15.10/4.69	YES
15.61/4.86	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
15.61/4.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.61/4.86	
15.61/4.86	
15.61/4.86	Termination w.r.t. Q of the given QTRS could be proven:
15.61/4.86	
15.61/4.86	(0) QTRS
15.61/4.86	(1) DependencyPairsProof [EQUIVALENT, 1 ms]
15.61/4.86	(2) QDP
15.61/4.86	(3) QDPOrderProof [EQUIVALENT, 180 ms]
15.61/4.86	(4) QDP
15.61/4.86	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
15.61/4.86	(6) TRUE
15.61/4.86	
15.61/4.86	
15.61/4.86	----------------------------------------
15.61/4.86	
15.61/4.86	(0)
15.61/4.86	Obligation:
15.61/4.86	Q restricted rewrite system:
15.61/4.86	The TRS R consists of the following rules:
15.61/4.86	
15.61/4.86	   a(x1) -> x1
15.61/4.86	   a(a(b(a(x1)))) -> b(b(a(a(a(x1)))))
15.61/4.86	   b(x1) -> a(x1)
15.61/4.86	
15.61/4.86	Q is empty.
15.61/4.86	
15.61/4.86	----------------------------------------
15.61/4.86	
15.61/4.86	(1) DependencyPairsProof (EQUIVALENT)
15.61/4.86	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
15.61/4.86	----------------------------------------
15.61/4.86	
15.61/4.86	(2)
15.61/4.86	Obligation:
15.61/4.86	Q DP problem:
15.61/4.86	The TRS P consists of the following rules:
15.61/4.86	
15.61/4.86	   A(a(b(a(x1)))) -> B(b(a(a(a(x1)))))
15.61/4.86	   A(a(b(a(x1)))) -> B(a(a(a(x1))))
15.61/4.86	   A(a(b(a(x1)))) -> A(a(a(x1)))
15.61/4.86	   A(a(b(a(x1)))) -> A(a(x1))
15.61/4.86	   B(x1) -> A(x1)
15.61/4.86	
15.61/4.86	The TRS R consists of the following rules:
15.61/4.86	
15.61/4.86	   a(x1) -> x1
15.61/4.86	   a(a(b(a(x1)))) -> b(b(a(a(a(x1)))))
15.61/4.86	   b(x1) -> a(x1)
15.61/4.86	
15.61/4.86	Q is empty.
15.61/4.86	We have to consider all minimal (P,Q,R)-chains.
15.61/4.86	----------------------------------------
15.61/4.86	
15.61/4.86	(3) QDPOrderProof (EQUIVALENT)
15.61/4.86	We use the reduction pair processor [LPAR04,JAR06].
15.61/4.86	
15.61/4.86	
15.61/4.86	The following pairs can be oriented strictly and are deleted.
15.61/4.86	
15.61/4.86	   A(a(b(a(x1)))) -> B(b(a(a(a(x1)))))
15.61/4.86	   A(a(b(a(x1)))) -> B(a(a(a(x1))))
15.61/4.86	   A(a(b(a(x1)))) -> A(a(a(x1)))
15.61/4.86	   A(a(b(a(x1)))) -> A(a(x1))
15.61/4.86	The remaining pairs can at least be oriented weakly.
15.61/4.86	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
15.61/4.86	
15.61/4.86	   <<<
15.61/4.86	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
15.61/4.86	>>>
15.61/4.86	
15.61/4.86	   <<<
15.61/4.86	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
15.61/4.86	>>>
15.61/4.86	
15.61/4.86	   <<<
15.61/4.86	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
15.61/4.86	>>>
15.61/4.86	
15.61/4.86	   <<<
15.61/4.86	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
15.61/4.86	>>>
15.61/4.86	
15.61/4.86	
15.61/4.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
15.61/4.86	
15.61/4.86	   a(x1) -> x1
15.61/4.86	   a(a(b(a(x1)))) -> b(b(a(a(a(x1)))))
15.61/4.86	   b(x1) -> a(x1)
15.61/4.86	
15.61/4.86	
15.61/4.86	----------------------------------------
15.61/4.86	
15.61/4.86	(4)
15.61/4.86	Obligation:
15.61/4.86	Q DP problem:
15.61/4.86	The TRS P consists of the following rules:
15.61/4.86	
15.61/4.86	   B(x1) -> A(x1)
15.61/4.86	
15.61/4.86	The TRS R consists of the following rules:
15.61/4.86	
15.61/4.86	   a(x1) -> x1
15.61/4.86	   a(a(b(a(x1)))) -> b(b(a(a(a(x1)))))
15.61/4.86	   b(x1) -> a(x1)
15.61/4.86	
15.61/4.86	Q is empty.
15.61/4.86	We have to consider all minimal (P,Q,R)-chains.
15.61/4.86	----------------------------------------
15.61/4.86	
15.61/4.86	(5) DependencyGraphProof (EQUIVALENT)
15.61/4.86	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
15.61/4.86	----------------------------------------
15.61/4.86	
15.61/4.86	(6)
15.61/4.86	TRUE
15.89/4.92	EOF