YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 44 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(a(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   d(d(d(x1))) -> a(c(x1))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a(x_1)) = 30 + x_1
   POL(b(x_1)) = 27 + x_1
   POL(c(x_1)) = 32 + x_1
   POL(d(x_1)) = 21 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a(a(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   d(d(d(x1))) -> a(c(x1))




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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