	O
addition	O
of	O
constant	O
or	O
function	O
symbols	O
,	O
meant	O
to	O
be	O
reduced	O
by	O
additional	O
delta	O
rule	O
)	O
,	O
head	B-Application
normal	I-Application
forms	I-Application
are	O
the	O
terms	O
of	O
the	O
following	O
shape	O
:	O
</s>
<s>
A	O
head	B-Application
normal	I-Application
form	I-Application
is	O
not	O
always	O
a	O
normal	O
form	O
,	O
because	O
the	O
applied	O
arguments	O
need	O
not	O
be	O
normal	O
.	O
</s>
<s>
However	O
,	O
the	O
converse	O
is	O
true	O
:	O
any	O
normal	O
form	O
is	O
also	O
a	O
head	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
normal	O
forms	O
are	O
exactly	O
the	O
head	B-Application
normal	I-Application
forms	I-Application
in	O
which	O
the	O
subterms	O
are	O
themselves	O
normal	O
forms	O
.	O
</s>
<s>
There	O
is	O
also	O
the	O
notion	O
of	O
weak	O
head	B-Application
normal	I-Application
form	I-Application
:	O
a	O
term	O
in	O
weak	O
head	B-Application
normal	I-Application
form	I-Application
is	O
either	O
a	O
term	O
in	O
head	B-Application
normal	I-Application
form	I-Application
or	O
a	O
lambda	B-Language
abstraction	I-Language
.	O
</s>
<s>
At	O
that	O
point	O
,	O
the	O
resulting	O
term	O
is	O
in	O
head	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Normal-order	O
reduction	O
is	O
complete	O
,	O
in	O
the	O
sense	O
that	O
if	O
a	O
term	O
has	O
a	O
head	B-Application
normal	I-Application
form	I-Application
,	O
then	O
normal‐order	O
reduction	O
will	O
eventually	O
reach	O
it	O
.	O
</s>
