YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 10 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 7665 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) TRUE


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

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

Q is empty.

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(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

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

   b(b(b(x1))) -> b(a(b(a(b(a(b(b(x1))))))))
   b(b(a(b(x1)))) -> b(a(a(b(a(b(a(b(x1))))))))
   b(a(a(b(a(b(a(b(b(x1))))))))) -> b(a(b(b(b(b(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:

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

Q is empty.

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))
   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))
   B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes.
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(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))

The TRS R consists of the following rules:

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

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


The following pairs can be oriented strictly and are deleted.

   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

   <<<
 POL(B_{B_1}(x_1)) =  	[[1]] 	 +  	[[1, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[0, 0, 1], [0, 1, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[0, 1, 0], [1, 0, 0], [0, 0, 1]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[1, 0, 0], [1, 1, 0], [1, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(B_{A_1}(x_1)) =  	[[0]] 	 +  	[[0, 0, 1]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[0, 0, 0], [1, 0, 0], [1, 0, 0]] 	* 	x_1
>>>


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

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


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

   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
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(12)
TRUE