/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 113 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:

   *(x, 1) -> x
   *(1, y) -> y
   *(i(x), x) -> 1
   *(x, i(x)) -> 1
   *(x, *(y, z)) -> *(*(x, y), z)
   i(1) -> 1
   *(*(x, y), i(y)) -> x
   *(*(x, i(y)), y) -> x
   i(i(x)) -> x
   i(*(x, y)) -> *(i(y), i(x))
   k(x, 1) -> 1
   k(x, x) -> 1
   *(k(x, y), k(y, x)) -> 1
   *(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
   k(*(x, i(y)), *(y, i(x))) -> 1

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
i_1 > *_2 > [1, k_2]


Status:
*_2: [2,1]
1: multiset status
i_1: [1]
k_2: [1,2]

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

   *(x, 1) -> x
   *(1, y) -> y
   *(i(x), x) -> 1
   *(x, i(x)) -> 1
   *(x, *(y, z)) -> *(*(x, y), z)
   i(1) -> 1
   *(*(x, y), i(y)) -> x
   *(*(x, i(y)), y) -> x
   i(i(x)) -> x
   i(*(x, y)) -> *(i(y), i(x))
   k(x, 1) -> 1
   k(x, x) -> 1
   *(k(x, y), k(y, x)) -> 1
   *(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
   k(*(x, i(y)), *(y, i(x))) -> 1




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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