/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 70 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:

   i(0) -> 0
   +(0, y) -> y
   +(x, 0) -> x
   i(i(x)) -> x
   +(i(x), x) -> 0
   +(x, i(x)) -> 0
   i(+(x, y)) -> +(i(x), i(y))
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, i(y)), y) -> x
   +(+(x, y), i(y)) -> x

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
i_1 > [0, +_2]


Status:
i_1: [1]
0: multiset status
+_2: [2,1]

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

   i(0) -> 0
   +(0, y) -> y
   +(x, 0) -> x
   i(i(x)) -> x
   +(i(x), x) -> 0
   +(x, i(x)) -> 0
   i(+(x, y)) -> +(i(x), i(y))
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, i(y)), y) -> x
   +(+(x, y), i(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