YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 48 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 3 ms]
(8) CSR
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


----------------------------------------

(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_3(x) -> cons_0(x, from_2(s_0(x)))
   from_4(x) -> cons_0(x, from_2(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
from_3: empty set
from_4: empty set
cons_1: empty set
nil_0: empty set

----------------------------------------

(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_3(x) -> cons_0(x, from_2(s_0(x)))
   from_4(x) -> cons_0(x, from_2(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
from_3: empty set
from_4: empty set
cons_1: empty set
nil_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons_0(x_1, x_2)) = x_1 + x_2
   POL(cons_1(x_1, x_2)) = 0
   POL(from_1(x_1)) = 0
   POL(from_2(x_1)) = x_1
   POL(from_3(x_1)) = 2 + 2*x_1
   POL(from_4(x_1)) = 2 + 2*x_1
   POL(nil_0) = 0
   POL(s_0(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   from_3(x) -> cons_0(x, from_2(s_0(x)))
   from_4(x) -> cons_0(x, from_2(s_0(x)))




----------------------------------------

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
cons_1: empty set
nil_0: empty set

----------------------------------------

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))
   cons_1(s_0(s_0(x)), xs) -> nil_0

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
cons_1: empty set
nil_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons_0(x_1, x_2)) = x_1 + x_2
   POL(cons_1(x_1, x_2)) = 1
   POL(from_1(x_1)) = 1
   POL(from_2(x_1)) = 1 + x_1
   POL(nil_0) = 0
   POL(s_0(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   cons_1(s_0(s_0(x)), xs) -> nil_0




----------------------------------------

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
cons_1: empty set

----------------------------------------

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))
   from_1(x) -> cons_1(x, from_1(s_0(x)))

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
cons_1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons_0(x_1, x_2)) = 2*x_1 + x_2
   POL(cons_1(x_1, x_2)) = 0
   POL(from_1(x_1)) = 1
   POL(from_2(x_1)) = 1 + 2*x_1
   POL(s_0(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   from_1(x) -> cons_1(x, from_1(s_0(x)))




----------------------------------------

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}

----------------------------------------

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   from_2(x) -> cons_0(x, from_1(s_0(x)))

The replacement map contains the following entries:

from_2: empty set
cons_0: {1, 2}
from_1: empty set
s_0: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(cons_0(x_1, x_2)) = x_1 + x_2
   POL(from_1(x_1)) = 0
   POL(from_2(x_1)) = 1 + x_1
   POL(s_0(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   from_2(x) -> cons_0(x, from_1(s_0(x)))




----------------------------------------

(8)
Obligation:
Context-sensitive rewrite system:
R is empty.

----------------------------------------

(9) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(10)
YES