42.92/12.69	YES
43.11/12.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
43.11/12.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
43.11/12.78	
43.11/12.78	
43.11/12.78	Termination w.r.t. Q of the given QTRS could be proven:
43.11/12.78	
43.11/12.78	(0) QTRS
43.11/12.78	(1) QTRS Reverse [EQUIVALENT, 0 ms]
43.11/12.78	(2) QTRS
43.11/12.78	(3) DependencyPairsProof [EQUIVALENT, 22 ms]
43.11/12.78	(4) QDP
43.11/12.78	(5) QDPOrderProof [EQUIVALENT, 121 ms]
43.11/12.78	(6) QDP
43.11/12.78	(7) QDPOrderProof [EQUIVALENT, 34 ms]
43.11/12.78	(8) QDP
43.11/12.78	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
43.11/12.78	(10) QDP
43.11/12.78	(11) QDPOrderProof [EQUIVALENT, 34 ms]
43.11/12.78	(12) QDP
43.11/12.78	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
43.11/12.78	(14) TRUE
43.11/12.78	
43.11/12.78	
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(0)
43.11/12.78	Obligation:
43.11/12.78	Q restricted rewrite system:
43.11/12.78	The TRS R consists of the following rules:
43.11/12.78	
43.11/12.78	   a(x1) -> x1
43.11/12.78	   a(a(x1)) -> b(a(c(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   b(c(x1)) -> a(x1)
43.11/12.78	
43.11/12.78	Q is empty.
43.11/12.78	
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(1) QTRS Reverse (EQUIVALENT)
43.11/12.78	We applied the QTRS Reverse Processor [REVERSE].
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(2)
43.11/12.78	Obligation:
43.11/12.78	Q restricted rewrite system:
43.11/12.78	The TRS R consists of the following rules:
43.11/12.78	
43.11/12.78	   a(x1) -> x1
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	
43.11/12.78	Q is empty.
43.11/12.78	
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(3) DependencyPairsProof (EQUIVALENT)
43.11/12.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(4)
43.11/12.78	Obligation:
43.11/12.78	Q DP problem:
43.11/12.78	The TRS P consists of the following rules:
43.11/12.78	
43.11/12.78	   A(a(x1)) -> B(c(a(b(x1))))
43.11/12.78	   A(a(x1)) -> C(a(b(x1)))
43.11/12.78	   A(a(x1)) -> A(b(x1))
43.11/12.78	   A(a(x1)) -> B(x1)
43.11/12.78	   B(x1) -> C(x1)
43.11/12.78	   C(b(x1)) -> A(x1)
43.11/12.78	
43.11/12.78	The TRS R consists of the following rules:
43.11/12.78	
43.11/12.78	   a(x1) -> x1
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	
43.11/12.78	Q is empty.
43.11/12.78	We have to consider all minimal (P,Q,R)-chains.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(5) QDPOrderProof (EQUIVALENT)
43.11/12.78	We use the reduction pair processor [LPAR04,JAR06].
43.11/12.78	
43.11/12.78	
43.11/12.78	The following pairs can be oriented strictly and are deleted.
43.11/12.78	
43.11/12.78	   A(a(x1)) -> A(b(x1))
43.11/12.78	The remaining pairs can at least be oriented weakly.
43.11/12.78	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[1A, 0A, -I], [0A, 1A, 1A], [1A, 1A, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(c(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [0A, -I, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [1A, 1A, 1A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	
43.11/12.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.11/12.78	
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	   a(x1) -> x1
43.11/12.78	
43.11/12.78	
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(6)
43.11/12.78	Obligation:
43.11/12.78	Q DP problem:
43.11/12.78	The TRS P consists of the following rules:
43.11/12.78	
43.11/12.78	   A(a(x1)) -> B(c(a(b(x1))))
43.11/12.78	   A(a(x1)) -> C(a(b(x1)))
43.11/12.78	   A(a(x1)) -> B(x1)
43.11/12.78	   B(x1) -> C(x1)
43.11/12.78	   C(b(x1)) -> A(x1)
43.11/12.78	
43.11/12.78	The TRS R consists of the following rules:
43.11/12.78	
43.11/12.78	   a(x1) -> x1
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	
43.11/12.78	Q is empty.
43.11/12.78	We have to consider all minimal (P,Q,R)-chains.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(7) QDPOrderProof (EQUIVALENT)
43.11/12.78	We use the reduction pair processor [LPAR04,JAR06].
43.11/12.78	
43.11/12.78	
43.11/12.78	The following pairs can be oriented strictly and are deleted.
43.11/12.78	
43.11/12.78	   B(x1) -> C(x1)
43.11/12.78	The remaining pairs can at least be oriented weakly.
43.11/12.78	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, 1A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [1A, 1A, 1A], [-I, -I, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(B(x_1)) =  	[[1A]] 	 +  	[[1A, 1A, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(c(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[1A, 1A, 1A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	
43.11/12.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.11/12.78	
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	   a(x1) -> x1
43.11/12.78	
43.11/12.78	
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(8)
43.11/12.78	Obligation:
43.11/12.78	Q DP problem:
43.11/12.78	The TRS P consists of the following rules:
43.11/12.78	
43.11/12.78	   A(a(x1)) -> B(c(a(b(x1))))
43.11/12.78	   A(a(x1)) -> C(a(b(x1)))
43.11/12.78	   A(a(x1)) -> B(x1)
43.11/12.78	   C(b(x1)) -> A(x1)
43.11/12.78	
43.11/12.78	The TRS R consists of the following rules:
43.11/12.78	
43.11/12.78	   a(x1) -> x1
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	
43.11/12.78	Q is empty.
43.11/12.78	We have to consider all minimal (P,Q,R)-chains.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(9) DependencyGraphProof (EQUIVALENT)
43.11/12.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(10)
43.11/12.78	Obligation:
43.11/12.78	Q DP problem:
43.11/12.78	The TRS P consists of the following rules:
43.11/12.78	
43.11/12.78	   A(a(x1)) -> C(a(b(x1)))
43.11/12.78	   C(b(x1)) -> A(x1)
43.11/12.78	
43.11/12.78	The TRS R consists of the following rules:
43.11/12.78	
43.11/12.78	   a(x1) -> x1
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	
43.11/12.78	Q is empty.
43.11/12.78	We have to consider all minimal (P,Q,R)-chains.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(11) QDPOrderProof (EQUIVALENT)
43.11/12.78	We use the reduction pair processor [LPAR04,JAR06].
43.11/12.78	
43.11/12.78	
43.11/12.78	The following pairs can be oriented strictly and are deleted.
43.11/12.78	
43.11/12.78	   C(b(x1)) -> A(x1)
43.11/12.78	The remaining pairs can at least be oriented weakly.
43.11/12.78	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(A(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 1A, 0A], [1A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, -I], [1A, 1A, 0A], [1A, 0A, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	   <<<
43.11/12.78	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, -I], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
43.11/12.78	>>>
43.11/12.78	
43.11/12.78	
43.11/12.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.11/12.78	
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	   a(x1) -> x1
43.11/12.78	
43.11/12.78	
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(12)
43.11/12.78	Obligation:
43.11/12.78	Q DP problem:
43.11/12.78	The TRS P consists of the following rules:
43.11/12.78	
43.11/12.78	   A(a(x1)) -> C(a(b(x1)))
43.11/12.78	
43.11/12.78	The TRS R consists of the following rules:
43.11/12.78	
43.11/12.78	   a(x1) -> x1
43.11/12.78	   a(a(x1)) -> b(c(a(b(x1))))
43.11/12.78	   b(x1) -> c(x1)
43.11/12.78	   c(b(x1)) -> a(x1)
43.11/12.78	
43.11/12.78	Q is empty.
43.11/12.78	We have to consider all minimal (P,Q,R)-chains.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(13) DependencyGraphProof (EQUIVALENT)
43.11/12.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
43.11/12.78	----------------------------------------
43.11/12.78	
43.11/12.78	(14)
43.11/12.78	TRUE
43.53/12.88	EOF