|
minus#(
s(
x
)
,
s(
y
)
)
|
→ |
minus#(
x
,
y
)
|
|
quot#(
s(
x
)
,
s(
y
)
)
|
→ |
quot#(
minus(
x
,
y
)
,
s(
y
)
)
|
|
quot#(
s(
x
)
,
s(
y
)
)
|
→ |
minus#(
x
,
y
)
|
|
plus#(
s(
x
)
,
y
)
|
→ |
plus#(
x
,
y
)
|
|
minus#(
minus(
x
,
y
)
,
z
)
|
→ |
minus#(
x
,
plus(
y
,
z
)
)
|
|
minus#(
minus(
x
,
y
)
,
z
)
|
→ |
plus#(
y
,
z
)
|
|
app#(
cons(
x
,
l
)
,
k
)
|
→ |
app#(
l
,
k
)
|
|
sum#(
cons(
x
,
cons(
y
,
l
)
)
)
|
→ |
sum#(
cons(
plus(
x
,
y
)
,
l
)
)
|
|
sum#(
cons(
x
,
cons(
y
,
l
)
)
)
|
→ |
plus#(
x
,
y
)
|
|
sum#(
app(
l
,
cons(
x
,
cons(
y
,
k
)
)
)
)
|
→ |
sum#(
app(
l
,
sum(
cons(
x
,
cons(
y
,
k
)
)
)
)
)
|
|
sum#(
app(
l
,
cons(
x
,
cons(
y
,
k
)
)
)
)
|
→ |
app#(
l
,
sum(
cons(
x
,
cons(
y
,
k
)
)
)
)
|
|
sum#(
app(
l
,
cons(
x
,
cons(
y
,
k
)
)
)
)
|
→ |
sum#(
cons(
x
,
cons(
y
,
k
)
)
)
|
The dependency pairs are split into 6 component(s).
-
The
1st
component contains the
pair(s)
|
quot#(
s(
x
)
,
s(
y
)
)
|
→ |
quot#(
minus(
x
,
y
)
,
s(
y
)
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[quot#
(x1, x2)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
2
x1 +
2
x2
+
2
|
|
[app
(x1, x2)
]
|
= |
x1 +
2
x2
+
3
|
|
[sum
(x1)
]
|
= |
0
|
|
[quot
(x1, x2)
]
|
= |
2
x1 + x2
|
|
[s
(x1)
]
|
= |
x1
+
1
|
|
[0]
|
= |
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).
1.1.1.1: P is empty
All dependency pairs have been removed.
-
The
2nd
component contains the
pair(s)
|
minus#(
minus(
x
,
y
)
,
z
)
|
→ |
minus#(
x
,
plus(
y
,
z
)
)
|
|
minus#(
s(
x
)
,
s(
y
)
)
|
→ |
minus#(
x
,
y
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
+
2
|
|
[plus
(x1, x2)
]
|
= |
x1
|
|
[app
(x1, x2)
]
|
= |
2
x1 +
2
x2
|
|
[sum
(x1)
]
|
= |
0
|
|
[quot
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[minus#
(x1, x2)
]
|
= |
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).
|
minus#(
s(
x
)
,
s(
y
)
)
|
→ |
minus#(
x
,
y
)
|
1.1.2.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
2
x1 + x2
+
1
|
|
[app
(x1, x2)
]
|
= |
2
x1 + x2
|
|
[sum
(x1)
]
|
= |
0
|
|
[quot
(x1, x2)
]
|
= |
x1
|
|
[s
(x1)
]
|
= |
x1
+
1
|
|
[0]
|
= |
0
|
|
[minus#
(x1, x2)
]
|
= |
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).
1.1.2.1.1: P is empty
All dependency pairs have been removed.
-
The
3rd
component contains the
pair(s)
|
sum#(
app(
l
,
cons(
x
,
cons(
y
,
k
)
)
)
)
|
→ |
sum#(
app(
l
,
sum(
cons(
x
,
cons(
y
,
k
)
)
)
)
)
|
1.1.3: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[sum#
(x1)
]
|
= |
x1
|
|
[app
(x1, x2)
]
|
= |
2
x1 +
2
x2
+
2
|
|
[sum
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
2
x1 +
2
x2
|
|
[quot
(x1, x2)
]
|
= |
3
x1
+
3
|
|
[0]
|
= |
1
|
|
[s
(x1)
]
|
= |
x1
+
2
|
|
[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).
1.1.3.1: P is empty
All dependency pairs have been removed.
-
The
4th
component contains the
pair(s)
|
sum#(
cons(
x
,
cons(
y
,
l
)
)
)
|
→ |
sum#(
cons(
plus(
x
,
y
)
,
l
)
)
|
1.1.4: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[sum#
(x1)
]
|
= |
2
x1
|
|
[plus
(x1, x2)
]
|
= |
x1
|
|
[app
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[sum
(x1)
]
|
= |
2
|
|
[quot
(x1, x2)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
x1
|
|
[nil]
|
= |
1
|
|
[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).
1.1.4.1: P is empty
All dependency pairs have been removed.
-
The
5th
component contains the
pair(s)
|
plus#(
s(
x
)
,
y
)
|
→ |
plus#(
x
,
y
)
|
1.1.5: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
3
x1 +
2
x2
|
|
[app
(x1, x2)
]
|
= |
x1 + x2
+
1
|
|
[sum
(x1)
]
|
= |
0
|
|
[quot
(x1, x2)
]
|
= |
x1
|
|
[s
(x1)
]
|
= |
x1
+
3
|
|
[0]
|
= |
0
|
|
[plus#
(x1, x2)
]
|
= |
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: P is empty
All dependency pairs have been removed.
-
The
6th
component contains the
pair(s)
|
app#(
cons(
x
,
l
)
,
k
)
|
→ |
app#(
l
,
k
)
|
1.1.6: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[minus
(x1, x2)
]
|
= |
2
x1
|
|
[plus
(x1, x2)
]
|
= |
x1
|
|
[app
(x1, x2)
]
|
= |
2
x1 + x2
|
|
[sum
(x1)
]
|
= |
2
|
|
[quot
(x1, x2)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[s
(x1)
]
|
= |
x1
|
|
[app#
(x1, x2)
]
|
= |
x1
|
|
[nil]
|
= |
0
|
|
[cons
(x1, x2)
]
|
= |
x1
+
2
|
|
[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.