|
active#(
fst(
0
,
Z
)
)
|
→ |
mark#(
nil
)
|
|
active#(
fst(
s(
X
)
,
cons(
Y
,
Z
)
)
)
|
→ |
mark#(
cons(
Y
,
fst(
X
,
Z
)
)
)
|
|
active#(
fst(
s(
X
)
,
cons(
Y
,
Z
)
)
)
|
→ |
cons#(
Y
,
fst(
X
,
Z
)
)
|
|
active#(
fst(
s(
X
)
,
cons(
Y
,
Z
)
)
)
|
→ |
fst#(
X
,
Z
)
|
|
active#(
from(
X
)
)
|
→ |
mark#(
cons(
X
,
from(
s(
X
)
)
)
)
|
|
active#(
from(
X
)
)
|
→ |
cons#(
X
,
from(
s(
X
)
)
)
|
|
active#(
from(
X
)
)
|
→ |
from#(
s(
X
)
)
|
|
active#(
from(
X
)
)
|
→ |
s#(
X
)
|
|
active#(
add(
0
,
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
add(
s(
X
)
,
Y
)
)
|
→ |
mark#(
s(
add(
X
,
Y
)
)
)
|
|
active#(
add(
s(
X
)
,
Y
)
)
|
→ |
s#(
add(
X
,
Y
)
)
|
|
active#(
add(
s(
X
)
,
Y
)
)
|
→ |
add#(
X
,
Y
)
|
|
active#(
len(
nil
)
)
|
→ |
mark#(
0
)
|
|
active#(
len(
cons(
X
,
Z
)
)
)
|
→ |
mark#(
s(
len(
Z
)
)
)
|
|
active#(
len(
cons(
X
,
Z
)
)
)
|
→ |
s#(
len(
Z
)
)
|
|
active#(
len(
cons(
X
,
Z
)
)
)
|
→ |
len#(
Z
)
|
|
mark#(
fst(
X1
,
X2
)
)
|
→ |
active#(
fst(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
fst(
X1
,
X2
)
)
|
→ |
fst#(
mark(
X1
)
,
mark(
X2
)
)
|
|
mark#(
fst(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
fst(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
|
mark#(
0
)
|
→ |
active#(
0
)
|
|
mark#(
nil
)
|
→ |
active#(
nil
)
|
|
mark#(
s(
X
)
)
|
→ |
active#(
s(
X
)
)
|
|
mark#(
s(
X
)
)
|
→ |
s#(
X
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
active#(
cons(
mark(
X1
)
,
X2
)
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
cons#(
mark(
X1
)
,
X2
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
from(
X
)
)
|
→ |
active#(
from(
mark(
X
)
)
)
|
|
mark#(
from(
X
)
)
|
→ |
from#(
mark(
X
)
)
|
|
mark#(
from(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
add(
X1
,
X2
)
)
|
→ |
active#(
add(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
add(
X1
,
X2
)
)
|
→ |
add#(
mark(
X1
)
,
mark(
X2
)
)
|
|
mark#(
add(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
add(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
|
mark#(
len(
X
)
)
|
→ |
active#(
len(
mark(
X
)
)
)
|
|
mark#(
len(
X
)
)
|
→ |
len#(
mark(
X
)
)
|
|
mark#(
len(
X
)
)
|
→ |
mark#(
X
)
|
|
fst#(
mark(
X1
)
,
X2
)
|
→ |
fst#(
X1
,
X2
)
|
|
fst#(
X1
,
mark(
X2
)
)
|
→ |
fst#(
X1
,
X2
)
|
|
fst#(
active(
X1
)
,
X2
)
|
→ |
fst#(
X1
,
X2
)
|
|
fst#(
X1
,
active(
X2
)
)
|
→ |
fst#(
X1
,
X2
)
|
|
s#(
mark(
X
)
)
|
→ |
s#(
X
)
|
|
s#(
active(
X
)
)
|
→ |
s#(
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
)
|
|
from#(
mark(
X
)
)
|
→ |
from#(
X
)
|
|
from#(
active(
X
)
)
|
→ |
from#(
X
)
|
|
add#(
mark(
X1
)
,
X2
)
|
→ |
add#(
X1
,
X2
)
|
|
add#(
X1
,
mark(
X2
)
)
|
→ |
add#(
X1
,
X2
)
|
|
add#(
active(
X1
)
,
X2
)
|
→ |
add#(
X1
,
X2
)
|
|
add#(
X1
,
active(
X2
)
)
|
→ |
add#(
X1
,
X2
)
|
|
len#(
mark(
X
)
)
|
→ |
len#(
X
)
|
|
len#(
active(
X
)
)
|
→ |
len#(
X
)
|
The dependency pairs are split into 7 component(s).
