|
+#(
0(
x
)
,
0(
y
)
)
|
→ |
0#(
+(
x
,
y
)
)
|
|
+#(
0(
x
)
,
0(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
0(
x
)
,
1(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
1(
x
)
,
0(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
1(
x
)
,
1(
y
)
)
|
→ |
0#(
+(
+(
x
,
y
)
,
1(
#
)
)
)
|
|
+#(
1(
x
)
,
1(
y
)
)
|
→ |
+#(
+(
x
,
y
)
,
1(
#
)
)
|
|
+#(
1(
x
)
,
1(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
+(
x
,
y
)
,
z
)
|
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
x
,
y
)
|
|
-#(
0(
x
)
,
0(
y
)
)
|
→ |
0#(
-(
x
,
y
)
)
|
|
-#(
0(
x
)
,
0(
y
)
)
|
→ |
-#(
x
,
y
)
|
|
-#(
0(
x
)
,
1(
y
)
)
|
→ |
-#(
-(
x
,
y
)
,
1(
#
)
)
|
|
-#(
0(
x
)
,
1(
y
)
)
|
→ |
-#(
x
,
y
)
|
|
-#(
1(
x
)
,
0(
y
)
)
|
→ |
-#(
x
,
y
)
|
|
-#(
1(
x
)
,
1(
y
)
)
|
→ |
0#(
-(
x
,
y
)
)
|
|
-#(
1(
x
)
,
1(
y
)
)
|
→ |
-#(
x
,
y
)
|
|
ge#(
0(
x
)
,
0(
y
)
)
|
→ |
ge#(
x
,
y
)
|
|
ge#(
0(
x
)
,
1(
y
)
)
|
→ |
not#(
ge(
y
,
x
)
)
|
|
ge#(
0(
x
)
,
1(
y
)
)
|
→ |
ge#(
y
,
x
)
|
|
ge#(
1(
x
)
,
0(
y
)
)
|
→ |
ge#(
x
,
y
)
|
|
ge#(
1(
x
)
,
1(
y
)
)
|
→ |
ge#(
x
,
y
)
|
|
ge#(
#
,
0(
x
)
)
|
→ |
ge#(
#
,
x
)
|
|
min#(
n(
x
,
y
,
z
)
)
|
→ |
min#(
y
)
|
|
max#(
n(
x
,
y
,
z
)
)
|
→ |
max#(
z
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
and#(
and(
ge(
x
,
max(
y
)
)
,
ge(
min(
z
)
,
x
)
)
,
and(
bs(
y
)
,
bs(
z
)
)
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
and#(
ge(
x
,
max(
y
)
)
,
ge(
min(
z
)
,
x
)
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
ge#(
x
,
max(
y
)
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
max#(
y
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
ge#(
min(
z
)
,
x
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
min#(
z
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
and#(
bs(
y
)
,
bs(
z
)
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
bs#(
y
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
bs#(
z
)
|
|
size#(
n(
x
,
y
,
z
)
)
|
→ |
+#(
+(
size(
x
)
,
size(
y
)
)
,
1(
#
)
)
|
|
size#(
n(
x
,
y
,
z
)
)
|
→ |
+#(
size(
x
)
,
size(
y
)
)
|
|
size#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
x
)
|
|
size#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
y
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
and#(
if(
ge(
size(
y
)
,
size(
z
)
)
,
ge(
1(
#
)
,
-(
size(
y
)
,
size(
z
)
)
)
,
ge(
1(
#
)
,
-(
size(
z
)
,
size(
y
)
)
)
)
,
and(
wb(
y
)
,
wb(
z
)
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
if#(
ge(
size(
y
)
,
size(
z
)
)
,
ge(
1(
#
)
,
-(
size(
y
)
,
size(
z
)
)
)
,
ge(
1(
#
)
,
-(
size(
z
)
,
size(
y
)
)
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
ge#(
size(
y
)
,
size(
z
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
y
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
z
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
ge#(
1(
#
)
,
-(
size(
y
)
,
size(
z
)
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
-#(
size(
y
)
,
size(
z
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
y
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
z
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
ge#(
1(
#
)
,
-(
size(
z
)
,
size(
y
)
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
-#(
size(
z
)
,
size(
y
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
z
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
y
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
and#(
wb(
y
)
,
wb(
z
)
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
wb#(
y
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
wb#(
z
)
|
The dependency pairs are split into 9 component(s).
