14.27/4.51	YES
15.18/4.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
15.18/4.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.18/4.71	
15.18/4.71	
15.18/4.71	Termination w.r.t. Q of the given QTRS could be proven:
15.18/4.71	
15.18/4.71	(0) QTRS
15.18/4.71	(1) QTRS Reverse [EQUIVALENT, 0 ms]
15.18/4.71	(2) QTRS
15.18/4.71	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
15.18/4.71	(4) QDP
15.18/4.71	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
15.18/4.71	(6) QDP
15.18/4.71	(7) QDPOrderProof [EQUIVALENT, 112 ms]
15.18/4.71	(8) QDP
15.18/4.71	(9) PisEmptyProof [EQUIVALENT, 0 ms]
15.18/4.71	(10) YES
15.18/4.71	
15.18/4.71	
15.18/4.71	----------------------------------------
15.18/4.71	
15.18/4.71	(0)
15.18/4.71	Obligation:
15.18/4.71	Q restricted rewrite system:
15.18/4.71	The TRS R consists of the following rules:
15.18/4.71	
15.18/4.71	   a(x1) -> x1
15.18/4.71	   a(a(b(x1))) -> b(b(a(a(x1))))
15.18/4.71	   b(x1) -> c(a(x1))
15.18/4.71	
15.18/4.71	Q is empty.
15.18/4.71	
15.18/4.71	----------------------------------------
15.18/4.71	
15.18/4.71	(1) QTRS Reverse (EQUIVALENT)
15.18/4.71	We applied the QTRS Reverse Processor [REVERSE].
15.18/4.71	----------------------------------------
15.18/4.71	
15.18/4.71	(2)
15.18/4.71	Obligation:
15.18/4.71	Q restricted rewrite system:
15.18/4.71	The TRS R consists of the following rules:
15.18/4.71	
15.18/4.71	   a(x1) -> x1
15.18/4.71	   b(a(a(x1))) -> a(a(b(b(x1))))
15.18/4.71	   b(x1) -> a(c(x1))
15.18/4.71	
15.18/4.71	Q is empty.
15.18/4.71	
15.18/4.71	----------------------------------------
15.18/4.72	
15.18/4.72	(3) DependencyPairsProof (EQUIVALENT)
15.18/4.72	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
15.18/4.72	----------------------------------------
15.18/4.72	
15.18/4.72	(4)
15.18/4.72	Obligation:
15.18/4.72	Q DP problem:
15.18/4.72	The TRS P consists of the following rules:
15.18/4.72	
15.18/4.72	   B(a(a(x1))) -> A(a(b(b(x1))))
15.18/4.72	   B(a(a(x1))) -> A(b(b(x1)))
15.18/4.72	   B(a(a(x1))) -> B(b(x1))
15.18/4.72	   B(a(a(x1))) -> B(x1)
15.18/4.72	   B(x1) -> A(c(x1))
15.18/4.72	
15.18/4.72	The TRS R consists of the following rules:
15.18/4.72	
15.18/4.72	   a(x1) -> x1
15.18/4.72	   b(a(a(x1))) -> a(a(b(b(x1))))
15.18/4.72	   b(x1) -> a(c(x1))
15.18/4.72	
15.18/4.72	Q is empty.
15.18/4.72	We have to consider all minimal (P,Q,R)-chains.
15.18/4.72	----------------------------------------
15.18/4.72	
15.18/4.72	(5) DependencyGraphProof (EQUIVALENT)
15.18/4.72	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
15.18/4.72	----------------------------------------
15.18/4.72	
15.18/4.72	(6)
15.18/4.72	Obligation:
15.18/4.72	Q DP problem:
15.18/4.72	The TRS P consists of the following rules:
15.18/4.72	
15.18/4.72	   B(a(a(x1))) -> B(x1)
15.18/4.72	   B(a(a(x1))) -> B(b(x1))
15.18/4.72	
15.18/4.72	The TRS R consists of the following rules:
15.18/4.72	
15.18/4.72	   a(x1) -> x1
15.18/4.72	   b(a(a(x1))) -> a(a(b(b(x1))))
15.18/4.72	   b(x1) -> a(c(x1))
15.18/4.72	
15.18/4.72	Q is empty.
15.18/4.72	We have to consider all minimal (P,Q,R)-chains.
15.18/4.72	----------------------------------------
15.18/4.72	
15.18/4.72	(7) QDPOrderProof (EQUIVALENT)
15.18/4.72	We use the reduction pair processor [LPAR04,JAR06].
15.18/4.72	
15.18/4.72	
15.18/4.72	The following pairs can be oriented strictly and are deleted.
15.18/4.72	
15.18/4.72	   B(a(a(x1))) -> B(x1)
15.18/4.72	   B(a(a(x1))) -> B(b(x1))
15.18/4.72	The remaining pairs can at least be oriented weakly.
15.18/4.72	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
15.18/4.72	
15.18/4.72	   <<<
15.18/4.72	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
15.18/4.72	>>>
15.18/4.72	
15.18/4.72	   <<<
15.18/4.72	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 1A], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
15.18/4.72	>>>
15.18/4.72	
15.18/4.72	   <<<
15.18/4.72	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[-I, 0A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
15.18/4.72	>>>
15.18/4.72	
15.18/4.72	   <<<
15.18/4.72	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
15.18/4.72	>>>
15.18/4.72	
15.18/4.72	
15.18/4.72	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
15.18/4.72	
15.18/4.72	   b(a(a(x1))) -> a(a(b(b(x1))))
15.18/4.72	   b(x1) -> a(c(x1))
15.18/4.72	   a(x1) -> x1
15.18/4.72	
15.18/4.72	
15.18/4.72	----------------------------------------
15.18/4.72	
15.18/4.72	(8)
15.18/4.72	Obligation:
15.18/4.72	Q DP problem:
15.18/4.72	P is empty.
15.18/4.72	The TRS R consists of the following rules:
15.18/4.72	
15.18/4.72	   a(x1) -> x1
15.18/4.72	   b(a(a(x1))) -> a(a(b(b(x1))))
15.18/4.72	   b(x1) -> a(c(x1))
15.18/4.72	
15.18/4.72	Q is empty.
15.18/4.72	We have to consider all minimal (P,Q,R)-chains.
15.18/4.72	----------------------------------------
15.18/4.72	
15.18/4.72	(9) PisEmptyProof (EQUIVALENT)
15.18/4.72	The TRS P is empty. Hence, there is no (P,Q,R) chain.
15.18/4.72	----------------------------------------
15.18/4.72	
15.18/4.72	(10)
15.18/4.72	YES
15.43/4.84	EOF