/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 19 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 1111 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 1102 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 13 ms]
(12) QDP
(13) PisEmptyProof [EQUIVALENT, 0 ms]
(14) YES


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

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

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

Q is empty.

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

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

Q is empty.

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

(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.

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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) 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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

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

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

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

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

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


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

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


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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(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_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{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] with arctic natural numbers [ARCTIC]:

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

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

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

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

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


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

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


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

(10)
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_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) 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_{B_1}(b_{b_1}(b_{b_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B_{B_1}(x_1)) = x_1
   POL(a_{a_1}(x_1)) = 0
   POL(a_{b_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = 1 + x_1
   POL(b_{b_1}(x_1)) = x_1

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

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


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

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(14)
YES