-
The
1st
component contains the
pair(s)
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
wb#(
z
)
|
|
wb#(
n(
x
,
y
,
z
)
)
|
→ |
wb#(
y
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
x1 +
2
x2 + x3
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
2
x1
+
2
|
|
[n
(x1, x2, x3)
]
|
= |
3
x1 +
3
x2 +
2
x3
+
3
|
|
[size
(x1)
]
|
= |
x1
|
|
[1
(x1)
]
|
= |
2
|
|
[wb#
(x1)
]
|
= |
2
x1
|
|
[ge
(x1, x2)
]
|
= |
1
|
|
[max
(x1)
]
|
= |
x1
|
|
[true]
|
= |
1
|
|
[and
(x1, x2)
]
|
= |
2
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 + x2
|
|
[-
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[l
(x1)
]
|
= |
x1
+
2
|
|
[min
(x1)
]
|
= |
3
x1
|
|
[0
(x1)
]
|
= |
0
|
|
[bs
(x1)
]
|
= |
3
x1
|
|
[not
(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)
|
size#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
y
)
|
|
size#(
n(
x
,
y
,
z
)
)
|
→ |
size#(
x
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
x1 +
2
x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
1
|
|
[n
(x1, x2, x3)
]
|
= |
3
x1 +
2
x2 +
2
x3
+
3
|
|
[size
(x1)
]
|
= |
2
x1
+
2
|
|
[1
(x1)
]
|
= |
2
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
2
x1
|
|
[size#
(x1)
]
|
= |
2
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 + x2
|
|
[-
(x1, x2)
]
|
= |
2
x1
|
|
[l
(x1)
]
|
= |
x1
+
3
|
|
[min
(x1)
]
|
= |
3
x1
|
|
[0
(x1)
]
|
= |
2
|
|
[bs
(x1)
]
|
= |
3
x1
+
3
|
|
[not
(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: P is empty
All dependency pairs have been removed.
-
The
3rd
component contains the
pair(s)
|
+#(
0(
x
)
,
1(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
0(
x
)
,
0(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
1(
x
)
,
0(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
1(
x
)
,
1(
y
)
)
|
→ |
+#(
+(
x
,
y
)
,
1(
#
)
)
|
|
+#(
1(
x
)
,
1(
y
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
+(
x
,
y
)
,
z
)
|
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
x
,
y
)
|
1.1.3: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
0
|
|
[n
(x1, x2, x3)
]
|
= |
x1 + x2 +
2
x3
+
3
|
|
[size
(x1)
]
|
= |
2
x1
+
2
|
|
[1
(x1)
]
|
= |
x1
+
1
|
|
[+#
(x1, x2)
]
|
= |
2
x1
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
2
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
2
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 + x2
|
|
[-
(x1, x2)
]
|
= |
x1
|
|
[l
(x1)
]
|
= |
x1
+
3
|
|
[min
(x1)
]
|
= |
2
x1
|
|
[0
(x1)
]
|
= |
x1
+
1
|
|
[bs
(x1)
]
|
= |
0
|
|
[not
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
+#(
1(
x
)
,
1(
y
)
)
|
→ |
+#(
+(
x
,
y
)
,
1(
#
)
)
|
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
+(
x
,
y
)
,
z
)
|
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
x
,
y
)
|
1.1.3.1: dependency graph processor
The dependency pairs are split into 2 component(s).
-
The
1st
component contains the
pair(s)
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
x
,
y
)
|
|
+#(
x
,
+(
y
,
z
)
)
|
→ |
+#(
+(
x
,
y
)
,
z
)
|
1.1.3.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
x1 + x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
2
x1
+
1
|
|
[n
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2 +
2
x3
+
1
|
|
[size
(x1)
]
|
= |
3
x1
|
|
[1
(x1)
]
|
= |
0
|
|
[+#
(x1, x2)
]
|
= |
x1
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 +
2
x2
+
1
|
|
[-
(x1, x2)
]
|
= |
x1
+
1
|
|
[l
(x1)
]
|
= |
x1
|
|
[min
(x1)
]
|
= |
x1
|
|
[0
(x1)
]
|
= |
1
|
|
[bs
(x1)
]
|
= |
1
|
|
[not
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.3.1.1.1: P is empty
All dependency pairs have been removed.
