/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 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, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   *(*(x, y), z) -> *(x, *(y, z))
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(l))
   prod(app(l1, l2)) -> *(prod(l1), prod(l2))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[*_2, app_2, sum_1, prod_1] > +_2 > [s_1, nil]
[*_2, app_2, sum_1, prod_1] > 0
[*_2, app_2, sum_1, prod_1] > cons_2


Status:
+_2: [1,2]
0: multiset status
s_1: [1]
*_2: [1,2]
app_2: [1,2]
nil: multiset status
cons_2: multiset status
sum_1: multiset status
prod_1: [1]

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

   +(x, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   *(*(x, y), z) -> *(x, *(y, z))
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(l))
   prod(app(l1, l2)) -> *(prod(l1), prod(l2))




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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