YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyPairsProof [EQUIVALENT, 3 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 1 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 2415 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 36 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) TRUE


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

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

Q is empty.

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

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

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

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

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

The TRS R consists of the following rules:

   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(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(a(b(a(a(b(b(x1))))))) -> B(b(b(a(b(x1)))))
   B(a(b(a(a(b(b(x1))))))) -> B(b(a(b(x1))))
   B(a(b(a(a(b(b(x1))))))) -> B(a(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(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
>>>

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


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

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


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

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

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

The TRS R consists of the following rules:

   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(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(b(a(b(x1))))) -> B(a(b(b(a(b(x1))))))
   B(a(b(b(x1)))) -> B(a(b(a(b(x1)))))
   B(a(b(b(x1)))) -> B(a(b(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( B_1(x_1) ) = max{0, 2x_1 - 2}
POL( b_1(x_1) ) = x_1 + 2
POL( a_1(x_1) ) = max{0, x_1 - 2}

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

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


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

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

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

The TRS R consists of the following rules:

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

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

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

(10)
TRUE