43.23/11.85	YES
43.58/11.92	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
43.58/11.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
43.58/11.92	
43.58/11.92	
43.58/11.92	Termination w.r.t. Q of the given QTRS could be proven:
43.58/11.92	
43.58/11.92	(0) QTRS
43.58/11.92	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
43.58/11.92	(2) QDP
43.58/11.92	(3) QDPOrderProof [EQUIVALENT, 147 ms]
43.58/11.92	(4) QDP
43.58/11.92	(5) QDPOrderProof [EQUIVALENT, 0 ms]
43.58/11.92	(6) QDP
43.58/11.92	(7) QDPOrderProof [EQUIVALENT, 30 ms]
43.58/11.92	(8) QDP
43.58/11.92	(9) QDPOrderProof [EQUIVALENT, 34 ms]
43.58/11.92	(10) QDP
43.58/11.92	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
43.58/11.92	(12) TRUE
43.58/11.92	
43.58/11.92	
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(0)
43.58/11.92	Obligation:
43.58/11.92	Q restricted rewrite system:
43.58/11.92	The TRS R consists of the following rules:
43.58/11.92	
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	
43.58/11.92	Q is empty.
43.58/11.92	
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(1) DependencyPairsProof (EQUIVALENT)
43.58/11.92	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(2)
43.58/11.92	Obligation:
43.58/11.92	Q DP problem:
43.58/11.92	The TRS P consists of the following rules:
43.58/11.92	
43.58/11.92	   A(a(b(c(x1)))) -> B(b(a(a(x1))))
43.58/11.92	   A(a(b(c(x1)))) -> B(a(a(x1)))
43.58/11.92	   A(a(b(c(x1)))) -> A(a(x1))
43.58/11.92	   A(a(b(c(x1)))) -> A(x1)
43.58/11.92	   B(x1) -> A(a(x1))
43.58/11.92	   B(x1) -> A(x1)
43.58/11.92	   B(c(x1)) -> A(x1)
43.58/11.92	
43.58/11.92	The TRS R consists of the following rules:
43.58/11.92	
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	
43.58/11.92	Q is empty.
43.58/11.92	We have to consider all minimal (P,Q,R)-chains.
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(3) QDPOrderProof (EQUIVALENT)
43.58/11.92	We use the reduction pair processor [LPAR04,JAR06].
43.58/11.92	
43.58/11.92	
43.58/11.92	The following pairs can be oriented strictly and are deleted.
43.58/11.92	
43.58/11.92	   A(a(b(c(x1)))) -> B(a(a(x1)))
43.58/11.92	   A(a(b(c(x1)))) -> A(a(x1))
43.58/11.92	   A(a(b(c(x1)))) -> A(x1)
43.58/11.92	The remaining pairs can at least be oriented weakly.
43.58/11.92	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	
43.58/11.92	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.58/11.92	
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	
43.58/11.92	
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(4)
43.58/11.92	Obligation:
43.58/11.92	Q DP problem:
43.58/11.92	The TRS P consists of the following rules:
43.58/11.92	
43.58/11.92	   A(a(b(c(x1)))) -> B(b(a(a(x1))))
43.58/11.92	   B(x1) -> A(a(x1))
43.58/11.92	   B(x1) -> A(x1)
43.58/11.92	   B(c(x1)) -> A(x1)
43.58/11.92	
43.58/11.92	The TRS R consists of the following rules:
43.58/11.92	
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	
43.58/11.92	Q is empty.
43.58/11.92	We have to consider all minimal (P,Q,R)-chains.
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(5) QDPOrderProof (EQUIVALENT)
43.58/11.92	We use the reduction pair processor [LPAR04,JAR06].
43.58/11.92	
43.58/11.92	
43.58/11.92	The following pairs can be oriented strictly and are deleted.
43.58/11.92	
43.58/11.92	   B(c(x1)) -> A(x1)
43.58/11.92	The remaining pairs can at least be oriented weakly.
43.58/11.92	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	
43.58/11.92	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.58/11.92	
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	
43.58/11.92	
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(6)
43.58/11.92	Obligation:
43.58/11.92	Q DP problem:
43.58/11.92	The TRS P consists of the following rules:
43.58/11.92	
43.58/11.92	   A(a(b(c(x1)))) -> B(b(a(a(x1))))
43.58/11.92	   B(x1) -> A(a(x1))
43.58/11.92	   B(x1) -> A(x1)
43.58/11.92	
43.58/11.92	The TRS R consists of the following rules:
43.58/11.92	
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	
43.58/11.92	Q is empty.
43.58/11.92	We have to consider all minimal (P,Q,R)-chains.
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(7) QDPOrderProof (EQUIVALENT)
43.58/11.92	We use the reduction pair processor [LPAR04,JAR06].
43.58/11.92	
43.58/11.92	
43.58/11.92	The following pairs can be oriented strictly and are deleted.
43.58/11.92	
43.58/11.92	   B(x1) -> A(x1)
43.58/11.92	The remaining pairs can at least be oriented weakly.
43.58/11.92	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(b(x_1)) =  	[[1A], [1A], [1A]] 	 +  	[[0A, 1A, 1A], [0A, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(c(x_1)) =  	[[-I], [1A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, 1A, 1A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	
43.58/11.92	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.58/11.92	
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	
43.58/11.92	
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(8)
43.58/11.92	Obligation:
43.58/11.92	Q DP problem:
43.58/11.92	The TRS P consists of the following rules:
43.58/11.92	
43.58/11.92	   A(a(b(c(x1)))) -> B(b(a(a(x1))))
43.58/11.92	   B(x1) -> A(a(x1))
43.58/11.92	
43.58/11.92	The TRS R consists of the following rules:
43.58/11.92	
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	
43.58/11.92	Q is empty.
43.58/11.92	We have to consider all minimal (P,Q,R)-chains.
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(9) QDPOrderProof (EQUIVALENT)
43.58/11.92	We use the reduction pair processor [LPAR04,JAR06].
43.58/11.92	
43.58/11.92	
43.58/11.92	The following pairs can be oriented strictly and are deleted.
43.58/11.92	
43.58/11.92	   B(x1) -> A(a(x1))
43.58/11.92	The remaining pairs can at least be oriented weakly.
43.58/11.92	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, 1A, 1A], [-I, -I, 0A], [1A, 1A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [1A, 1A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	   <<<
43.58/11.92	 POL(B(x_1)) =  	[[-I]] 	 +  	[[1A, 1A, 0A]] 	* 	x_1
43.58/11.92	>>>
43.58/11.92	
43.58/11.92	
43.58/11.92	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.58/11.92	
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	
43.58/11.92	
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(10)
43.58/11.92	Obligation:
43.58/11.92	Q DP problem:
43.58/11.92	The TRS P consists of the following rules:
43.58/11.92	
43.58/11.92	   A(a(b(c(x1)))) -> B(b(a(a(x1))))
43.58/11.92	
43.58/11.92	The TRS R consists of the following rules:
43.58/11.92	
43.58/11.92	   a(a(b(c(x1)))) -> b(b(a(a(x1))))
43.58/11.92	   b(x1) -> c(c(a(a(x1))))
43.58/11.92	   b(c(x1)) -> a(x1)
43.58/11.92	   a(a(c(x1))) -> x1
43.58/11.92	
43.58/11.92	Q is empty.
43.58/11.92	We have to consider all minimal (P,Q,R)-chains.
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(11) DependencyGraphProof (EQUIVALENT)
43.58/11.92	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
43.58/11.92	----------------------------------------
43.58/11.92	
43.58/11.92	(12)
43.58/11.92	TRUE
43.93/12.01	EOF