|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
U12#(
isNat(
V2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
isNat#(
V2
)
|
|
active#(
U12(
tt
)
)
|
→ |
mark#(
tt
)
|
|
active#(
U21(
tt
)
)
|
→ |
mark#(
tt
)
|
|
active#(
U31(
tt
,
N
)
)
|
→ |
mark#(
N
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
U42#(
isNat(
N
)
,
M
,
N
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
isNat#(
N
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
s#(
plus(
N
,
M
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
plus#(
N
,
M
)
|
|
active#(
isNat(
0
)
)
|
→ |
mark#(
tt
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
U11#(
isNat(
V1
)
,
V2
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
isNat#(
V1
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
U21#(
isNat(
V1
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
isNat#(
V1
)
|
|
active#(
plus(
N
,
0
)
)
|
→ |
mark#(
U31(
isNat(
N
)
,
N
)
)
|
|
active#(
plus(
N
,
0
)
)
|
→ |
U31#(
isNat(
N
)
,
N
)
|
|
active#(
plus(
N
,
0
)
)
|
→ |
isNat#(
N
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
mark#(
U41(
isNat(
M
)
,
M
,
N
)
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
U41#(
isNat(
M
)
,
M
,
N
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
isNat#(
M
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
U11#(
mark(
X1
)
,
X2
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
tt
)
|
→ |
active#(
tt
)
|
|
mark#(
U12(
X
)
)
|
→ |
active#(
U12(
mark(
X
)
)
)
|
|
mark#(
U12(
X
)
)
|
→ |
U12#(
mark(
X
)
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
mark#(
isNat(
X
)
)
|
→ |
isNat#(
X
)
|
|
mark#(
U21(
X
)
)
|
→ |
active#(
U21(
mark(
X
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
U21#(
mark(
X
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
active#(
U31(
mark(
X1
)
,
X2
)
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
U31#(
mark(
X1
)
,
X2
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
U41#(
mark(
X1
)
,
X2
,
X3
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
U42#(
mark(
X1
)
,
X2
,
X3
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
s(
X
)
)
|
→ |
active#(
s(
mark(
X
)
)
)
|
|
mark#(
s(
X
)
)
|
→ |
s#(
mark(
X
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
plus#(
mark(
X1
)
,
mark(
X2
)
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
|
mark#(
0
)
|
→ |
active#(
0
)
|
|
U11#(
mark(
X1
)
,
X2
)
|
→ |
U11#(
X1
,
X2
)
|
|
U11#(
X1
,
mark(
X2
)
)
|
→ |
U11#(
X1
,
X2
)
|
|
U11#(
active(
X1
)
,
X2
)
|
→ |
U11#(
X1
,
X2
)
|
|
U11#(
X1
,
active(
X2
)
)
|
→ |
U11#(
X1
,
X2
)
|
|
U12#(
mark(
X
)
)
|
→ |
U12#(
X
)
|
|
U12#(
active(
X
)
)
|
→ |
U12#(
X
)
|
|
isNat#(
mark(
X
)
)
|
→ |
isNat#(
X
)
|
|
isNat#(
active(
X
)
)
|
→ |
isNat#(
X
)
|
|
U21#(
mark(
X
)
)
|
→ |
U21#(
X
)
|
|
U21#(
active(
X
)
)
|
→ |
U21#(
X
)
|
|
U31#(
mark(
X1
)
,
X2
)
|
→ |
U31#(
X1
,
X2
)
|
|
U31#(
X1
,
mark(
X2
)
)
|
→ |
U31#(
X1
,
X2
)
|
|
U31#(
active(
X1
)
,
X2
)
|
→ |
U31#(
X1
,
X2
)
|
|
U31#(
X1
,
active(
X2
)
)
|
→ |
U31#(
X1
,
X2
)
|
|
U41#(
mark(
X1
)
,
X2
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
mark(
X2
)
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
X2
,
mark(
X3
)
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
active(
X1
)
,
X2
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U42#(
mark(
X1
)
,
X2
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
mark(
X2
)
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
X2
,
mark(
X3
)
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
active(
X1
)
,
X2
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
s#(
mark(
X
)
)
|
→ |
s#(
X
)
|
|
s#(
active(
X
)
)
|
→ |
s#(
X
)
|
|
plus#(
mark(
X1
)
,
X2
)
|
→ |
plus#(
X1
,
X2
)
|
|
plus#(
X1
,
mark(
X2
)
)
|
→ |
plus#(
X1
,
X2
)
|
|
plus#(
active(
X1
)
,
X2
)
|
→ |
plus#(
X1
,
X2
)
|
|
plus#(
X1
,
active(
X2
)
)
|
→ |
plus#(
X1
,
X2
)
|
The dependency pairs are split into 10 component(s).
