/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) DependencyPairsProof [EQUIVALENT, 34 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 523 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 142 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 302 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 127 ms]
(12) QDP
(13) QDPOrderProof [EQUIVALENT, 259 ms]
(14) QDP
(15) QDPOrderProof [EQUIVALENT, 1505 ms]
(16) QDP
(17) DependencyGraphProof [EQUIVALENT, 0 ms]
(18) TRUE


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

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

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

Q is empty.

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

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

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1))))))))
   B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1)))))))
   B(b(a(a(a(x1))))) -> A(b(b(a(b(b(x1))))))
   B(b(a(a(a(x1))))) -> B(b(a(b(b(x1)))))
   B(b(a(a(a(x1))))) -> B(a(b(b(x1))))
   B(b(a(a(a(x1))))) -> A(b(b(x1)))
   B(b(a(a(a(x1))))) -> B(b(x1))
   B(b(a(a(a(x1))))) -> B(x1)
   A(x1) -> B(b(b(x1)))
   A(x1) -> B(b(x1))
   A(x1) -> B(x1)

The TRS R consists of the following rules:

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

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

(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B(b(a(a(a(x1))))) -> B(b(a(b(b(x1)))))
   B(b(a(a(a(x1))))) -> B(b(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>


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

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


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

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

   B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1))))))))
   B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1)))))))
   B(b(a(a(a(x1))))) -> A(b(b(a(b(b(x1))))))
   B(b(a(a(a(x1))))) -> B(a(b(b(x1))))
   B(b(a(a(a(x1))))) -> A(b(b(x1)))
   B(b(a(a(a(x1))))) -> B(x1)
   A(x1) -> B(b(b(x1)))
   A(x1) -> B(b(x1))
   A(x1) -> B(x1)

The TRS R consists of the following rules:

   b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1))))))))
   a(b(b(x1))) -> x1
   a(x1) -> b(b(b(x1)))
   a(x1) -> b(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(a(a(a(x1))))) -> A(b(b(a(b(b(x1))))))
   B(b(a(a(a(x1))))) -> A(b(b(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>

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


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

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


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

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

   B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1))))))))
   B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1)))))))
   B(b(a(a(a(x1))))) -> B(a(b(b(x1))))
   B(b(a(a(a(x1))))) -> B(x1)
   A(x1) -> B(b(b(x1)))
   A(x1) -> B(b(x1))
   A(x1) -> B(x1)

The TRS R consists of the following rules:

   b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1))))))))
   a(b(b(x1))) -> x1
   a(x1) -> b(b(b(x1)))
   a(x1) -> b(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(a(a(a(x1))))) -> B(a(b(b(x1))))
   B(b(a(a(a(x1))))) -> B(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [1A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>


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

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


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

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

   B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1))))))))
   B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1)))))))
   A(x1) -> B(b(b(x1)))
   A(x1) -> B(b(x1))
   A(x1) -> B(x1)

The TRS R consists of the following rules:

   b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1))))))))
   a(b(b(x1))) -> x1
   a(x1) -> b(b(b(x1)))
   a(x1) -> b(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(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1)))))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>

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


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

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


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

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

   B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1))))))))
   A(x1) -> B(b(b(x1)))
   A(x1) -> B(b(x1))
   A(x1) -> B(x1)

The TRS R consists of the following rules:

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

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

(13) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A(x1) -> B(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [1A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(A(x_1)) =  	[[1A]] 	 +  	[[1A, 1A, 1A]] 	* 	x_1
>>>


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

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


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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1))))))))
   A(x1) -> B(b(b(x1)))
   A(x1) -> B(b(x1))

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A(x1) -> B(b(b(x1)))
   A(x1) -> B(b(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:

   <<<
 POL(B(x_1)) =  	[[0A]] 	 +  	[[-1A, -I, -1A]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[0A], [-1A], [-I]] 	 +  	[[-I, -I, -1A], [0A, -I, 0A], [-I, -1A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[2A], [0A], [-1A]] 	 +  	[[-1A, -1A, 1A], [2A, -1A, 0A], [-1A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(A(x_1)) =  	[[1A]] 	 +  	[[-1A, -1A, 1A]] 	* 	x_1
>>>


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

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


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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1))))))))

The TRS R consists of the following rules:

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

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

(17) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(18)
TRUE