<s>
Bar	B-Algorithm
recursion	I-Algorithm
is	O
a	O
generalized	O
form	O
of	O
recursion	O
developed	O
by	O
C	O
.	O
Spector	O
in	O
his	O
1962	O
paper	O
.	O
</s>
<s>
It	O
is	O
related	O
to	O
bar	O
induction	B-Algorithm
in	O
the	O
same	O
fashion	O
that	O
primitive	B-Architecture
recursion	I-Architecture
is	O
related	O
to	O
ordinary	O
induction	B-Algorithm
,	O
or	O
transfinite	O
recursion	O
is	O
related	O
to	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
Let	O
V	O
,	O
R	O
,	O
and	O
O	O
be	O
types	O
,	O
and	O
i	O
be	O
any	O
natural	O
number	O
,	O
representing	O
a	O
sequence	O
of	O
parameters	O
taken	O
from	O
V	O
.	O
Then	O
the	O
function	O
sequence	O
f	O
of	O
functions	O
fn	O
from	O
Vi+n	O
→	O
R	O
to	O
O	O
is	O
defined	O
by	O
bar	B-Algorithm
recursion	I-Algorithm
from	O
the	O
functions	O
Ln	O
:	O
R	O
→	O
O	O
and	O
B	O
with	O
Bn	O
:	O
( ( 	O
Vi+n	O
→	O
R	O
)	O
x	O
(	O
Vn	O
→	O
R	O
)	O
)	O
→	O
O	O
if	O
:	O
</s>
<s>
Provided	O
that	O
for	O
every	O
sufficiently	O
long	O
function	O
( λα	O
)	O
r	O
of	O
type	O
Vi	O
→	O
R	O
,	O
there	O
is	O
some	O
n	O
with	O
Ln(r )	O
=	O
Bn((λα )	O
r	O
,	O
(	O
λx:V	O
)	O
Ln+	O
1(r )	O
)	O
,	O
the	O
bar	O
induction	B-Algorithm
rule	O
ensures	O
that	O
f	O
is	O
well-defined	O
.	O
</s>
<s>
The	O
idea	O
is	O
that	O
one	O
extends	O
the	O
sequence	O
arbitrarily	O
,	O
using	O
the	O
recursion	O
term	O
B	O
to	O
determine	O
the	O
effect	O
,	O
until	O
a	O
sufficiently	O
long	O
node	O
of	O
the	O
tree	O
of	O
sequences	O
over	O
V	O
is	O
reached	O
;	O
then	O
the	O
base	O
term	O
L	O
determines	O
the	O
final	O
value	O
of	O
f	O
.	O
The	O
well-definedness	O
condition	O
corresponds	O
to	O
the	O
requirement	O
that	O
every	O
infinite	O
path	O
must	O
eventually	O
pass	O
through	O
a	O
sufficiently	O
long	O
node	O
:	O
the	O
same	O
requirement	O
that	O
is	O
needed	O
to	O
invoke	O
a	O
bar	O
induction	B-Algorithm
.	O
</s>
<s>
The	O
principles	O
of	O
bar	O
induction	B-Algorithm
and	O
bar	B-Algorithm
recursion	I-Algorithm
are	O
the	O
intuitionistic	O
equivalents	O
of	O
the	O
axiom	O
of	O
dependent	O
choices	O
.	O
</s>
