21.46/6.32	YES
22.00/6.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.00/6.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.00/6.43	
22.00/6.43	
22.00/6.43	Termination w.r.t. Q of the given QTRS could be proven:
22.00/6.43	
22.00/6.43	(0) QTRS
22.00/6.43	(1) QTRS Reverse [EQUIVALENT, 0 ms]
22.00/6.43	(2) QTRS
22.00/6.43	(3) DependencyPairsProof [EQUIVALENT, 10 ms]
22.00/6.43	(4) QDP
22.00/6.43	(5) MRRProof [EQUIVALENT, 39 ms]
22.00/6.43	(6) QDP
22.00/6.43	(7) QDPOrderProof [EQUIVALENT, 57 ms]
22.00/6.43	(8) QDP
22.00/6.43	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
22.00/6.43	(10) TRUE
22.00/6.43	
22.00/6.43	
22.00/6.43	----------------------------------------
22.00/6.43	
22.00/6.43	(0)
22.00/6.43	Obligation:
22.00/6.43	Q restricted rewrite system:
22.00/6.43	The TRS R consists of the following rules:
22.00/6.43	
22.00/6.43	   a(a(x1)) -> b(c(x1))
22.00/6.43	   a(b(x1)) -> c(a(x1))
22.00/6.43	   c(c(x1)) -> a(b(x1))
22.00/6.43	
22.00/6.43	Q is empty.
22.00/6.43	
22.00/6.43	----------------------------------------
22.00/6.43	
22.00/6.43	(1) QTRS Reverse (EQUIVALENT)
22.00/6.43	We applied the QTRS Reverse Processor [REVERSE].
22.00/6.43	----------------------------------------
22.00/6.43	
22.00/6.43	(2)
22.00/6.43	Obligation:
22.00/6.43	Q restricted rewrite system:
22.00/6.43	The TRS R consists of the following rules:
22.00/6.43	
22.00/6.43	   a(a(x1)) -> c(b(x1))
22.00/6.43	   b(a(x1)) -> a(c(x1))
22.00/6.43	   c(c(x1)) -> b(a(x1))
22.00/6.43	
22.00/6.43	Q is empty.
22.00/6.43	
22.00/6.43	----------------------------------------
22.00/6.43	
22.00/6.43	(3) DependencyPairsProof (EQUIVALENT)
22.00/6.43	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.00/6.43	----------------------------------------
22.00/6.43	
22.00/6.43	(4)
22.00/6.43	Obligation:
22.00/6.43	Q DP problem:
22.00/6.43	The TRS P consists of the following rules:
22.00/6.43	
22.00/6.43	   A(a(x1)) -> C(b(x1))
22.00/6.43	   A(a(x1)) -> B(x1)
22.00/6.43	   B(a(x1)) -> A(c(x1))
22.00/6.43	   B(a(x1)) -> C(x1)
22.00/6.43	   C(c(x1)) -> B(a(x1))
22.00/6.43	   C(c(x1)) -> A(x1)
22.00/6.43	
22.00/6.43	The TRS R consists of the following rules:
22.00/6.43	
22.00/6.43	   a(a(x1)) -> c(b(x1))
22.00/6.43	   b(a(x1)) -> a(c(x1))
22.00/6.43	   c(c(x1)) -> b(a(x1))
22.00/6.43	
22.00/6.43	Q is empty.
22.00/6.43	We have to consider all minimal (P,Q,R)-chains.
