29.30/8.41	YES
31.29/9.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.29/9.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.29/9.13	
31.29/9.13	
31.29/9.13	Termination w.r.t. Q of the given QTRS could be proven:
31.29/9.13	
31.29/9.13	(0) QTRS
31.29/9.13	(1) DependencyPairsProof [EQUIVALENT, 22 ms]
31.29/9.13	(2) QDP
31.29/9.13	(3) QDPOrderProof [EQUIVALENT, 38 ms]
31.29/9.13	(4) QDP
31.29/9.13	(5) QDPOrderProof [EQUIVALENT, 0 ms]
31.29/9.13	(6) QDP
31.29/9.13	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
31.29/9.13	(8) TRUE
31.29/9.13	
31.29/9.13	
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(0)
31.29/9.13	Obligation:
31.29/9.13	Q restricted rewrite system:
31.29/9.13	The TRS R consists of the following rules:
31.29/9.13	
31.29/9.13	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
31.29/9.13	   b(a(b(x1))) -> a(b(a(x1)))
31.29/9.13	
31.29/9.13	Q is empty.
31.29/9.13	
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(1) DependencyPairsProof (EQUIVALENT)
31.29/9.13	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(2)
31.29/9.13	Obligation:
31.29/9.13	Q DP problem:
31.29/9.13	The TRS P consists of the following rules:
31.29/9.13	
31.29/9.13	   A(a(a(a(x1)))) -> B(a(a(b(x1))))
31.29/9.13	   A(a(a(a(x1)))) -> A(a(b(x1)))
31.29/9.13	   A(a(a(a(x1)))) -> A(b(x1))
31.29/9.13	   A(a(a(a(x1)))) -> B(x1)
31.29/9.13	   B(a(b(x1))) -> A(b(a(x1)))
31.29/9.13	   B(a(b(x1))) -> B(a(x1))
31.29/9.13	   B(a(b(x1))) -> A(x1)
31.29/9.13	
31.29/9.13	The TRS R consists of the following rules:
31.29/9.13	
31.29/9.13	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
31.29/9.13	   b(a(b(x1))) -> a(b(a(x1)))
31.29/9.13	
31.29/9.13	Q is empty.
31.29/9.13	We have to consider all minimal (P,Q,R)-chains.
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(3) QDPOrderProof (EQUIVALENT)
31.29/9.13	We use the reduction pair processor [LPAR04,JAR06].
31.29/9.13	
31.29/9.13	
31.29/9.13	The following pairs can be oriented strictly and are deleted.
31.29/9.13	
31.29/9.13	   A(a(a(a(x1)))) -> A(a(b(x1)))
31.29/9.13	   A(a(a(a(x1)))) -> A(b(x1))
31.29/9.13	   A(a(a(a(x1)))) -> B(x1)
31.29/9.13	   B(a(b(x1))) -> B(a(x1))
31.29/9.13	   B(a(b(x1))) -> A(x1)
31.29/9.13	The remaining pairs can at least be oriented weakly.
31.29/9.13	Used ordering:  Polynomial interpretation [POLO]:
31.29/9.13	
31.29/9.13	   POL(A(x_1)) = 1 + x_1
31.29/9.13	   POL(B(x_1)) = 1 + x_1
31.29/9.13	   POL(a(x_1)) = 1 + x_1
31.29/9.13	   POL(b(x_1)) = 1 + x_1
31.29/9.13	
31.29/9.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.29/9.13	
31.29/9.13	   b(a(b(x1))) -> a(b(a(x1)))
31.29/9.13	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
31.29/9.13	
31.29/9.13	
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(4)
31.29/9.13	Obligation:
31.29/9.13	Q DP problem:
31.29/9.13	The TRS P consists of the following rules:
31.29/9.13	
31.29/9.13	   A(a(a(a(x1)))) -> B(a(a(b(x1))))
31.29/9.13	   B(a(b(x1))) -> A(b(a(x1)))
31.29/9.13	
31.29/9.13	The TRS R consists of the following rules:
31.29/9.13	
31.29/9.13	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
31.29/9.13	   b(a(b(x1))) -> a(b(a(x1)))
31.29/9.13	
31.29/9.13	Q is empty.
31.29/9.13	We have to consider all minimal (P,Q,R)-chains.
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(5) QDPOrderProof (EQUIVALENT)
31.29/9.13	We use the reduction pair processor [LPAR04,JAR06].
31.29/9.13	
31.29/9.13	
31.29/9.13	The following pairs can be oriented strictly and are deleted.
31.29/9.13	
31.29/9.13	   B(a(b(x1))) -> A(b(a(x1)))
31.29/9.13	The remaining pairs can at least be oriented weakly.
31.29/9.13	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.29/9.13	
31.29/9.13	   <<<
31.29/9.13	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
31.29/9.13	>>>
31.29/9.13	
31.29/9.13	   <<<
31.29/9.13	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 1A], [0A, -I, -I]] 	* 	x_1
31.29/9.13	>>>
31.29/9.13	
31.29/9.13	   <<<
31.29/9.13	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
31.29/9.13	>>>
31.29/9.13	
31.29/9.13	   <<<
31.29/9.13	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [-I, -I, 1A], [0A, -I, 1A]] 	* 	x_1
31.29/9.13	>>>
31.29/9.13	
31.29/9.13	
31.29/9.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.29/9.13	
31.29/9.13	   b(a(b(x1))) -> a(b(a(x1)))
31.29/9.13	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
31.29/9.13	
31.29/9.13	
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(6)
31.29/9.13	Obligation:
31.29/9.13	Q DP problem:
31.29/9.13	The TRS P consists of the following rules:
31.29/9.13	
31.29/9.13	   A(a(a(a(x1)))) -> B(a(a(b(x1))))
31.29/9.13	
31.29/9.13	The TRS R consists of the following rules:
31.29/9.13	
31.29/9.13	   a(a(a(a(x1)))) -> b(a(a(b(x1))))
31.29/9.13	   b(a(b(x1))) -> a(b(a(x1)))
31.29/9.13	
31.29/9.13	Q is empty.
31.29/9.13	We have to consider all minimal (P,Q,R)-chains.
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(7) DependencyGraphProof (EQUIVALENT)
31.29/9.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
31.29/9.13	----------------------------------------
31.29/9.13	
31.29/9.13	(8)
31.29/9.13	TRUE
31.47/10.28	EOF