17.30/5.22	YES
17.30/5.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.30/5.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.30/5.23	
17.30/5.23	
17.30/5.23	Termination w.r.t. Q of the given QTRS could be proven:
17.30/5.23	
17.30/5.23	(0) QTRS
17.30/5.23	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
17.30/5.23	(2) QDP
17.30/5.23	(3) QDPOrderProof [EQUIVALENT, 94 ms]
17.30/5.23	(4) QDP
17.30/5.23	(5) PisEmptyProof [EQUIVALENT, 0 ms]
17.30/5.23	(6) YES
17.30/5.23	
17.30/5.23	
17.30/5.23	----------------------------------------
17.30/5.23	
17.30/5.23	(0)
17.30/5.23	Obligation:
17.30/5.23	Q restricted rewrite system:
17.30/5.23	The TRS R consists of the following rules:
17.30/5.23	
17.30/5.23	   a(x1) -> x1
17.30/5.23	   a(a(x1)) -> b(c(x1))
17.30/5.23	   a(b(b(x1))) -> b(b(a(a(x1))))
17.30/5.23	
17.30/5.23	Q is empty.
17.30/5.23	
17.30/5.23	----------------------------------------
17.30/5.23	
17.30/5.23	(1) DependencyPairsProof (EQUIVALENT)
17.30/5.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.30/5.23	----------------------------------------
17.30/5.23	
17.30/5.23	(2)
17.30/5.23	Obligation:
17.30/5.23	Q DP problem:
17.30/5.23	The TRS P consists of the following rules:
17.30/5.23	
17.30/5.23	   A(b(b(x1))) -> A(a(x1))
17.30/5.23	   A(b(b(x1))) -> A(x1)
17.30/5.23	
17.30/5.23	The TRS R consists of the following rules:
17.30/5.23	
17.30/5.23	   a(x1) -> x1
17.30/5.23	   a(a(x1)) -> b(c(x1))
17.30/5.23	   a(b(b(x1))) -> b(b(a(a(x1))))
17.30/5.23	
17.30/5.23	Q is empty.
17.30/5.23	We have to consider all minimal (P,Q,R)-chains.
17.30/5.23	----------------------------------------
17.30/5.23	
17.30/5.23	(3) QDPOrderProof (EQUIVALENT)
17.30/5.23	We use the reduction pair processor [LPAR04,JAR06].
17.30/5.23	
17.30/5.23	
17.30/5.23	The following pairs can be oriented strictly and are deleted.
17.30/5.23	
17.30/5.23	   A(b(b(x1))) -> A(a(x1))
17.30/5.23	   A(b(b(x1))) -> A(x1)
17.30/5.23	The remaining pairs can at least be oriented weakly.
17.30/5.23	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
17.30/5.23	
17.30/5.23	   <<<
17.30/5.23	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
17.30/5.23	>>>
17.30/5.23	
17.30/5.23	   <<<
17.30/5.23	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
17.30/5.23	>>>
17.30/5.23	
17.30/5.23	   <<<
17.30/5.23	 POL(a(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
17.30/5.23	>>>
17.30/5.23	
17.30/5.23	   <<<
17.30/5.23	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
17.30/5.23	>>>
17.30/5.23	
17.30/5.23	
17.30/5.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.30/5.23	
17.30/5.23	   a(x1) -> x1
17.30/5.23	   a(a(x1)) -> b(c(x1))
17.30/5.23	   a(b(b(x1))) -> b(b(a(a(x1))))
17.30/5.23	
17.30/5.23	
17.30/5.23	----------------------------------------
17.30/5.23	
17.30/5.23	(4)
17.30/5.23	Obligation:
17.30/5.23	Q DP problem:
17.30/5.23	P is empty.
17.30/5.23	The TRS R consists of the following rules:
17.30/5.23	
17.30/5.23	   a(x1) -> x1
17.30/5.23	   a(a(x1)) -> b(c(x1))
17.30/5.23	   a(b(b(x1))) -> b(b(a(a(x1))))
17.30/5.23	
17.30/5.23	Q is empty.
17.30/5.23	We have to consider all minimal (P,Q,R)-chains.
17.30/5.23	----------------------------------------
17.30/5.23	
17.30/5.23	(5) PisEmptyProof (EQUIVALENT)
17.30/5.23	The TRS P is empty. Hence, there is no (P,Q,R) chain.
17.30/5.23	----------------------------------------
17.30/5.23	
17.30/5.23	(6)
17.30/5.23	YES
17.44/5.33	EOF