This program is a wellknown example in computer science, since the func
tion computed by this program grows extremely rapidly. We wish to prove
that this program always terminates, and therefore deFnes a total function.
Counting down from
x
is not good enough, since the third equation does
not decrease
x
+ 1
, because of the embedded
Ack
(
x
+ 1
,y
)
. We will devise
a different way of counting down, by deFning a wellfounded partial order
with the property that it always decreases to a terminating state.
D
EFINITION
5.9 (W
ELL

FOUNDED
P
ARTIAL
O
RDERS
)
A
partial order
(
A,
≤
)
is
wellfounded
if and only if it has no inFnite de
creasing chain of elements: that is, for
every
inFnite sequence
a
1
,a
2
,a
3
,...
of elements in
A
with
a
1
≥
a
2
≥
a
3
≥
...
, there exists
m
∈
N
such that
a
n
=
a
m
for every
n
≥
m
.
±or example, the conventional numerical order
≤
on
N
is a wellfounded
partial order. This is
not
the case for
≤
on
Z
, which can decrease for ever.
P
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 Spring '09
 Koskesh
 Math, Computer Science, Counting, Order theory, Partially ordered set, wellfounded partial order

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