20.52/6.04	YES
20.62/6.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.62/6.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.62/6.08	
20.62/6.08	
20.62/6.08	Termination w.r.t. Q of the given QTRS could be proven:
20.62/6.08	
20.62/6.08	(0) QTRS
20.62/6.08	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.62/6.08	(2) QTRS
20.62/6.08	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
20.62/6.08	(4) QDP
20.62/6.08	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
20.62/6.08	(6) QDP
20.62/6.08	(7) QDPOrderProof [EQUIVALENT, 17 ms]
20.62/6.08	(8) QDP
20.62/6.08	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
20.62/6.08	(10) QDP
20.62/6.08	(11) QDPOrderProof [EQUIVALENT, 64 ms]
20.62/6.08	(12) QDP
20.62/6.08	(13) PisEmptyProof [EQUIVALENT, 0 ms]
20.62/6.08	(14) YES
20.62/6.08	
20.62/6.08	
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(0)
20.62/6.08	Obligation:
20.62/6.08	Q restricted rewrite system:
20.62/6.08	The TRS R consists of the following rules:
20.62/6.08	
20.62/6.08	   a(b(b(a(x1)))) -> a(c(a(b(x1))))
20.62/6.08	   a(c(x1)) -> c(c(a(x1)))
20.62/6.08	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.08	
20.62/6.08	Q is empty.
20.62/6.08	
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(1) QTRS Reverse (EQUIVALENT)
20.62/6.08	We applied the QTRS Reverse Processor [REVERSE].
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(2)
20.62/6.08	Obligation:
20.62/6.08	Q restricted rewrite system:
20.62/6.08	The TRS R consists of the following rules:
20.62/6.08	
20.62/6.08	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.08	   c(a(x1)) -> a(c(c(x1)))
20.62/6.08	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.08	
20.62/6.08	Q is empty.
20.62/6.08	
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(3) DependencyPairsProof (EQUIVALENT)
20.62/6.08	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(4)
20.62/6.08	Obligation:
20.62/6.08	Q DP problem:
20.62/6.08	The TRS P consists of the following rules:
20.62/6.08	
20.62/6.08	   A(b(b(a(x1)))) -> A(c(a(x1)))
20.62/6.08	   A(b(b(a(x1)))) -> C(a(x1))
20.62/6.08	   C(a(x1)) -> A(c(c(x1)))
20.62/6.08	   C(a(x1)) -> C(c(x1))
20.62/6.08	   C(a(x1)) -> C(x1)
20.62/6.08	   C(c(c(x1))) -> C(b(x1))
20.62/6.08	
20.62/6.08	The TRS R consists of the following rules:
20.62/6.08	
20.62/6.08	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.08	   c(a(x1)) -> a(c(c(x1)))
20.62/6.08	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.08	
20.62/6.08	Q is empty.
20.62/6.08	We have to consider all minimal (P,Q,R)-chains.
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(5) DependencyGraphProof (EQUIVALENT)
20.62/6.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(6)
20.62/6.08	Obligation:
20.62/6.08	Q DP problem:
20.62/6.08	The TRS P consists of the following rules:
20.62/6.08	
20.62/6.08	   A(b(b(a(x1)))) -> C(a(x1))
20.62/6.08	   C(a(x1)) -> A(c(c(x1)))
20.62/6.08	   A(b(b(a(x1)))) -> A(c(a(x1)))
20.62/6.08	   C(a(x1)) -> C(c(x1))
20.62/6.08	   C(a(x1)) -> C(x1)
20.62/6.08	
20.62/6.08	The TRS R consists of the following rules:
20.62/6.08	
20.62/6.08	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.08	   c(a(x1)) -> a(c(c(x1)))
20.62/6.08	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.08	
20.62/6.08	Q is empty.
20.62/6.08	We have to consider all minimal (P,Q,R)-chains.
20.62/6.08	----------------------------------------
20.62/6.08	
20.62/6.08	(7) QDPOrderProof (EQUIVALENT)
20.62/6.08	We use the reduction pair processor [LPAR04,JAR06].
20.62/6.08	
20.62/6.08	
20.62/6.08	The following pairs can be oriented strictly and are deleted.
20.62/6.08	
20.62/6.08	   C(a(x1)) -> A(c(c(x1)))
20.62/6.08	   C(a(x1)) -> C(c(x1))
20.62/6.08	   C(a(x1)) -> C(x1)
20.62/6.08	The remaining pairs can at least be oriented weakly.
