/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 75 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 17 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 12 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

Q is empty.

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

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

   a__if(false, X, Y) -> mark(Y)




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

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

Q is empty.

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

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

   mark(false) -> false




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

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

Q is empty.

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

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

   mark(f(X)) -> a__f(mark(X))




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(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

Q is empty.

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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:mark_1 > a__if_3 > if_3 > f_1 > true > c > a__f_1

and weight map:

   c=1
   true=1
   a__f_1=4
   mark_1=0
   f_1=1
   a__if_3=0
   if_3=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:

   a__f(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)




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(8)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(9) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(10)
YES