/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 60 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 8 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(X) -> if(X, c, n__f(n__true))
   if(true, X, Y) -> X
   if(false, X, Y) -> activate(Y)
   f(X) -> n__f(X)
   true -> n__true
   activate(n__f(X)) -> f(activate(X))
   activate(n__true) -> true
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(activate(x_1)) = x_1
   POL(c) = 0
   POL(f(x_1)) = 2*x_1
   POL(false) = 1
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(n__f(x_1)) = 2*x_1
   POL(n__true) = 0
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   if(false, X, Y) -> activate(Y)




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

   f(X) -> if(X, c, n__f(n__true))
   if(true, X, Y) -> X
   f(X) -> n__f(X)
   true -> n__true
   activate(n__f(X)) -> f(activate(X))
   activate(n__true) -> true
   activate(X) -> X

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(activate(x_1)) = 2 + 2*x_1
   POL(c) = 0
   POL(f(x_1)) = 2 + 2*x_1
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(n__f(x_1)) = 2 + 2*x_1
   POL(n__true) = 0
   POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   if(true, X, Y) -> X
   true -> n__true
   activate(n__true) -> true
   activate(X) -> X




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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(X) -> if(X, c, n__f(n__true))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(activate(X))

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
activate_1 > [f_1, n__true] > n__f_1 > [if_3, c]


Status:
f_1: multiset status
if_3: [2,3,1]
c: multiset status
n__f_1: multiset status
n__true: multiset status
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) -> if(X, c, n__f(n__true))
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(activate(X))




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(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(7) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES