YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 27 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

The set Q consists of the following terms:

   a__zeros
   mark(zeros)
   mark(tail(x0))
   mark(cons(x0, x1))
   mark(0)
   a__tail(x0)


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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:mark_1 > a__tail_1 > tail_1 > a__zeros > zeros > cons_2 > 0

and weight map:

   a__zeros=4
   0=1
   zeros=1
   a__tail_1=2
   mark_1=3
   tail_1=2
   cons_2=1

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
The set Q consists of the following terms:

   a__zeros
   mark(zeros)
   mark(tail(x0))
   mark(cons(x0, x1))
   mark(0)
   a__tail(x0)


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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES