27.27/7.84	YES
28.43/8.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
28.43/8.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
28.43/8.08	
28.43/8.08	
28.43/8.08	Termination w.r.t. Q of the given QTRS could be proven:
28.43/8.08	
28.43/8.08	(0) QTRS
28.43/8.08	(1) DependencyPairsProof [EQUIVALENT, 6 ms]
28.43/8.08	(2) QDP
28.43/8.08	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
28.43/8.08	(4) QDP
28.43/8.08	(5) QDPOrderProof [EQUIVALENT, 126 ms]
28.43/8.08	(6) QDP
28.43/8.08	(7) QDPOrderProof [EQUIVALENT, 61 ms]
28.43/8.08	(8) QDP
28.43/8.08	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
28.43/8.08	(10) TRUE
28.43/8.08	
28.43/8.08	
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(0)
28.43/8.08	Obligation:
28.43/8.08	Q restricted rewrite system:
28.43/8.08	The TRS R consists of the following rules:
28.43/8.08	
28.43/8.08	   a(x1) -> x1
28.43/8.08	   a(a(x1)) -> b(c(x1))
28.43/8.08	   c(b(b(x1))) -> a(a(b(a(x1))))
28.43/8.08	
28.43/8.08	Q is empty.
28.43/8.08	
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(1) DependencyPairsProof (EQUIVALENT)
28.43/8.08	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(2)
28.43/8.08	Obligation:
28.43/8.08	Q DP problem:
28.43/8.08	The TRS P consists of the following rules:
28.43/8.08	
28.43/8.08	   A(a(x1)) -> C(x1)
28.43/8.08	   C(b(b(x1))) -> A(a(b(a(x1))))
28.43/8.08	   C(b(b(x1))) -> A(b(a(x1)))
28.43/8.08	   C(b(b(x1))) -> A(x1)
28.43/8.08	
28.43/8.08	The TRS R consists of the following rules:
28.43/8.08	
28.43/8.08	   a(x1) -> x1
28.43/8.08	   a(a(x1)) -> b(c(x1))
28.43/8.08	   c(b(b(x1))) -> a(a(b(a(x1))))
28.43/8.08	
28.43/8.08	Q is empty.
28.43/8.08	We have to consider all minimal (P,Q,R)-chains.
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(3) DependencyGraphProof (EQUIVALENT)
28.43/8.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(4)
28.43/8.08	Obligation:
28.43/8.08	Q DP problem:
28.43/8.08	The TRS P consists of the following rules:
28.43/8.08	
28.43/8.08	   C(b(b(x1))) -> A(a(b(a(x1))))
28.43/8.08	   A(a(x1)) -> C(x1)
28.43/8.08	   C(b(b(x1))) -> A(x1)
28.43/8.08	
28.43/8.08	The TRS R consists of the following rules:
28.43/8.08	
28.43/8.08	   a(x1) -> x1
28.43/8.08	   a(a(x1)) -> b(c(x1))
28.43/8.08	   c(b(b(x1))) -> a(a(b(a(x1))))
28.43/8.08	
28.43/8.08	Q is empty.
28.43/8.08	We have to consider all minimal (P,Q,R)-chains.
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(5) QDPOrderProof (EQUIVALENT)
28.43/8.08	We use the reduction pair processor [LPAR04,JAR06].
28.43/8.08	
28.43/8.08	
28.43/8.08	The following pairs can be oriented strictly and are deleted.
28.43/8.08	
28.43/8.08	   C(b(b(x1))) -> A(x1)
28.43/8.08	The remaining pairs can at least be oriented weakly.
28.43/8.08	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(C(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [1A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	
28.43/8.08	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.43/8.08	
28.43/8.08	   a(x1) -> x1
28.43/8.08	   a(a(x1)) -> b(c(x1))
28.43/8.08	   c(b(b(x1))) -> a(a(b(a(x1))))
28.43/8.08	
28.43/8.08	
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(6)
28.43/8.08	Obligation:
28.43/8.08	Q DP problem:
28.43/8.08	The TRS P consists of the following rules:
28.43/8.08	
28.43/8.08	   C(b(b(x1))) -> A(a(b(a(x1))))
28.43/8.08	   A(a(x1)) -> C(x1)
28.43/8.08	
28.43/8.08	The TRS R consists of the following rules:
28.43/8.08	
28.43/8.08	   a(x1) -> x1
28.43/8.08	   a(a(x1)) -> b(c(x1))
28.43/8.08	   c(b(b(x1))) -> a(a(b(a(x1))))
28.43/8.08	
28.43/8.08	Q is empty.
28.43/8.08	We have to consider all minimal (P,Q,R)-chains.
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(7) QDPOrderProof (EQUIVALENT)
28.43/8.08	We use the reduction pair processor [LPAR04,JAR06].
28.43/8.08	
28.43/8.08	
28.43/8.08	The following pairs can be oriented strictly and are deleted.
28.43/8.08	
28.43/8.08	   A(a(x1)) -> C(x1)
28.43/8.08	The remaining pairs can at least be oriented weakly.
28.43/8.08	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [1A, 1A, 1A], [0A, 0A, -I]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	   <<<
28.43/8.08	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
28.43/8.08	>>>
28.43/8.08	
28.43/8.08	
28.43/8.08	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.43/8.08	
28.43/8.08	   a(x1) -> x1
28.43/8.08	   a(a(x1)) -> b(c(x1))
28.43/8.08	   c(b(b(x1))) -> a(a(b(a(x1))))
28.43/8.08	
28.43/8.08	
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(8)
28.43/8.08	Obligation:
28.43/8.08	Q DP problem:
28.43/8.08	The TRS P consists of the following rules:
28.43/8.08	
28.43/8.08	   C(b(b(x1))) -> A(a(b(a(x1))))
28.43/8.08	
28.43/8.08	The TRS R consists of the following rules:
28.43/8.08	
28.43/8.08	   a(x1) -> x1
28.43/8.08	   a(a(x1)) -> b(c(x1))
28.43/8.08	   c(b(b(x1))) -> a(a(b(a(x1))))
28.43/8.08	
28.43/8.08	Q is empty.
28.43/8.08	We have to consider all minimal (P,Q,R)-chains.
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(9) DependencyGraphProof (EQUIVALENT)
28.43/8.08	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
28.43/8.08	----------------------------------------
28.43/8.08	
28.43/8.08	(10)
28.43/8.08	TRUE
28.71/8.18	EOF