-
The
1st
component contains the
pair(s)
|
active#(
fst(
s(
X
)
,
cons(
Y
,
Z
)
)
)
|
→ |
mark#(
cons(
Y
,
fst(
X
,
Z
)
)
)
|
|
mark#(
fst(
X1
,
X2
)
)
|
→ |
active#(
fst(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
active#(
from(
X
)
)
|
→ |
mark#(
cons(
X
,
from(
s(
X
)
)
)
)
|
|
mark#(
fst(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
fst(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
|
mark#(
s(
X
)
)
|
→ |
active#(
s(
X
)
)
|
|
active#(
add(
0
,
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
active#(
cons(
mark(
X1
)
,
X2
)
)
|
|
active#(
add(
s(
X
)
,
Y
)
)
|
→ |
mark#(
s(
add(
X
,
Y
)
)
)
|
|
mark#(
cons(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
from(
X
)
)
|
→ |
active#(
from(
mark(
X
)
)
)
|
|
active#(
len(
cons(
X
,
Z
)
)
)
|
→ |
mark#(
s(
len(
Z
)
)
)
|
|
mark#(
from(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
add(
X1
,
X2
)
)
|
→ |
active#(
add(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
add(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
add(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
|
mark#(
len(
X
)
)
|
→ |
active#(
len(
mark(
X
)
)
)
|
|
mark#(
len(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
2
x1
+
2
|
|
[len
(x1)
]
|
= |
x1
+
2
|
|
[fst
(x1, x2)
]
|
= |
2
x1 +
3
x2
+
3
|
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
2
x1
|
|
[active
(x1)
]
|
= |
x1
|
|
[add
(x1, x2)
]
|
= |
2
x1 + x2
+
3
|
|
[mark#
(x1)
]
|
= |
2
x1
+
1
|
|
[s
(x1)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
x1
+
1
|
|
[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)
|
fst#(
X1
,
mark(
X2
)
)
|
→ |
fst#(
X1
,
X2
)
|
|
fst#(
mark(
X1
)
,
X2
)
|
→ |
fst#(
X1
,
X2
)
|
|
fst#(
active(
X1
)
,
X2
)
|
→ |
fst#(
X1
,
X2
)
|
|
fst#(
X1
,
active(
X2
)
)
|
→ |
fst#(
X1
,
X2
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
1
|
|
[len
(x1)
]
|
= |
3
|
|
[fst
(x1, x2)
]
|
= |
x1
|
|
[mark
(x1)
]
|
= |
2
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[fst#
(x1, x2)
]
|
= |
3
x1 +
3
x2
|
|
[add
(x1, x2)
]
|
= |
x1 +
3
x2
+
3
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
1
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
fst#(
active(
X1
)
,
X2
)
|
→ |
fst#(
X1
,
X2
)
|
|
fst#(
X1
,
active(
X2
)
)
|
→ |
fst#(
X1
,
X2
)
|
1.1.2.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
0
|
|
[len
(x1)
]
|
= |
0
|
|
[fst
(x1, x2)
]
|
= |
0
|
|
[mark
(x1)
]
|
= |
1
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[fst#
(x1, x2)
]
|
= |
3
x1 +
2
x2
|
|
[add
(x1, x2)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
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)
|
s#(
active(
X
)
)
|
→ |
s#(
X
)
|
|
s#(
mark(
X
)
)
|
→ |
s#(
X
)
|
1.1.3: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
2
|
|
[len
(x1)
]
|
= |
2
x1
+
1
|
|
[fst
(x1, x2)
]
|
= |
3
|
|
[mark
(x1)
]
|
= |
2
x1
|
|
[active
(x1)
]
|
= |
x1
+
1
|
|
[add
(x1, x2)
]
|
= |
2
x1 +
2
x2
+
3
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
2
|
|
[nil]
|
= |
2
|
|
[cons
(x1, x2)
]
|
= |
1
|
|
[s#
(x1)
]
|
= |
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
s#(
mark(
X
)
)
|
→ |
s#(
X
)
|
1.1.3.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
2
|
|
[len
(x1)
]
|
= |
3
|
|
[fst
(x1, x2)
]
|
= |
2
x1
+
1
|
|
[mark
(x1)
]
|
= |
2
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[add
(x1, x2)
]
|
= |
2
x1
+
2
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[s#
(x1)
]
|
= |
2
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.3.1.1: P is empty
All dependency pairs have been removed.
