/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 58 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 9 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

Q is empty.

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

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

   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))




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

   f(s(0)) -> *(s(s(0)), f(0))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
f_1 > [0, *_2] > s_1


Status:
f_1: multiset status
s_1: multiset status
0: multiset status
*_2: multiset status

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

   f(s(0)) -> *(s(s(0)), f(0))




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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