-
The
1st
component contains the
pair(s)
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
active#(
U12(
mark(
X
)
)
)
|
|
active#(
U31(
tt
,
N
)
)
|
→ |
mark#(
N
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
active#(
U21(
mark(
X
)
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
active#(
U31(
mark(
X1
)
,
X2
)
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
plus(
N
,
0
)
)
|
→ |
mark#(
U31(
isNat(
N
)
,
N
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
s(
X
)
)
|
→ |
active#(
s(
mark(
X
)
)
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
mark#(
U41(
isNat(
M
)
,
M
,
N
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
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
|
|
[mark#
(x1)
]
|
= |
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
active#(
U12(
mark(
X
)
)
)
|
|
active#(
U31(
tt
,
N
)
)
|
→ |
mark#(
N
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
active#(
U31(
mark(
X1
)
,
X2
)
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
plus(
N
,
0
)
)
|
→ |
mark#(
U31(
isNat(
N
)
,
N
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
mark#(
U41(
isNat(
M
)
,
M
,
N
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
0
|
|
[active#
(x1)
]
|
= |
2
x1
|
|
[active
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
2
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
active#(
U31(
tt
,
N
)
)
|
→ |
mark#(
N
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
active#(
U31(
mark(
X1
)
,
X2
)
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
plus(
N
,
0
)
)
|
→ |
mark#(
U31(
isNat(
N
)
,
N
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
mark#(
U41(
isNat(
M
)
,
M
,
N
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
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
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
x1
|
|
[0]
|
= |
2
|
|
[U41
(x1, x2, x3)
]
|
= |
2
x1 + x2 +
2
x3
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
x1
|
|
[U31
(x1, x2)
]
|
= |
x1 +
2
x2
|
|
[U21
(x1)
]
|
= |
2
x1
|
|
[plus
(x1, x2)
]
|
= |
2
x1 + x2
|
|
[s
(x1)
]
|
= |
x1
|
|
[U42
(x1, x2, x3)
]
|
= |
2
x1 + x2 +
2
x3
|
|
[U12
(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#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
active#(
U31(
tt
,
N
)
)
|
→ |
mark#(
N
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
active#(
U31(
mark(
X1
)
,
X2
)
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
mark#(
X1
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
mark#(
U41(
isNat(
M
)
,
M
,
N
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
mark#(
X2
)
|
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)
]
|
= |
3
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
3
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1 + x2 + x3
+
3
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
2
x1
|
|
[U31
(x1, x2)
]
|
= |
x1 + x2
+
1
|
|
[U21
(x1)
]
|
= |
3
x1
|
|
[plus
(x1, x2)
]
|
= |
x1 + x2
+
3
|
|
[s
(x1)
]
|
= |
x1
|
|
[U42
(x1, x2, x3)
]
|
= |
2
x1 + x2 + x3
+
3
|
|
[U12
(x1)
]
|
= |
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
U31(
X1
,
X2
)
)
|
→ |
active#(
U31(
mark(
X1
)
,
X2
)
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
mark#(
U41(
isNat(
M
)
,
M
,
N
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
0
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
2
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
2
|
|
[U11
(x1, x2)
]
|
= |
2
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
2
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
2
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
plus(
N
,
s(
M
)
)
)
|
→ |
mark#(
U41(
isNat(
M
)
,
M
,
N
)
)
|
|
mark#(
s(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
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
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
2
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
2
x1 +
3
x2 +
2
x3
|
|
[tt]
|
= |
2
|
|
[isNat
(x1)
]
|
= |
2
|
|
[U11
(x1, x2)
]
|
= |
x1
|
|
[U31
(x1, x2)
]
|
= |
x1
+
2
|
|
[U21
(x1)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
2
x1 +
3
x2
+
2
|
|
[s
(x1)
]
|
= |
x1
+
2
|
|
[U42
(x1, x2, x3)
]
|
= |
2
x1 +
3
x2 +
2
x3
|
|
[U12
(x1)
]
|
= |
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
|
mark#(
plus(
X1
,
X2
)
)
|
→ |
active#(
plus(
mark(
X1
)
,
mark(
X2
)
)
)
|
1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
2
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
2
|
|
[mark#
(x1)
]
|
= |
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
mark#(
U41(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U41(
mark(
X1
)
,
X2
,
X3
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
0
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
x1
|
|
[0]
|
= |
3
|
|
[U41
(x1, x2, x3)
]
|
= |
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
x1
|
|
[U31
(x1, x2)
]
