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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 16 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 1 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 3 ms]
(8) YES


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

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

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

   POL(a(x_1)) = 66 + x_1
   POL(b(x_1)) = 44 + x_1
   POL(c(x_1)) = 29 + x_1
   POL(d(x_1)) = 19 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(a(a(x1)))) -> b(b(b(b(b(b(x1))))))

Q is empty.

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

   POL(a(x_1)) = 1 + x_1
   POL(b(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a(a(a(a(x1)))) -> b(b(b(b(b(b(x1))))))




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(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(7) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES