17.00/5.37	YES
17.81/5.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.81/5.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.81/5.54	
17.81/5.54	
17.81/5.54	Termination w.r.t. Q of the given QTRS could be proven:
17.81/5.54	
17.81/5.54	(0) QTRS
17.81/5.54	(1) DependencyPairsProof [EQUIVALENT, 17 ms]
17.81/5.54	(2) QDP
17.81/5.54	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
17.81/5.54	(4) QDP
17.81/5.54	(5) QDPOrderProof [EQUIVALENT, 110 ms]
17.81/5.54	(6) QDP
17.81/5.54	(7) PisEmptyProof [EQUIVALENT, 0 ms]
17.81/5.54	(8) YES
17.81/5.54	
17.81/5.54	
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(0)
17.81/5.54	Obligation:
17.81/5.54	Q restricted rewrite system:
17.81/5.54	The TRS R consists of the following rules:
17.81/5.54	
17.81/5.54	   a(x1) -> x1
17.81/5.54	   a(x1) -> b(c(x1))
17.81/5.54	   a(b(b(x1))) -> b(b(a(a(x1))))
17.81/5.54	   b(x1) -> x1
17.81/5.54	
17.81/5.54	Q is empty.
17.81/5.54	
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(1) DependencyPairsProof (EQUIVALENT)
17.81/5.54	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(2)
17.81/5.54	Obligation:
17.81/5.54	Q DP problem:
17.81/5.54	The TRS P consists of the following rules:
17.81/5.54	
17.81/5.54	   A(x1) -> B(c(x1))
17.81/5.54	   A(b(b(x1))) -> B(b(a(a(x1))))
17.81/5.54	   A(b(b(x1))) -> B(a(a(x1)))
17.81/5.54	   A(b(b(x1))) -> A(a(x1))
17.81/5.54	   A(b(b(x1))) -> A(x1)
17.81/5.54	
17.81/5.54	The TRS R consists of the following rules:
17.81/5.54	
17.81/5.54	   a(x1) -> x1
17.81/5.54	   a(x1) -> b(c(x1))
17.81/5.54	   a(b(b(x1))) -> b(b(a(a(x1))))
17.81/5.54	   b(x1) -> x1
17.81/5.54	
17.81/5.54	Q is empty.
17.81/5.54	We have to consider all minimal (P,Q,R)-chains.
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(3) DependencyGraphProof (EQUIVALENT)
17.81/5.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(4)
17.81/5.54	Obligation:
17.81/5.54	Q DP problem:
17.81/5.54	The TRS P consists of the following rules:
17.81/5.54	
17.81/5.54	   A(b(b(x1))) -> A(x1)
17.81/5.54	   A(b(b(x1))) -> A(a(x1))
17.81/5.54	
17.81/5.54	The TRS R consists of the following rules:
17.81/5.54	
17.81/5.54	   a(x1) -> x1
17.81/5.54	   a(x1) -> b(c(x1))
17.81/5.54	   a(b(b(x1))) -> b(b(a(a(x1))))
17.81/5.54	   b(x1) -> x1
17.81/5.54	
17.81/5.54	Q is empty.
17.81/5.54	We have to consider all minimal (P,Q,R)-chains.
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(5) QDPOrderProof (EQUIVALENT)
17.81/5.54	We use the reduction pair processor [LPAR04,JAR06].
17.81/5.54	
17.81/5.54	
17.81/5.54	The following pairs can be oriented strictly and are deleted.
17.81/5.54	
17.81/5.54	   A(b(b(x1))) -> A(x1)
17.81/5.54	   A(b(b(x1))) -> A(a(x1))
17.81/5.54	The remaining pairs can at least be oriented weakly.
17.81/5.54	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
17.81/5.54	
17.81/5.54	   <<<
17.81/5.54	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
17.81/5.54	>>>
17.81/5.54	
17.81/5.54	   <<<
17.81/5.54	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
17.81/5.54	>>>
17.81/5.54	
17.81/5.54	   <<<
17.81/5.54	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
17.81/5.54	>>>
17.81/5.54	
17.81/5.54	   <<<
17.81/5.54	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [-I, -I, -I], [-I, 0A, 0A]] 	* 	x_1
17.81/5.54	>>>
17.81/5.54	
17.81/5.54	
17.81/5.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.81/5.54	
17.81/5.54	   a(x1) -> x1
17.81/5.54	   a(x1) -> b(c(x1))
17.81/5.54	   a(b(b(x1))) -> b(b(a(a(x1))))
17.81/5.54	   b(x1) -> x1
17.81/5.54	
17.81/5.54	
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(6)
17.81/5.54	Obligation:
17.81/5.54	Q DP problem:
17.81/5.54	P is empty.
17.81/5.54	The TRS R consists of the following rules:
17.81/5.54	
17.81/5.54	   a(x1) -> x1
17.81/5.54	   a(x1) -> b(c(x1))
17.81/5.54	   a(b(b(x1))) -> b(b(a(a(x1))))
17.81/5.54	   b(x1) -> x1
17.81/5.54	
17.81/5.54	Q is empty.
17.81/5.54	We have to consider all minimal (P,Q,R)-chains.
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(7) PisEmptyProof (EQUIVALENT)
17.81/5.54	The TRS P is empty. Hence, there is no (P,Q,R) chain.
17.81/5.54	----------------------------------------
17.81/5.54	
17.81/5.54	(8)
17.81/5.54	YES
19.88/9.27	EOF