28.87/8.27	YES
29.92/8.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
29.92/8.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.92/8.78	
29.92/8.78	
29.92/8.78	Termination w.r.t. Q of the given QTRS could be proven:
29.92/8.78	
29.92/8.78	(0) QTRS
29.92/8.78	(1) QTRS Reverse [EQUIVALENT, 0 ms]
29.92/8.78	(2) QTRS
29.92/8.78	(3) DependencyPairsProof [EQUIVALENT, 30 ms]
29.92/8.78	(4) QDP
29.92/8.78	(5) DependencyGraphProof [EQUIVALENT, 4 ms]
29.92/8.78	(6) QDP
29.92/8.78	(7) QDPOrderProof [EQUIVALENT, 71 ms]
29.92/8.78	(8) QDP
29.92/8.78	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
29.92/8.78	(10) QDP
29.92/8.78	(11) QDPOrderProof [EQUIVALENT, 205 ms]
29.92/8.78	(12) QDP
29.92/8.78	(13) PisEmptyProof [EQUIVALENT, 0 ms]
29.92/8.78	(14) YES
29.92/8.78	
29.92/8.78	
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(0)
29.92/8.78	Obligation:
29.92/8.78	Q restricted rewrite system:
29.92/8.78	The TRS R consists of the following rules:
29.92/8.78	
29.92/8.78	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
29.92/8.78	   a(a(b(b(x1)))) -> a(b(a(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> b(a(a(a(x1))))
29.92/8.78	   a(b(a(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	
29.92/8.78	Q is empty.
29.92/8.78	
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(1) QTRS Reverse (EQUIVALENT)
29.92/8.78	We applied the QTRS Reverse Processor [REVERSE].
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(2)
29.92/8.78	Obligation:
29.92/8.78	Q restricted rewrite system:
29.92/8.78	The TRS R consists of the following rules:
29.92/8.78	
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	
29.92/8.78	Q is empty.
29.92/8.78	
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(3) DependencyPairsProof (EQUIVALENT)
29.92/8.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(4)
29.92/8.78	Obligation:
29.92/8.78	Q DP problem:
29.92/8.78	The TRS P consists of the following rules:
29.92/8.78	
29.92/8.78	   B(b(a(b(x1)))) -> A(a(b(b(x1))))
29.92/8.78	   B(b(a(b(x1)))) -> A(b(b(x1)))
29.92/8.78	   B(b(a(b(x1)))) -> B(b(x1))
29.92/8.78	   B(b(a(a(x1)))) -> A(a(b(a(x1))))
29.92/8.78	   B(b(a(a(x1)))) -> A(b(a(x1)))
29.92/8.78	   B(b(a(a(x1)))) -> B(a(x1))
29.92/8.78	   B(b(b(b(x1)))) -> A(a(a(b(x1))))
29.92/8.78	   B(b(b(b(x1)))) -> A(a(b(x1)))
29.92/8.78	   B(b(b(b(x1)))) -> A(b(x1))
29.92/8.78	   A(a(b(a(x1)))) -> A(b(b(a(x1))))
29.92/8.78	   A(a(b(a(x1)))) -> B(b(a(x1)))
29.92/8.78	
29.92/8.78	The TRS R consists of the following rules:
29.92/8.78	
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	
29.92/8.78	Q is empty.
29.92/8.78	We have to consider all minimal (P,Q,R)-chains.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(5) DependencyGraphProof (EQUIVALENT)
29.92/8.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(6)
29.92/8.78	Obligation:
29.92/8.78	Q DP problem:
29.92/8.78	The TRS P consists of the following rules:
29.92/8.78	
29.92/8.78	   A(a(b(a(x1)))) -> A(b(b(a(x1))))
29.92/8.78	   A(a(b(a(x1)))) -> B(b(a(x1)))
29.92/8.78	   B(b(a(b(x1)))) -> A(a(b(b(x1))))
29.92/8.78	   B(b(a(b(x1)))) -> A(b(b(x1)))
29.92/8.78	   B(b(a(b(x1)))) -> B(b(x1))
29.92/8.78	   B(b(a(a(x1)))) -> A(a(b(a(x1))))
29.92/8.78	   B(b(a(a(x1)))) -> A(b(a(x1)))
29.92/8.78	   B(b(b(b(x1)))) -> A(a(a(b(x1))))
29.92/8.78	   B(b(b(b(x1)))) -> A(a(b(x1)))
29.92/8.78	   B(b(b(b(x1)))) -> A(b(x1))
29.92/8.78	
29.92/8.78	The TRS R consists of the following rules:
29.92/8.78	
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	
29.92/8.78	Q is empty.
