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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) FlatCCProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 9 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(a) -> f(b)
   g(b) -> g(a)
   f(x) -> g(x)

Q is empty.

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(1) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]
As Q is empty the flat context closure was sound AND complete.

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

   f(a) -> f(b)
   g(b) -> g(a)
   f(f(x)) -> f(g(x))
   g(f(x)) -> g(g(x))

Q is empty.

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(3) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled

As Q is empty the root labeling was sound AND complete.

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

   f_{a}(a) -> f_{b}(b)
   g_{b}(b) -> g_{a}(a)
   f_{f_1}(f_{f_1}(x)) -> f_{g_1}(g_{f_1}(x))
   f_{f_1}(f_{a}(x)) -> f_{g_1}(g_{a}(x))
   f_{f_1}(f_{b}(x)) -> f_{g_1}(g_{b}(x))
   f_{f_1}(f_{g_1}(x)) -> f_{g_1}(g_{g_1}(x))
   g_{f_1}(f_{f_1}(x)) -> g_{g_1}(g_{f_1}(x))
   g_{f_1}(f_{a}(x)) -> g_{g_1}(g_{a}(x))
   g_{f_1}(f_{b}(x)) -> g_{g_1}(g_{b}(x))
   g_{f_1}(f_{g_1}(x)) -> g_{g_1}(g_{g_1}(x))

Q is empty.

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

   POL(a) = 0
   POL(b) = 1
   POL(f_{a}(x_1)) = 2 + x_1
   POL(f_{b}(x_1)) = x_1
   POL(f_{f_1}(x_1)) = 1 + x_1
   POL(f_{g_1}(x_1)) = x_1
   POL(g_{a}(x_1)) = x_1
   POL(g_{b}(x_1)) = x_1
   POL(g_{f_1}(x_1)) = 1 + x_1
   POL(g_{g_1}(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f_{a}(a) -> f_{b}(b)
   g_{b}(b) -> g_{a}(a)
   f_{f_1}(f_{f_1}(x)) -> f_{g_1}(g_{f_1}(x))
   f_{f_1}(f_{a}(x)) -> f_{g_1}(g_{a}(x))
   f_{f_1}(f_{b}(x)) -> f_{g_1}(g_{b}(x))
   f_{f_1}(f_{g_1}(x)) -> f_{g_1}(g_{g_1}(x))
   g_{f_1}(f_{f_1}(x)) -> g_{g_1}(g_{f_1}(x))
   g_{f_1}(f_{a}(x)) -> g_{g_1}(g_{a}(x))
   g_{f_1}(f_{b}(x)) -> g_{g_1}(g_{b}(x))
   g_{f_1}(f_{g_1}(x)) -> g_{g_1}(g_{g_1}(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