21.66/6.38	YES
22.38/6.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
22.38/6.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.38/6.60	
22.38/6.60	
22.38/6.60	Termination w.r.t. Q of the given QTRS could be proven:
22.38/6.60	
22.38/6.60	(0) QTRS
22.38/6.60	(1) QTRS Reverse [EQUIVALENT, 0 ms]
22.38/6.60	(2) QTRS
22.38/6.60	(3) DependencyPairsProof [EQUIVALENT, 10 ms]
22.38/6.60	(4) QDP
22.38/6.60	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
22.38/6.60	(6) QDP
22.38/6.60	(7) QDPOrderProof [EQUIVALENT, 118 ms]
22.38/6.60	(8) QDP
22.38/6.60	(9) PisEmptyProof [EQUIVALENT, 0 ms]
22.38/6.60	(10) YES
22.38/6.60	
22.38/6.60	
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(0)
22.38/6.60	Obligation:
22.38/6.60	Q restricted rewrite system:
22.38/6.60	The TRS R consists of the following rules:
22.38/6.60	
22.38/6.60	   a(x1) -> x1
22.38/6.60	   a(b(x1)) -> b(b(b(a(a(c(x1))))))
22.38/6.60	   b(x1) -> x1
22.38/6.60	   c(c(x1)) -> x1
22.38/6.60	
22.38/6.60	Q is empty.
22.38/6.60	
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(1) QTRS Reverse (EQUIVALENT)
22.38/6.60	We applied the QTRS Reverse Processor [REVERSE].
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(2)
22.38/6.60	Obligation:
22.38/6.60	Q restricted rewrite system:
22.38/6.60	The TRS R consists of the following rules:
22.38/6.60	
22.38/6.60	   a(x1) -> x1
22.38/6.60	   b(a(x1)) -> c(a(a(b(b(b(x1))))))
22.38/6.60	   b(x1) -> x1
22.38/6.60	   c(c(x1)) -> x1
22.38/6.60	
22.38/6.60	Q is empty.
22.38/6.60	
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(3) DependencyPairsProof (EQUIVALENT)
22.38/6.60	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(4)
22.38/6.60	Obligation:
22.38/6.60	Q DP problem:
22.38/6.60	The TRS P consists of the following rules:
22.38/6.60	
22.38/6.60	   B(a(x1)) -> C(a(a(b(b(b(x1))))))
22.38/6.60	   B(a(x1)) -> A(a(b(b(b(x1)))))
22.38/6.60	   B(a(x1)) -> A(b(b(b(x1))))
22.38/6.60	   B(a(x1)) -> B(b(b(x1)))
22.38/6.60	   B(a(x1)) -> B(b(x1))
22.38/6.60	   B(a(x1)) -> B(x1)
22.38/6.60	
22.38/6.60	The TRS R consists of the following rules:
22.38/6.60	
22.38/6.60	   a(x1) -> x1
22.38/6.60	   b(a(x1)) -> c(a(a(b(b(b(x1))))))
22.38/6.60	   b(x1) -> x1
22.38/6.60	   c(c(x1)) -> x1
22.38/6.60	
22.38/6.60	Q is empty.
22.38/6.60	We have to consider all minimal (P,Q,R)-chains.
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(5) DependencyGraphProof (EQUIVALENT)
22.38/6.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(6)
22.38/6.60	Obligation:
22.38/6.60	Q DP problem:
22.38/6.60	The TRS P consists of the following rules:
22.38/6.60	
22.38/6.60	   B(a(x1)) -> B(b(x1))
22.38/6.60	   B(a(x1)) -> B(b(b(x1)))
22.38/6.60	   B(a(x1)) -> B(x1)
22.38/6.60	
22.38/6.60	The TRS R consists of the following rules:
22.38/6.60	
22.38/6.60	   a(x1) -> x1
22.38/6.60	   b(a(x1)) -> c(a(a(b(b(b(x1))))))
22.38/6.60	   b(x1) -> x1
22.38/6.60	   c(c(x1)) -> x1
22.38/6.60	
22.38/6.60	Q is empty.
22.38/6.60	We have to consider all minimal (P,Q,R)-chains.
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(7) QDPOrderProof (EQUIVALENT)
22.38/6.60	We use the reduction pair processor [LPAR04,JAR06].
22.38/6.60	
22.38/6.60	
22.38/6.60	The following pairs can be oriented strictly and are deleted.
22.38/6.60	
22.38/6.60	   B(a(x1)) -> B(b(x1))
22.38/6.60	   B(a(x1)) -> B(b(b(x1)))
22.38/6.60	   B(a(x1)) -> B(x1)
22.38/6.60	The remaining pairs can at least be oriented weakly.
22.38/6.60	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.38/6.60	
22.38/6.60	   <<<
22.38/6.60	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
22.38/6.60	>>>
22.38/6.60	
22.38/6.60	   <<<
22.38/6.60	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 1A, -I], [-I, 0A, 0A]] 	* 	x_1
22.38/6.60	>>>
22.38/6.60	
22.38/6.60	   <<<
22.38/6.60	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
22.38/6.60	>>>
22.38/6.60	
22.38/6.60	   <<<
22.38/6.60	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
22.38/6.60	>>>
22.38/6.60	
22.38/6.60	
22.38/6.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.38/6.60	
22.38/6.60	   b(a(x1)) -> c(a(a(b(b(b(x1))))))
22.38/6.60	   b(x1) -> x1
22.38/6.60	   a(x1) -> x1
22.38/6.60	   c(c(x1)) -> x1
22.38/6.60	
22.38/6.60	
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(8)
22.38/6.60	Obligation:
22.38/6.60	Q DP problem:
22.38/6.60	P is empty.
22.38/6.60	The TRS R consists of the following rules:
22.38/6.60	
22.38/6.60	   a(x1) -> x1
22.38/6.60	   b(a(x1)) -> c(a(a(b(b(b(x1))))))
22.38/6.60	   b(x1) -> x1
22.38/6.60	   c(c(x1)) -> x1
22.38/6.60	
22.38/6.60	Q is empty.
22.38/6.60	We have to consider all minimal (P,Q,R)-chains.
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(9) PisEmptyProof (EQUIVALENT)
22.38/6.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
22.38/6.60	----------------------------------------
22.38/6.60	
22.38/6.60	(10)
22.38/6.60	YES
22.62/6.67	EOF