46.99/12.87	YES
47.49/12.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
47.49/12.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
47.49/12.95	
47.49/12.95	
47.49/12.95	Termination w.r.t. Q of the given QTRS could be proven:
47.49/12.95	
47.49/12.95	(0) QTRS
47.49/12.95	(1) QTRS Reverse [EQUIVALENT, 0 ms]
47.49/12.95	(2) QTRS
47.49/12.95	(3) DependencyPairsProof [EQUIVALENT, 23 ms]
47.49/12.95	(4) QDP
47.49/12.95	(5) QDPOrderProof [EQUIVALENT, 46 ms]
47.49/12.95	(6) QDP
47.49/12.95	(7) QDPOrderProof [EQUIVALENT, 193 ms]
47.49/12.95	(8) QDP
47.49/12.95	(9) QDPOrderProof [EQUIVALENT, 67 ms]
47.49/12.95	(10) QDP
47.49/12.95	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
47.49/12.95	(12) TRUE
47.49/12.95	
47.49/12.95	
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(0)
47.49/12.95	Obligation:
47.49/12.95	Q restricted rewrite system:
47.49/12.95	The TRS R consists of the following rules:
47.49/12.95	
47.49/12.95	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
47.49/12.95	   a(a(b(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(b(a(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	Q is empty.
47.49/12.95	
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(1) QTRS Reverse (EQUIVALENT)
47.49/12.95	We applied the QTRS Reverse Processor [REVERSE].
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(2)
47.49/12.95	Obligation:
47.49/12.95	Q restricted rewrite system:
47.49/12.95	The TRS R consists of the following rules:
47.49/12.95	
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	Q is empty.
47.49/12.95	
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(3) DependencyPairsProof (EQUIVALENT)
47.49/12.95	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(4)
47.49/12.95	Obligation:
47.49/12.95	Q DP problem:
47.49/12.95	The TRS P consists of the following rules:
47.49/12.95	
47.49/12.95	   B(a(a(a(x1)))) -> A(a(b(a(x1))))
47.49/12.95	   B(a(a(a(x1)))) -> A(b(a(x1)))
47.49/12.95	   B(a(a(a(x1)))) -> B(a(x1))
47.49/12.95	   A(b(a(a(x1)))) -> B(b(b(b(x1))))
47.49/12.95	   A(b(a(a(x1)))) -> B(b(b(x1)))
47.49/12.95	   A(b(a(a(x1)))) -> B(b(x1))
47.49/12.95	   A(b(a(a(x1)))) -> B(x1)
47.49/12.95	   B(a(b(b(x1)))) -> A(a(a(a(x1))))
47.49/12.95	   B(a(b(b(x1)))) -> A(a(a(x1)))
47.49/12.95	   B(a(b(b(x1)))) -> A(a(x1))
47.49/12.95	   B(a(b(b(x1)))) -> A(x1)
47.49/12.95	
47.49/12.95	The TRS R consists of the following rules:
47.49/12.95	
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	Q is empty.
47.49/12.95	We have to consider all minimal (P,Q,R)-chains.
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(5) QDPOrderProof (EQUIVALENT)
47.49/12.95	We use the reduction pair processor [LPAR04,JAR06].
47.49/12.95	
47.49/12.95	
47.49/12.95	The following pairs can be oriented strictly and are deleted.
47.49/12.95	
47.49/12.95	   B(a(a(a(x1)))) -> A(b(a(x1)))
47.49/12.95	   B(a(a(a(x1)))) -> B(a(x1))
47.49/12.95	   A(b(a(a(x1)))) -> B(b(b(x1)))
47.49/12.95	   A(b(a(a(x1)))) -> B(b(x1))
47.49/12.95	   A(b(a(a(x1)))) -> B(x1)
47.49/12.95	   B(a(b(b(x1)))) -> A(a(a(x1)))
47.49/12.95	   B(a(b(b(x1)))) -> A(a(x1))
47.49/12.95	   B(a(b(b(x1)))) -> A(x1)
47.49/12.95	The remaining pairs can at least be oriented weakly.