-
The
2nd
component contains the
pair(s)
|
+#(
1(
x
)
,
1(
y
)
)
|
→ |
+#(
+(
x
,
y
)
,
1(
#
)
)
|
1.1.3.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
1
|
|
[n
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2 +
2
x3
+
3
|
|
[size
(x1)
]
|
= |
2
x1
|
|
[1
(x1)
]
|
= |
x1
+
2
|
|
[+#
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
2
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[-
(x1, x2)
]
|
= |
x1
|
|
[l
(x1)
]
|
= |
x1
+
1
|
|
[min
(x1)
]
|
= |
3
x1
|
|
[0
(x1)
]
|
= |
x1
+
2
|
|
[bs
(x1)
]
|
= |
3
|
|
[not
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.3.1.2.1: P is empty
All dependency pairs have been removed.
-
The
4th
component contains the
pair(s)
|
-#(
0(
x
)
,
1(
y
)
)
|
→ |
-#(
-(
x
,
y
)
,
1(
#
)
)
|
|
-#(
0(
x
)
,
1(
y
)
)
|
→ |
-#(
x
,
y
)
|
|
-#(
0(
x
)
,
0(
y
)
)
|
→ |
-#(
x
,
y
)
|
|
-#(
1(
x
)
,
0(
y
)
)
|
→ |
-#(
x
,
y
)
|
|
-#(
1(
x
)
,
1(
y
)
)
|
→ |
-#(
x
,
y
)
|
1.1.4: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
2
|
|
[n
(x1, x2, x3)
]
|
= |
x1 +
2
x2 +
2
x3
+
2
|
|
[size
(x1)
]
|
= |
2
x1
+
1
|
|
[1
(x1)
]
|
= |
x1
+
2
|
|
[-#
(x1, x2)
]
|
= |
2
x1
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
2
x1
+
1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 + x2
|
|
[-
(x1, x2)
]
|
= |
x1
|
|
[l
(x1)
]
|
= |
x1
+
2
|
|
[min
(x1)
]
|
= |
3
x1
|
|
[0
(x1)
]
|
= |
x1
+
2
|
|
[bs
(x1)
]
|
= |
2
x1
|
|
[not
(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: P is empty
All dependency pairs have been removed.
-
The
5th
component contains the
pair(s)
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
bs#(
z
)
|
|
bs#(
n(
x
,
y
,
z
)
)
|
→ |
bs#(
y
)
|
1.1.5: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
x1 +
2
x2 + x3
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
2
x1
+
2
|
|
[n
(x1, x2, x3)
]
|
= |
3
x1 +
3
x2 +
2
x3
+
3
|
|
[size
(x1)
]
|
= |
x1
|
|
[1
(x1)
]
|
= |
2
|
|
[bs#
(x1)
]
|
= |
2
x1
|
|
[ge
(x1, x2)
]
|
= |
1
|
|
[max
(x1)
]
|
= |
x1
|
|
[true]
|
= |
1
|
|
[and
(x1, x2)
]
|
= |
2
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 + x2
|
|
[-
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[l
(x1)
]
|
= |
x1
+
2
|
|
[min
(x1)
]
|
= |
3
x1
|
|
[0
(x1)
]
|
= |
0
|
|
[bs
(x1)
]
|
= |
3
x1
|
|
[not
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.5.1: P is empty
All dependency pairs have been removed.
