/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 77 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 6 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(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

Q is empty.

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

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

   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(a) -> a
   a__b -> b




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

   a__f(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(b) -> a__b
   a__f(X1, X2, X3) -> f(X1, X2, X3)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[a, a__b, mark_1] > a__f_3 > [b, f_3]


Status:
a__f_3: multiset status
a: multiset status
a__b: multiset status
b: multiset status
mark_1: multiset status
f_3: [3,1,2]

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

   mark(b) -> a__b
   a__f(X1, X2, X3) -> f(X1, X2, X3)




----------------------------------------

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

   a__f(a, X, X) -> a__f(X, a__b, b)
   a__b -> a

Q is empty.

----------------------------------------

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

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

   a__f(a, X, X) -> a__f(X, a__b, b)




----------------------------------------

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

   a__b -> a

Q is empty.

----------------------------------------

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

   POL(a) = 0
   POL(a__b) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a__b -> a




----------------------------------------

(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