26.07/7.59	YES
26.07/7.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
26.07/7.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.07/7.60	
26.07/7.60	
26.07/7.60	Termination w.r.t. Q of the given QTRS could be proven:
26.07/7.60	
26.07/7.60	(0) QTRS
26.07/7.60	(1) DependencyPairsProof [EQUIVALENT, 27 ms]
26.07/7.60	(2) QDP
26.07/7.60	(3) DependencyGraphProof [EQUIVALENT, 6 ms]
26.07/7.60	(4) QDP
26.07/7.60	(5) QDPOrderProof [EQUIVALENT, 70 ms]
26.07/7.60	(6) QDP
26.07/7.60	(7) PisEmptyProof [EQUIVALENT, 0 ms]
26.07/7.60	(8) YES
26.07/7.60	
26.07/7.60	
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(0)
26.07/7.60	Obligation:
26.07/7.60	Q restricted rewrite system:
26.07/7.60	The TRS R consists of the following rules:
26.07/7.60	
26.07/7.60	   a(x1) -> x1
26.07/7.60	   a(b(x1)) -> c(b(b(a(a(x1)))))
26.07/7.60	   b(b(x1)) -> x1
26.07/7.60	   c(c(x1)) -> x1
26.07/7.60	
26.07/7.60	Q is empty.
26.07/7.60	
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(1) DependencyPairsProof (EQUIVALENT)
26.07/7.60	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(2)
26.07/7.60	Obligation:
26.07/7.60	Q DP problem:
26.07/7.60	The TRS P consists of the following rules:
26.07/7.60	
26.07/7.60	   A(b(x1)) -> C(b(b(a(a(x1)))))
26.07/7.60	   A(b(x1)) -> B(b(a(a(x1))))
26.07/7.60	   A(b(x1)) -> B(a(a(x1)))
26.07/7.60	   A(b(x1)) -> A(a(x1))
26.07/7.60	   A(b(x1)) -> A(x1)
26.07/7.60	
26.07/7.60	The TRS R consists of the following rules:
26.07/7.60	
26.07/7.60	   a(x1) -> x1
26.07/7.60	   a(b(x1)) -> c(b(b(a(a(x1)))))
26.07/7.60	   b(b(x1)) -> x1
26.07/7.60	   c(c(x1)) -> x1
26.07/7.60	
26.07/7.60	Q is empty.
26.07/7.60	We have to consider all minimal (P,Q,R)-chains.
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(3) DependencyGraphProof (EQUIVALENT)
26.07/7.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(4)
26.07/7.60	Obligation:
26.07/7.60	Q DP problem:
26.07/7.60	The TRS P consists of the following rules:
26.07/7.60	
26.07/7.60	   A(b(x1)) -> A(x1)
26.07/7.60	   A(b(x1)) -> A(a(x1))
26.07/7.60	
26.07/7.60	The TRS R consists of the following rules:
26.07/7.60	
26.07/7.60	   a(x1) -> x1
26.07/7.60	   a(b(x1)) -> c(b(b(a(a(x1)))))
26.07/7.60	   b(b(x1)) -> x1
26.07/7.60	   c(c(x1)) -> x1
26.07/7.60	
26.07/7.60	Q is empty.
26.07/7.60	We have to consider all minimal (P,Q,R)-chains.
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(5) QDPOrderProof (EQUIVALENT)
26.07/7.60	We use the reduction pair processor [LPAR04,JAR06].
26.07/7.60	
26.07/7.60	
26.07/7.60	The following pairs can be oriented strictly and are deleted.
26.07/7.60	
26.07/7.60	   A(b(x1)) -> A(x1)
26.07/7.60	   A(b(x1)) -> A(a(x1))
26.07/7.60	The remaining pairs can at least be oriented weakly.
26.07/7.60	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.07/7.60	
26.07/7.60	   <<<
26.07/7.60	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
26.07/7.60	>>>
26.07/7.60	
26.07/7.60	   <<<
26.07/7.60	 POL(b(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 1A]] 	* 	x_1
26.07/7.60	>>>
26.07/7.60	
26.07/7.60	   <<<
26.07/7.60	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
26.07/7.60	>>>
26.07/7.60	
26.07/7.60	   <<<
26.07/7.60	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [-I, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
26.07/7.60	>>>
26.07/7.60	
26.07/7.60	
26.07/7.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.07/7.60	
26.07/7.60	   a(x1) -> x1
26.07/7.60	   a(b(x1)) -> c(b(b(a(a(x1)))))
26.07/7.60	   b(b(x1)) -> x1
26.07/7.60	   c(c(x1)) -> x1
26.07/7.60	
26.07/7.60	
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(6)
26.07/7.60	Obligation:
26.07/7.60	Q DP problem:
26.07/7.60	P is empty.
26.07/7.60	The TRS R consists of the following rules:
26.07/7.60	
26.07/7.60	   a(x1) -> x1
26.07/7.60	   a(b(x1)) -> c(b(b(a(a(x1)))))
26.07/7.60	   b(b(x1)) -> x1
26.07/7.60	   c(c(x1)) -> x1
26.07/7.60	
26.07/7.60	Q is empty.
26.07/7.60	We have to consider all minimal (P,Q,R)-chains.
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(7) PisEmptyProof (EQUIVALENT)
26.07/7.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.07/7.60	----------------------------------------
26.07/7.60	
26.07/7.60	(8)
26.07/7.60	YES
26.45/7.69	EOF