/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 49 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:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, x)))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[f_1, g_2] > +_2 > s_1
[0, 1] > s_1


Status:
f_1: [1]
0: multiset status
1: multiset status
s_1: [1]
g_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:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, x)))




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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