22.00/6.43	----------------------------------------
22.00/6.43	
22.00/6.43	(5) MRRProof (EQUIVALENT)
22.00/6.43	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
22.00/6.43	
22.00/6.43	Strictly oriented dependency pairs:
22.00/6.43	
22.00/6.43	   A(a(x1)) -> B(x1)
22.00/6.43	   B(a(x1)) -> C(x1)
22.00/6.43	   C(c(x1)) -> A(x1)
22.00/6.43	
22.00/6.43	
22.00/6.43	Used ordering: Polynomial interpretation [POLO]:
22.00/6.43	
22.00/6.43	   POL(A(x_1)) = 2*x_1
22.00/6.43	   POL(B(x_1)) = 2*x_1
22.04/6.43	   POL(C(x_1)) = 2*x_1
22.04/6.43	   POL(a(x_1)) = 2 + 2*x_1
22.04/6.43	   POL(b(x_1)) = 2 + 2*x_1
22.04/6.43	   POL(c(x_1)) = 2 + 2*x_1
22.04/6.43	
22.04/6.43	
22.04/6.43	----------------------------------------
22.04/6.43	
22.04/6.43	(6)
22.04/6.43	Obligation:
22.04/6.43	Q DP problem:
22.04/6.43	The TRS P consists of the following rules:
22.04/6.43	
22.04/6.43	   A(a(x1)) -> C(b(x1))
22.04/6.43	   B(a(x1)) -> A(c(x1))
22.04/6.43	   C(c(x1)) -> B(a(x1))
22.04/6.43	
22.04/6.43	The TRS R consists of the following rules:
22.04/6.43	
22.04/6.43	   a(a(x1)) -> c(b(x1))
22.04/6.43	   b(a(x1)) -> a(c(x1))
22.04/6.43	   c(c(x1)) -> b(a(x1))
22.04/6.43	
22.04/6.43	Q is empty.
22.04/6.43	We have to consider all minimal (P,Q,R)-chains.
22.04/6.43	----------------------------------------
22.04/6.43	
22.04/6.43	(7) QDPOrderProof (EQUIVALENT)
22.04/6.43	We use the reduction pair processor [LPAR04,JAR06].
22.04/6.43	
22.04/6.43	
22.04/6.43	The following pairs can be oriented strictly and are deleted.
22.04/6.43	
22.04/6.43	   B(a(x1)) -> A(c(x1))
22.04/6.43	The remaining pairs can at least be oriented weakly.
22.04/6.43	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.04/6.43	
22.04/6.43	   <<<
22.04/6.43	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
22.04/6.43	>>>
22.04/6.43	
22.04/6.43	   <<<
22.04/6.43	 POL(a(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [1A, -I, 0A], [0A, 1A, 0A]] 	* 	x_1
22.04/6.43	>>>
22.04/6.43	
22.04/6.43	   <<<
22.04/6.43	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
22.04/6.43	>>>
22.04/6.43	
22.04/6.43	   <<<
22.04/6.43	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
22.04/6.43	>>>
22.04/6.43	
22.04/6.43	   <<<
22.04/6.43	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
22.04/6.43	>>>
22.04/6.43	
22.04/6.43	   <<<
22.04/6.43	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [0A, 1A, 0A]] 	* 	x_1
22.04/6.43	>>>
22.04/6.43	
22.04/6.43	
22.04/6.43	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.04/6.43	
22.04/6.43	   c(c(x1)) -> b(a(x1))
22.04/6.43	   b(a(x1)) -> a(c(x1))
22.04/6.43	   a(a(x1)) -> c(b(x1))
22.04/6.43	
22.04/6.43	
22.04/6.43	----------------------------------------
22.04/6.43	
22.04/6.43	(8)
22.04/6.43	Obligation:
22.04/6.43	Q DP problem:
22.04/6.43	The TRS P consists of the following rules:
22.04/6.43	
22.04/6.43	   A(a(x1)) -> C(b(x1))
22.04/6.43	   C(c(x1)) -> B(a(x1))
22.04/6.43	
22.04/6.43	The TRS R consists of the following rules:
22.04/6.43	
22.04/6.43	   a(a(x1)) -> c(b(x1))
22.04/6.43	   b(a(x1)) -> a(c(x1))
22.04/6.43	   c(c(x1)) -> b(a(x1))
22.04/6.43	
22.04/6.43	Q is empty.
22.04/6.43	We have to consider all minimal (P,Q,R)-chains.
22.04/6.43	----------------------------------------
22.04/6.43	
22.04/6.43	(9) DependencyGraphProof (EQUIVALENT)
22.04/6.43	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
22.04/6.43	----------------------------------------
22.04/6.43	
22.04/6.43	(10)
22.04/6.43	TRUE
22.13/8.80	EOF