29.70/8.45	YES
29.86/8.49	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
29.86/8.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.86/8.49	
29.86/8.49	
29.86/8.49	Termination w.r.t. Q of the given QTRS could be proven:
29.86/8.49	
29.86/8.49	(0) QTRS
29.86/8.49	(1) QTRS Reverse [EQUIVALENT, 0 ms]
29.86/8.49	(2) QTRS
29.86/8.49	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
29.86/8.49	(4) QDP
29.86/8.49	(5) QDPOrderProof [EQUIVALENT, 122 ms]
29.86/8.49	(6) QDP
29.86/8.49	(7) QDPOrderProof [EQUIVALENT, 33 ms]
29.86/8.49	(8) QDP
29.86/8.49	(9) QDPOrderProof [EQUIVALENT, 0 ms]
29.86/8.49	(10) QDP
29.86/8.49	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
29.86/8.49	(12) TRUE
29.86/8.49	
29.86/8.49	
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(0)
29.86/8.49	Obligation:
29.86/8.49	Q restricted rewrite system:
29.86/8.49	The TRS R consists of the following rules:
29.86/8.49	
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	   a(b(x1)) -> c(c(x1))
29.86/8.49	   b(c(x1)) -> b(a(a(x1)))
29.86/8.49	
29.86/8.49	Q is empty.
29.86/8.49	
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(1) QTRS Reverse (EQUIVALENT)
29.86/8.49	We applied the QTRS Reverse Processor [REVERSE].
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(2)
29.86/8.49	Obligation:
29.86/8.49	Q restricted rewrite system:
29.86/8.49	The TRS R consists of the following rules:
29.86/8.49	
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	
29.86/8.49	Q is empty.
29.86/8.49	
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(3) DependencyPairsProof (EQUIVALENT)
29.86/8.49	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(4)
29.86/8.49	Obligation:
29.86/8.49	Q DP problem:
29.86/8.49	The TRS P consists of the following rules:
29.86/8.49	
29.86/8.49	   A(a(x1)) -> B(x1)
29.86/8.49	   B(a(x1)) -> C(c(x1))
29.86/8.49	   B(a(x1)) -> C(x1)
29.86/8.49	   C(b(x1)) -> A(a(b(x1)))
29.86/8.49	   C(b(x1)) -> A(b(x1))
29.86/8.49	
29.86/8.49	The TRS R consists of the following rules:
29.86/8.49	
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	
29.86/8.49	Q is empty.
29.86/8.49	We have to consider all minimal (P,Q,R)-chains.
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(5) QDPOrderProof (EQUIVALENT)
29.86/8.49	We use the reduction pair processor [LPAR04,JAR06].
29.86/8.49	
29.86/8.49	
29.86/8.49	The following pairs can be oriented strictly and are deleted.
29.86/8.49	
29.86/8.49	   B(a(x1)) -> C(x1)
29.86/8.49	The remaining pairs can at least be oriented weakly.
29.86/8.49	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [1A, 1A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 1A, 0A], [0A, -I, -I], [0A, 1A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	
29.86/8.49	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.86/8.49	
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	
29.86/8.49	
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(6)
29.86/8.49	Obligation:
29.86/8.49	Q DP problem:
29.86/8.49	The TRS P consists of the following rules:
29.86/8.49	
29.86/8.49	   A(a(x1)) -> B(x1)
29.86/8.49	   B(a(x1)) -> C(c(x1))
29.86/8.49	   C(b(x1)) -> A(a(b(x1)))
29.86/8.49	   C(b(x1)) -> A(b(x1))
29.86/8.49	
29.86/8.49	The TRS R consists of the following rules:
29.86/8.49	
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	
29.86/8.49	Q is empty.
29.86/8.49	We have to consider all minimal (P,Q,R)-chains.
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(7) QDPOrderProof (EQUIVALENT)
29.86/8.49	We use the reduction pair processor [LPAR04,JAR06].
29.86/8.49	
29.86/8.49	
29.86/8.49	The following pairs can be oriented strictly and are deleted.
29.86/8.49	
29.86/8.49	   C(b(x1)) -> A(b(x1))
29.86/8.49	The remaining pairs can at least be oriented weakly.
29.86/8.49	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(c(x_1)) =  	[[1A], [1A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 1A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	
29.86/8.49	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.86/8.49	
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	
29.86/8.49	
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(8)
29.86/8.49	Obligation:
29.86/8.49	Q DP problem:
29.86/8.49	The TRS P consists of the following rules:
29.86/8.49	
29.86/8.49	   A(a(x1)) -> B(x1)
29.86/8.49	   B(a(x1)) -> C(c(x1))
29.86/8.49	   C(b(x1)) -> A(a(b(x1)))
29.86/8.49	
29.86/8.49	The TRS R consists of the following rules:
29.86/8.49	
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	
29.86/8.49	Q is empty.
29.86/8.49	We have to consider all minimal (P,Q,R)-chains.
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(9) QDPOrderProof (EQUIVALENT)
29.86/8.49	We use the reduction pair processor [LPAR04,JAR06].
29.86/8.49	
29.86/8.49	
29.86/8.49	The following pairs can be oriented strictly and are deleted.
29.86/8.49	
29.86/8.49	   B(a(x1)) -> C(c(x1))
29.86/8.49	The remaining pairs can at least be oriented weakly.
29.86/8.49	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 1A, 1A], [-I, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	   <<<
29.86/8.49	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
29.86/8.49	>>>
29.86/8.49	
29.86/8.49	
29.86/8.49	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.86/8.49	
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	
29.86/8.49	
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(10)
29.86/8.49	Obligation:
29.86/8.49	Q DP problem:
29.86/8.49	The TRS P consists of the following rules:
29.86/8.49	
29.86/8.49	   A(a(x1)) -> B(x1)
29.86/8.49	   C(b(x1)) -> A(a(b(x1)))
29.86/8.49	
29.86/8.49	The TRS R consists of the following rules:
29.86/8.49	
29.86/8.49	   a(a(x1)) -> b(x1)
29.86/8.49	   b(a(x1)) -> c(c(x1))
29.86/8.49	   c(b(x1)) -> a(a(b(x1)))
29.86/8.49	
29.86/8.49	Q is empty.
29.86/8.49	We have to consider all minimal (P,Q,R)-chains.
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(11) DependencyGraphProof (EQUIVALENT)
29.86/8.49	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
29.86/8.49	----------------------------------------
29.86/8.49	
29.86/8.49	(12)
29.86/8.49	TRUE
30.19/8.57	EOF