40.17/11.22	YES
40.91/11.32	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
40.91/11.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.91/11.32	
40.91/11.32	
40.91/11.32	Termination w.r.t. Q of the given QTRS could be proven:
40.91/11.32	
40.91/11.32	(0) QTRS
40.91/11.32	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
40.91/11.32	(2) QDP
40.91/11.32	(3) QDPOrderProof [EQUIVALENT, 37 ms]
40.91/11.32	(4) QDP
40.91/11.32	(5) QDPOrderProof [EQUIVALENT, 88 ms]
40.91/11.32	(6) QDP
40.91/11.32	(7) QDPOrderProof [EQUIVALENT, 537 ms]
40.91/11.32	(8) QDP
40.91/11.32	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
40.91/11.32	(10) TRUE
40.91/11.32	
40.91/11.32	
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(0)
40.91/11.32	Obligation:
40.91/11.32	Q restricted rewrite system:
40.91/11.32	The TRS R consists of the following rules:
40.91/11.32	
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	
40.91/11.32	Q is empty.
40.91/11.32	
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(1) DependencyPairsProof (EQUIVALENT)
40.91/11.32	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(2)
40.91/11.32	Obligation:
40.91/11.32	Q DP problem:
40.91/11.32	The TRS P consists of the following rules:
40.91/11.32	
40.91/11.32	   A(a(a(a(x1)))) -> A(b(b(a(x1))))
40.91/11.32	   A(a(a(a(x1)))) -> B(b(a(x1)))
40.91/11.32	   A(a(a(a(x1)))) -> B(a(x1))
40.91/11.32	   B(b(b(a(x1)))) -> A(a(b(a(x1))))
40.91/11.32	   B(b(b(a(x1)))) -> A(b(a(x1)))
40.91/11.32	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
40.91/11.32	   A(b(b(a(x1)))) -> A(a(b(x1)))
40.91/11.32	   A(b(b(a(x1)))) -> A(b(x1))
40.91/11.32	   A(b(b(a(x1)))) -> B(x1)
40.91/11.32	
40.91/11.32	The TRS R consists of the following rules:
40.91/11.32	
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	
40.91/11.32	Q is empty.
40.91/11.32	We have to consider all minimal (P,Q,R)-chains.
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(3) QDPOrderProof (EQUIVALENT)
40.91/11.32	We use the reduction pair processor [LPAR04,JAR06].
40.91/11.32	
40.91/11.32	
40.91/11.32	The following pairs can be oriented strictly and are deleted.
40.91/11.32	
40.91/11.32	   A(a(a(a(x1)))) -> B(b(a(x1)))
40.91/11.32	   A(a(a(a(x1)))) -> B(a(x1))
40.91/11.32	   B(b(b(a(x1)))) -> A(b(a(x1)))
40.91/11.32	   A(b(b(a(x1)))) -> A(a(b(x1)))
40.91/11.32	   A(b(b(a(x1)))) -> A(b(x1))
40.91/11.32	   A(b(b(a(x1)))) -> B(x1)
40.91/11.32	The remaining pairs can at least be oriented weakly.
40.91/11.32	Used ordering:  Polynomial interpretation [POLO]:
40.91/11.32	
40.91/11.32	   POL(A(x_1)) = 1 + x_1
40.91/11.32	   POL(B(x_1)) = 1 + x_1
40.91/11.32	   POL(a(x_1)) = 1 + x_1
40.91/11.32	   POL(b(x_1)) = 1 + x_1
40.91/11.32	
40.91/11.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.91/11.32	
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	
40.91/11.32	
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(4)
40.91/11.32	Obligation:
40.91/11.32	Q DP problem:
40.91/11.32	The TRS P consists of the following rules:
40.91/11.32	
40.91/11.32	   A(a(a(a(x1)))) -> A(b(b(a(x1))))
40.91/11.32	   B(b(b(a(x1)))) -> A(a(b(a(x1))))
40.91/11.32	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
40.91/11.32	
40.91/11.32	The TRS R consists of the following rules:
40.91/11.32	
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	
40.91/11.32	Q is empty.
40.91/11.32	We have to consider all minimal (P,Q,R)-chains.
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(5) QDPOrderProof (EQUIVALENT)
40.91/11.32	We use the reduction pair processor [LPAR04,JAR06].
40.91/11.32	
40.91/11.32	
40.91/11.32	The following pairs can be oriented strictly and are deleted.
40.91/11.32	
40.91/11.32	   A(a(a(a(x1)))) -> A(b(b(a(x1))))
40.91/11.32	The remaining pairs can at least be oriented weakly.
40.91/11.32	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, -I]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [-I, 0A, -I], [0A, -I, 0A]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, 1A, 0A]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	
40.91/11.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.91/11.32	
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	
40.91/11.32	
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(6)
40.91/11.32	Obligation:
40.91/11.32	Q DP problem:
40.91/11.32	The TRS P consists of the following rules:
40.91/11.32	
40.91/11.32	   B(b(b(a(x1)))) -> A(a(b(a(x1))))
40.91/11.32	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
40.91/11.32	
40.91/11.32	The TRS R consists of the following rules:
40.91/11.32	
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	
40.91/11.32	Q is empty.
40.91/11.32	We have to consider all minimal (P,Q,R)-chains.
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(7) QDPOrderProof (EQUIVALENT)
40.91/11.32	We use the reduction pair processor [LPAR04,JAR06].
40.91/11.32	
40.91/11.32	
40.91/11.32	The following pairs can be oriented strictly and are deleted.
40.91/11.32	
40.91/11.32	   B(b(b(a(x1)))) -> A(a(b(a(x1))))
40.91/11.32	The remaining pairs can at least be oriented weakly.
40.91/11.32	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 1A], [-I, -I, 0A], [1A, -I, -I]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	   <<<
40.91/11.32	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
40.91/11.32	>>>
40.91/11.32	
40.91/11.32	
40.91/11.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.91/11.32	
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	
40.91/11.32	
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(8)
40.91/11.32	Obligation:
40.91/11.32	Q DP problem:
40.91/11.32	The TRS P consists of the following rules:
40.91/11.32	
40.91/11.32	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
40.91/11.32	
40.91/11.32	The TRS R consists of the following rules:
40.91/11.32	
40.91/11.32	   a(a(a(a(x1)))) -> a(b(b(a(x1))))
40.91/11.32	   b(b(b(a(x1)))) -> a(a(b(a(x1))))
40.91/11.32	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
40.91/11.32	
40.91/11.32	Q is empty.
40.91/11.32	We have to consider all minimal (P,Q,R)-chains.
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(9) DependencyGraphProof (EQUIVALENT)
40.91/11.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
40.91/11.32	----------------------------------------
40.91/11.32	
40.91/11.32	(10)
40.91/11.32	TRUE
41.25/11.50	EOF