23.28/6.95	YES
23.28/6.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.28/6.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.28/6.96	
23.28/6.96	
23.28/6.96	Termination w.r.t. Q of the given QTRS could be proven:
23.28/6.96	
23.28/6.96	(0) QTRS
23.28/6.96	(1) DependencyPairsProof [EQUIVALENT, 4 ms]
23.28/6.96	(2) QDP
23.28/6.96	(3) QDPOrderProof [EQUIVALENT, 108 ms]
23.28/6.96	(4) QDP
23.28/6.96	(5) PisEmptyProof [EQUIVALENT, 0 ms]
23.28/6.96	(6) YES
23.28/6.96	
23.28/6.96	
23.28/6.96	----------------------------------------
23.28/6.96	
23.28/6.96	(0)
23.28/6.96	Obligation:
23.28/6.96	Q restricted rewrite system:
23.28/6.96	The TRS R consists of the following rules:
23.28/6.96	
23.28/6.96	   a(x1) -> b(c(x1))
23.28/6.96	   a(b(x1)) -> x1
23.28/6.96	   a(b(b(x1))) -> b(b(a(a(x1))))
23.28/6.96	
23.28/6.96	Q is empty.
23.28/6.96	
23.28/6.96	----------------------------------------
23.28/6.96	
23.28/6.96	(1) DependencyPairsProof (EQUIVALENT)
23.28/6.96	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.28/6.96	----------------------------------------
23.28/6.96	
23.28/6.96	(2)
23.28/6.96	Obligation:
23.28/6.96	Q DP problem:
23.28/6.96	The TRS P consists of the following rules:
23.28/6.96	
23.28/6.96	   A(b(b(x1))) -> A(a(x1))
23.28/6.96	   A(b(b(x1))) -> A(x1)
23.28/6.96	
23.28/6.96	The TRS R consists of the following rules:
23.28/6.96	
23.28/6.96	   a(x1) -> b(c(x1))
23.28/6.96	   a(b(x1)) -> x1
23.28/6.96	   a(b(b(x1))) -> b(b(a(a(x1))))
23.28/6.96	
23.28/6.96	Q is empty.
23.28/6.96	We have to consider all minimal (P,Q,R)-chains.
23.28/6.96	----------------------------------------
23.28/6.96	
23.28/6.96	(3) QDPOrderProof (EQUIVALENT)
23.28/6.96	We use the reduction pair processor [LPAR04,JAR06].
23.28/6.96	
23.28/6.96	
23.28/6.96	The following pairs can be oriented strictly and are deleted.
23.28/6.96	
23.28/6.96	   A(b(b(x1))) -> A(a(x1))
23.28/6.96	   A(b(b(x1))) -> A(x1)
23.28/6.96	The remaining pairs can at least be oriented weakly.
23.28/6.96	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.28/6.96	
23.28/6.96	   <<<
23.28/6.96	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
23.28/6.96	>>>
23.28/6.96	
23.28/6.96	   <<<
23.28/6.96	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
23.28/6.96	>>>
23.28/6.96	
23.28/6.96	   <<<
23.28/6.96	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
23.28/6.96	>>>
23.28/6.96	
23.28/6.96	   <<<
23.28/6.96	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [-I, -I, -I]] 	* 	x_1
23.28/6.96	>>>
23.28/6.96	
23.28/6.96	
23.28/6.96	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.28/6.96	
23.28/6.96	   a(x1) -> b(c(x1))
23.28/6.96	   a(b(x1)) -> x1
23.28/6.96	   a(b(b(x1))) -> b(b(a(a(x1))))
23.28/6.96	
23.28/6.96	
23.28/6.96	----------------------------------------
23.28/6.96	
23.28/6.96	(4)
23.28/6.96	Obligation:
23.28/6.96	Q DP problem:
23.28/6.96	P is empty.
23.28/6.96	The TRS R consists of the following rules:
23.28/6.96	
23.28/6.96	   a(x1) -> b(c(x1))
23.28/6.96	   a(b(x1)) -> x1
23.28/6.96	   a(b(b(x1))) -> b(b(a(a(x1))))
23.28/6.96	
23.28/6.96	Q is empty.
23.28/6.96	We have to consider all minimal (P,Q,R)-chains.
23.28/6.96	----------------------------------------
23.28/6.96	
23.28/6.96	(5) PisEmptyProof (EQUIVALENT)
23.28/6.96	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.28/6.96	----------------------------------------
23.28/6.96	
23.28/6.96	(6)
23.28/6.96	YES
23.62/7.04	EOF