|
= |
x1
+
2
|
|
[U21
(x1)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
x1 + x2
+
2
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(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#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U41(
tt
,
M
,
N
)
)
|
→ |
mark#(
U42(
isNat(
N
)
,
M
,
N
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
1.1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
0
|
|
[active#
(x1)
]
|
= |
2
x1
|
|
[active
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
0
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
active#(
U42(
tt
,
M
,
N
)
)
|
→ |
mark#(
s(
plus(
N
,
M
)
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
2
x1
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
2
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1 + x2
+
1
|
|
[tt]
|
= |
3
|
|
[isNat
(x1)
]
|
= |
3
|
|
[U11
(x1, x2)
]
|
= |
x1
|
|
[U31
(x1, x2)
]
|
= |
x1
+
1
|
|
[U21
(x1)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
x1 +
3
x2
+
2
|
|
[s
(x1)
]
|
= |
2
|
|
[U42
(x1, x2, x3)
]
|
= |
x1 + x2
+
1
|
|
[U12
(x1)
]
|
= |
x1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
|
mark#(
U42(
X1
,
X2
,
X3
)
)
|
→ |
active#(
U42(
mark(
X1
)
,
X2
,
X3
)
)
|
1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
0
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
mark#(
X1
)
|
|
mark#(
U12(
X
)
)
|
→ |
mark#(
X
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
plus(
V1
,
V2
)
)
)
|
→ |
mark#(
U11(
isNat(
V1
)
,
V2
)
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1 + x2
+
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
2
x1
|
|
[U11
(x1, x2)
]
|
= |
x1 +
2
x2
+
2
|
|
[U31
(x1, x2)
]
|
= |
x1
|
|
[U21
(x1)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
x1 + x2
+
2
|
|
[s
(x1)
]
|
= |
x1
|
|
[U42
(x1, x2, x3)
]
|
= |
x1 + x2
+
2
|
|
[U12
(x1)
]
|
= |
x1
+
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
active#(
U11(
tt
,
V2
)
)
|
→ |
mark#(
U12(
isNat(
V2
)
)
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1
+
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
x1
+
2
|
|
[U21
(x1)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
2
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
2
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U11(
X1
,
X2
)
)
|
→ |
active#(
U11(
mark(
X1
)
,
X2
)
)
|
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
0
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
0
|
|
[mark#
(x1)
]
|
= |
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
isNat(
X
)
)
|
→ |
active#(
isNat(
X
)
)
|
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
|
|
[active#
(x1)
]
|
= |
x1
|
|
[active
(x1)
]
|
= |
x1
|
|
[mark#
(x1)
]
|
= |
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1 + x2
+
1
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
x1
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
x1
|
|
[U21
(x1)
]
|
= |
x1
|
|
[plus
(x1, x2)
]
|
= |
x1 + x2
|
|
[s
(x1)
]
|
= |
x1
+
1
|
|
[U42
(x1, x2, x3)
]
|
= |
x1 + x2
+
1
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
mark#(
U21(
X
)
)
|
→ |
mark#(
X
)
|
|
active#(
isNat(
s(
V1
)
)
)
|
→ |
mark#(
U21(
isNat(
V1
)
)
)
|
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1: dependency graph processor
The dependency pairs are split into 1 component(s).
-
The
2nd
component contains the
pair(s)
|
U11#(
X1
,
mark(
X2
)
)
|
→ |
U11#(
X1
,
X2
)
|
|
U11#(
mark(
X1
)
,
X2
)
|
→ |
U11#(
X1
,
X2
)
|
|
U11#(
active(
X1
)
,
X2
)
|
→ |
U11#(
X1
,
X2
)
|
|
U11#(
X1
,
active(
X2
)
)
|
→ |
U11#(
X1
,
X2
)
|
1.1.2: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1
+
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
2
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
2
|
|
[U31
(x1, x2)
]
|
= |
x1
+
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U11#
(x1, x2)
]
|
= |
3
x1 +
3
x2
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U11#(
active(
X1
)
,
X2
)
|
→ |
U11#(
X1
,
X2
)
|
|
U11#(
X1
,
active(
X2
)
)
|
→ |
U11#(
X1
,
X2
)
|
1.1.2.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
3
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U11#
(x1, x2)
]
|
= |
2
x1 +
2
x2
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(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)
|
U12#(
active(
X
)
)
|
→ |
U12#(
X
)
|
|
U12#(
mark(
X
)
)
|
→ |
U12#(
X
)
|
1.1.3: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
3
|
|
[U31
(x1, x2)
]
|
= |
x1
+
2
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U12#
(x1)
]
|
= |
x1
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U12#(
active(
X
)
)
|
→ |
U12#(
X
)
|
1.1.3.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[tt]
|
= |
1
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U12#
(x1)
]
|
= |
3
x1
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.3.1.1: P is empty
All dependency pairs have been removed.
