/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 73 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:

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

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
exp_2 > *_2 > [0, s_1, -_2] > +_2


Status:
exp_2: [1,2]
0: multiset status
s_1: [1]
*_2: [1,2]
+_2: [1,2]
-_2: [2,1]

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

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