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proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt2-5.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) FlatCCProof [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 74 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 118 ms]
(8) YES


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

   f(f(x)) -> x

The relative TRS consists of the following S rules:

   f(x) -> g(f(g(f(x))))


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

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

   f(f(f(x))) -> f(x)
   g(f(f(x))) -> g(x)

The relative TRS consists of the following S rules:

   f(f(x)) -> f(g(f(g(f(x)))))
   g(f(x)) -> g(g(f(g(f(x)))))


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


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

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

The relative TRS consists of the following S rules:

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


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

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

   f_{f_1}(f_{f_1}(f_{f_1}(x))) -> f_{f_1}(x)
   f_{f_1}(f_{f_1}(f_{g_1}(x))) -> f_{g_1}(x)
   g_{f_1}(f_{f_1}(f_{f_1}(x))) -> g_{f_1}(x)
   g_{f_1}(f_{f_1}(f_{g_1}(x))) -> g_{g_1}(x)
Rules from S:

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




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

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


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(7) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES