|
active#(
f(
0
)
)
|
→ |
mark#(
cons(
0
,
f(
s(
0
)
)
)
)
|
|
active#(
f(
0
)
)
|
→ |
cons#(
0
,
f(
s(
0
)
)
)
|
|
active#(
f(
0
)
)
|
→ |
f#(
s(
0
)
)
|
|
active#(
f(
0
)
)
|
→ |
s#(
0
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
mark#(
f(
p(
s(
0
)
)
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
f#(
p(
s(
0
)
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
p#(
s(
0
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
s#(
0
)
|
|
active#(
p(
s(
0
)
)
)
|
→ |
mark#(
0
)
|
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
mark#(
f(
X
)
)
|
→ |
f#(
mark(
X
)
)
|
|
mark#(
f(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
0
)
|
→ |
active#(
0
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
active#(
cons(
mark(
X1
)
,
X2
)
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
cons#(
mark(
X1
)
,
X2
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
s(
X
)
)
|
→ |
active#(
s(
mark(
X
)
)
)
|
|
mark#(
s(
X
)
)
|
→ |
s#(
mark(
X
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
p(
X
)
)
|
→ |
active#(
p(
mark(
X
)
)
)
|
|
mark#(
p(
X
)
)
|
→ |
p#(
mark(
X
)
)
|
|
mark#(
p(
X
)
)
|
→ |
mark#(
X
)
|
|
f#(
mark(
X
)
)
|
→ |
f#(
X
)
|
|
f#(
active(
X
)
)
|
→ |
f#(
X
)
|
|
cons#(
mark(
X1
)
,
X2
)
|
→ |
cons#(
X1
,
X2
)
|
|
cons#(
X1
,
mark(
X2
)
)
|
→ |
cons#(
X1
,
X2
)
|
|
cons#(
active(
X1
)
,
X2
)
|
→ |
cons#(
X1
,
X2
)
|
|
cons#(
X1
,
active(
X2
)
)
|
→ |
cons#(
X1
,
X2
)
|
|
s#(
mark(
X
)
)
|
→ |
s#(
X
)
|
|
s#(
active(
X
)
)
|
→ |
s#(
X
)
|
|
p#(
mark(
X
)
)
|
→ |
p#(
X
)
|
|
p#(
active(
X
)
)
|
→ |
p#(
X
)
|
The dependency pairs are split into 5 component(s).
-
The
1st
component contains the
pair(s)
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
active#(
f(
0
)
)
|
→ |
mark#(
cons(
0
,
f(
s(
0
)
)
)
)
|
|
mark#(
f(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
active#(
cons(
mark(
X1
)
,
X2
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
mark#(
f(
p(
s(
0
)
)
)
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
s(
X
)
)
|
→ |
active#(
s(
mark(
X
)
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
p(
X
)
)
|
→ |
active#(
p(
mark(
X
)
)
)
|
|
mark#(
p(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
2
x1
|
|
[mark#
(x1)
]
|
= |
x1
|
|
[0]
|
= |
2
|
|
[s
(x1)
]
|
= |
2
x1
|
|
[cons
(x1, x2)
]
|
= |
x1
|
|
[p
(x1)
]
|
= |
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
mark#(
f(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
active#(
cons(
mark(
X1
)
,
X2
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
mark#(
f(
p(
s(
0
)
)
)
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
s(
X
)
)
|
→ |
active#(
s(
mark(
X
)
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
p(
X
)
)
|
→ |
active#(
p(
mark(
X
)
)
)
|
|
mark#(
p(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
0
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
0
|
|
[f
(x1)
]
|
= |
1
|
|
[mark#
(x1)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[p
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
mark#(
f(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
mark#(
f(
p(
s(
0
)
)
)
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
p(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
x1
+
2
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
x1
+
2
|
|
[mark#
(x1)
]
|
= |
x1
+
2
|
|
[s
(x1)
]
|
= |
2
x1
|
|
[0]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
x1
|
|
[p
(x1)
]
|
= |
2
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
mark#(
f(
p(
s(
0
)
)
)
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
p(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
2
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
2
x1
+
1
|
|
[mark#
(x1)
]
|
= |
2
x1
|
|
[s
(x1)
]
|
= |
2
x1
|
|
[0]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
x1
+
1
|
|
[p
(x1)
]
|
= |
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
mark#(
f(
p(
s(
0
)
)
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
p(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
2
x1
|
|
[active#
(x1)
]
|
= |
0
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
2
x1
|
|
[s
(x1)
]
|
= |
2
x1
+
3
|
|
[0]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
x1
|
|
[p
(x1)
]
|
= |
x1
+
3
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
|
active#(
f(
s(
0
)
)
)
|
→ |
mark#(
f(
p(
s(
0
)
)
)
)
|
1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
2
x1
+
2
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
2
x1
+
3
|
|
[mark#
(x1)
]
|
= |
2
x1
+
2
|
|
[s
(x1)
]
|
= |
2
|
|
[0]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
2
x1
|
|
[p
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
f(
X
)
)
|
→ |
active#(
f(
mark(
X
)
)
)
|
1.1.1.1.1.1.1.1.1: 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
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
x1
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
3
|
|
[f#
(x1)
]
|
= |
2
x1
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[p
(x1)
]
|
= |
2
|
|
[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
|
[mark
(x1)
]
|
= |
1
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[f
(x1)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[f#
(x1)
]
|
= |
3
x1
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[p
(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)
|
cons#(
X1
,
mark(
X2
)
)
|
→ |
cons#(
X1
,
X2
)
|
|
cons#(
mark(
X1
)
,
X2
)
|
→ |
cons#(
X1
,
X2
)
|
|
cons#(
active(
X1
)
,
X2
)
|
→ |
cons#(
X1
,
X2
)
|
|
cons#(
X1
,
active(
X2
)
)
|
→ |
cons#(
X1
,
X2
)
|
1.1.3: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
2
x1
+
1
|
|
[cons#
(x1, x2)
]
|
= |
x1 + x2
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
2
x1
+
1
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
2
x1
+
3
|
|
[cons
(x1, x2)
]
|
= |
x1
|
|
[p
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
cons#(
active(
X1
)
,
X2
)
|
→ |
cons#(
X1
,
X2
)
|
|
cons#(
X1
,
active(
X2
)
)
|
→ |
cons#(
X1
,
X2
)
|
1.1.3.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[cons#
(x1, x2)
]
|
= |
x1 +
3
x2
|
|
[active
(x1)
]
|
= |
x1
+
3
|
|
[f
(x1)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[p
(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)
|
s#(
active(
X
)
)
|
→ |
s#(
X
)
|
|
s#(
mark(
X
)
)
|
→ |
s#(
X
)
|
1.1.4: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
x1
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
3
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[s#
(x1)
]
|
= |
2
x1
|
|
[p
(x1)
]
|
= |
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
s#(
active(
X
)
)
|
→ |
s#(
X
)
|
1.1.4.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
1
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[f
(x1)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[s#
(x1)
]
|
= |
3
x1
|
|
[p
(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)
|
p#(
active(
X
)
)
|
→ |
p#(
X
)
|
|
p#(
mark(
X
)
)
|
→ |
p#(
X
)
|
1.1.5: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[f
(x1)
]
|
= |
x1
|
|
[p#
(x1)
]
|
= |
2
x1
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
3
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[p
(x1)
]
|
= |
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
p#(
active(
X
)
)
|
→ |
p#(
X
)
|
1.1.5.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
1
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[f
(x1)
]
|
= |
0
|
|
[p#
(x1)
]
|
= |
3
x1
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[p
(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.