21.76/6.40	YES
21.76/6.44	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
21.76/6.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.76/6.44	
21.76/6.44	
21.76/6.44	Termination w.r.t. Q of the given QTRS could be proven:
21.76/6.44	
21.76/6.44	(0) QTRS
21.76/6.44	(1) QTRS Reverse [EQUIVALENT, 0 ms]
21.76/6.44	(2) QTRS
21.76/6.44	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
21.76/6.44	(4) QDP
21.76/6.44	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
21.76/6.44	(6) QDP
21.76/6.44	(7) QDPOrderProof [EQUIVALENT, 120 ms]
21.76/6.44	(8) QDP
21.76/6.44	(9) QDPOrderProof [EQUIVALENT, 47 ms]
21.76/6.44	(10) QDP
21.76/6.44	(11) PisEmptyProof [EQUIVALENT, 0 ms]
21.76/6.44	(12) YES
21.76/6.44	
21.76/6.44	
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(0)
21.76/6.44	Obligation:
21.76/6.44	Q restricted rewrite system:
21.76/6.44	The TRS R consists of the following rules:
21.76/6.44	
21.76/6.44	   a(x1) -> x1
21.76/6.44	   a(b(c(x1))) -> c(c(a(b(b(x1)))))
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	Q is empty.
21.76/6.44	
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(1) QTRS Reverse (EQUIVALENT)
21.76/6.44	We applied the QTRS Reverse Processor [REVERSE].
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(2)
21.76/6.44	Obligation:
21.76/6.44	Q restricted rewrite system:
21.76/6.44	The TRS R consists of the following rules:
21.76/6.44	
21.76/6.44	   a(x1) -> x1
21.76/6.44	   c(b(a(x1))) -> b(b(a(c(c(x1)))))
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	Q is empty.
21.76/6.44	
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(3) DependencyPairsProof (EQUIVALENT)
21.76/6.44	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(4)
21.76/6.44	Obligation:
21.76/6.44	Q DP problem:
21.76/6.44	The TRS P consists of the following rules:
21.76/6.44	
21.76/6.44	   C(b(a(x1))) -> B(b(a(c(c(x1)))))
21.76/6.44	   C(b(a(x1))) -> B(a(c(c(x1))))
21.76/6.44	   C(b(a(x1))) -> A(c(c(x1)))
21.76/6.44	   C(b(a(x1))) -> C(c(x1))
21.76/6.44	   C(b(a(x1))) -> C(x1)
21.76/6.44	   B(x1) -> A(x1)
21.76/6.44	
21.76/6.44	The TRS R consists of the following rules:
21.76/6.44	
21.76/6.44	   a(x1) -> x1
21.76/6.44	   c(b(a(x1))) -> b(b(a(c(c(x1)))))
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	Q is empty.
21.76/6.44	We have to consider all minimal (P,Q,R)-chains.
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(5) DependencyGraphProof (EQUIVALENT)
21.76/6.44	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(6)
21.76/6.44	Obligation:
21.76/6.44	Q DP problem:
21.76/6.44	The TRS P consists of the following rules:
21.76/6.44	
21.76/6.44	   C(b(a(x1))) -> C(x1)
21.76/6.44	   C(b(a(x1))) -> C(c(x1))
21.76/6.44	
21.76/6.44	The TRS R consists of the following rules:
21.76/6.44	
21.76/6.44	   a(x1) -> x1
21.76/6.44	   c(b(a(x1))) -> b(b(a(c(c(x1)))))
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	Q is empty.
21.76/6.44	We have to consider all minimal (P,Q,R)-chains.
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(7) QDPOrderProof (EQUIVALENT)
21.76/6.44	We use the reduction pair processor [LPAR04,JAR06].
21.76/6.44	
21.76/6.44	
21.76/6.44	The following pairs can be oriented strictly and are deleted.
21.76/6.44	
21.76/6.44	   C(b(a(x1))) -> C(x1)
21.76/6.44	The remaining pairs can at least be oriented weakly.
21.76/6.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	
21.76/6.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.76/6.44	
21.76/6.44	   c(b(a(x1))) -> b(b(a(c(c(x1)))))
21.76/6.44	   a(x1) -> x1
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(8)
21.76/6.44	Obligation:
21.76/6.44	Q DP problem:
21.76/6.44	The TRS P consists of the following rules:
21.76/6.44	
21.76/6.44	   C(b(a(x1))) -> C(c(x1))
21.76/6.44	
21.76/6.44	The TRS R consists of the following rules:
21.76/6.44	
21.76/6.44	   a(x1) -> x1
21.76/6.44	   c(b(a(x1))) -> b(b(a(c(c(x1)))))
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	Q is empty.
21.76/6.44	We have to consider all minimal (P,Q,R)-chains.
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(9) QDPOrderProof (EQUIVALENT)
21.76/6.44	We use the reduction pair processor [LPAR04,JAR06].
21.76/6.44	
21.76/6.44	
21.76/6.44	The following pairs can be oriented strictly and are deleted.
21.76/6.44	
21.76/6.44	   C(b(a(x1))) -> C(c(x1))
21.76/6.44	The remaining pairs can at least be oriented weakly.
21.76/6.44	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	   <<<
21.76/6.44	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
21.76/6.44	>>>
21.76/6.44	
21.76/6.44	
21.76/6.44	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.76/6.44	
21.76/6.44	   c(b(a(x1))) -> b(b(a(c(c(x1)))))
21.76/6.44	   a(x1) -> x1
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(10)
21.76/6.44	Obligation:
21.76/6.44	Q DP problem:
21.76/6.44	P is empty.
21.76/6.44	The TRS R consists of the following rules:
21.76/6.44	
21.76/6.44	   a(x1) -> x1
21.76/6.44	   c(b(a(x1))) -> b(b(a(c(c(x1)))))
21.76/6.44	   b(x1) -> a(x1)
21.76/6.44	
21.76/6.44	Q is empty.
21.76/6.44	We have to consider all minimal (P,Q,R)-chains.
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(11) PisEmptyProof (EQUIVALENT)
21.76/6.44	The TRS P is empty. Hence, there is no (P,Q,R) chain.
21.76/6.44	----------------------------------------
21.76/6.44	
21.76/6.44	(12)
21.76/6.44	YES
22.21/6.52	EOF