/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 68 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   +(x, 0) -> x
   +(minus(x), x) -> 0
   minus(0) -> 0
   minus(minus(x)) -> x
   minus(+(x, y)) -> +(minus(y), minus(x))
   *(x, 1) -> x
   *(x, 0) -> 0
   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(x, minus(y)) -> minus(*(x, y))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
+/2(YES,YES)
0/0)
minus/1)YES(
*/2(YES,YES)
1/0)

Quasi precedence:
*_2 > +_2 > 0


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

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

   +(x, 0) -> x
   +(minus(x), x) -> 0
   *(x, 1) -> x
   *(x, 0) -> 0
   *(x, +(y, z)) -> +(*(x, y), *(x, z))




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
*_2 > minus_1 > 0
*_2 > minus_1 > +_2


Status:
minus_1: multiset status
0: multiset status
+_2: multiset status
*_2: [1,2]

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

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




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES