46.30/12.66	YES
46.30/12.73	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
46.30/12.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
46.30/12.73	
46.30/12.73	
46.30/12.73	Termination w.r.t. Q of the given QTRS could be proven:
46.30/12.73	
46.30/12.73	(0) QTRS
46.30/12.73	(1) DependencyPairsProof [EQUIVALENT, 11 ms]
46.30/12.73	(2) QDP
46.30/12.73	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
46.30/12.73	(4) QDP
46.30/12.73	(5) QDPOrderProof [EQUIVALENT, 131 ms]
46.30/12.73	(6) QDP
46.30/12.73	(7) QDPOrderProof [EQUIVALENT, 0 ms]
46.30/12.73	(8) QDP
46.30/12.73	(9) QDPOrderProof [EQUIVALENT, 0 ms]
46.30/12.73	(10) QDP
46.30/12.73	(11) PisEmptyProof [EQUIVALENT, 0 ms]
46.30/12.73	(12) YES
46.30/12.73	
46.30/12.73	
46.30/12.73	----------------------------------------
46.30/12.73	
46.30/12.73	(0)
46.30/12.73	Obligation:
46.30/12.73	Q restricted rewrite system:
46.30/12.73	The TRS R consists of the following rules:
46.30/12.73	
46.30/12.73	   a(x1) -> x1
46.30/12.73	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.73	   b(c(x1)) -> x1
46.30/12.73	
46.30/12.73	Q is empty.
46.30/12.73	
46.30/12.73	----------------------------------------
46.30/12.73	
46.30/12.73	(1) DependencyPairsProof (EQUIVALENT)
46.30/12.73	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
46.30/12.73	----------------------------------------
46.30/12.73	
46.30/12.73	(2)
46.30/12.73	Obligation:
46.30/12.73	Q DP problem:
46.30/12.73	The TRS P consists of the following rules:
46.30/12.73	
46.30/12.73	   A(a(b(x1))) -> A(b(b(a(a(x1)))))
46.30/12.73	   A(a(b(x1))) -> B(b(a(a(x1))))
46.30/12.73	   A(a(b(x1))) -> B(a(a(x1)))
46.30/12.73	   A(a(b(x1))) -> A(a(x1))
46.30/12.73	   A(a(b(x1))) -> A(x1)
46.30/12.73	
46.30/12.73	The TRS R consists of the following rules:
46.30/12.73	
46.30/12.73	   a(x1) -> x1
46.30/12.73	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.73	   b(c(x1)) -> x1
46.30/12.73	
46.30/12.73	Q is empty.
46.30/12.73	We have to consider all minimal (P,Q,R)-chains.
46.30/12.73	----------------------------------------
46.30/12.73	
46.30/12.73	(3) DependencyGraphProof (EQUIVALENT)
46.30/12.73	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
46.30/12.73	----------------------------------------
46.30/12.73	
46.30/12.73	(4)
46.30/12.73	Obligation:
46.30/12.73	Q DP problem:
46.30/12.73	The TRS P consists of the following rules:
46.30/12.73	
46.30/12.73	   A(a(b(x1))) -> A(a(x1))
46.30/12.73	   A(a(b(x1))) -> A(b(b(a(a(x1)))))
46.30/12.73	   A(a(b(x1))) -> A(x1)
46.30/12.73	
46.30/12.73	The TRS R consists of the following rules:
46.30/12.73	
46.30/12.73	   a(x1) -> x1
46.30/12.73	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.73	   b(c(x1)) -> x1
46.30/12.73	
46.30/12.73	Q is empty.
46.30/12.73	We have to consider all minimal (P,Q,R)-chains.
46.30/12.73	----------------------------------------
46.30/12.73	
46.30/12.73	(5) QDPOrderProof (EQUIVALENT)
46.30/12.73	We use the reduction pair processor [LPAR04,JAR06].
46.30/12.73	
46.30/12.73	
46.30/12.73	The following pairs can be oriented strictly and are deleted.
46.30/12.74	
46.30/12.74	   A(a(b(x1))) -> A(x1)
46.30/12.74	The remaining pairs can at least be oriented weakly.
46.30/12.74	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(a(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, -I, 1A], [1A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [0A, -I, 1A]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, -I]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	
46.30/12.74	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
46.30/12.74	
46.30/12.74	   a(x1) -> x1
46.30/12.74	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.74	   b(c(x1)) -> x1
46.30/12.74	
46.30/12.74	
46.30/12.74	----------------------------------------
46.30/12.74	
46.30/12.74	(6)
46.30/12.74	Obligation:
46.30/12.74	Q DP problem:
46.30/12.74	The TRS P consists of the following rules:
46.30/12.74	
46.30/12.74	   A(a(b(x1))) -> A(a(x1))
46.30/12.74	   A(a(b(x1))) -> A(b(b(a(a(x1)))))
46.30/12.74	
46.30/12.74	The TRS R consists of the following rules:
46.30/12.74	
46.30/12.74	   a(x1) -> x1
46.30/12.74	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.74	   b(c(x1)) -> x1
46.30/12.74	
46.30/12.74	Q is empty.
46.30/12.74	We have to consider all minimal (P,Q,R)-chains.
46.30/12.74	----------------------------------------
46.30/12.74	
46.30/12.74	(7) QDPOrderProof (EQUIVALENT)
46.30/12.74	We use the reduction pair processor [LPAR04,JAR06].
46.30/12.74	
46.30/12.74	
46.30/12.74	The following pairs can be oriented strictly and are deleted.
46.30/12.74	
46.30/12.74	   A(a(b(x1))) -> A(a(x1))
46.30/12.74	The remaining pairs can at least be oriented weakly.
46.30/12.74	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[1A, 0A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [-I, -I, -I]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	
46.30/12.74	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
46.30/12.74	
46.30/12.74	   a(x1) -> x1
46.30/12.74	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.74	   b(c(x1)) -> x1
46.30/12.74	
46.30/12.74	
46.30/12.74	----------------------------------------
46.30/12.74	
46.30/12.74	(8)
46.30/12.74	Obligation:
46.30/12.74	Q DP problem:
46.30/12.74	The TRS P consists of the following rules:
46.30/12.74	
46.30/12.74	   A(a(b(x1))) -> A(b(b(a(a(x1)))))
46.30/12.74	
46.30/12.74	The TRS R consists of the following rules:
46.30/12.74	
46.30/12.74	   a(x1) -> x1
46.30/12.74	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.74	   b(c(x1)) -> x1
46.30/12.74	
46.30/12.74	Q is empty.
46.30/12.74	We have to consider all minimal (P,Q,R)-chains.
46.30/12.74	----------------------------------------
46.30/12.74	
46.30/12.74	(9) QDPOrderProof (EQUIVALENT)
46.30/12.74	We use the reduction pair processor [LPAR04,JAR06].
46.30/12.74	
46.30/12.74	
46.30/12.74	The following pairs can be oriented strictly and are deleted.
46.30/12.74	
46.30/12.74	   A(a(b(x1))) -> A(b(b(a(a(x1)))))
46.30/12.74	The remaining pairs can at least be oriented weakly.
46.30/12.74	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [1A, -I, 0A]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, 0A], [1A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	   <<<
46.30/12.74	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
46.30/12.74	>>>
46.30/12.74	
46.30/12.74	
46.30/12.74	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
46.30/12.74	
46.30/12.74	   a(x1) -> x1
46.30/12.74	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.74	   b(c(x1)) -> x1
46.30/12.74	
46.30/12.74	
46.30/12.74	----------------------------------------
46.30/12.74	
46.30/12.74	(10)
46.30/12.74	Obligation:
46.30/12.74	Q DP problem:
46.30/12.74	P is empty.
46.30/12.74	The TRS R consists of the following rules:
46.30/12.74	
46.30/12.74	   a(x1) -> x1
46.30/12.74	   a(a(b(x1))) -> c(a(b(b(a(a(x1))))))
46.30/12.74	   b(c(x1)) -> x1
46.30/12.74	
46.30/12.74	Q is empty.
46.30/12.74	We have to consider all minimal (P,Q,R)-chains.
46.30/12.74	----------------------------------------
46.30/12.74	
46.30/12.74	(11) PisEmptyProof (EQUIVALENT)
46.30/12.74	The TRS P is empty. Hence, there is no (P,Q,R) chain.
46.30/12.74	----------------------------------------
46.30/12.74	
46.30/12.74	(12)
46.30/12.74	YES
46.79/12.84	EOF