/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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, 46 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 0 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) TRUE


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

(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:

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

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:

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

Q is empty.

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

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

   POL(a_{a_1}(x_1)) = 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)) = x_1
   POL(c_{b_1}(x_1)) = 1 + x_1
   POL(c_{c_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:

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




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(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(7) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
   A_{A_1}(a_{a_1}(x1)) -> C_{A_1}(a_{a_1}(x1))
   A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
   A_{A_1}(a_{b_1}(x1)) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
   A_{A_1}(a_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1))
   A_{A_1}(a_{c_1}(x1)) -> B_{A_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
   A_{A_1}(a_{c_1}(x1)) -> C_{A_1}(a_{c_1}(x1))
   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{c_1}(c_{a_1}(x1)))
   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1)
   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{c_1}(c_{b_1}(x1)))
   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{A_1}(a_{c_1}(c_{c_1}(x1)))
   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{c_1}(x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{c_1}(c_{a_1}(x1)))
   A_{A_1}(a_{b_1}(x1)) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{c_1}(c_{b_1}(x1)))
   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))
   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{A_1}(a_{c_1}(c_{c_1}(x1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(11) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{c_1}(c_{a_1}(x1)))
   A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{a_1}(x1))
   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{c_1}(c_{b_1}(x1)))
   A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(x1))
   A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{A_1}(a_{c_1}(c_{c_1}(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A_{A_1}(x_1)) = 1 + x_1
   POL(A_{C_1}(x_1)) = 1 + x_1
   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = 1 + x_1
   POL(b_{b_1}(x_1)) = 1 + x_1
   POL(b_{c_1}(x_1)) = 1 + x_1
   POL(c_{a_1}(x_1)) = x_1
   POL(c_{b_1}(x_1)) = x_1
   POL(c_{c_1}(x_1)) = x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


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(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{a_1}(a_{a_1}(x1)))
   A_{A_1}(a_{b_1}(x1)) -> A_{C_1}(c_{a_1}(a_{b_1}(x1)))
   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{a_1}(a_{c_1}(x1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(13) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
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(14)
TRUE