/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 129 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:

   from(X) -> cons(X, n__from(s(X)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(activate(XS))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   from(X) -> n__from(X)
   take(X1, X2) -> n__take(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
from/1(YES)
cons/2(YES,YES)
n__from/1)YES(
s/1(YES)
head/1(YES)
2nd/1(YES)
activate/1(YES)
take/2(YES,YES)
0/0)
nil/0)
n__take/2(YES,YES)
sel/2(YES,YES)

Quasi precedence:
2nd_1 > [from_1, activate_1, take_2] > s_1 > cons_2 > n__take_2
2nd_1 > [from_1, activate_1, take_2] > [0, nil] > n__take_2
2nd_1 > head_1 > n__take_2
sel_2 > [from_1, activate_1, take_2] > s_1 > cons_2 > n__take_2
sel_2 > [from_1, activate_1, take_2] > [0, nil] > n__take_2


Status:
from_1: multiset status
cons_2: multiset status
s_1: multiset status
head_1: multiset status
2nd_1: multiset status
activate_1: multiset status
take_2: multiset status
0: multiset status
nil: multiset status
n__take_2: multiset status
sel_2: [1,2]

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

   from(X) -> cons(X, n__from(s(X)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(activate(XS))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   from(X) -> n__from(X)
   take(X1, X2) -> n__take(X1, X2)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(X) -> X




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

   activate(n__from(X)) -> from(X)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:activate_1 > from_1 > n__from_1

and weight map:

   activate_1=1
   n__from_1=1
   from_1=2

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

   activate(n__from(X)) -> from(X)




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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