35.05/9.98	YES
35.68/10.04	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
35.68/10.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.68/10.04	
35.68/10.04	
35.68/10.04	Termination w.r.t. Q of the given QTRS could be proven:
35.68/10.04	
35.68/10.04	(0) QTRS
35.68/10.04	(1) QTRS Reverse [EQUIVALENT, 0 ms]
35.68/10.04	(2) QTRS
35.68/10.04	(3) DependencyPairsProof [EQUIVALENT, 12 ms]
35.68/10.04	(4) QDP
35.68/10.04	(5) DependencyGraphProof [EQUIVALENT, 5 ms]
35.68/10.04	(6) QDP
35.68/10.04	(7) QDPOrderProof [EQUIVALENT, 87 ms]
35.68/10.04	(8) QDP
35.68/10.04	(9) QDPOrderProof [EQUIVALENT, 28 ms]
35.68/10.04	(10) QDP
35.68/10.04	(11) PisEmptyProof [EQUIVALENT, 0 ms]
35.68/10.04	(12) YES
35.68/10.04	
35.68/10.04	
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(0)
35.68/10.04	Obligation:
35.68/10.04	Q restricted rewrite system:
35.68/10.04	The TRS R consists of the following rules:
35.68/10.04	
35.68/10.04	   a(x1) -> x1
35.68/10.04	   a(a(b(x1))) -> b(b(a(a(x1))))
35.68/10.04	   b(b(x1)) -> c(a(x1))
35.68/10.04	
35.68/10.04	Q is empty.
35.68/10.04	
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(1) QTRS Reverse (EQUIVALENT)
35.68/10.04	We applied the QTRS Reverse Processor [REVERSE].
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(2)
35.68/10.04	Obligation:
35.68/10.04	Q restricted rewrite system:
35.68/10.04	The TRS R consists of the following rules:
35.68/10.04	
35.68/10.04	   a(x1) -> x1
35.68/10.04	   b(a(a(x1))) -> a(a(b(b(x1))))
35.68/10.04	   b(b(x1)) -> a(c(x1))
35.68/10.04	
35.68/10.04	Q is empty.
35.68/10.04	
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(3) DependencyPairsProof (EQUIVALENT)
35.68/10.04	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(4)
35.68/10.04	Obligation:
35.68/10.04	Q DP problem:
35.68/10.04	The TRS P consists of the following rules:
35.68/10.04	
35.68/10.04	   B(a(a(x1))) -> A(a(b(b(x1))))
35.68/10.04	   B(a(a(x1))) -> A(b(b(x1)))
35.68/10.04	   B(a(a(x1))) -> B(b(x1))
35.68/10.04	   B(a(a(x1))) -> B(x1)
35.68/10.04	   B(b(x1)) -> A(c(x1))
35.68/10.04	
35.68/10.04	The TRS R consists of the following rules:
35.68/10.04	
35.68/10.04	   a(x1) -> x1
35.68/10.04	   b(a(a(x1))) -> a(a(b(b(x1))))
35.68/10.04	   b(b(x1)) -> a(c(x1))
35.68/10.04	
35.68/10.04	Q is empty.
35.68/10.04	We have to consider all minimal (P,Q,R)-chains.
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(5) DependencyGraphProof (EQUIVALENT)
35.68/10.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(6)
35.68/10.04	Obligation:
35.68/10.04	Q DP problem:
35.68/10.04	The TRS P consists of the following rules:
35.68/10.04	
35.68/10.04	   B(a(a(x1))) -> B(x1)
35.68/10.04	   B(a(a(x1))) -> B(b(x1))
35.68/10.04	
35.68/10.04	The TRS R consists of the following rules:
35.68/10.04	
35.68/10.04	   a(x1) -> x1
35.68/10.04	   b(a(a(x1))) -> a(a(b(b(x1))))
35.68/10.04	   b(b(x1)) -> a(c(x1))
35.68/10.04	
35.68/10.04	Q is empty.
35.68/10.04	We have to consider all minimal (P,Q,R)-chains.
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(7) QDPOrderProof (EQUIVALENT)
35.68/10.04	We use the reduction pair processor [LPAR04,JAR06].
35.68/10.04	
35.68/10.04	
35.68/10.04	The following pairs can be oriented strictly and are deleted.
35.68/10.04	
35.68/10.04	   B(a(a(x1))) -> B(x1)
35.68/10.04	The remaining pairs can at least be oriented weakly.
35.68/10.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [-I, 1A, 0A]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [-I, 0A, -I], [-I, 1A, 0A]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	
35.68/10.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.68/10.04	
35.68/10.04	   b(a(a(x1))) -> a(a(b(b(x1))))
35.68/10.04	   b(b(x1)) -> a(c(x1))
35.68/10.04	   a(x1) -> x1
35.68/10.04	
35.68/10.04	
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(8)
35.68/10.04	Obligation:
35.68/10.04	Q DP problem:
35.68/10.04	The TRS P consists of the following rules:
35.68/10.04	
35.68/10.04	   B(a(a(x1))) -> B(b(x1))
35.68/10.04	
35.68/10.04	The TRS R consists of the following rules:
35.68/10.04	
35.68/10.04	   a(x1) -> x1
35.68/10.04	   b(a(a(x1))) -> a(a(b(b(x1))))
35.68/10.04	   b(b(x1)) -> a(c(x1))
35.68/10.04	
35.68/10.04	Q is empty.
35.68/10.04	We have to consider all minimal (P,Q,R)-chains.
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(9) QDPOrderProof (EQUIVALENT)
35.68/10.04	We use the reduction pair processor [LPAR04,JAR06].
35.68/10.04	
35.68/10.04	
35.68/10.04	The following pairs can be oriented strictly and are deleted.
35.68/10.04	
35.68/10.04	   B(a(a(x1))) -> B(b(x1))
35.68/10.04	The remaining pairs can at least be oriented weakly.
35.68/10.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(a(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, -I, -I], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	   <<<
35.68/10.04	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, -I], [-I, -I, -I], [-I, 0A, -I]] 	* 	x_1
35.68/10.04	>>>
35.68/10.04	
35.68/10.04	
35.68/10.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.68/10.04	
35.68/10.04	   b(a(a(x1))) -> a(a(b(b(x1))))
35.68/10.04	   b(b(x1)) -> a(c(x1))
35.68/10.04	   a(x1) -> x1
35.68/10.04	
35.68/10.04	
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(10)
35.68/10.04	Obligation:
35.68/10.04	Q DP problem:
35.68/10.04	P is empty.
35.68/10.04	The TRS R consists of the following rules:
35.68/10.04	
35.68/10.04	   a(x1) -> x1
35.68/10.04	   b(a(a(x1))) -> a(a(b(b(x1))))
35.68/10.04	   b(b(x1)) -> a(c(x1))
35.68/10.04	
35.68/10.04	Q is empty.
35.68/10.04	We have to consider all minimal (P,Q,R)-chains.
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(11) PisEmptyProof (EQUIVALENT)
35.68/10.04	The TRS P is empty. Hence, there is no (P,Q,R) chain.
35.68/10.04	----------------------------------------
35.68/10.04	
35.68/10.04	(12)
35.68/10.04	YES
36.14/10.37	EOF