Termination proof
1: switching to dependency pairs
The following set of initial dependency pairs has been identified.
|
f#(
g(
x
)
)
|
→ |
f#(
f(
x
)
)
|
|
f#(
g(
x
)
)
|
→ |
f#(
x
)
|
|
f'#(
s(
x
)
,
y
,
y
)
|
→ |
f'#(
y
,
x
,
s(
x
)
)
|
1.1: dependency graph processor
The dependency pairs are split into 2 component(s).
-
The
1st
component contains the
pair(s)
|
f#(
g(
x
)
)
|
→ |
f#(
x
)
|
|
f#(
g(
x
)
)
|
→ |
f#(
f(
x
)
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
0
|
|
[f'
(x1, x2, x3)
]
|
= |
0
|
|
[f
(x1)
]
|
= |
x1
|
|
[s
(x1)
]
|
= |
0
|
|
[f#
(x1)
]
|
= |
x1
|
|
[g
(x1)
]
|
= |
2
x1
+
3
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.1.1: P is empty
All dependency pairs have been removed.
-
The
2nd
component contains the
pair(s)
|
f'#(
s(
x
)
,
y
,
y
)
|
→ |
f'#(
y
,
x
,
s(
x
)
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
0
|
|
[f'
(x1, x2, x3)
]
|
= |
0
|
|
[f
(x1)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
3
x1
+
3
|
|
[f'#
(x1, x2, x3)
]
|
= |
3
x1 +
2
x2 +
2
x3
|
|
[g
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.2.1: P is empty
All dependency pairs have been removed.