20.62/6.08	Used ordering:  Polynomial interpretation [POLO]:
20.62/6.08	
20.62/6.08	   POL(A(x_1)) = x_1
20.62/6.08	   POL(C(x_1)) = x_1
20.62/6.08	   POL(a(x_1)) = 1 + x_1
20.62/6.08	   POL(b(x_1)) = x_1
20.62/6.08	   POL(c(x_1)) = x_1
20.62/6.08	
20.62/6.08	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.62/6.08	
20.62/6.08	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.08	   c(a(x1)) -> a(c(c(x1)))
20.62/6.09	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.09	
20.62/6.09	
20.62/6.09	----------------------------------------
20.62/6.09	
20.62/6.09	(8)
20.62/6.09	Obligation:
20.62/6.09	Q DP problem:
20.62/6.09	The TRS P consists of the following rules:
20.62/6.09	
20.62/6.09	   A(b(b(a(x1)))) -> C(a(x1))
20.62/6.09	   A(b(b(a(x1)))) -> A(c(a(x1)))
20.62/6.09	
20.62/6.09	The TRS R consists of the following rules:
20.62/6.09	
20.62/6.09	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.09	   c(a(x1)) -> a(c(c(x1)))
20.62/6.09	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.09	
20.62/6.09	Q is empty.
20.62/6.09	We have to consider all minimal (P,Q,R)-chains.
20.62/6.09	----------------------------------------
20.62/6.09	
20.62/6.09	(9) DependencyGraphProof (EQUIVALENT)
20.62/6.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
20.62/6.09	----------------------------------------
20.62/6.09	
20.62/6.09	(10)
20.62/6.09	Obligation:
20.62/6.09	Q DP problem:
20.62/6.09	The TRS P consists of the following rules:
20.62/6.09	
20.62/6.09	   A(b(b(a(x1)))) -> A(c(a(x1)))
20.62/6.09	
20.62/6.09	The TRS R consists of the following rules:
20.62/6.09	
20.62/6.09	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.09	   c(a(x1)) -> a(c(c(x1)))
20.62/6.09	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.09	
20.62/6.09	Q is empty.
20.62/6.09	We have to consider all minimal (P,Q,R)-chains.
20.62/6.09	----------------------------------------
20.62/6.09	
20.62/6.09	(11) QDPOrderProof (EQUIVALENT)
20.62/6.09	We use the reduction pair processor [LPAR04,JAR06].
20.62/6.09	
20.62/6.09	
20.62/6.09	The following pairs can be oriented strictly and are deleted.
20.62/6.09	
20.62/6.09	   A(b(b(a(x1)))) -> A(c(a(x1)))
20.62/6.09	The remaining pairs can at least be oriented weakly.
20.62/6.09	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.62/6.09	
20.62/6.09	   <<<
20.62/6.09	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
20.62/6.09	>>>
20.62/6.09	
20.62/6.09	   <<<
20.62/6.09	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, 0A], [-I, -I, -I], [-I, 0A, 1A]] 	* 	x_1
20.62/6.09	>>>
20.62/6.09	
20.62/6.09	   <<<
20.62/6.09	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
20.62/6.09	>>>
20.62/6.09	
20.62/6.09	   <<<
20.62/6.09	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [-I, 0A, -I]] 	* 	x_1
20.62/6.09	>>>
20.62/6.09	
20.62/6.09	
20.62/6.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.62/6.09	
20.62/6.09	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.09	   c(a(x1)) -> a(c(c(x1)))
20.62/6.09	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.09	
20.62/6.09	
20.62/6.09	----------------------------------------
20.62/6.09	
20.62/6.09	(12)
20.62/6.09	Obligation:
20.62/6.09	Q DP problem:
20.62/6.09	P is empty.
20.62/6.09	The TRS R consists of the following rules:
20.62/6.09	
20.62/6.09	   a(b(b(a(x1)))) -> b(a(c(a(x1))))
20.62/6.09	   c(a(x1)) -> a(c(c(x1)))
20.62/6.09	   c(c(c(x1))) -> b(c(b(x1)))
20.62/6.09	
20.62/6.09	Q is empty.
20.62/6.09	We have to consider all minimal (P,Q,R)-chains.
20.62/6.09	----------------------------------------
20.62/6.09	
20.62/6.09	(13) PisEmptyProof (EQUIVALENT)
20.62/6.09	The TRS P is empty. Hence, there is no (P,Q,R) chain.
20.62/6.09	----------------------------------------
20.62/6.09	
20.62/6.09	(14)
20.62/6.09	YES
20.86/6.18	EOF