The
1st
component contains the
pair(s)
|
mark#(
adx(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
incr(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
hd(
X
)
)
|
→ |
a__hd#(
mark(
X
)
)
|
|
a__hd#(
cons(
X
,
Y
)
)
|
→ |
mark#(
X
)
|
|
mark#(
hd(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
tl(
X
)
)
|
→ |
a__tl#(
mark(
X
)
)
|
|
a__tl#(
cons(
X
,
Y
)
)
|
→ |
mark#(
Y
)
|
|
mark#(
tl(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[zeros]
|
= |
0
|
|
[mark
(x1)
]
|
= |
2
x1
|
|
[hd
(x1)
]
|
= |
2
x1
+
3
|
|
[a__zeros]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
2
x1
+
2
|
|
[a__hd#
(x1)
]
|
= |
x1
+
3
|
|
[a__nats]
|
= |
0
|
|
[0]
|
= |
0
|
|
[adx
(x1)
]
|
= |
x1
|
|
[a__hd
(x1)
]
|
= |
2
x1
+
3
|
|
[cons
(x1, x2)
]
|
= |
2
x1 +
2
x2
|
|
[a__adx
(x1)
]
|
= |
x1
|
|
[incr
(x1)
]
|
= |
x1
|
|
[a__tl#
(x1)
]
|
= |
2
x1
+
3
|
|
[nats]
|
= |
0
|
|
[tl
(x1)
]
|
= |
2
x1
+
2
|
|
[a__incr
(x1)
]
|
= |
x1
|
|
[s
(x1)
]
|
= |
0
|
|
[a__tl
(x1)
]
|
= |
2
x1
+
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
adx(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
incr(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[zeros]
|
= |
0
|
|
[mark
(x1)
]
|
= |
2
x1
|
|
[hd
(x1)
]
|
= |
x1
+
1
|
|
[a__zeros]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
2
x1
|
|
[a__nats]
|
= |
1
|
|
[0]
|
= |
0
|
|
[adx
(x1)
]
|
= |
x1
+
1
|
|
[cons
(x1, x2)
]
|
= |
2
x1 + x2
|
|
[a__hd
(x1)
]
|
= |
x1
+
2
|
|
[a__adx
(x1)
]
|
= |
x1
+
1
|
|
[incr
(x1)
]
|
= |
x1
|
|
[nats]
|
= |
1
|
|
[a__incr
(x1)
]
|
= |
x1
|
|
[tl
(x1)
]
|
= |
3
x1
|
|
[s
(x1)
]
|
= |
0
|
|
[a__tl
(x1)
]
|
= |
3
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
incr(
X
)
)
|
→ |
mark#(
X
)
|
1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[zeros]
|
= |
0
|
|
[mark
(x1)
]
|
= |
3
x1
|
|
[hd
(x1)
]
|
= |
2
x1
|
|
[a__zeros]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
x1
|
|
[a__nats]
|
= |
3
|
|
[0]
|
= |
0
|
|
[adx
(x1)
]
|
= |
x1
+
1
|
|
[cons
(x1, x2)
]
|
= |
2
x1 + x2
|
|
[a__hd
(x1)
]
|
= |
2
x1
|
|
[a__adx
(x1)
]
|
= |
x1
+
3
|
|
[incr
(x1)
]
|
= |
x1
+
1
|
|
[nats]
|
= |
1
|
|
[a__incr
(x1)
]
|
= |
x1
+
2
|
|
[tl
(x1)
]
|
= |
3
x1
+
2
|
|
[s
(x1)
]
|
= |
x1
|
|
[a__tl
(x1)
]
|
= |
3
x1
+
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.1.1.1.1: P is empty
All dependency pairs have been removed.