The dependency pairs are split into 6 component(s).
-
The
1st
component contains the
pair(s)
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
active#(
f(
f(
X
)
)
)
|
→ |
mark#(
c(
f(
g(
f(
X
)
)
)
)
)
|
|
mark#(
f(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
c(
X
)
)
|
→ |
active#(
c(
X
)
)
|
|
active#(
c(
X
)
)
|
→ |
mark#(
d(
X
)
)
|
|
mark#(
g(
X
)
)
|
→ |
active#(
g(
X
)
)
|
|
active#(
h(
X
)
)
|
→ |
mark#(
c(
d(
X
)
)
)
|
|
mark#(
d(
X
)
)
|
→ |
active#(
d(
X
)
)
|
|
mark#(
h(
X
)
)
|
→ |
active#(
h(
mark(
X
)
)
)
|
|
mark#(
h(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
2
x1
|
|
[mark
(x1)
]
|
= |
2
x1
|
|
[active#
(x1)
]
|
= |
0
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
2
x1
+
3
|
|
[d
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
x1
|
|
[c
(x1)
]
|
= |
0
|
|
[g
(x1)
]
|
= |
x1
+
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
active#(
f(
f(
X
)
)
)
|
→ |
mark#(
c(
f(
g(
f(
X
)
)
)
)
)
|
|
mark#(
c(
X
)
)
|
→ |
active#(
c(
X
)
)
|
|
active#(
c(
X
)
)
|
→ |
mark#(
d(
X
)
)
|
|
active#(
h(
X
)
)
|
→ |
mark#(
c(
d(
X
)
)
)
|
|
mark#(
d(
X
)
)
|
→ |
active#(
d(
X
)
)
|
|
mark#(
h(
X
)
)
|
→ |
active#(
h(
mark(
X
)
)
)
|
|
mark#(
h(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
2
x1
+
3
|
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
2
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
2
x1
+
1
|
|
[d
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
3
x1
+
1
|
|
[c
(x1)
]
|
= |
1
|
|
[g
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
d(
X
)
)
|
→ |
active#(
d(
X
)
)
|
1.1.1.1.1: dependency graph processor
The dependency pairs are split into 0 component(s).
-
The
2nd
component contains the
pair(s)
|
f#(
active(
X
)
)
|
→ |
f#(
X
)
|
|
f#(
mark(
X
)
)
|
→ |
f#(
X
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
1
|
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
2
x1
|
|
[f
(x1)
]
|
= |
1
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
1
|
|
[f#
(x1)
]
|
= |
x1
|
|
[g
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
f#(
active(
X
)
)
|
→ |
f#(
X
)
|
1.1.2.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
0
|
|
[mark
(x1)
]
|
= |
2
|
|
[active
(x1)
]
|
= |
2
x1
+
2
|
|
[f
(x1)
]
|
= |
0
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
0
|
|
[f#
(x1)
]
|
= |
3
x1
|
|
[g
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.2.1.1: P is empty
All dependency pairs have been removed.
-
The
3rd
component contains the
pair(s)
|
c#(
active(
X
)
)
|
→ |
c#(
X
)
|
|
c#(
mark(
X
)
)
|
→ |
c#(
X
)
|
1.1.3: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
1
|
|
[c#
(x1)
]
|
= |
x1
|
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
2
x1
|
|
[f
(x1)
]
|
= |
1
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
1
|
|
[g
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
c#(
active(
X
)
)
|
→ |
c#(
X
)
|
1.1.3.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
0
|
|
[c#
(x1)
]
|
= |
3
x1
|
|
[mark
(x1)
]
|
= |
2
|
|
[active
(x1)
]
|
= |
2
x1
+
2
|
|
[f
(x1)
]
|
= |
0
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
0
|
|
[g
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.3.1.1: P is empty
All dependency pairs have been removed.
-
The
4th
component contains the
pair(s)
|
g#(
active(
X
)
)
|
→ |
g#(
X
)
|
|
g#(
mark(
X
)
)
|
→ |
g#(
X
)
|
1.1.4: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
1
|
|
[g#
(x1)
]
|
= |
x1
|
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
2
x1
|
|
[f
(x1)
]
|
= |
1
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
1
|
|
[g
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
g#(
active(
X
)
)
|
→ |
g#(
X
)
|
1.1.4.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
0
|
|
[g#
(x1)
]
|
= |
3
x1
|
|
[mark
(x1)
]
|
= |
2
|
|
[active
(x1)
]
|
= |
2
x1
+
2
|
|
[f
(x1)
]
|
= |
0
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
0
|
|
[g
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.4.1.1: P is empty
All dependency pairs have been removed.
-
The
5th
component contains the
pair(s)
|
d#(
active(
X
)
)
|
→ |
d#(
X
)
|
|
d#(
mark(
X
)
)
|
→ |
d#(
X
)
|
1.1.5: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
1
|
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
2
x1
|
|
[f
(x1)
]
|
= |
1
|
|
[d
(x1)
]
|
= |
0
|
|
[d#
(x1)
]
|
= |
x1
|
|
[c
(x1)
]
|
= |
1
|
|
[g
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
d#(
active(
X
)
)
|
→ |
d#(
X
)
|
1.1.5.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
0
|
|
[mark
(x1)
]
|
= |
2
|
|
[active
(x1)
]
|
= |
2
x1
+
2
|
|
[f
(x1)
]
|
= |
0
|
|
[d
(x1)
]
|
= |
0
|
|
[d#
(x1)
]
|
= |
3
x1
|
|
[c
(x1)
]
|
= |
0
|
|
[g
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.5.1.1: P is empty
All dependency pairs have been removed.
-
The
6th
component contains the
pair(s)
|
h#(
active(
X
)
)
|
→ |
h#(
X
)
|
|
h#(
mark(
X
)
)
|
→ |
h#(
X
)
|
1.1.6: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
1
|
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
2
x1
|
|
[h#
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
1
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
1
|
|
[g
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
h#(
active(
X
)
)
|
→ |
h#(
X
)
|
1.1.6.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[h
(x1)
]
|
= |
0
|
|
[mark
(x1)
]
|
= |
2
|
|
[active
(x1)
]
|
= |
2
x1
+
2
|
|
[h#
(x1)
]
|
= |
3
x1
|
|
[f
(x1)
]
|
= |
0
|
|
[d
(x1)
]
|
= |
0
|
|
[c
(x1)
]
|
= |
0
|
|
[g
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.6.1.1: P is empty
All dependency pairs have been removed.