29.56/8.42	YES
29.56/8.44	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
29.56/8.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.56/8.44	
29.56/8.44	
29.56/8.44	Termination w.r.t. Q of the given QTRS could be proven:
29.56/8.44	
29.56/8.44	(0) QTRS
29.56/8.44	(1) DependencyPairsProof [EQUIVALENT, 12 ms]
29.56/8.44	(2) QDP
29.56/8.44	(3) QDPOrderProof [EQUIVALENT, 89 ms]
29.56/8.44	(4) QDP
29.56/8.44	(5) QDPOrderProof [EQUIVALENT, 0 ms]
29.56/8.44	(6) QDP
29.56/8.44	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
29.56/8.44	(8) TRUE
29.56/8.44	
29.56/8.44	
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(0)
29.56/8.44	Obligation:
29.56/8.44	Q restricted rewrite system:
29.56/8.44	The TRS R consists of the following rules:
29.56/8.44	
29.56/8.44	   a(x1) -> x1
29.56/8.44	   a(b(b(x1))) -> b(b(b(c(x1))))
29.56/8.44	   c(b(x1)) -> a(a(x1))
29.56/8.44	
29.56/8.44	Q is empty.
29.56/8.44	
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(1) DependencyPairsProof (EQUIVALENT)
29.56/8.44	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(2)
29.56/8.44	Obligation:
29.56/8.44	Q DP problem:
29.56/8.44	The TRS P consists of the following rules:
29.56/8.44	
29.56/8.44	   A(b(b(x1))) -> C(x1)
29.56/8.44	   C(b(x1)) -> A(a(x1))
29.56/8.44	   C(b(x1)) -> A(x1)
29.56/8.44	
29.56/8.44	The TRS R consists of the following rules:
29.56/8.44	
29.56/8.44	   a(x1) -> x1
29.56/8.44	   a(b(b(x1))) -> b(b(b(c(x1))))
29.56/8.44	   c(b(x1)) -> a(a(x1))
29.56/8.44	
29.56/8.44	Q is empty.
29.56/8.44	We have to consider all minimal (P,Q,R)-chains.
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(3) QDPOrderProof (EQUIVALENT)
29.56/8.44	We use the reduction pair processor [LPAR04,JAR06].
29.56/8.44	
29.56/8.44	
29.56/8.44	The following pairs can be oriented strictly and are deleted.
29.56/8.44	
29.56/8.44	   C(b(x1)) -> A(x1)
29.56/8.44	The remaining pairs can at least be oriented weakly.
29.56/8.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	
29.56/8.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.56/8.44	
29.56/8.44	   a(x1) -> x1
29.56/8.44	   a(b(b(x1))) -> b(b(b(c(x1))))
29.56/8.44	   c(b(x1)) -> a(a(x1))
29.56/8.44	
29.56/8.44	
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(4)
29.56/8.44	Obligation:
29.56/8.44	Q DP problem:
29.56/8.44	The TRS P consists of the following rules:
29.56/8.44	
29.56/8.44	   A(b(b(x1))) -> C(x1)
29.56/8.44	   C(b(x1)) -> A(a(x1))
29.56/8.44	
29.56/8.44	The TRS R consists of the following rules:
29.56/8.44	
29.56/8.44	   a(x1) -> x1
29.56/8.44	   a(b(b(x1))) -> b(b(b(c(x1))))
29.56/8.44	   c(b(x1)) -> a(a(x1))
29.56/8.44	
29.56/8.44	Q is empty.
29.56/8.44	We have to consider all minimal (P,Q,R)-chains.
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(5) QDPOrderProof (EQUIVALENT)
29.56/8.44	We use the reduction pair processor [LPAR04,JAR06].
29.56/8.44	
29.56/8.44	
29.56/8.44	The following pairs can be oriented strictly and are deleted.
29.56/8.44	
29.56/8.44	   C(b(x1)) -> A(a(x1))
29.56/8.44	The remaining pairs can at least be oriented weakly.
29.56/8.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(C(x_1)) =  	[[1A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	   <<<
29.56/8.44	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
29.56/8.44	>>>
29.56/8.44	
29.56/8.44	
29.56/8.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.56/8.44	
29.56/8.44	   a(x1) -> x1
29.56/8.44	   a(b(b(x1))) -> b(b(b(c(x1))))
29.56/8.44	   c(b(x1)) -> a(a(x1))
29.56/8.44	
29.56/8.44	
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(6)
29.56/8.44	Obligation:
29.56/8.44	Q DP problem:
29.56/8.44	The TRS P consists of the following rules:
29.56/8.44	
29.56/8.44	   A(b(b(x1))) -> C(x1)
29.56/8.44	
29.56/8.44	The TRS R consists of the following rules:
29.56/8.44	
29.56/8.44	   a(x1) -> x1
29.56/8.44	   a(b(b(x1))) -> b(b(b(c(x1))))
29.56/8.44	   c(b(x1)) -> a(a(x1))
29.56/8.44	
29.56/8.44	Q is empty.
29.56/8.44	We have to consider all minimal (P,Q,R)-chains.
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(7) DependencyGraphProof (EQUIVALENT)
29.56/8.44	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
29.56/8.44	----------------------------------------
29.56/8.44	
29.56/8.44	(8)
29.56/8.44	TRUE
30.02/8.55	EOF