/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 70 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 3 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 2 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

Q is empty.

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

   POL(a(x_1)) = 3 + x_1
   POL(b(x_1)) = 5 + x_1
   POL(c(x_1)) = 13 + 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:

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




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

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

Q is empty.

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

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

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




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

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

Q is empty.

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

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

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