/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 35 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   b(b(b(x1))) -> x1
   c(c(c(x1))) -> a(a(x1))

The relative TRS consists of the following S rules:

   a(x1) -> a(c(b(x1)))


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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:

   b(b(b(b(x1)))) -> b(x1)
   c(b(b(b(x1)))) -> c(x1)
   a(b(b(b(x1)))) -> a(x1)
   b(c(c(c(x1)))) -> b(a(a(x1)))
   c(c(c(c(x1)))) -> c(a(a(x1)))
   a(c(c(c(x1)))) -> a(a(a(x1)))

The relative TRS consists of the following S rules:

   a(x1) -> a(c(b(x1)))


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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:

   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(x1)
   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(x1)
   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(x1)
   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(x1)
   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(x1)
   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(x1)
   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(x1)
   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1)))
   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(x1)))
   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))

The relative TRS consists of the following S rules:

   a_{b_1}(x1) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   a_{c_1}(x1) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(x1) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))


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

   POL(a_{a_1}(x_1)) = x_1
   POL(a_{b_1}(x_1)) = 1 + x_1
   POL(a_{c_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = 1 + x_1
   POL(b_{c_1}(x_1)) = x_1
   POL(c_{a_1}(x_1)) = 1 + x_1
   POL(c_{b_1}(x_1)) = x_1
   POL(c_{c_1}(x_1)) = 1 + x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(x1)
   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(x1)
   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(x1)
   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(x1)
   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(x1)
   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(x1)
   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(x1)
   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1)))
   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(x1)))
   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
Rules from S:
none




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

   a_{b_1}(x1) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   a_{c_1}(x1) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(x1) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))


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