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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 38 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   zeros -> cons(0, zeros)
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tail: {1}

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

   zeros -> cons(0, zeros)
   tail(cons(X, XS)) -> XS

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

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

   tail(cons(X, XS)) -> XS




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

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

   zeros -> cons(0, zeros)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set

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

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

   zeros -> cons(0, zeros)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(cons(x_1, x_2)) = x_1
   POL(zeros) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   zeros -> cons(0, zeros)




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

(4)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(5) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(6)
YES