29.92/8.78	We have to consider all minimal (P,Q,R)-chains.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(7) QDPOrderProof (EQUIVALENT)
29.92/8.78	We use the reduction pair processor [LPAR04,JAR06].
29.92/8.78	
29.92/8.78	
29.92/8.78	The following pairs can be oriented strictly and are deleted.
29.92/8.78	
29.92/8.78	   A(a(b(a(x1)))) -> B(b(a(x1)))
29.92/8.78	   B(b(a(b(x1)))) -> A(b(b(x1)))
29.92/8.78	   B(b(a(b(x1)))) -> B(b(x1))
29.92/8.78	   B(b(a(a(x1)))) -> A(b(a(x1)))
29.92/8.78	   B(b(b(b(x1)))) -> A(a(b(x1)))
29.92/8.78	   B(b(b(b(x1)))) -> A(b(x1))
29.92/8.78	The remaining pairs can at least be oriented weakly.
29.92/8.78	Used ordering:  Polynomial interpretation [POLO]:
29.92/8.78	
29.92/8.78	   POL(A(x_1)) = x_1
29.92/8.78	   POL(B(x_1)) = x_1
29.92/8.78	   POL(a(x_1)) = 1 + x_1
29.92/8.78	   POL(b(x_1)) = 1 + x_1
29.92/8.78	
29.92/8.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.92/8.78	
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	
29.92/8.78	
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(8)
29.92/8.78	Obligation:
29.92/8.78	Q DP problem:
29.92/8.78	The TRS P consists of the following rules:
29.92/8.78	
29.92/8.78	   A(a(b(a(x1)))) -> A(b(b(a(x1))))
29.92/8.78	   B(b(a(b(x1)))) -> A(a(b(b(x1))))
29.92/8.78	   B(b(a(a(x1)))) -> A(a(b(a(x1))))
29.92/8.78	   B(b(b(b(x1)))) -> A(a(a(b(x1))))
29.92/8.78	
29.92/8.78	The TRS R consists of the following rules:
29.92/8.78	
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	
29.92/8.78	Q is empty.
29.92/8.78	We have to consider all minimal (P,Q,R)-chains.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(9) DependencyGraphProof (EQUIVALENT)
29.92/8.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(10)
29.92/8.78	Obligation:
29.92/8.78	Q DP problem:
29.92/8.78	The TRS P consists of the following rules:
29.92/8.78	
29.92/8.78	   A(a(b(a(x1)))) -> A(b(b(a(x1))))
29.92/8.78	
29.92/8.78	The TRS R consists of the following rules:
29.92/8.78	
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	
29.92/8.78	Q is empty.
29.92/8.78	We have to consider all minimal (P,Q,R)-chains.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(11) QDPOrderProof (EQUIVALENT)
29.92/8.78	We use the reduction pair processor [LPAR04,JAR06].
29.92/8.78	
29.92/8.78	
29.92/8.78	The following pairs can be oriented strictly and are deleted.
29.92/8.78	
29.92/8.78	   A(a(b(a(x1)))) -> A(b(b(a(x1))))
29.92/8.78	The remaining pairs can at least be oriented weakly.
29.92/8.78	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.92/8.78	
29.92/8.78	   <<<
29.92/8.78	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
29.92/8.78	>>>
29.92/8.78	
29.92/8.78	   <<<
29.92/8.78	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [-I, -I, -I]] 	* 	x_1
29.92/8.78	>>>
29.92/8.78	
29.92/8.78	   <<<
29.92/8.78	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 1A, 0A], [-I, -I, 0A], [0A, -I, 0A]] 	* 	x_1
29.92/8.78	>>>
29.92/8.78	
29.92/8.78	
29.92/8.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.92/8.78	
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	
29.92/8.78	
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(12)
29.92/8.78	Obligation:
29.92/8.78	Q DP problem:
29.92/8.78	P is empty.
29.92/8.78	The TRS R consists of the following rules:
29.92/8.78	
29.92/8.78	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
29.92/8.78	   b(b(a(a(x1)))) -> a(a(b(a(x1))))
29.92/8.78	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
29.92/8.78	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
29.92/8.78	
29.92/8.78	Q is empty.
29.92/8.78	We have to consider all minimal (P,Q,R)-chains.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(13) PisEmptyProof (EQUIVALENT)
29.92/8.78	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.92/8.78	----------------------------------------
29.92/8.78	
29.92/8.78	(14)
29.92/8.78	YES
29.99/9.22	EOF