-
The
4th
component contains the
pair(s)
|
isNat#(
active(
X
)
)
|
→ |
isNat#(
X
)
|
|
isNat#(
mark(
X
)
)
|
→ |
isNat#(
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
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
2
|
|
[isNat#
(x1)
]
|
= |
x1
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
3
|
|
[U31
(x1, x2)
]
|
= |
x1
+
2
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
isNat#(
active(
X
)
)
|
→ |
isNat#(
X
)
|
1.1.4.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[isNat#
(x1)
]
|
= |
3
x1
|
|
[tt]
|
= |
1
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.4.1.1: P is empty
All dependency pairs have been removed.
-
The
5th
component contains the
pair(s)
|
U21#(
active(
X
)
)
|
→ |
U21#(
X
)
|
|
U21#(
mark(
X
)
)
|
→ |
U21#(
X
)
|
1.1.5: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[U21#
(x1)
]
|
= |
x1
|
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
3
|
|
[U31
(x1, x2)
]
|
= |
x1
+
2
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
2
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U21#(
active(
X
)
)
|
→ |
U21#(
X
)
|
1.1.5.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[U21#
(x1)
]
|
= |
3
x1
|
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[tt]
|
= |
1
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.5.1.1: P is empty
All dependency pairs have been removed.
-
The
6th
component contains the
pair(s)
|
U31#(
X1
,
mark(
X2
)
)
|
→ |
U31#(
X1
,
X2
)
|
|
U31#(
mark(
X1
)
,
X2
)
|
→ |
U31#(
X1
,
X2
)
|
|
U31#(
active(
X1
)
,
X2
)
|
→ |
U31#(
X1
,
X2
)
|
|
U31#(
X1
,
active(
X2
)
)
|
→ |
U31#(
X1
,
X2
)
|
1.1.6: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1
+
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
2
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
2
|
|
[U31
(x1, x2)
]
|
= |
x1
+
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U31#
(x1, x2)
]
|
= |
3
x1 +
3
x2
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U31#(
active(
X1
)
,
X2
)
|
→ |
U31#(
X1
,
X2
)
|
|
U31#(
X1
,
active(
X2
)
)
|
→ |
U31#(
X1
,
X2
)
|
1.1.6.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
3
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U31#
(x1, x2)
]
|
= |
2
x1 +
2
x2
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.6.1.1: P is empty
All dependency pairs have been removed.
-
The
7th
component contains the
pair(s)
|
U41#(
X1
,
mark(
X2
)
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
mark(
X1
)
,
X2
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
X2
,
mark(
X3
)
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
active(
X1
)
,
X2
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
1.1.7: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41#
(x1, x2, x3)
]
|
= |
3
x1 +
3
x2 +
3
x3
|
|
[U41
(x1, x2, x3)
]
|
= |
x1
+
2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
2
|
|
[U31
(x1, x2)
]
|
= |
x1
+
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
x1
+
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U41#(
active(
X1
)
,
X2
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
1.1.7.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
3
|
|
[0]
|
= |
0
|
|
[U41#
(x1, x2, x3)
]
|
= |
2
x1
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U41#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
|
U41#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
1.1.7.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
x1
+
3
|
|
[0]
|
= |
0
|
|
[U41#
(x1, x2, x3)
]
|
= |
3
x1
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U41#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U41#(
X1
,
X2
,
X3
)
|
1.1.7.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41#
(x1, x2, x3)
]
|
= |
3
x1
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.7.1.1.1.1: P is empty
All dependency pairs have been removed.
