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 defines 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 defining 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 infinite de
creasing chain of elements: that is, for
every
infinite 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
.
For 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.
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 Spring '09
 Koskesh
 Math, Computer Science, Counting, Order theory, Partially ordered set

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