29.56/8.44	YES
29.98/8.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
29.98/8.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.98/8.48	
29.98/8.48	
29.98/8.48	Termination w.r.t. Q of the given QTRS could be proven:
29.98/8.48	
29.98/8.48	(0) QTRS
29.98/8.48	(1) QTRS Reverse [EQUIVALENT, 0 ms]
29.98/8.48	(2) QTRS
29.98/8.48	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
29.98/8.48	(4) QDP
29.98/8.48	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
29.98/8.48	(6) QDP
29.98/8.48	(7) QDPOrderProof [EQUIVALENT, 129 ms]
29.98/8.48	(8) QDP
29.98/8.48	(9) QDPOrderProof [EQUIVALENT, 39 ms]
29.98/8.48	(10) QDP
29.98/8.48	(11) PisEmptyProof [EQUIVALENT, 0 ms]
29.98/8.48	(12) YES
29.98/8.48	
29.98/8.48	
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(0)
29.98/8.48	Obligation:
29.98/8.48	Q restricted rewrite system:
29.98/8.48	The TRS R consists of the following rules:
29.98/8.48	
29.98/8.48	   a(x1) -> x1
29.98/8.48	   a(a(b(x1))) -> b(a(b(c(a(x1)))))
29.98/8.48	   c(b(x1)) -> a(x1)
29.98/8.48	
29.98/8.48	Q is empty.
29.98/8.48	
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(1) QTRS Reverse (EQUIVALENT)
29.98/8.48	We applied the QTRS Reverse Processor [REVERSE].
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(2)
29.98/8.48	Obligation:
29.98/8.48	Q restricted rewrite system:
29.98/8.48	The TRS R consists of the following rules:
29.98/8.48	
29.98/8.48	   a(x1) -> x1
29.98/8.48	   b(a(a(x1))) -> a(c(b(a(b(x1)))))
29.98/8.48	   b(c(x1)) -> a(x1)
29.98/8.48	
29.98/8.48	Q is empty.
29.98/8.48	
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(3) DependencyPairsProof (EQUIVALENT)
29.98/8.48	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(4)
29.98/8.48	Obligation:
29.98/8.48	Q DP problem:
29.98/8.48	The TRS P consists of the following rules:
29.98/8.48	
29.98/8.48	   B(a(a(x1))) -> A(c(b(a(b(x1)))))
29.98/8.48	   B(a(a(x1))) -> B(a(b(x1)))
29.98/8.48	   B(a(a(x1))) -> A(b(x1))
29.98/8.48	   B(a(a(x1))) -> B(x1)
29.98/8.48	   B(c(x1)) -> A(x1)
29.98/8.48	
29.98/8.48	The TRS R consists of the following rules:
29.98/8.48	
29.98/8.48	   a(x1) -> x1
29.98/8.48	   b(a(a(x1))) -> a(c(b(a(b(x1)))))
29.98/8.48	   b(c(x1)) -> a(x1)
29.98/8.48	
29.98/8.48	Q is empty.
29.98/8.48	We have to consider all minimal (P,Q,R)-chains.
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(5) DependencyGraphProof (EQUIVALENT)
29.98/8.48	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(6)
29.98/8.48	Obligation:
29.98/8.48	Q DP problem:
29.98/8.48	The TRS P consists of the following rules:
29.98/8.48	
29.98/8.48	   B(a(a(x1))) -> B(x1)
29.98/8.48	   B(a(a(x1))) -> B(a(b(x1)))
29.98/8.48	
29.98/8.48	The TRS R consists of the following rules:
29.98/8.48	
29.98/8.48	   a(x1) -> x1
29.98/8.48	   b(a(a(x1))) -> a(c(b(a(b(x1)))))
29.98/8.48	   b(c(x1)) -> a(x1)
29.98/8.48	
29.98/8.48	Q is empty.
29.98/8.48	We have to consider all minimal (P,Q,R)-chains.
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(7) QDPOrderProof (EQUIVALENT)
29.98/8.48	We use the reduction pair processor [LPAR04,JAR06].
29.98/8.48	
29.98/8.48	
29.98/8.48	The following pairs can be oriented strictly and are deleted.
29.98/8.48	
29.98/8.48	   B(a(a(x1))) -> B(x1)
29.98/8.48	The remaining pairs can at least be oriented weakly.
29.98/8.48	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [1A, 0A, -I]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	
29.98/8.48	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.98/8.48	
29.98/8.48	   b(a(a(x1))) -> a(c(b(a(b(x1)))))
29.98/8.48	   b(c(x1)) -> a(x1)
29.98/8.48	   a(x1) -> x1
29.98/8.48	
29.98/8.48	
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(8)
29.98/8.48	Obligation:
29.98/8.48	Q DP problem:
29.98/8.48	The TRS P consists of the following rules:
29.98/8.48	
29.98/8.48	   B(a(a(x1))) -> B(a(b(x1)))
29.98/8.48	
29.98/8.48	The TRS R consists of the following rules:
29.98/8.48	
29.98/8.48	   a(x1) -> x1
29.98/8.48	   b(a(a(x1))) -> a(c(b(a(b(x1)))))
29.98/8.48	   b(c(x1)) -> a(x1)
29.98/8.48	
29.98/8.48	Q is empty.
29.98/8.48	We have to consider all minimal (P,Q,R)-chains.
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(9) QDPOrderProof (EQUIVALENT)
29.98/8.48	We use the reduction pair processor [LPAR04,JAR06].
29.98/8.48	
29.98/8.48	
29.98/8.48	The following pairs can be oriented strictly and are deleted.
29.98/8.48	
29.98/8.48	   B(a(a(x1))) -> B(a(b(x1)))
29.98/8.48	The remaining pairs can at least be oriented weakly.
29.98/8.48	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 1A, -I], [1A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	   <<<
29.98/8.48	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, 1A, 0A]] 	* 	x_1
29.98/8.48	>>>
29.98/8.48	
29.98/8.48	
29.98/8.48	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.98/8.48	
29.98/8.48	   b(a(a(x1))) -> a(c(b(a(b(x1)))))
29.98/8.48	   b(c(x1)) -> a(x1)
29.98/8.48	   a(x1) -> x1
29.98/8.48	
29.98/8.48	
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(10)
29.98/8.48	Obligation:
29.98/8.48	Q DP problem:
29.98/8.48	P is empty.
29.98/8.48	The TRS R consists of the following rules:
29.98/8.48	
29.98/8.48	   a(x1) -> x1
29.98/8.48	   b(a(a(x1))) -> a(c(b(a(b(x1)))))
29.98/8.48	   b(c(x1)) -> a(x1)
29.98/8.48	
29.98/8.48	Q is empty.
29.98/8.48	We have to consider all minimal (P,Q,R)-chains.
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(11) PisEmptyProof (EQUIVALENT)
29.98/8.48	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.98/8.48	----------------------------------------
29.98/8.48	
29.98/8.48	(12)
29.98/8.48	YES
30.21/8.58	EOF