30.99/8.83	YES
32.13/9.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.13/9.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.13/9.13	
32.13/9.13	
32.13/9.13	Termination w.r.t. Q of the given QTRS could be proven:
32.13/9.13	
32.13/9.13	(0) QTRS
32.13/9.13	(1) DependencyPairsProof [EQUIVALENT, 3 ms]
32.13/9.13	(2) QDP
32.13/9.13	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
32.13/9.13	(4) QDP
32.13/9.13	(5) QDPOrderProof [EQUIVALENT, 118 ms]
32.13/9.13	(6) QDP
32.13/9.13	(7) QDPOrderProof [EQUIVALENT, 72 ms]
32.13/9.13	(8) QDP
32.13/9.13	(9) PisEmptyProof [EQUIVALENT, 0 ms]
32.13/9.13	(10) YES
32.13/9.13	
32.13/9.13	
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(0)
32.13/9.13	Obligation:
32.13/9.13	Q restricted rewrite system:
32.13/9.13	The TRS R consists of the following rules:
32.13/9.13	
32.13/9.13	   a(b(c(x1))) -> c(b(a(a(c(b(x1))))))
32.13/9.13	   a(x1) -> x1
32.13/9.13	   b(x1) -> x1
32.13/9.13	   c(x1) -> x1
32.13/9.13	
32.13/9.13	Q is empty.
32.13/9.13	
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(1) DependencyPairsProof (EQUIVALENT)
32.13/9.13	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(2)
32.13/9.13	Obligation:
32.13/9.13	Q DP problem:
32.13/9.13	The TRS P consists of the following rules:
32.13/9.13	
32.13/9.13	   A(b(c(x1))) -> C(b(a(a(c(b(x1))))))
32.13/9.13	   A(b(c(x1))) -> B(a(a(c(b(x1)))))
32.13/9.13	   A(b(c(x1))) -> A(a(c(b(x1))))
32.13/9.13	   A(b(c(x1))) -> A(c(b(x1)))
32.13/9.13	   A(b(c(x1))) -> C(b(x1))
32.13/9.13	   A(b(c(x1))) -> B(x1)
32.13/9.13	
32.13/9.13	The TRS R consists of the following rules:
32.13/9.13	
32.13/9.13	   a(b(c(x1))) -> c(b(a(a(c(b(x1))))))
32.13/9.13	   a(x1) -> x1
32.13/9.13	   b(x1) -> x1
32.13/9.13	   c(x1) -> x1
32.13/9.13	
32.13/9.13	Q is empty.
32.13/9.13	We have to consider all minimal (P,Q,R)-chains.
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(3) DependencyGraphProof (EQUIVALENT)
32.13/9.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(4)
32.13/9.13	Obligation:
32.13/9.13	Q DP problem:
32.13/9.13	The TRS P consists of the following rules:
32.13/9.13	
32.13/9.13	   A(b(c(x1))) -> A(c(b(x1)))
32.13/9.13	   A(b(c(x1))) -> A(a(c(b(x1))))
32.13/9.13	
32.13/9.13	The TRS R consists of the following rules:
32.13/9.13	
32.13/9.13	   a(b(c(x1))) -> c(b(a(a(c(b(x1))))))
32.13/9.13	   a(x1) -> x1
32.13/9.13	   b(x1) -> x1
32.13/9.13	   c(x1) -> x1
32.13/9.13	
32.13/9.13	Q is empty.
32.13/9.13	We have to consider all minimal (P,Q,R)-chains.
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(5) QDPOrderProof (EQUIVALENT)
32.13/9.13	We use the reduction pair processor [LPAR04,JAR06].
32.13/9.13	
32.13/9.13	
32.13/9.13	The following pairs can be oriented strictly and are deleted.
32.13/9.13	
32.13/9.13	   A(b(c(x1))) -> A(c(b(x1)))
32.13/9.13	The remaining pairs can at least be oriented weakly.
32.13/9.13	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(c(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 1A, 1A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	
32.13/9.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.13/9.13	
32.13/9.13	   b(x1) -> x1
32.13/9.13	   c(x1) -> x1
32.13/9.13	   a(b(c(x1))) -> c(b(a(a(c(b(x1))))))
32.13/9.13	   a(x1) -> x1
32.13/9.13	
32.13/9.13	
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(6)
32.13/9.13	Obligation:
32.13/9.13	Q DP problem:
32.13/9.13	The TRS P consists of the following rules:
32.13/9.13	
32.13/9.13	   A(b(c(x1))) -> A(a(c(b(x1))))
32.13/9.13	
32.13/9.13	The TRS R consists of the following rules:
32.13/9.13	
32.13/9.13	   a(b(c(x1))) -> c(b(a(a(c(b(x1))))))
32.13/9.13	   a(x1) -> x1
32.13/9.13	   b(x1) -> x1
32.13/9.13	   c(x1) -> x1
32.13/9.13	
32.13/9.13	Q is empty.
32.13/9.13	We have to consider all minimal (P,Q,R)-chains.
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(7) QDPOrderProof (EQUIVALENT)
32.13/9.13	We use the reduction pair processor [LPAR04,JAR06].
32.13/9.13	
32.13/9.13	
32.13/9.13	The following pairs can be oriented strictly and are deleted.
32.13/9.13	
32.13/9.13	   A(b(c(x1))) -> A(a(c(b(x1))))
32.13/9.13	The remaining pairs can at least be oriented weakly.
32.13/9.13	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, -I], [-I, 0A, -I], [0A, -I, 0A]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	   <<<
32.13/9.13	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, -I, 0A]] 	* 	x_1
32.13/9.13	>>>
32.13/9.13	
32.13/9.13	
32.13/9.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.13/9.13	
32.13/9.13	   b(x1) -> x1
32.13/9.13	   c(x1) -> x1
32.13/9.13	   a(b(c(x1))) -> c(b(a(a(c(b(x1))))))
32.13/9.13	   a(x1) -> x1
32.13/9.13	
32.13/9.13	
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(8)
32.13/9.13	Obligation:
32.13/9.13	Q DP problem:
32.13/9.13	P is empty.
32.13/9.13	The TRS R consists of the following rules:
32.13/9.13	
32.13/9.13	   a(b(c(x1))) -> c(b(a(a(c(b(x1))))))
32.13/9.13	   a(x1) -> x1
32.13/9.13	   b(x1) -> x1
32.13/9.13	   c(x1) -> x1
32.13/9.13	
32.13/9.13	Q is empty.
32.13/9.13	We have to consider all minimal (P,Q,R)-chains.
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(9) PisEmptyProof (EQUIVALENT)
32.13/9.13	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.13/9.13	----------------------------------------
32.13/9.13	
32.13/9.13	(10)
32.13/9.13	YES
32.35/9.21	EOF