47.49/12.95	Used ordering:  Polynomial interpretation [POLO]:
47.49/12.95	
47.49/12.95	   POL(A(x_1)) = 1 + x_1
47.49/12.95	   POL(B(x_1)) = 1 + x_1
47.49/12.95	   POL(a(x_1)) = 1 + x_1
47.49/12.95	   POL(b(x_1)) = 1 + x_1
47.49/12.95	
47.49/12.95	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
47.49/12.95	
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(6)
47.49/12.95	Obligation:
47.49/12.95	Q DP problem:
47.49/12.95	The TRS P consists of the following rules:
47.49/12.95	
47.49/12.95	   B(a(a(a(x1)))) -> A(a(b(a(x1))))
47.49/12.95	   A(b(a(a(x1)))) -> B(b(b(b(x1))))
47.49/12.95	   B(a(b(b(x1)))) -> A(a(a(a(x1))))
47.49/12.95	
47.49/12.95	The TRS R consists of the following rules:
47.49/12.95	
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	Q is empty.
47.49/12.95	We have to consider all minimal (P,Q,R)-chains.
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(7) QDPOrderProof (EQUIVALENT)
47.49/12.95	We use the reduction pair processor [LPAR04,JAR06].
47.49/12.95	
47.49/12.95	
47.49/12.95	The following pairs can be oriented strictly and are deleted.
47.49/12.95	
47.49/12.95	   B(a(b(b(x1)))) -> A(a(a(a(x1))))
47.49/12.95	The remaining pairs can at least be oriented weakly.
47.49/12.95	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(b(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[0A, 1A, -I], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	
47.49/12.95	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
47.49/12.95	
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(8)
47.49/12.95	Obligation:
47.49/12.95	Q DP problem:
47.49/12.95	The TRS P consists of the following rules:
47.49/12.95	
47.49/12.95	   B(a(a(a(x1)))) -> A(a(b(a(x1))))
47.49/12.95	   A(b(a(a(x1)))) -> B(b(b(b(x1))))
47.49/12.95	
47.49/12.95	The TRS R consists of the following rules:
47.49/12.95	
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	Q is empty.
47.49/12.95	We have to consider all minimal (P,Q,R)-chains.
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(9) QDPOrderProof (EQUIVALENT)
47.49/12.95	We use the reduction pair processor [LPAR04,JAR06].
47.49/12.95	
47.49/12.95	
47.49/12.95	The following pairs can be oriented strictly and are deleted.
47.49/12.95	
47.49/12.95	   A(b(a(a(x1)))) -> B(b(b(b(x1))))
47.49/12.95	The remaining pairs can at least be oriented weakly.
47.49/12.95	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [-I, -I, 0A], [0A, 1A, -I]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	   <<<
47.49/12.95	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
47.49/12.95	>>>
47.49/12.95	
47.49/12.95	
47.49/12.95	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
47.49/12.95	
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(10)
47.49/12.95	Obligation:
47.49/12.95	Q DP problem:
47.49/12.95	The TRS P consists of the following rules:
47.49/12.95	
47.49/12.95	   B(a(a(a(x1)))) -> A(a(b(a(x1))))
47.49/12.95	
47.49/12.95	The TRS R consists of the following rules:
47.49/12.95	
47.49/12.95	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
47.49/12.95	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
47.49/12.95	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
47.49/12.95	
47.49/12.95	Q is empty.
47.49/12.95	We have to consider all minimal (P,Q,R)-chains.
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(11) DependencyGraphProof (EQUIVALENT)
47.49/12.95	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
47.49/12.95	----------------------------------------
47.49/12.95	
47.49/12.95	(12)
47.49/12.95	TRUE
47.60/13.01	EOF