18.81/5.75	YES
19.00/5.76	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
19.00/5.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.00/5.76	
19.00/5.76	
19.00/5.76	Termination w.r.t. Q of the given QTRS could be proven:
19.00/5.76	
19.00/5.76	(0) QTRS
19.00/5.76	(1) QTRS Reverse [EQUIVALENT, 0 ms]
19.00/5.76	(2) QTRS
19.00/5.76	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
19.00/5.76	(4) QDP
19.00/5.76	(5) QDPOrderProof [EQUIVALENT, 22 ms]
19.00/5.76	(6) QDP
19.00/5.76	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
19.00/5.76	(8) QDP
19.00/5.76	(9) QDPOrderProof [EQUIVALENT, 37 ms]
19.00/5.76	(10) QDP
19.00/5.76	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
19.00/5.76	(12) TRUE
19.00/5.76	
19.00/5.76	
19.00/5.76	----------------------------------------
19.00/5.76	
19.00/5.76	(0)
19.00/5.76	Obligation:
19.00/5.76	Q restricted rewrite system:
19.00/5.76	The TRS R consists of the following rules:
19.00/5.76	
19.00/5.76	   a(a(x1)) -> c(b(x1))
19.00/5.76	   b(b(x1)) -> c(a(x1))
19.00/5.76	   c(c(x1)) -> b(a(x1))
19.00/5.76	
19.00/5.76	Q is empty.
19.00/5.76	
19.00/5.76	----------------------------------------
19.00/5.76	
19.00/5.76	(1) QTRS Reverse (EQUIVALENT)
19.00/5.76	We applied the QTRS Reverse Processor [REVERSE].
19.00/5.76	----------------------------------------
19.00/5.76	
19.00/5.76	(2)
19.00/5.76	Obligation:
19.00/5.76	Q restricted rewrite system:
19.00/5.76	The TRS R consists of the following rules:
19.00/5.76	
19.00/5.76	   a(a(x1)) -> b(c(x1))
19.00/5.76	   b(b(x1)) -> a(c(x1))
19.00/5.76	   c(c(x1)) -> a(b(x1))
19.00/5.76	
19.00/5.76	Q is empty.
19.00/5.76	
19.00/5.76	----------------------------------------
19.00/5.76	
19.00/5.76	(3) DependencyPairsProof (EQUIVALENT)
19.00/5.76	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.00/5.76	----------------------------------------
19.00/5.76	
19.00/5.76	(4)
19.00/5.76	Obligation:
19.00/5.76	Q DP problem:
19.00/5.76	The TRS P consists of the following rules:
19.00/5.76	
19.00/5.76	   A(a(x1)) -> B(c(x1))
19.00/5.76	   A(a(x1)) -> C(x1)
19.00/5.76	   B(b(x1)) -> A(c(x1))
19.00/5.76	   B(b(x1)) -> C(x1)
19.00/5.76	   C(c(x1)) -> A(b(x1))
19.00/5.76	   C(c(x1)) -> B(x1)
19.00/5.76	
19.00/5.76	The TRS R consists of the following rules:
19.00/5.76	
19.00/5.76	   a(a(x1)) -> b(c(x1))
19.00/5.76	   b(b(x1)) -> a(c(x1))
19.00/5.76	   c(c(x1)) -> a(b(x1))
19.00/5.76	
19.00/5.76	Q is empty.
19.00/5.76	We have to consider all minimal (P,Q,R)-chains.
19.00/5.76	----------------------------------------
19.00/5.76	
19.00/5.76	(5) QDPOrderProof (EQUIVALENT)
19.00/5.76	We use the reduction pair processor [LPAR04,JAR06].
19.00/5.76	
19.00/5.76	
19.00/5.76	The following pairs can be oriented strictly and are deleted.
19.00/5.76	
19.00/5.76	   A(a(x1)) -> C(x1)
19.00/5.76	   B(b(x1)) -> C(x1)
19.00/5.76	   C(c(x1)) -> B(x1)
19.00/5.76	The remaining pairs can at least be oriented weakly.
