/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 105 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:

   f(X) -> cons(X, n__f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/1(YES)
cons/2(YES,YES)
n__f/1)YES(
g/1(YES)
0/0)
s/1(YES)
sel/2(YES,YES)
activate/1(YES)

Quasi precedence:
sel_2 > [f_1, activate_1] > g_1 > 0 > cons_2
sel_2 > [f_1, activate_1] > g_1 > s_1 > cons_2


Status:
f_1: multiset status
cons_2: multiset status
g_1: multiset status
0: multiset status
s_1: multiset status
sel_2: [1,2]
activate_1: multiset status

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

   f(X) -> cons(X, n__f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   f(X) -> n__f(X)
   activate(X) -> X




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

   activate(n__f(X)) -> f(X)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
n__f_1 > [activate_1, f_1]


Status:
activate_1: multiset status
n__f_1: multiset status
f_1: multiset status

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

   activate(n__f(X)) -> f(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