39.99/11.22	YES
40.56/11.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
40.56/11.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.56/11.39	
40.56/11.39	
40.56/11.39	Termination w.r.t. Q of the given QTRS could be proven:
40.56/11.39	
40.56/11.39	(0) QTRS
40.56/11.39	(1) DependencyPairsProof [EQUIVALENT, 1 ms]
40.56/11.39	(2) QDP
40.56/11.39	(3) QDPOrderProof [EQUIVALENT, 22 ms]
40.56/11.39	(4) QDP
40.56/11.39	(5) QDPOrderProof [EQUIVALENT, 95 ms]
40.56/11.39	(6) QDP
40.56/11.39	(7) QDPOrderProof [EQUIVALENT, 26 ms]
40.56/11.39	(8) QDP
40.56/11.39	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
40.56/11.39	(10) TRUE
40.56/11.39	
40.56/11.39	
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(0)
40.56/11.39	Obligation:
40.56/11.39	Q restricted rewrite system:
40.56/11.39	The TRS R consists of the following rules:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	Q is empty.
40.56/11.39	
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(1) DependencyPairsProof (EQUIVALENT)
40.56/11.39	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(2)
40.56/11.39	Obligation:
40.56/11.39	Q DP problem:
40.56/11.39	The TRS P consists of the following rules:
40.56/11.39	
40.56/11.39	   A(a(b(x1))) -> B(a(a(x1)))
40.56/11.39	   A(a(b(x1))) -> A(a(x1))
40.56/11.39	   A(a(b(x1))) -> A(x1)
40.56/11.39	   B(c(x1)) -> A(b(x1))
40.56/11.39	   B(c(x1)) -> B(x1)
40.56/11.39	
40.56/11.39	The TRS R consists of the following rules:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	Q is empty.
40.56/11.39	We have to consider all minimal (P,Q,R)-chains.
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(3) QDPOrderProof (EQUIVALENT)
40.56/11.39	We use the reduction pair processor [LPAR04,JAR06].
40.56/11.39	
40.56/11.39	
40.56/11.39	The following pairs can be oriented strictly and are deleted.
40.56/11.39	
40.56/11.39	   A(a(b(x1))) -> A(a(x1))
40.56/11.39	   A(a(b(x1))) -> A(x1)
40.56/11.39	The remaining pairs can at least be oriented weakly.
40.56/11.39	Used ordering:  Polynomial interpretation [POLO]:
40.56/11.39	
40.56/11.39	   POL(A(x_1)) = x_1
40.56/11.39	   POL(B(x_1)) = 1 + x_1
40.56/11.39	   POL(a(x_1)) = x_1
40.56/11.39	   POL(b(x_1)) = 1 + x_1
40.56/11.39	   POL(c(x_1)) = x_1
40.56/11.39	
40.56/11.39	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(4)
40.56/11.39	Obligation:
40.56/11.39	Q DP problem:
40.56/11.39	The TRS P consists of the following rules:
40.56/11.39	
40.56/11.39	   A(a(b(x1))) -> B(a(a(x1)))
40.56/11.39	   B(c(x1)) -> A(b(x1))
40.56/11.39	   B(c(x1)) -> B(x1)
40.56/11.39	
40.56/11.39	The TRS R consists of the following rules:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	Q is empty.
40.56/11.39	We have to consider all minimal (P,Q,R)-chains.
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(5) QDPOrderProof (EQUIVALENT)
40.56/11.39	We use the reduction pair processor [LPAR04,JAR06].
40.56/11.39	
40.56/11.39	
40.56/11.39	The following pairs can be oriented strictly and are deleted.
40.56/11.39	
40.56/11.39	   B(c(x1)) -> B(x1)
40.56/11.39	The remaining pairs can at least be oriented weakly.
40.56/11.39	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [1A, -I, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, 0A], [1A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(c(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[-I, 0A, 1A], [0A, 0A, 0A], [1A, 1A, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	
40.56/11.39	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(6)
40.56/11.39	Obligation:
40.56/11.39	Q DP problem:
40.56/11.39	The TRS P consists of the following rules:
40.56/11.39	
40.56/11.39	   A(a(b(x1))) -> B(a(a(x1)))
40.56/11.39	   B(c(x1)) -> A(b(x1))
40.56/11.39	
40.56/11.39	The TRS R consists of the following rules:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	Q is empty.
40.56/11.39	We have to consider all minimal (P,Q,R)-chains.
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(7) QDPOrderProof (EQUIVALENT)
40.56/11.39	We use the reduction pair processor [LPAR04,JAR06].
40.56/11.39	
40.56/11.39	
40.56/11.39	The following pairs can be oriented strictly and are deleted.
40.56/11.39	
40.56/11.39	   A(a(b(x1))) -> B(a(a(x1)))
40.56/11.39	The remaining pairs can at least be oriented weakly.
40.56/11.39	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	   <<<
40.56/11.39	 POL(c(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[0A, -I, 1A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
40.56/11.39	>>>
40.56/11.39	
40.56/11.39	
40.56/11.39	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(8)
40.56/11.39	Obligation:
40.56/11.39	Q DP problem:
40.56/11.39	The TRS P consists of the following rules:
40.56/11.39	
40.56/11.39	   B(c(x1)) -> A(b(x1))
40.56/11.39	
40.56/11.39	The TRS R consists of the following rules:
40.56/11.39	
40.56/11.39	   a(x1) -> x1
40.56/11.39	   a(a(b(x1))) -> c(b(a(a(x1))))
40.56/11.39	   b(c(x1)) -> a(b(x1))
40.56/11.39	
40.56/11.39	Q is empty.
40.56/11.39	We have to consider all minimal (P,Q,R)-chains.
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(9) DependencyGraphProof (EQUIVALENT)
40.56/11.39	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
40.56/11.39	----------------------------------------
40.56/11.39	
40.56/11.39	(10)
40.56/11.39	TRUE
40.94/11.46	EOF