29.39/8.44	YES
29.79/8.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
29.79/8.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.79/8.47	
29.79/8.47	
29.79/8.47	Termination w.r.t. Q of the given QTRS could be proven:
29.79/8.47	
29.79/8.47	(0) QTRS
29.79/8.47	(1) DependencyPairsProof [EQUIVALENT, 8 ms]
29.79/8.47	(2) QDP
29.79/8.47	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
29.79/8.47	(4) QDP
29.79/8.47	(5) QDPOrderProof [EQUIVALENT, 119 ms]
29.79/8.47	(6) QDP
29.79/8.47	(7) QDPOrderProof [EQUIVALENT, 0 ms]
29.79/8.47	(8) QDP
29.79/8.47	(9) PisEmptyProof [EQUIVALENT, 0 ms]
29.79/8.47	(10) YES
29.79/8.47	
29.79/8.47	
29.79/8.47	----------------------------------------
29.79/8.47	
29.79/8.47	(0)
29.79/8.47	Obligation:
29.79/8.47	Q restricted rewrite system:
29.79/8.47	The TRS R consists of the following rules:
29.79/8.47	
29.79/8.47	   a(x1) -> x1
29.79/8.47	   a(x1) -> b(x1)
29.79/8.47	   a(c(b(x1))) -> c(c(a(a(x1))))
29.79/8.47	   c(x1) -> b(x1)
29.79/8.47	
29.79/8.47	Q is empty.
29.79/8.47	
29.79/8.47	----------------------------------------
29.79/8.47	
29.79/8.47	(1) DependencyPairsProof (EQUIVALENT)
29.79/8.47	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.79/8.47	----------------------------------------
29.79/8.47	
29.79/8.47	(2)
29.79/8.47	Obligation:
29.79/8.47	Q DP problem:
29.79/8.47	The TRS P consists of the following rules:
29.79/8.47	
29.79/8.47	   A(c(b(x1))) -> C(c(a(a(x1))))
29.79/8.47	   A(c(b(x1))) -> C(a(a(x1)))
29.79/8.48	   A(c(b(x1))) -> A(a(x1))
29.79/8.48	   A(c(b(x1))) -> A(x1)
29.79/8.48	
29.79/8.48	The TRS R consists of the following rules:
29.79/8.48	
29.79/8.48	   a(x1) -> x1
29.79/8.48	   a(x1) -> b(x1)
29.79/8.48	   a(c(b(x1))) -> c(c(a(a(x1))))
29.79/8.48	   c(x1) -> b(x1)
29.79/8.48	
29.79/8.48	Q is empty.
29.79/8.48	We have to consider all minimal (P,Q,R)-chains.
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(3) DependencyGraphProof (EQUIVALENT)
29.79/8.48	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(4)
29.79/8.48	Obligation:
29.79/8.48	Q DP problem:
29.79/8.48	The TRS P consists of the following rules:
29.79/8.48	
29.79/8.48	   A(c(b(x1))) -> A(x1)
29.79/8.48	   A(c(b(x1))) -> A(a(x1))
29.79/8.48	
29.79/8.48	The TRS R consists of the following rules:
29.79/8.48	
29.79/8.48	   a(x1) -> x1
29.79/8.48	   a(x1) -> b(x1)
29.79/8.48	   a(c(b(x1))) -> c(c(a(a(x1))))
29.79/8.48	   c(x1) -> b(x1)
29.79/8.48	
29.79/8.48	Q is empty.
29.79/8.48	We have to consider all minimal (P,Q,R)-chains.
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(5) QDPOrderProof (EQUIVALENT)
29.79/8.48	We use the reduction pair processor [LPAR04,JAR06].
29.79/8.48	
29.79/8.48	
29.79/8.48	The following pairs can be oriented strictly and are deleted.
29.79/8.48	
29.79/8.48	   A(c(b(x1))) -> A(x1)
29.79/8.48	The remaining pairs can at least be oriented weakly.
29.79/8.48	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(c(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	
29.79/8.48	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.79/8.48	
29.79/8.48	   a(x1) -> x1
29.79/8.48	   a(x1) -> b(x1)
29.79/8.48	   a(c(b(x1))) -> c(c(a(a(x1))))
29.79/8.48	   c(x1) -> b(x1)
29.79/8.48	
29.79/8.48	
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(6)
29.79/8.48	Obligation:
29.79/8.48	Q DP problem:
29.79/8.48	The TRS P consists of the following rules:
29.79/8.48	
29.79/8.48	   A(c(b(x1))) -> A(a(x1))
29.79/8.48	
29.79/8.48	The TRS R consists of the following rules:
29.79/8.48	
29.79/8.48	   a(x1) -> x1
29.79/8.48	   a(x1) -> b(x1)
29.79/8.48	   a(c(b(x1))) -> c(c(a(a(x1))))
29.79/8.48	   c(x1) -> b(x1)
29.79/8.48	
29.79/8.48	Q is empty.
29.79/8.48	We have to consider all minimal (P,Q,R)-chains.
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(7) QDPOrderProof (EQUIVALENT)
29.79/8.48	We use the reduction pair processor [LPAR04,JAR06].
29.79/8.48	
29.79/8.48	
29.79/8.48	The following pairs can be oriented strictly and are deleted.
29.79/8.48	
29.79/8.48	   A(c(b(x1))) -> A(a(x1))
29.79/8.48	The remaining pairs can at least be oriented weakly.
29.79/8.48	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, 1A], [0A, 0A, 0A]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	   <<<
29.79/8.48	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.79/8.48	>>>
29.79/8.48	
29.79/8.48	
29.79/8.48	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.79/8.48	
29.79/8.48	   a(x1) -> x1
29.79/8.48	   a(x1) -> b(x1)
29.79/8.48	   a(c(b(x1))) -> c(c(a(a(x1))))
29.79/8.48	   c(x1) -> b(x1)
29.79/8.48	
29.79/8.48	
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(8)
29.79/8.48	Obligation:
29.79/8.48	Q DP problem:
29.79/8.48	P is empty.
29.79/8.48	The TRS R consists of the following rules:
29.79/8.48	
29.79/8.48	   a(x1) -> x1
29.79/8.48	   a(x1) -> b(x1)
29.79/8.48	   a(c(b(x1))) -> c(c(a(a(x1))))
29.79/8.48	   c(x1) -> b(x1)
29.79/8.48	
29.79/8.48	Q is empty.
29.79/8.48	We have to consider all minimal (P,Q,R)-chains.
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(9) PisEmptyProof (EQUIVALENT)
29.79/8.48	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.79/8.48	----------------------------------------
29.79/8.48	
29.79/8.48	(10)
29.79/8.48	YES
30.17/8.58	EOF