26.39/7.74	YES
26.39/7.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
26.39/7.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.39/7.75	
26.39/7.75	
26.39/7.75	Termination w.r.t. Q of the given QTRS could be proven:
26.39/7.75	
26.39/7.75	(0) QTRS
26.39/7.75	(1) DependencyPairsProof [EQUIVALENT, 20 ms]
26.39/7.75	(2) QDP
26.39/7.75	(3) DependencyGraphProof [EQUIVALENT, 1 ms]
26.39/7.75	(4) QDP
26.39/7.75	(5) QDPOrderProof [EQUIVALENT, 11 ms]
26.39/7.75	(6) QDP
26.39/7.75	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
26.39/7.75	(8) TRUE
26.39/7.75	
26.39/7.75	
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(0)
26.39/7.75	Obligation:
26.39/7.75	Q restricted rewrite system:
26.39/7.75	The TRS R consists of the following rules:
26.39/7.75	
26.39/7.75	   a(x1) -> x1
26.39/7.75	   a(b(x1)) -> c(x1)
26.39/7.75	   b(x1) -> x1
26.39/7.75	   c(c(x1)) -> b(b(a(a(c(x1)))))
26.39/7.75	
26.39/7.75	Q is empty.
26.39/7.75	
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(1) DependencyPairsProof (EQUIVALENT)
26.39/7.75	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(2)
26.39/7.75	Obligation:
26.39/7.75	Q DP problem:
26.39/7.75	The TRS P consists of the following rules:
26.39/7.75	
26.39/7.75	   A(b(x1)) -> C(x1)
26.39/7.75	   C(c(x1)) -> B(b(a(a(c(x1)))))
26.39/7.75	   C(c(x1)) -> B(a(a(c(x1))))
26.39/7.75	   C(c(x1)) -> A(a(c(x1)))
26.39/7.75	   C(c(x1)) -> A(c(x1))
26.39/7.75	
26.39/7.75	The TRS R consists of the following rules:
26.39/7.75	
26.39/7.75	   a(x1) -> x1
26.39/7.75	   a(b(x1)) -> c(x1)
26.39/7.75	   b(x1) -> x1
26.39/7.75	   c(c(x1)) -> b(b(a(a(c(x1)))))
26.39/7.75	
26.39/7.75	Q is empty.
26.39/7.75	We have to consider all minimal (P,Q,R)-chains.
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(3) DependencyGraphProof (EQUIVALENT)
26.39/7.75	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(4)
26.39/7.75	Obligation:
26.39/7.75	Q DP problem:
26.39/7.75	The TRS P consists of the following rules:
26.39/7.75	
26.39/7.75	   C(c(x1)) -> A(a(c(x1)))
26.39/7.75	   A(b(x1)) -> C(x1)
26.39/7.75	   C(c(x1)) -> A(c(x1))
26.39/7.75	
26.39/7.75	The TRS R consists of the following rules:
26.39/7.75	
26.39/7.75	   a(x1) -> x1
26.39/7.75	   a(b(x1)) -> c(x1)
26.39/7.75	   b(x1) -> x1
26.39/7.75	   c(c(x1)) -> b(b(a(a(c(x1)))))
26.39/7.75	
26.39/7.75	Q is empty.
26.39/7.75	We have to consider all minimal (P,Q,R)-chains.
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(5) QDPOrderProof (EQUIVALENT)
26.39/7.75	We use the reduction pair processor [LPAR04,JAR06].
26.39/7.75	
26.39/7.75	
26.39/7.75	The following pairs can be oriented strictly and are deleted.
26.39/7.75	
26.39/7.75	   A(b(x1)) -> C(x1)
26.39/7.75	The remaining pairs can at least be oriented weakly.
26.39/7.75	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.39/7.75	
26.39/7.75	   <<<
26.39/7.75	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
26.39/7.75	>>>
26.39/7.75	
26.39/7.75	   <<<
26.39/7.75	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, 1A, 0A], [0A, 1A, 0A]] 	* 	x_1
26.39/7.75	>>>
26.39/7.75	
26.39/7.75	   <<<
26.39/7.75	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
26.39/7.75	>>>
26.39/7.75	
26.39/7.75	   <<<
26.39/7.75	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
26.39/7.75	>>>
26.39/7.75	
26.39/7.75	   <<<
26.39/7.75	 POL(b(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, 0A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
26.39/7.75	>>>
26.39/7.75	
26.39/7.75	
26.39/7.75	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.39/7.75	
26.39/7.75	   c(c(x1)) -> b(b(a(a(c(x1)))))
26.39/7.75	   a(x1) -> x1
26.39/7.75	   a(b(x1)) -> c(x1)
26.39/7.75	   b(x1) -> x1
26.39/7.75	
26.39/7.75	
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(6)
26.39/7.75	Obligation:
26.39/7.75	Q DP problem:
26.39/7.75	The TRS P consists of the following rules:
26.39/7.75	
26.39/7.75	   C(c(x1)) -> A(a(c(x1)))
26.39/7.75	   C(c(x1)) -> A(c(x1))
26.39/7.75	
26.39/7.75	The TRS R consists of the following rules:
26.39/7.75	
26.39/7.75	   a(x1) -> x1
26.39/7.75	   a(b(x1)) -> c(x1)
26.39/7.75	   b(x1) -> x1
26.39/7.75	   c(c(x1)) -> b(b(a(a(c(x1)))))
26.39/7.75	
26.39/7.75	Q is empty.
26.39/7.75	We have to consider all minimal (P,Q,R)-chains.
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(7) DependencyGraphProof (EQUIVALENT)
26.39/7.75	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
26.39/7.75	----------------------------------------
26.39/7.75	
26.39/7.75	(8)
26.39/7.75	TRUE
26.79/7.90	EOF