34.46/9.82	YES
34.68/9.83	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
34.68/9.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.68/9.83	
34.68/9.83	
34.68/9.83	Termination w.r.t. Q of the given QTRS could be proven:
34.68/9.83	
34.68/9.83	(0) QTRS
34.68/9.83	(1) QTRS Reverse [EQUIVALENT, 0 ms]
34.68/9.83	(2) QTRS
34.68/9.83	(3) DependencyPairsProof [EQUIVALENT, 13 ms]
34.68/9.83	(4) QDP
34.68/9.83	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
34.68/9.83	(6) QDP
34.68/9.83	(7) QDPOrderProof [EQUIVALENT, 112 ms]
34.68/9.83	(8) QDP
34.68/9.83	(9) QDPOrderProof [EQUIVALENT, 54 ms]
34.68/9.83	(10) QDP
34.68/9.83	(11) QDPOrderProof [EQUIVALENT, 26 ms]
34.68/9.83	(12) QDP
34.68/9.83	(13) PisEmptyProof [EQUIVALENT, 0 ms]
34.68/9.83	(14) YES
34.68/9.83	
34.68/9.83	
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(0)
34.68/9.83	Obligation:
34.68/9.83	Q restricted rewrite system:
34.68/9.83	The TRS R consists of the following rules:
34.68/9.83	
34.68/9.83	   a(x1) -> x1
34.68/9.83	   a(a(b(x1))) -> b(b(b(a(a(x1)))))
34.68/9.83	   b(x1) -> c(a(x1))
34.68/9.83	
34.68/9.83	Q is empty.
34.68/9.83	
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(1) QTRS Reverse (EQUIVALENT)
34.68/9.83	We applied the QTRS Reverse Processor [REVERSE].
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(2)
34.68/9.83	Obligation:
34.68/9.83	Q restricted rewrite system:
34.68/9.83	The TRS R consists of the following rules:
34.68/9.83	
34.68/9.83	   a(x1) -> x1
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	
34.68/9.83	Q is empty.
34.68/9.83	
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(3) DependencyPairsProof (EQUIVALENT)
34.68/9.83	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(4)
34.68/9.83	Obligation:
34.68/9.83	Q DP problem:
34.68/9.83	The TRS P consists of the following rules:
34.68/9.83	
34.68/9.83	   B(a(a(x1))) -> A(a(b(b(b(x1)))))
34.68/9.83	   B(a(a(x1))) -> A(b(b(b(x1))))
34.68/9.83	   B(a(a(x1))) -> B(b(b(x1)))
34.68/9.83	   B(a(a(x1))) -> B(b(x1))
34.68/9.83	   B(a(a(x1))) -> B(x1)
34.68/9.83	   B(x1) -> A(c(x1))
34.68/9.83	
34.68/9.83	The TRS R consists of the following rules:
34.68/9.83	
34.68/9.83	   a(x1) -> x1
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	
34.68/9.83	Q is empty.
34.68/9.83	We have to consider all minimal (P,Q,R)-chains.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(5) DependencyGraphProof (EQUIVALENT)
34.68/9.83	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(6)
34.68/9.83	Obligation:
34.68/9.83	Q DP problem:
34.68/9.83	The TRS P consists of the following rules:
34.68/9.83	
34.68/9.83	   B(a(a(x1))) -> B(b(x1))
34.68/9.83	   B(a(a(x1))) -> B(b(b(x1)))
34.68/9.83	   B(a(a(x1))) -> B(x1)
34.68/9.83	
34.68/9.83	The TRS R consists of the following rules:
34.68/9.83	
34.68/9.83	   a(x1) -> x1
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	
34.68/9.83	Q is empty.
34.68/9.83	We have to consider all minimal (P,Q,R)-chains.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(7) QDPOrderProof (EQUIVALENT)
34.68/9.83	We use the reduction pair processor [LPAR04,JAR06].
34.68/9.83	
34.68/9.83	
34.68/9.83	The following pairs can be oriented strictly and are deleted.
34.68/9.83	
34.68/9.83	   B(a(a(x1))) -> B(b(b(x1)))
34.68/9.83	The remaining pairs can at least be oriented weakly.
34.68/9.83	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 1A, -I], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, -I, -I]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	
34.68/9.83	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.68/9.83	
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	   a(x1) -> x1
34.68/9.83	
34.68/9.83	
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(8)
34.68/9.83	Obligation:
34.68/9.83	Q DP problem:
34.68/9.83	The TRS P consists of the following rules:
34.68/9.83	
34.68/9.83	   B(a(a(x1))) -> B(b(x1))
34.68/9.83	   B(a(a(x1))) -> B(x1)
34.68/9.83	
34.68/9.83	The TRS R consists of the following rules:
34.68/9.83	
34.68/9.83	   a(x1) -> x1
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	
34.68/9.83	Q is empty.
34.68/9.83	We have to consider all minimal (P,Q,R)-chains.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(9) QDPOrderProof (EQUIVALENT)
34.68/9.83	We use the reduction pair processor [LPAR04,JAR06].
34.68/9.83	
34.68/9.83	
34.68/9.83	The following pairs can be oriented strictly and are deleted.
34.68/9.83	
34.68/9.83	   B(a(a(x1))) -> B(x1)
34.68/9.83	The remaining pairs can at least be oriented weakly.
34.68/9.83	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(b(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	
34.68/9.83	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.68/9.83	
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	   a(x1) -> x1
34.68/9.83	
34.68/9.83	
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(10)
34.68/9.83	Obligation:
34.68/9.83	Q DP problem:
34.68/9.83	The TRS P consists of the following rules:
34.68/9.83	
34.68/9.83	   B(a(a(x1))) -> B(b(x1))
34.68/9.83	
34.68/9.83	The TRS R consists of the following rules:
34.68/9.83	
34.68/9.83	   a(x1) -> x1
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	
34.68/9.83	Q is empty.
34.68/9.83	We have to consider all minimal (P,Q,R)-chains.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(11) QDPOrderProof (EQUIVALENT)
34.68/9.83	We use the reduction pair processor [LPAR04,JAR06].
34.68/9.83	
34.68/9.83	
34.68/9.83	The following pairs can be oriented strictly and are deleted.
34.68/9.83	
34.68/9.83	   B(a(a(x1))) -> B(b(x1))
34.68/9.83	The remaining pairs can at least be oriented weakly.
34.68/9.83	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 1A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(b(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	   <<<
34.68/9.83	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
34.68/9.83	>>>
34.68/9.83	
34.68/9.83	
34.68/9.83	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.68/9.83	
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	   a(x1) -> x1
34.68/9.83	
34.68/9.83	
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(12)
34.68/9.83	Obligation:
34.68/9.83	Q DP problem:
34.68/9.83	P is empty.
34.68/9.83	The TRS R consists of the following rules:
34.68/9.83	
34.68/9.83	   a(x1) -> x1
34.68/9.83	   b(a(a(x1))) -> a(a(b(b(b(x1)))))
34.68/9.83	   b(x1) -> a(c(x1))
34.68/9.83	
34.68/9.83	Q is empty.
34.68/9.83	We have to consider all minimal (P,Q,R)-chains.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(13) PisEmptyProof (EQUIVALENT)
34.68/9.83	The TRS P is empty. Hence, there is no (P,Q,R) chain.
34.68/9.83	----------------------------------------
34.68/9.83	
34.68/9.83	(14)
34.68/9.83	YES
34.99/9.95	EOF