-
The
6th
component contains the
pair(s)
|
ge#(
0(
x
)
,
1(
y
)
)
|
→ |
ge#(
y
,
x
)
|
|
ge#(
0(
x
)
,
0(
y
)
)
|
→ |
ge#(
x
,
y
)
|
|
ge#(
1(
x
)
,
0(
y
)
)
|
→ |
ge#(
x
,
y
)
|
|
ge#(
1(
x
)
,
1(
y
)
)
|
→ |
ge#(
x
,
y
)
|
1.1.6: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
1
|
|
[n
(x1, x2, x3)
]
|
= |
x1 +
2
x2 +
2
x3
+
3
|
|
[size
(x1)
]
|
= |
2
x1
+
1
|
|
[1
(x1)
]
|
= |
x1
+
1
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
2
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[-
(x1, x2)
]
|
= |
x1
|
|
[l
(x1)
]
|
= |
x1
|
|
[min
(x1)
]
|
= |
3
x1
|
|
[0
(x1)
]
|
= |
x1
+
1
|
|
[ge#
(x1, x2)
]
|
= |
3
x1 +
3
x2
|
|
[bs
(x1)
]
|
= |
3
|
|
[not
(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: P is empty
All dependency pairs have been removed.
-
The
7th
component contains the
pair(s)
|
ge#(
#
,
0(
x
)
)
|
→ |
ge#(
#
,
x
)
|
1.1.7: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
0
|
|
[n
(x1, x2, x3)
]
|
= |
3
x1 +
2
x2 +
2
x3
+
2
|
|
[size
(x1)
]
|
= |
2
x1
|
|
[1
(x1)
]
|
= |
x1
+
1
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
3
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
2
x1 + x2
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[-
(x1, x2)
]
|
= |
x1
|
|
[l
(x1)
]
|
= |
x1
+
1
|
|
[min
(x1)
]
|
= |
2
x1
|
|
[0
(x1)
]
|
= |
x1
+
1
|
|
[ge#
(x1, x2)
]
|
= |
2
x1
|
|
[bs
(x1)
]
|
= |
x1
|
|
[not
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.7.1: P is empty
All dependency pairs have been removed.
-
The
8th
component contains the
pair(s)
|
min#(
n(
x
,
y
,
z
)
)
|
→ |
min#(
y
)
|
1.1.8: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
2
x1 + x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
2
x1
+
2
|
|
[n
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2 +
2
x3
+
3
|
|
[size
(x1)
]
|
= |
x1
|
|
[1
(x1)
]
|
= |
x1
+
1
|
|
[ge
(x1, x2)
]
|
= |
x1
|
|
[min#
(x1)
]
|
= |
3
x1
|
|
[max
(x1)
]
|
= |
3
x1
+
1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
x1
+
1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 + x2
|
|
[-
(x1, x2)
]
|
= |
x1
|
|
[l
(x1)
]
|
= |
x1
+
3
|
|
[min
(x1)
]
|
= |
2
x1
|
|
[0
(x1)
]
|
= |
x1
+
1
|
|
[bs
(x1)
]
|
= |
2
x1
+
2
|
|
[not
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.8.1: P is empty
All dependency pairs have been removed.
-
The
9th
component contains the
pair(s)
|
max#(
n(
x
,
y
,
z
)
)
|
→ |
max#(
z
)
|
1.1.9: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[val
(x1)
]
|
= |
x1
|
|
[if
(x1, x2, x3)
]
|
= |
2
x1 +
2
x2
|
|
[#]
|
= |
0
|
|
[wb
(x1)
]
|
= |
0
|
|
[n
(x1, x2, x3)
]
|
= |
x1 + x2 +
3
x3
+
1
|
|
[size
(x1)
]
|
= |
0
|
|
[1
(x1)
]
|
= |
0
|
|
[ge
(x1, x2)
]
|
= |
0
|
|
[max
(x1)
]
|
= |
3
x1
|
|
[true]
|
= |
0
|
|
[and
(x1, x2)
]
|
= |
2
x1
|
|
[false]
|
= |
0
|
|
[+
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[-
(x1, x2)
]
|
= |
2
x1
|
|
[l
(x1)
]
|
= |
x1
|
|
[min
(x1)
]
|
= |
x1
|
|
[0
(x1)
]
|
= |
0
|
|
[max#
(x1)
]
|
= |
2
x1
|
|
[bs
(x1)
]
|
= |
0
|
|
[not
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.9.1: P is empty
All dependency pairs have been removed.