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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [SOUND, 0 ms]
(2) RelTRS
(3) FlatCCProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) RootLabelingProof [EQUIVALENT, 0 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 57 ms]
(8) RelTRS
(9) RIsEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   f(g(x)) -> x

The relative TRS consists of the following S rules:

   a -> h(g(f(a)))


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(1) RelTRS Reverse (SOUND)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   f(g(x)) -> x

The set of rules S is 
   a -> h(g(f(a)))

We have obtained the following relative TRS:
The set of rules R is 
   g(f(x)) -> x

The set of rules S is 
   a'(x) -> a'(f(g(h(x))))


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(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   g(f(x)) -> x

The relative TRS consists of the following S rules:

   a'(x) -> a'(f(g(h(x))))


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(3) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]

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(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   g(g(f(x))) -> g(x)
   f(g(f(x))) -> f(x)
   a'(g(f(x))) -> a'(x)
   h(g(f(x))) -> h(x)

The relative TRS consists of the following S rules:

   a'(x) -> a'(f(g(h(x))))


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


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(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   g_{g_1}(g_{f_1}(f_{g_1}(x))) -> g_{g_1}(x)
   g_{g_1}(g_{f_1}(f_{f_1}(x))) -> g_{f_1}(x)
   g_{g_1}(g_{f_1}(f_{a'_1}(x))) -> g_{a'_1}(x)
   g_{g_1}(g_{f_1}(f_{h_1}(x))) -> g_{h_1}(x)
   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(x)
   f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{f_1}(x)
   f_{g_1}(g_{f_1}(f_{a'_1}(x))) -> f_{a'_1}(x)
   f_{g_1}(g_{f_1}(f_{h_1}(x))) -> f_{h_1}(x)
   a'_{g_1}(g_{f_1}(f_{g_1}(x))) -> a'_{g_1}(x)
   a'_{g_1}(g_{f_1}(f_{f_1}(x))) -> a'_{f_1}(x)
   a'_{g_1}(g_{f_1}(f_{a'_1}(x))) -> a'_{a'_1}(x)
   a'_{g_1}(g_{f_1}(f_{h_1}(x))) -> a'_{h_1}(x)
   h_{g_1}(g_{f_1}(f_{g_1}(x))) -> h_{g_1}(x)
   h_{g_1}(g_{f_1}(f_{f_1}(x))) -> h_{f_1}(x)
   h_{g_1}(g_{f_1}(f_{a'_1}(x))) -> h_{a'_1}(x)
   h_{g_1}(g_{f_1}(f_{h_1}(x))) -> h_{h_1}(x)

The relative TRS consists of the following S rules:

   a'_{g_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{g_1}(x))))
   a'_{f_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{f_1}(x))))
   a'_{a'_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{a'_1}(x))))
   a'_{h_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{h_1}(x))))


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(7) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a'_{a'_1}(x_1)) = 1 + x_1
   POL(a'_{f_1}(x_1)) = x_1
   POL(a'_{g_1}(x_1)) = x_1
   POL(a'_{h_1}(x_1)) = x_1
   POL(f_{a'_1}(x_1)) = 1 + x_1
   POL(f_{f_1}(x_1)) = 1 + x_1
   POL(f_{g_1}(x_1)) = x_1
   POL(f_{h_1}(x_1)) = 1 + x_1
   POL(g_{a'_1}(x_1)) = x_1
   POL(g_{f_1}(x_1)) = 1 + x_1
   POL(g_{g_1}(x_1)) = x_1
   POL(g_{h_1}(x_1)) = x_1
   POL(h_{a'_1}(x_1)) = 1 + x_1
   POL(h_{f_1}(x_1)) = x_1
   POL(h_{g_1}(x_1)) = x_1
   POL(h_{h_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   g_{g_1}(g_{f_1}(f_{g_1}(x))) -> g_{g_1}(x)
   g_{g_1}(g_{f_1}(f_{f_1}(x))) -> g_{f_1}(x)
   g_{g_1}(g_{f_1}(f_{a'_1}(x))) -> g_{a'_1}(x)
   g_{g_1}(g_{f_1}(f_{h_1}(x))) -> g_{h_1}(x)
   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(x)
   f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{f_1}(x)
   f_{g_1}(g_{f_1}(f_{a'_1}(x))) -> f_{a'_1}(x)
   f_{g_1}(g_{f_1}(f_{h_1}(x))) -> f_{h_1}(x)
   a'_{g_1}(g_{f_1}(f_{g_1}(x))) -> a'_{g_1}(x)
   a'_{g_1}(g_{f_1}(f_{f_1}(x))) -> a'_{f_1}(x)
   a'_{g_1}(g_{f_1}(f_{a'_1}(x))) -> a'_{a'_1}(x)
   a'_{g_1}(g_{f_1}(f_{h_1}(x))) -> a'_{h_1}(x)
   h_{g_1}(g_{f_1}(f_{g_1}(x))) -> h_{g_1}(x)
   h_{g_1}(g_{f_1}(f_{f_1}(x))) -> h_{f_1}(x)
   h_{g_1}(g_{f_1}(f_{a'_1}(x))) -> h_{a'_1}(x)
   h_{g_1}(g_{f_1}(f_{h_1}(x))) -> h_{h_1}(x)
Rules from S:
none




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(8)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   a'_{g_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{g_1}(x))))
   a'_{f_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{f_1}(x))))
   a'_{a'_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{a'_1}(x))))
   a'_{h_1}(x) -> a'_{f_1}(f_{g_1}(g_{h_1}(h_{h_1}(x))))


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