23.10/6.77	YES
23.25/6.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
23.25/6.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.25/6.78	
23.25/6.78	
23.25/6.78	Termination w.r.t. Q of the given QTRS could be proven:
23.25/6.78	
23.25/6.78	(0) QTRS
23.25/6.78	(1) QTRS Reverse [EQUIVALENT, 0 ms]
23.25/6.78	(2) QTRS
23.25/6.78	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
23.25/6.78	(4) QDP
23.25/6.78	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
23.25/6.78	(6) QDP
23.25/6.78	(7) QDPOrderProof [EQUIVALENT, 128 ms]
23.25/6.78	(8) QDP
23.25/6.78	(9) QDPOrderProof [EQUIVALENT, 0 ms]
23.25/6.78	(10) QDP
23.25/6.78	(11) PisEmptyProof [EQUIVALENT, 0 ms]
23.25/6.78	(12) YES
23.25/6.78	
23.25/6.78	
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(0)
23.25/6.78	Obligation:
23.25/6.78	Q restricted rewrite system:
23.25/6.78	The TRS R consists of the following rules:
23.25/6.78	
23.25/6.78	   a(x1) -> x1
23.25/6.78	   a(a(b(x1))) -> b(b(a(a(x1))))
23.25/6.78	   b(x1) -> c(c(a(x1)))
23.25/6.78	
23.25/6.78	Q is empty.
23.25/6.78	
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(1) QTRS Reverse (EQUIVALENT)
23.25/6.78	We applied the QTRS Reverse Processor [REVERSE].
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(2)
23.25/6.78	Obligation:
23.25/6.78	Q restricted rewrite system:
23.25/6.78	The TRS R consists of the following rules:
23.25/6.78	
23.25/6.78	   a(x1) -> x1
23.25/6.78	   b(a(a(x1))) -> a(a(b(b(x1))))
23.25/6.78	   b(x1) -> a(c(c(x1)))
23.25/6.78	
23.25/6.78	Q is empty.
23.25/6.78	
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(3) DependencyPairsProof (EQUIVALENT)
23.25/6.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(4)
23.25/6.78	Obligation:
23.25/6.78	Q DP problem:
23.25/6.78	The TRS P consists of the following rules:
23.25/6.78	
23.25/6.78	   B(a(a(x1))) -> A(a(b(b(x1))))
23.25/6.78	   B(a(a(x1))) -> A(b(b(x1)))
23.25/6.78	   B(a(a(x1))) -> B(b(x1))
23.25/6.78	   B(a(a(x1))) -> B(x1)
23.25/6.78	   B(x1) -> A(c(c(x1)))
23.25/6.78	
23.25/6.78	The TRS R consists of the following rules:
23.25/6.78	
23.25/6.78	   a(x1) -> x1
23.25/6.78	   b(a(a(x1))) -> a(a(b(b(x1))))
23.25/6.78	   b(x1) -> a(c(c(x1)))
23.25/6.78	
23.25/6.78	Q is empty.
23.25/6.78	We have to consider all minimal (P,Q,R)-chains.
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(5) DependencyGraphProof (EQUIVALENT)
23.25/6.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(6)
23.25/6.78	Obligation:
23.25/6.78	Q DP problem:
23.25/6.78	The TRS P consists of the following rules:
23.25/6.78	
23.25/6.78	   B(a(a(x1))) -> B(x1)
23.25/6.78	   B(a(a(x1))) -> B(b(x1))
23.25/6.78	
23.25/6.78	The TRS R consists of the following rules:
23.25/6.78	
23.25/6.78	   a(x1) -> x1
23.25/6.78	   b(a(a(x1))) -> a(a(b(b(x1))))
23.25/6.78	   b(x1) -> a(c(c(x1)))
23.25/6.78	
23.25/6.78	Q is empty.
23.25/6.78	We have to consider all minimal (P,Q,R)-chains.
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(7) QDPOrderProof (EQUIVALENT)
23.25/6.78	We use the reduction pair processor [LPAR04,JAR06].
23.25/6.78	
23.25/6.78	
23.25/6.78	The following pairs can be oriented strictly and are deleted.
23.25/6.78	
23.25/6.78	   B(a(a(x1))) -> B(x1)
23.25/6.78	The remaining pairs can at least be oriented weakly.
23.25/6.78	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 0A]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [1A, 1A, 0A]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [-I, 0A, -I]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	
23.25/6.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.25/6.78	
23.25/6.78	   b(a(a(x1))) -> a(a(b(b(x1))))
23.25/6.78	   b(x1) -> a(c(c(x1)))
23.25/6.78	   a(x1) -> x1
23.25/6.78	
23.25/6.78	
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(8)
23.25/6.78	Obligation:
23.25/6.78	Q DP problem:
23.25/6.78	The TRS P consists of the following rules:
23.25/6.78	
23.25/6.78	   B(a(a(x1))) -> B(b(x1))
23.25/6.78	
23.25/6.78	The TRS R consists of the following rules:
23.25/6.78	
23.25/6.78	   a(x1) -> x1
23.25/6.78	   b(a(a(x1))) -> a(a(b(b(x1))))
23.25/6.78	   b(x1) -> a(c(c(x1)))
23.25/6.78	
23.25/6.78	Q is empty.
23.25/6.78	We have to consider all minimal (P,Q,R)-chains.
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(9) QDPOrderProof (EQUIVALENT)
23.25/6.78	We use the reduction pair processor [LPAR04,JAR06].
23.25/6.78	
23.25/6.78	
23.25/6.78	The following pairs can be oriented strictly and are deleted.
23.25/6.78	
23.25/6.78	   B(a(a(x1))) -> B(b(x1))
23.25/6.78	The remaining pairs can at least be oriented weakly.
23.25/6.78	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(a(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(b(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[0A, -I, -I], [1A, -I, -I], [0A, -I, -I]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	   <<<
23.25/6.78	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
23.25/6.78	>>>
23.25/6.78	
23.25/6.78	
23.25/6.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.25/6.78	
23.25/6.78	   b(a(a(x1))) -> a(a(b(b(x1))))
23.25/6.78	   b(x1) -> a(c(c(x1)))
23.25/6.78	   a(x1) -> x1
23.25/6.78	
23.25/6.78	
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(10)
23.25/6.78	Obligation:
23.25/6.78	Q DP problem:
23.25/6.78	P is empty.
23.25/6.78	The TRS R consists of the following rules:
23.25/6.78	
23.25/6.78	   a(x1) -> x1
23.25/6.78	   b(a(a(x1))) -> a(a(b(b(x1))))
23.25/6.78	   b(x1) -> a(c(c(x1)))
23.25/6.78	
23.25/6.78	Q is empty.
23.25/6.78	We have to consider all minimal (P,Q,R)-chains.
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(11) PisEmptyProof (EQUIVALENT)
23.25/6.78	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.25/6.78	----------------------------------------
23.25/6.78	
23.25/6.78	(12)
23.25/6.78	YES
23.25/6.83	EOF