/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 58 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)))
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   sel(0, cons(X, Z)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__first(X1, X2)) -> first(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)
first/2(YES,YES)
0/0)
nil/0)
n__first/2(YES,YES)
activate/1(YES)
sel/2(YES,YES)

Quasi precedence:
0 > nil > cons_2
sel_2 > [from_1, first_2, activate_1] > s_1 > n__first_2 > cons_2
sel_2 > [from_1, first_2, activate_1] > nil > cons_2


Status:
from_1: [1]
cons_2: multiset status
s_1: multiset status
first_2: [2,1]
0: multiset status
nil: multiset status
n__first_2: [1,2]
activate_1: [1]
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)))
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   sel(0, cons(X, Z)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__first(X1, X2)) -> first(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