YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 47 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 4 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 0 ms]
(10) QTRS
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) 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

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1, x2))
   mark(b)
   mark(a)
   a__f(x0, x1, x2)


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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




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

(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)

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1, x2))
   mark(b)
   mark(a)
   a__f(x0, x1, x2)


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

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

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

   mark(b) -> a__b




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

(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
   a__f(X1, X2, X3) -> f(X1, X2, X3)

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1, x2))
   mark(b)
   mark(a)
   a__f(x0, x1, x2)


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

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

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

   a__f(X1, X2, X3) -> f(X1, X2, X3)




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

(6)
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

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1, x2))
   mark(b)
   mark(a)
   a__f(x0, x1, x2)


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

(7) 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)




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

(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a__b -> a

The set Q consists of the following terms:

   a__b
   mark(f(x0, x1, x2))
   mark(b)
   mark(a)
   a__f(x0, x1, x2)


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(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:a__b > a

and weight map:

   a__b=1
   a=1

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__b -> a




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

(10)
Obligation:
Q restricted rewrite system:
R is empty.
The set Q consists of the following terms:

   a__b
   mark(f(x0, x1, x2))
   mark(b)
   mark(a)
   a__f(x0, x1, x2)


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(11) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(12)
YES