<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
coinduction	B-Application
is	O
a	O
technique	O
for	O
defining	O
and	O
proving	O
properties	O
of	O
systems	O
of	O
concurrent	O
interacting	O
objects	O
.	O
</s>
<s>
Coinduction	B-Application
is	O
the	O
mathematical	O
dual	O
to	O
structural	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
Coinductively	O
defined	O
types	O
are	O
known	O
as	O
codata	O
and	O
are	O
typically	O
infinite	B-Application
data	B-General_Concept
structures	I-General_Concept
,	O
such	O
as	O
streams	B-Architecture
.	O
</s>
<s>
As	O
a	O
definition	O
or	O
specification	B-Application
,	O
coinduction	B-Application
describes	O
how	O
an	O
object	O
may	O
be	O
"	O
observed	O
"	O
,	O
"	O
broken	O
down	O
"	O
or	O
"	O
destructed	O
"	O
into	O
simpler	O
objects	O
.	O
</s>
<s>
As	O
a	O
proof	O
technique	O
,	O
it	O
may	O
be	O
used	O
to	O
show	O
that	O
an	O
equation	O
is	O
satisfied	O
by	O
all	O
possible	O
implementations	O
of	O
such	O
a	O
specification	B-Application
.	O
</s>
<s>
To	O
generate	O
and	O
manipulate	O
codata	O
,	O
one	O
typically	O
uses	O
corecursive	B-Application
functions	O
,	O
in	O
conjunction	O
with	O
lazy	O
evaluation	O
.	O
</s>
<s>
In	O
programming	O
,	O
co-logic	O
programming	O
(	O
co-LP	O
for	O
brevity	O
)	O
"	O
is	O
a	O
natural	O
generalization	O
of	O
logic	O
programming	O
and	O
coinductive	B-Application
logic	O
programming	O
,	O
which	O
in	O
turn	O
generalizes	O
other	O
extensions	O
of	O
logic	O
programming	O
,	O
such	O
as	O
infinite	B-Application
trees	O
,	O
lazy	O
predicates	O
,	O
and	O
concurrent	O
communicating	O
predicates	O
.	O
</s>
<s>
Co-LP	O
has	O
applications	O
to	O
rational	O
trees	O
,	O
verifying	O
infinitary	O
properties	O
,	O
lazy	O
evaluation	O
,	O
concurrent	O
logic	O
programming	O
,	O
model	O
checking	O
,	O
bisimilarity	B-Application
proofs	O
,	O
etc.	O
"	O
</s>
<s>
Experimental	O
implementations	O
of	O
co-LP	O
are	O
available	O
from	O
The	O
University	O
of	O
Texas	O
at	O
Dallas	O
and	O
in	O
Logtalk	B-Language
(	O
for	O
examples	O
see	O
)	O
and	O
SWI-Prolog	B-Application
.	O
</s>
<s>
In	O
a	O
concise	O
statement	O
is	O
given	O
of	O
both	O
the	O
principle	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
and	O
the	O
principle	O
of	O
coinduction	B-Application
.	O
</s>
<s>
While	O
this	O
article	O
is	O
not	O
primarily	O
concerned	O
with	O
induction	B-Algorithm
,	O
it	O
is	O
useful	O
to	O
consider	O
their	O
somewhat	O
generalized	O
forms	O
at	O
once	O
.	O
</s>
<s>
We	O
can	O
now	O
state	O
the	O
principles	O
of	O
induction	B-Algorithm
and	O
coinduction	B-Application
.	O
</s>
<s>
By	O
the	O
principle	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
,	O
it	O
suffices	O
to	O
exhibit	O
an	O
F-closed	O
set	O
for	O
which	O
the	O
property	O
holds	O
.	O
</s>
<s>
Using	O
the	O
principle	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
,	O
we	O
can	O
prove	O
the	O
following	O
claim	O
:	O
</s>
<s>
Clearly	O
cannot	O
produce	O
an	O
infinite	B-Application
string	O
,	O
so	O
it	O
turns	O
out	O
this	O
set	O
is	O
F-closed	O
and	O
the	O
conclusion	O
follows	O
.	O
</s>
<s>
We	O
would	O
like	O
to	O
use	O
the	O
principle	O
of	O
coinduction	B-Application
to	O
prove	O
the	O
following	O
claim	O
:	O
</s>
<s>
Here	O
denotes	O
the	O
infinite	B-Application
list	O
consisting	O
of	O
all	O
.	O
</s>
<s>
To	O
use	O
the	O
principle	O
of	O
coinduction	B-Application
,	O
consider	O
the	O
set	O
:	O
</s>
<s>
In	O
any	O
case	O
,	O
a	O
stream	O
is	O
an	O
infinite	B-Application
list	O
of	O
elements	O
from	O
which	O
you	O
may	O
observe	O
the	O
first	O
element	O
,	O
or	O
place	O
an	O
element	O
in	O
front	O
of	O
to	O
get	O
another	O
stream	O
.	O
</s>
<s>
We	O
will	O
demonstrate	O
how	O
the	O
principle	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
subsumes	O
mathematical	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
We	O
will	O
take	O
the	O
following	O
definition	O
of	O
mathematical	B-Algorithm
induction	I-Algorithm
:	O
</s>
<s>
Therefore	O
,	O
by	O
the	O
principle	B-Algorithm
of	I-Algorithm
induction	I-Algorithm
,	O
if	O
we	O
wish	O
to	O
prove	O
some	O
property	O
of	O
,	O
it	O
suffices	O
to	O
show	O
that	O
is	O
F-closed	O
.	O
</s>
<s>
This	O
is	O
precisely	O
mathematical	B-Algorithm
induction	I-Algorithm
as	O
stated	O
.	O
</s>
