Termination proof
1: switching to dependency pairs
The following set of initial dependency pairs has been identified.
|
f#(
a
,
a
)
|
→ |
f#(
a
,
b
)
|
|
f#(
a
,
b
)
|
→ |
f#(
s(
a
)
,
c
)
|
|
f#(
s(
X
)
,
c
)
|
→ |
f#(
X
,
c
)
|
|
f#(
c
,
c
)
|
→ |
f#(
a
,
a
)
|
1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[b]
|
= |
0
|
|
[a]
|
= |
0
|
|
[f
(x1, x2)
]
|
= |
3
x1
|
|
[c]
|
= |
1
|
|
[s
(x1)
]
|
= |
2
x1
|
|
[f#
(x1, x2)
]
|
= |
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
f#(
a
,
a
)
|
→ |
f#(
a
,
b
)
|
|
f#(
a
,
b
)
|
→ |
f#(
s(
a
)
,
c
)
|
|
f#(
s(
X
)
,
c
)
|
→ |
f#(
X
,
c
)
|
1.1.1: dependency graph processor
The dependency pairs are split into 1 component(s).