22.70/6.70	YES
22.70/6.75	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
22.70/6.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.70/6.75	
22.70/6.75	
22.70/6.75	Termination w.r.t. Q of the given QTRS could be proven:
22.70/6.75	
22.70/6.75	(0) QTRS
22.70/6.75	(1) QTRS Reverse [EQUIVALENT, 0 ms]
22.70/6.75	(2) QTRS
22.70/6.75	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
22.70/6.75	(4) QDP
22.70/6.75	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
22.70/6.75	(6) QDP
22.70/6.75	(7) QDPOrderProof [EQUIVALENT, 172 ms]
22.70/6.75	(8) QDP
22.70/6.75	(9) PisEmptyProof [EQUIVALENT, 0 ms]
22.70/6.75	(10) YES
22.70/6.75	
22.70/6.75	
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(0)
22.70/6.75	Obligation:
22.70/6.75	Q restricted rewrite system:
22.70/6.75	The TRS R consists of the following rules:
22.70/6.75	
22.70/6.75	   a(x1) -> x1
22.70/6.75	   a(b(x1)) -> b(b(a(a(c(x1)))))
22.70/6.75	   b(x1) -> x1
22.70/6.75	   c(a(c(x1))) -> x1
22.70/6.75	
22.70/6.75	Q is empty.
22.70/6.75	
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(1) QTRS Reverse (EQUIVALENT)
22.70/6.75	We applied the QTRS Reverse Processor [REVERSE].
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(2)
22.70/6.75	Obligation:
22.70/6.75	Q restricted rewrite system:
22.70/6.75	The TRS R consists of the following rules:
22.70/6.75	
22.70/6.75	   a(x1) -> x1
22.70/6.75	   b(a(x1)) -> c(a(a(b(b(x1)))))
22.70/6.75	   b(x1) -> x1
22.70/6.75	   c(a(c(x1))) -> x1
22.70/6.75	
22.70/6.75	Q is empty.
22.70/6.75	
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(3) DependencyPairsProof (EQUIVALENT)
22.70/6.75	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(4)
22.70/6.75	Obligation:
22.70/6.75	Q DP problem:
22.70/6.75	The TRS P consists of the following rules:
22.70/6.75	
22.70/6.75	   B(a(x1)) -> C(a(a(b(b(x1)))))
22.70/6.75	   B(a(x1)) -> A(a(b(b(x1))))
22.70/6.75	   B(a(x1)) -> A(b(b(x1)))
22.70/6.75	   B(a(x1)) -> B(b(x1))
22.70/6.75	   B(a(x1)) -> B(x1)
22.70/6.75	
22.70/6.75	The TRS R consists of the following rules:
22.70/6.75	
22.70/6.75	   a(x1) -> x1
22.70/6.75	   b(a(x1)) -> c(a(a(b(b(x1)))))
22.70/6.75	   b(x1) -> x1
22.70/6.75	   c(a(c(x1))) -> x1
22.70/6.75	
22.70/6.75	Q is empty.
22.70/6.75	We have to consider all minimal (P,Q,R)-chains.
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(5) DependencyGraphProof (EQUIVALENT)
22.70/6.75	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(6)
22.70/6.75	Obligation:
22.70/6.75	Q DP problem:
22.70/6.75	The TRS P consists of the following rules:
22.70/6.75	
22.70/6.75	   B(a(x1)) -> B(x1)
22.70/6.75	   B(a(x1)) -> B(b(x1))
22.70/6.75	
22.70/6.75	The TRS R consists of the following rules:
22.70/6.75	
22.70/6.75	   a(x1) -> x1
22.70/6.75	   b(a(x1)) -> c(a(a(b(b(x1)))))
22.70/6.75	   b(x1) -> x1
22.70/6.75	   c(a(c(x1))) -> x1
22.70/6.75	
22.70/6.75	Q is empty.
22.70/6.75	We have to consider all minimal (P,Q,R)-chains.
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(7) QDPOrderProof (EQUIVALENT)
22.70/6.75	We use the reduction pair processor [LPAR04,JAR06].
22.70/6.75	
22.70/6.75	
22.70/6.75	The following pairs can be oriented strictly and are deleted.
22.70/6.75	
22.70/6.75	   B(a(x1)) -> B(x1)
22.70/6.75	   B(a(x1)) -> B(b(x1))
22.70/6.75	The remaining pairs can at least be oriented weakly.
22.70/6.75	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.70/6.75	
22.70/6.75	   <<<
22.70/6.75	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
22.70/6.75	>>>
22.70/6.75	
22.70/6.75	   <<<
22.70/6.75	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
22.70/6.75	>>>
22.70/6.75	
22.70/6.75	   <<<
22.70/6.75	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 1A], [-I, 0A, -I], [-I, -I, 0A]] 	* 	x_1
22.70/6.75	>>>
22.70/6.75	
22.70/6.75	   <<<
22.70/6.75	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
22.70/6.75	>>>
22.70/6.75	
22.70/6.75	
22.70/6.75	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.70/6.75	
22.70/6.75	   b(a(x1)) -> c(a(a(b(b(x1)))))
22.70/6.75	   b(x1) -> x1
22.70/6.75	   a(x1) -> x1
22.70/6.75	   c(a(c(x1))) -> x1
22.70/6.75	
22.70/6.75	
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(8)
22.70/6.75	Obligation:
22.70/6.75	Q DP problem:
22.70/6.75	P is empty.
22.70/6.75	The TRS R consists of the following rules:
22.70/6.75	
22.70/6.75	   a(x1) -> x1
22.70/6.75	   b(a(x1)) -> c(a(a(b(b(x1)))))
22.70/6.75	   b(x1) -> x1
22.70/6.75	   c(a(c(x1))) -> x1
22.70/6.75	
22.70/6.75	Q is empty.
22.70/6.75	We have to consider all minimal (P,Q,R)-chains.
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(9) PisEmptyProof (EQUIVALENT)
22.70/6.75	The TRS P is empty. Hence, there is no (P,Q,R) chain.
22.70/6.75	----------------------------------------
22.70/6.75	
22.70/6.75	(10)
22.70/6.75	YES
23.09/6.88	EOF