/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 175 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:

   a(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), l))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
a/4(YES,YES,YES,YES)
h/0)
s/1(YES)
+/2(YES,YES)
1/0)
app/2(YES,YES)
nil/0)
cons/2(YES,YES)
sum/1)YES(

Quasi precedence:
+_2 > [s_1, 1] > h
app_2 > cons_2 > a_4 > [s_1, 1] > h
app_2 > cons_2 > nil


Status:
a_4: [1,2,3,4]
h: multiset status
s_1: [1]
+_2: [1,2]
1: multiset status
app_2: [2,1]
nil: multiset status
cons_2: [2,1]

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

   a(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), l))




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

   sum(cons(x, nil)) -> cons(x, nil)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:sum_1 > nil > cons_2

and weight map:

   nil=1
   sum_1=0
   cons_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   sum(cons(x, nil)) -> cons(x, nil)




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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