19.16/5.84	YES
19.32/5.88	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
19.32/5.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.32/5.88	
19.32/5.88	
19.32/5.88	Termination w.r.t. Q of the given QTRS could be proven:
19.32/5.88	
19.32/5.88	(0) QTRS
19.32/5.88	(1) QTRS Reverse [EQUIVALENT, 0 ms]
19.32/5.88	(2) QTRS
19.32/5.88	(3) DependencyPairsProof [EQUIVALENT, 9 ms]
19.32/5.88	(4) QDP
19.32/5.88	(5) MRRProof [EQUIVALENT, 66 ms]
19.32/5.88	(6) QDP
19.32/5.88	(7) QDPOrderProof [EQUIVALENT, 0 ms]
19.32/5.88	(8) QDP
19.32/5.88	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
19.32/5.88	(10) TRUE
19.32/5.88	
19.32/5.88	
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(0)
19.32/5.88	Obligation:
19.32/5.88	Q restricted rewrite system:
19.32/5.88	The TRS R consists of the following rules:
19.32/5.88	
19.32/5.88	   a(x1) -> b(x1)
19.32/5.88	   a(b(b(x1))) -> c(x1)
19.32/5.88	   c(c(x1)) -> a(b(c(a(x1))))
19.32/5.88	
19.32/5.88	Q is empty.
19.32/5.88	
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(1) QTRS Reverse (EQUIVALENT)
19.32/5.88	We applied the QTRS Reverse Processor [REVERSE].
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(2)
19.32/5.88	Obligation:
19.32/5.88	Q restricted rewrite system:
19.32/5.88	The TRS R consists of the following rules:
19.32/5.88	
19.32/5.88	   a(x1) -> b(x1)
19.32/5.88	   b(b(a(x1))) -> c(x1)
19.32/5.88	   c(c(x1)) -> a(c(b(a(x1))))
19.32/5.88	
19.32/5.88	Q is empty.
19.32/5.88	
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(3) DependencyPairsProof (EQUIVALENT)
19.32/5.88	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(4)
19.32/5.88	Obligation:
19.32/5.88	Q DP problem:
19.32/5.88	The TRS P consists of the following rules:
19.32/5.88	
19.32/5.88	   A(x1) -> B(x1)
19.32/5.88	   B(b(a(x1))) -> C(x1)
19.32/5.88	   C(c(x1)) -> A(c(b(a(x1))))
19.32/5.88	   C(c(x1)) -> C(b(a(x1)))
19.32/5.88	   C(c(x1)) -> B(a(x1))
19.32/5.88	   C(c(x1)) -> A(x1)
19.32/5.88	
19.32/5.88	The TRS R consists of the following rules:
19.32/5.88	
19.32/5.88	   a(x1) -> b(x1)
19.32/5.88	   b(b(a(x1))) -> c(x1)
19.32/5.88	   c(c(x1)) -> a(c(b(a(x1))))
19.32/5.88	
19.32/5.88	Q is empty.
19.32/5.88	We have to consider all minimal (P,Q,R)-chains.
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(5) MRRProof (EQUIVALENT)
19.32/5.88	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.
19.32/5.88	
19.32/5.88	Strictly oriented dependency pairs:
19.32/5.88	
19.32/5.88	   C(c(x1)) -> C(b(a(x1)))
19.32/5.88	   C(c(x1)) -> B(a(x1))
19.32/5.88	   C(c(x1)) -> A(x1)
19.32/5.88	
19.32/5.88	
19.32/5.88	Used ordering: Polynomial interpretation [POLO]:
19.32/5.88	
19.32/5.88	   POL(A(x_1)) = x_1
19.32/5.88	   POL(B(x_1)) = x_1
19.32/5.88	   POL(C(x_1)) = 2 + x_1
19.32/5.88	   POL(a(x_1)) = 1 + x_1
19.32/5.88	   POL(b(x_1)) = 1 + x_1
19.32/5.88	   POL(c(x_1)) = 3 + x_1
19.32/5.88	
19.32/5.88	
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(6)
19.32/5.88	Obligation:
19.32/5.88	Q DP problem:
19.32/5.88	The TRS P consists of the following rules:
19.32/5.88	
19.32/5.88	   A(x1) -> B(x1)
19.32/5.88	   B(b(a(x1))) -> C(x1)
19.32/5.88	   C(c(x1)) -> A(c(b(a(x1))))
19.32/5.88	
19.32/5.88	The TRS R consists of the following rules:
19.32/5.88	
19.32/5.88	   a(x1) -> b(x1)
19.32/5.88	   b(b(a(x1))) -> c(x1)
19.32/5.88	   c(c(x1)) -> a(c(b(a(x1))))
19.32/5.88	
19.32/5.88	Q is empty.
19.32/5.88	We have to consider all minimal (P,Q,R)-chains.
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(7) QDPOrderProof (EQUIVALENT)
19.32/5.88	We use the reduction pair processor [LPAR04,JAR06].
19.32/5.88	
19.32/5.88	
19.32/5.88	The following pairs can be oriented strictly and are deleted.
19.32/5.88	
19.32/5.88	   B(b(a(x1))) -> C(x1)
19.32/5.88	The remaining pairs can at least be oriented weakly.
19.32/5.88	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
19.32/5.88	
19.32/5.88	   <<<
19.32/5.88	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
19.32/5.88	>>>
19.32/5.88	
19.32/5.88	   <<<
19.32/5.88	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
19.32/5.88	>>>
19.32/5.88	
19.32/5.88	   <<<
19.32/5.88	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [0A, 0A, -I], [-I, 0A, -I]] 	* 	x_1
19.32/5.88	>>>
19.32/5.88	
19.32/5.88	   <<<
19.32/5.88	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[1A, 1A, 0A], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
19.32/5.88	>>>
19.32/5.88	
19.32/5.88	   <<<
19.32/5.88	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
19.32/5.88	>>>
19.32/5.88	
19.32/5.88	   <<<
19.32/5.88	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [-I, -I, 0A], [1A, 1A, 0A]] 	* 	x_1
19.32/5.88	>>>
19.32/5.88	
19.32/5.88	
19.32/5.88	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.32/5.88	
19.32/5.88	   b(b(a(x1))) -> c(x1)
19.32/5.88	   c(c(x1)) -> a(c(b(a(x1))))
19.32/5.88	   a(x1) -> b(x1)
19.32/5.88	
19.32/5.88	
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(8)
19.32/5.88	Obligation:
19.32/5.88	Q DP problem:
19.32/5.88	The TRS P consists of the following rules:
19.32/5.88	
19.32/5.88	   A(x1) -> B(x1)
19.32/5.88	   C(c(x1)) -> A(c(b(a(x1))))
19.32/5.88	
19.32/5.88	The TRS R consists of the following rules:
19.32/5.88	
19.32/5.88	   a(x1) -> b(x1)
19.32/5.88	   b(b(a(x1))) -> c(x1)
19.32/5.88	   c(c(x1)) -> a(c(b(a(x1))))
19.32/5.88	
19.32/5.88	Q is empty.
19.32/5.88	We have to consider all minimal (P,Q,R)-chains.
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(9) DependencyGraphProof (EQUIVALENT)
19.32/5.88	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
19.32/5.88	----------------------------------------
19.32/5.88	
19.32/5.88	(10)
19.32/5.88	TRUE
19.56/5.97	EOF