/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 67 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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(y))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[0, minus_1, *_2] > +_2 > p_1 > s_1


Status:
+_2: multiset status
0: multiset status
s_1: multiset status
p_1: multiset status
minus_1: 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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(y))




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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