-
The
8th
component contains the
pair(s)
|
U42#(
X1
,
mark(
X2
)
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
mark(
X1
)
,
X2
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
X2
,
mark(
X3
)
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
active(
X1
)
,
X2
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
1.1.8: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1
+
2
|
|
[tt]
|
= |
0
|
|
[U42#
(x1, x2, x3)
]
|
= |
3
x1 +
3
x2 +
3
x3
|
|
[isNat
(x1)
]
|
= |
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
2
|
|
[U31
(x1, x2)
]
|
= |
x1
+
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
x1
+
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U42#(
active(
X1
)
,
X2
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
1.1.8.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
3
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[U42#
(x1, x2, x3)
]
|
= |
2
x1
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U42#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
|
U42#(
X1
,
X2
,
active(
X3
)
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
1.1.8.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
x1
+
3
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[tt]
|
= |
0
|
|
[U42#
(x1, x2, x3)
]
|
= |
3
x1
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
U42#(
X1
,
active(
X2
)
,
X3
)
|
→ |
U42#(
X1
,
X2
,
X3
)
|
1.1.8.1.1.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[tt]
|
= |
0
|
|
[U42#
(x1, x2, x3)
]
|
= |
3
x1
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.8.1.1.1.1: P is empty
All dependency pairs have been removed.
-
The
9th
component contains the
pair(s)
|
s#(
active(
X
)
)
|
→ |
s#(
X
)
|
|
s#(
mark(
X
)
)
|
→ |
s#(
X
)
|
1.1.9: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
2
|
|
[s#
(x1)
]
|
= |
x1
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
3
|
|
[U31
(x1, x2)
]
|
= |
x1
+
2
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(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
)
|
1.1.9.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
1
|
|
[s#
(x1)
]
|
= |
3
x1
|
|
[tt]
|
= |
1
|
|
[isNat
(x1)
]
|
= |
1
|
|
[U11
(x1, x2)
]
|
= |
1
|
|
[U31
(x1, x2)
]
|
= |
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
1
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.9.1.1: P is empty
All dependency pairs have been removed.
-
The
10th
component contains the
pair(s)
|
plus#(
X1
,
mark(
X2
)
)
|
→ |
plus#(
X1
,
X2
)
|
|
plus#(
mark(
X1
)
,
X2
)
|
→ |
plus#(
X1
,
X2
)
|
|
plus#(
active(
X1
)
,
X2
)
|
→ |
plus#(
X1
,
X2
)
|
|
plus#(
X1
,
active(
X2
)
)
|
→ |
plus#(
X1
,
X2
)
|
1.1.10: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
x1
+
1
|
|
[active
(x1)
]
|
= |
x1
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
x1
+
2
|
|
[plus#
(x1, x2)
]
|
= |
3
x1 +
3
x2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
2
x1
+
2
|
|
[U11
(x1, x2)
]
|
= |
2
|
|
[U31
(x1, x2)
]
|
= |
x1
+
1
|
|
[U21
(x1)
]
|
= |
1
|
|
[plus
(x1, x2)
]
|
= |
x1
+
3
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
1
|
|
[U12
(x1)
]
|
= |
1
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
|
plus#(
active(
X1
)
,
X2
)
|
→ |
plus#(
X1
,
X2
)
|
|
plus#(
X1
,
active(
X2
)
)
|
→ |
plus#(
X1
,
X2
)
|
1.1.10.1: reduction pair processor
Using the following reduction pair
Linear polynomial
interpretation over
the naturals
|
[mark
(x1)
]
|
= |
3
|
|
[active
(x1)
]
|
= |
2
x1
+
3
|
|
[0]
|
= |
0
|
|
[U41
(x1, x2, x3)
]
|
= |
0
|
|
[plus#
(x1, x2)
]
|
= |
2
x1 +
2
x2
|
|
[tt]
|
= |
0
|
|
[isNat
(x1)
]
|
= |
0
|
|
[U11
(x1, x2)
]
|
= |
0
|
|
[U31
(x1, x2)
]
|
= |
0
|
|
[U21
(x1)
]
|
= |
0
|
|
[plus
(x1, x2)
]
|
= |
0
|
|
[s
(x1)
]
|
= |
0
|
|
[U42
(x1, x2, x3)
]
|
= |
0
|
|
[U12
(x1)
]
|
= |
0
|
|
[f(x1, ..., xn)]
|
= |
x1 + ... + xn + 1
|
for all other symbols f of arity n
|
one remains with the following pair(s).
1.1.10.1.1: P is empty
All dependency pairs have been removed.