-
The
4th
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
)
|
1.1.4: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
1
|
|
[len
(x1)
]
|
= |
3
|
|
[fst
(x1, x2)
]
|
= |
x1
|
|
[mark
(x1)
]
|
= |
2
x1
+
1
|
|
[cons#
(x1, x2)
]
|
= |
3
x1 +
3
x2
|
|
[active
(x1)
]
|
= |
x1
|
|
[add
(x1, x2)
]
|
= |
x1 +
3
x2
+
3
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
1
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[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
)
|
1.1.4.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
0
|
|
[len
(x1)
]
|
= |
0
|
|
[fst
(x1, x2)
]
|
= |
0
|
|
[mark
(x1)
]
|
= |
1
|
|
[cons#
(x1, x2)
]
|
= |
3
x1 +
2
x2
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[add
(x1, x2)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
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)
|
from#(
active(
X
)
)
|
→ |
from#(
X
)
|
|
from#(
mark(
X
)
)
|
→ |
from#(
X
)
|
1.1.5: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
2
|
|
[len
(x1)
]
|
= |
2
x1
+
1
|
|
[fst
(x1, x2)
]
|
= |
3
|
|
[mark
(x1)
]
|
= |
2
x1
|
|
[active
(x1)
]
|
= |
x1
+
1
|
|
[add
(x1, x2)
]
|
= |
2
x1 +
2
x2
+
3
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
2
|
|
[from#
(x1)
]
|
= |
x1
|
|
[nil]
|
= |
2
|
|
[cons
(x1, x2)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
from#(
mark(
X
)
)
|
→ |
from#(
X
)
|
1.1.5.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
2
|
|
[len
(x1)
]
|
= |
3
|
|
[fst
(x1, x2)
]
|
= |
2
x1
+
1
|
|
[mark
(x1)
]
|
= |
2
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[add
(x1, x2)
]
|
= |
2
x1
+
2
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[from#
(x1)
]
|
= |
2
x1
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
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)
|
add#(
X1
,
mark(
X2
)
)
|
→ |
add#(
X1
,
X2
)
|
|
add#(
mark(
X1
)
,
X2
)
|
→ |
add#(
X1
,
X2
)
|
|
add#(
active(
X1
)
,
X2
)
|
→ |
add#(
X1
,
X2
)
|
|
add#(
X1
,
active(
X2
)
)
|
→ |
add#(
X1
,
X2
)
|
1.1.6: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
1
|
|
[len
(x1)
]
|
= |
3
|
|
[fst
(x1, x2)
]
|
= |
x1
|
|
[mark
(x1)
]
|
= |
2
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[add
(x1, x2)
]
|
= |
x1 +
3
x2
+
3
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
1
|
|
[add#
(x1, x2)
]
|
= |
3
x1 +
3
x2
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
add#(
active(
X1
)
,
X2
)
|
→ |
add#(
X1
,
X2
)
|
|
add#(
X1
,
active(
X2
)
)
|
→ |
add#(
X1
,
X2
)
|
1.1.6.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
0
|
|
[len
(x1)
]
|
= |
0
|
|
[fst
(x1, x2)
]
|
= |
0
|
|
[mark
(x1)
]
|
= |
1
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[add
(x1, x2)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[add#
(x1, x2)
]
|
= |
3
x1 +
2
x2
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
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.
-
The
7th
component contains the
pair(s)
|
len#(
active(
X
)
)
|
→ |
len#(
X
)
|
|
len#(
mark(
X
)
)
|
→ |
len#(
X
)
|
1.1.7: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
2
|
|
[len
(x1)
]
|
= |
2
x1
+
1
|
|
[fst
(x1, x2)
]
|
= |
3
|
|
[mark
(x1)
]
|
= |
2
x1
|
|
[active
(x1)
]
|
= |
x1
+
1
|
|
[add
(x1, x2)
]
|
= |
2
x1 +
2
x2
+
3
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
2
|
|
[len#
(x1)
]
|
= |
x1
|
|
[nil]
|
= |
2
|
|
[cons
(x1, x2)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
len#(
mark(
X
)
)
|
→ |
len#(
X
)
|
1.1.7.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[from
(x1)
]
|
= |
2
|
|
[len
(x1)
]
|
= |
3
|
|
[fst
(x1, x2)
]
|
= |
2
x1
+
1
|
|
[mark
(x1)
]
|
= |
2
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[add
(x1, x2)
]
|
= |
2
x1
+
2
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[len#
(x1)
]
|
= |
2
x1
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.7.1.1: P is empty
All dependency pairs have been removed.