19.00/5.76	Used ordering:  Polynomial interpretation [POLO]:
19.00/5.76	
19.00/5.76	   POL(A(x_1)) = 1 + x_1
19.00/5.77	   POL(B(x_1)) = 1 + x_1
19.00/5.77	   POL(C(x_1)) = 1 + x_1
19.00/5.77	   POL(a(x_1)) = 1 + x_1
19.00/5.77	   POL(b(x_1)) = 1 + x_1
19.00/5.77	   POL(c(x_1)) = 1 + x_1
19.00/5.77	
19.00/5.77	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.00/5.77	
19.00/5.77	   c(c(x1)) -> a(b(x1))
19.00/5.77	   b(b(x1)) -> a(c(x1))
19.00/5.77	   a(a(x1)) -> b(c(x1))
19.00/5.77	
19.00/5.77	
19.00/5.77	----------------------------------------
19.00/5.77	
19.00/5.77	(6)
19.00/5.77	Obligation:
19.00/5.77	Q DP problem:
19.00/5.77	The TRS P consists of the following rules:
19.00/5.77	
19.00/5.77	   A(a(x1)) -> B(c(x1))
19.00/5.77	   B(b(x1)) -> A(c(x1))
19.00/5.77	   C(c(x1)) -> A(b(x1))
19.00/5.77	
19.00/5.77	The TRS R consists of the following rules:
19.00/5.77	
19.00/5.77	   a(a(x1)) -> b(c(x1))
19.00/5.77	   b(b(x1)) -> a(c(x1))
19.00/5.77	   c(c(x1)) -> a(b(x1))
19.00/5.77	
19.00/5.77	Q is empty.
19.00/5.77	We have to consider all minimal (P,Q,R)-chains.
19.00/5.77	----------------------------------------
19.00/5.77	
19.00/5.77	(7) DependencyGraphProof (EQUIVALENT)
19.00/5.77	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
19.00/5.77	----------------------------------------
19.00/5.77	
19.00/5.77	(8)
19.00/5.77	Obligation:
19.00/5.77	Q DP problem:
19.00/5.77	The TRS P consists of the following rules:
19.00/5.77	
19.00/5.77	   B(b(x1)) -> A(c(x1))
19.00/5.77	   A(a(x1)) -> B(c(x1))
19.00/5.77	
19.00/5.77	The TRS R consists of the following rules:
19.00/5.77	
19.00/5.77	   a(a(x1)) -> b(c(x1))
19.00/5.77	   b(b(x1)) -> a(c(x1))
19.00/5.77	   c(c(x1)) -> a(b(x1))
19.00/5.77	
19.00/5.77	Q is empty.
19.00/5.77	We have to consider all minimal (P,Q,R)-chains.
19.00/5.77	----------------------------------------
19.00/5.77	
19.00/5.77	(9) QDPOrderProof (EQUIVALENT)
19.00/5.77	We use the reduction pair processor [LPAR04,JAR06].
19.00/5.77	
19.00/5.77	
19.00/5.77	The following pairs can be oriented strictly and are deleted.
19.00/5.77	
19.00/5.77	   A(a(x1)) -> B(c(x1))
19.00/5.77	The remaining pairs can at least be oriented weakly.
19.00/5.77	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
19.00/5.77	
19.00/5.77	   <<<
19.00/5.77	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
19.00/5.77	>>>
19.00/5.77	
19.00/5.77	   <<<
19.00/5.77	 POL(b(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
19.00/5.77	>>>
19.00/5.77	
19.00/5.77	   <<<
19.00/5.77	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
19.00/5.77	>>>
19.00/5.77	
19.00/5.77	   <<<
19.00/5.77	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
19.00/5.77	>>>
19.00/5.77	
19.00/5.77	   <<<
19.00/5.77	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
19.00/5.77	>>>
19.00/5.77	
19.00/5.77	
19.00/5.77	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.00/5.77	
19.00/5.77	   c(c(x1)) -> a(b(x1))
19.00/5.77	   b(b(x1)) -> a(c(x1))
19.00/5.77	   a(a(x1)) -> b(c(x1))
19.00/5.77	
19.00/5.77	
19.00/5.77	----------------------------------------
19.00/5.77	
19.00/5.77	(10)
19.00/5.77	Obligation:
19.00/5.77	Q DP problem:
19.00/5.77	The TRS P consists of the following rules:
19.00/5.77	
19.00/5.77	   B(b(x1)) -> A(c(x1))
19.00/5.77	
19.00/5.77	The TRS R consists of the following rules:
19.00/5.77	
19.00/5.77	   a(a(x1)) -> b(c(x1))
19.00/5.77	   b(b(x1)) -> a(c(x1))
19.00/5.77	   c(c(x1)) -> a(b(x1))
19.00/5.77	
19.00/5.77	Q is empty.
19.00/5.77	We have to consider all minimal (P,Q,R)-chains.
19.00/5.77	----------------------------------------
19.00/5.77	
19.00/5.77	(11) DependencyGraphProof (EQUIVALENT)
19.00/5.77	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
19.00/5.77	----------------------------------------
19.00/5.77	
19.00/5.77	(12)
19.00/5.77	TRUE
19.00/5.84	EOF