Termination proof
1: switching to dependency pairs
The following set of initial dependency pairs has been identified.
|
v#(
a(
a(
x
)
)
)
|
→ |
u#(
v(
x
)
)
|
|
v#(
a(
a(
x
)
)
)
|
→ |
v#(
x
)
|
|
v#(
a(
c(
x
)
)
)
|
→ |
u#(
b(
d(
x
)
)
)
|
|
w#(
a(
a(
x
)
)
)
|
→ |
u#(
w(
x
)
)
|
|
w#(
a(
a(
x
)
)
)
|
→ |
w#(
x
)
|
|
w#(
a(
c(
x
)
)
)
|
→ |
u#(
b(
d(
x
)
)
)
|
1.1: dependency graph processor
The dependency pairs are split into 2 component(s).
-
The
1st
component contains the
pair(s)
|
v#(
a(
a(
x
)
)
)
|
→ |
v#(
x
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[v#
(x1)
]
|
= |
2
x1
|
|
[a
(x1)
]
|
= |
3
x1
+
2
|
|
[u
(x1)
]
|
= |
2
x1
+
1
|
|
[b
(x1)
]
|
= |
x1
+
3
|
|
[v
(x1)
]
|
= |
x1
+
1
|
|
[d
(x1)
]
|
= |
2
x1
|
|
[c
(x1)
]
|
= |
2
x1
+
3
|
|
[w
(x1)
]
|
= |
2
x1
+
2
|
|
[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)
|
w#(
a(
a(
x
)
)
)
|
→ |
w#(
x
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[a
(x1)
]
|
= |
3
x1
+
2
|
|
[u
(x1)
]
|
= |
2
x1
+
1
|
|
[b
(x1)
]
|
= |
x1
+
3
|
|
[v
(x1)
]
|
= |
x1
+
1
|
|
[d
(x1)
]
|
= |
2
x1
|
|
[c
(x1)
]
|
= |
2
x1
+
3
|
|
[w
(x1)
]
|
= |
2
x1
+
2
|
|
[w#
(x1)
]
|
= |
2
x1
|
|
[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.