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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (7) YES
    (8) QDP
        (9) QDPOrderProof [EQUIVALENT, 622 ms]
        (10) QDP
        (11) PisEmptyProof [EQUIVALENT, 0 ms]
        (12) YES


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

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

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

Q is empty.

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

The TRS R consists of the following rules:

   b(b(x1)) -> b(a(b(a(a(b(x1))))))
   b(b(a(b(x1)))) -> b(a(b(b(x1))))
   b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1))))))
   b(b(a(a(b(x1))))) -> b(a(a(b(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 2 SCCs with 5 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(6) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*B(b(a(a(b(x1))))) -> B(b(x1))
The graph contains the following edges 1 > 1


*B(b(a(b(x1)))) -> B(b(x1))
The graph contains the following edges 1 > 1


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

(7)
YES

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

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

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

The TRS R consists of the following rules:

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

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

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


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

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


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

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

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

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

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

(12)
YES