The dependency pairs are split into 3 component(s).
-
The
1st
component contains the
pair(s)
|
log#(
s(
s(
x
)
)
)
|
→ |
log#(
s(
quot(
x
,
s(
s(
0
)
)
)
)
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[log
(x1)
]
|
= |
3
x1
+
3
|
|
[log#
(x1)
]
|
= |
2
x1
|
|
[quot
(x1, x2)
]
|
= |
x1
|
|
[s
(x1)
]
|
= |
2
x1
+
2
|
|
[0]
|
= |
0
|
|
[pred
(x1)
]
|
= |
x1
|
|
[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)
|
quot#(
s(
x
)
,
s(
y
)
)
|
→ |
quot#(
minus(
x
,
y
)
,
s(
y
)
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[log
(x1)
]
|
= |
2
x1
|
|
[quot#
(x1, x2)
]
|
= |
2
x1
|
|
[quot
(x1, x2)
]
|
= |
x1
+
1
|
|
[s
(x1)
]
|
= |
x1
+
2
|
|
[0]
|
= |
0
|
|
[pred
(x1)
]
|
= |
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.
-
The
3rd
component contains the
pair(s)
|
minus#(
x
,
s(
y
)
)
|
→ |
minus#(
x
,
y
)
|
1.1.3: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[log
(x1)
]
|
= |
x1
|
|
[quot
(x1, x2)
]
|
= |
x1
|
|
[s
(x1)
]
|
= |
x1
+
1
|
|
[0]
|
= |
0
|
|
[pred
(x1)
]
|
= |
x1
|
|
[minus#
(x1, x2)
]
|
= |
2
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.3.1: P is empty
All dependency pairs have been removed.