38.45/10.65	YES
38.64/10.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
38.64/10.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
38.64/10.70	
38.64/10.70	
38.64/10.70	Termination w.r.t. Q of the given QTRS could be proven:
38.64/10.70	
38.64/10.70	(0) QTRS
38.64/10.70	(1) QTRS Reverse [EQUIVALENT, 0 ms]
38.64/10.70	(2) QTRS
38.64/10.70	(3) DependencyPairsProof [EQUIVALENT, 7 ms]
38.64/10.70	(4) QDP
38.64/10.70	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
38.64/10.70	(6) QDP
38.64/10.70	(7) QDPOrderProof [EQUIVALENT, 74 ms]
38.64/10.70	(8) QDP
38.64/10.70	(9) PisEmptyProof [EQUIVALENT, 0 ms]
38.64/10.70	(10) YES
38.64/10.70	
38.64/10.70	
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(0)
38.64/10.70	Obligation:
38.64/10.70	Q restricted rewrite system:
38.64/10.70	The TRS R consists of the following rules:
38.64/10.70	
38.64/10.70	   a(x1) -> x1
38.64/10.70	   a(b(x1)) -> b(b(a(a(c(x1)))))
38.64/10.70	   b(x1) -> x1
38.64/10.70	   c(c(x1)) -> x1
38.64/10.70	
38.64/10.70	Q is empty.
38.64/10.70	
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(1) QTRS Reverse (EQUIVALENT)
38.64/10.70	We applied the QTRS Reverse Processor [REVERSE].
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(2)
38.64/10.70	Obligation:
38.64/10.70	Q restricted rewrite system:
38.64/10.70	The TRS R consists of the following rules:
38.64/10.70	
38.64/10.70	   a(x1) -> x1
38.64/10.70	   b(a(x1)) -> c(a(a(b(b(x1)))))
38.64/10.70	   b(x1) -> x1
38.64/10.70	   c(c(x1)) -> x1
38.64/10.70	
38.64/10.70	Q is empty.
38.64/10.70	
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(3) DependencyPairsProof (EQUIVALENT)
38.64/10.70	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(4)
38.64/10.70	Obligation:
38.64/10.70	Q DP problem:
38.64/10.70	The TRS P consists of the following rules:
38.64/10.70	
38.64/10.70	   B(a(x1)) -> C(a(a(b(b(x1)))))
38.64/10.70	   B(a(x1)) -> A(a(b(b(x1))))
38.64/10.70	   B(a(x1)) -> A(b(b(x1)))
38.64/10.70	   B(a(x1)) -> B(b(x1))
38.64/10.70	   B(a(x1)) -> B(x1)
38.64/10.70	
38.64/10.70	The TRS R consists of the following rules:
38.64/10.70	
38.64/10.70	   a(x1) -> x1
38.64/10.70	   b(a(x1)) -> c(a(a(b(b(x1)))))
38.64/10.70	   b(x1) -> x1
38.64/10.70	   c(c(x1)) -> x1
38.64/10.70	
38.64/10.70	Q is empty.
38.64/10.70	We have to consider all minimal (P,Q,R)-chains.
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(5) DependencyGraphProof (EQUIVALENT)
38.64/10.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(6)
38.64/10.70	Obligation:
38.64/10.70	Q DP problem:
38.64/10.70	The TRS P consists of the following rules:
38.64/10.70	
38.64/10.70	   B(a(x1)) -> B(x1)
38.64/10.70	   B(a(x1)) -> B(b(x1))
38.64/10.70	
38.64/10.70	The TRS R consists of the following rules:
38.64/10.70	
38.64/10.70	   a(x1) -> x1
38.64/10.70	   b(a(x1)) -> c(a(a(b(b(x1)))))
38.64/10.70	   b(x1) -> x1
38.64/10.70	   c(c(x1)) -> x1
38.64/10.70	
38.64/10.70	Q is empty.
38.64/10.70	We have to consider all minimal (P,Q,R)-chains.
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(7) QDPOrderProof (EQUIVALENT)
38.64/10.70	We use the reduction pair processor [LPAR04,JAR06].
38.64/10.70	
38.64/10.70	
38.64/10.70	The following pairs can be oriented strictly and are deleted.
38.64/10.70	
38.64/10.70	   B(a(x1)) -> B(x1)
38.64/10.70	   B(a(x1)) -> B(b(x1))
38.64/10.70	The remaining pairs can at least be oriented weakly.
38.64/10.70	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.64/10.70	
38.64/10.70	   <<<
38.64/10.70	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
38.64/10.70	>>>
38.64/10.70	
38.64/10.70	   <<<
38.64/10.70	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, -I, 0A], [-I, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
38.64/10.70	>>>
38.64/10.70	
38.64/10.70	   <<<
38.64/10.70	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
38.64/10.70	>>>
38.64/10.70	
38.64/10.70	   <<<
38.64/10.70	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
38.64/10.70	>>>
38.64/10.70	
38.64/10.70	
38.64/10.70	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.64/10.70	
38.64/10.70	   b(a(x1)) -> c(a(a(b(b(x1)))))
38.64/10.70	   b(x1) -> x1
38.64/10.70	   a(x1) -> x1
38.64/10.70	   c(c(x1)) -> x1
38.64/10.70	
38.64/10.70	
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(8)
38.64/10.70	Obligation:
38.64/10.70	Q DP problem:
38.64/10.70	P is empty.
38.64/10.70	The TRS R consists of the following rules:
38.64/10.70	
38.64/10.70	   a(x1) -> x1
38.64/10.70	   b(a(x1)) -> c(a(a(b(b(x1)))))
38.64/10.70	   b(x1) -> x1
38.64/10.70	   c(c(x1)) -> x1
38.64/10.70	
38.64/10.70	Q is empty.
38.64/10.70	We have to consider all minimal (P,Q,R)-chains.
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(9) PisEmptyProof (EQUIVALENT)
38.64/10.70	The TRS P is empty. Hence, there is no (P,Q,R) chain.
38.64/10.70	----------------------------------------
38.64/10.70	
38.64/10.70	(10)
38.64/10.70	YES
38.88/10.81	EOF