23.24/6.84	YES
23.45/6.88	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
23.45/6.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.45/6.88	
23.45/6.88	
23.45/6.88	Termination w.r.t. Q of the given QTRS could be proven:
23.45/6.88	
23.45/6.88	(0) QTRS
23.45/6.88	(1) QTRS Reverse [EQUIVALENT, 0 ms]
23.45/6.88	(2) QTRS
23.45/6.88	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
23.45/6.88	(4) QDP
23.45/6.88	(5) QDPOrderProof [EQUIVALENT, 161 ms]
23.45/6.88	(6) QDP
23.45/6.88	(7) QDPOrderProof [EQUIVALENT, 0 ms]
23.45/6.88	(8) QDP
23.45/6.88	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
23.45/6.88	(10) TRUE
23.45/6.88	
23.45/6.88	
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(0)
23.45/6.88	Obligation:
23.45/6.88	Q restricted rewrite system:
23.45/6.88	The TRS R consists of the following rules:
23.45/6.88	
23.45/6.88	   a(x1) -> b(x1)
23.45/6.88	   b(a(c(x1))) -> c(b(a(a(x1))))
23.45/6.88	   b(b(x1)) -> c(x1)
23.45/6.88	
23.45/6.88	Q is empty.
23.45/6.88	
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(1) QTRS Reverse (EQUIVALENT)
23.45/6.88	We applied the QTRS Reverse Processor [REVERSE].
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(2)
23.45/6.88	Obligation:
23.45/6.88	Q restricted rewrite system:
23.45/6.88	The TRS R consists of the following rules:
23.45/6.88	
23.45/6.88	   a(x1) -> b(x1)
23.45/6.88	   c(a(b(x1))) -> a(a(b(c(x1))))
23.45/6.88	   b(b(x1)) -> c(x1)
23.45/6.88	
23.45/6.88	Q is empty.
23.45/6.88	
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(3) DependencyPairsProof (EQUIVALENT)
23.45/6.88	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(4)
23.45/6.88	Obligation:
23.45/6.88	Q DP problem:
23.45/6.88	The TRS P consists of the following rules:
23.45/6.88	
23.45/6.88	   A(x1) -> B(x1)
23.45/6.88	   C(a(b(x1))) -> A(a(b(c(x1))))
23.45/6.88	   C(a(b(x1))) -> A(b(c(x1)))
23.45/6.88	   C(a(b(x1))) -> B(c(x1))
23.45/6.88	   C(a(b(x1))) -> C(x1)
23.45/6.88	   B(b(x1)) -> C(x1)
23.45/6.88	
23.45/6.88	The TRS R consists of the following rules:
23.45/6.88	
23.45/6.88	   a(x1) -> b(x1)
23.45/6.88	   c(a(b(x1))) -> a(a(b(c(x1))))
23.45/6.88	   b(b(x1)) -> c(x1)
23.45/6.88	
23.45/6.88	Q is empty.
23.45/6.88	We have to consider all minimal (P,Q,R)-chains.
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(5) QDPOrderProof (EQUIVALENT)
23.45/6.88	We use the reduction pair processor [LPAR04,JAR06].
23.45/6.88	
23.45/6.88	
23.45/6.88	The following pairs can be oriented strictly and are deleted.
23.45/6.88	
23.45/6.88	   C(a(b(x1))) -> C(x1)
23.45/6.88	The remaining pairs can at least be oriented weakly.
23.45/6.88	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, -I]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	
23.45/6.88	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.45/6.88	
23.45/6.88	   b(b(x1)) -> c(x1)
23.45/6.88	   c(a(b(x1))) -> a(a(b(c(x1))))
23.45/6.88	   a(x1) -> b(x1)
23.45/6.88	
23.45/6.88	
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(6)
23.45/6.88	Obligation:
23.45/6.88	Q DP problem:
23.45/6.88	The TRS P consists of the following rules:
23.45/6.88	
23.45/6.88	   A(x1) -> B(x1)
23.45/6.88	   C(a(b(x1))) -> A(a(b(c(x1))))
23.45/6.88	   C(a(b(x1))) -> A(b(c(x1)))
23.45/6.88	   C(a(b(x1))) -> B(c(x1))
23.45/6.88	   B(b(x1)) -> C(x1)
23.45/6.88	
23.45/6.88	The TRS R consists of the following rules:
23.45/6.88	
23.45/6.88	   a(x1) -> b(x1)
23.45/6.88	   c(a(b(x1))) -> a(a(b(c(x1))))
23.45/6.88	   b(b(x1)) -> c(x1)
23.45/6.88	
23.45/6.88	Q is empty.
23.45/6.88	We have to consider all minimal (P,Q,R)-chains.
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(7) QDPOrderProof (EQUIVALENT)
23.45/6.88	We use the reduction pair processor [LPAR04,JAR06].
23.45/6.88	
23.45/6.88	
23.45/6.88	The following pairs can be oriented strictly and are deleted.
23.45/6.88	
23.45/6.88	   B(b(x1)) -> C(x1)
23.45/6.88	The remaining pairs can at least be oriented weakly.
23.45/6.88	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	   <<<
23.45/6.88	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.45/6.88	>>>
23.45/6.88	
23.45/6.88	
23.45/6.88	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.45/6.88	
23.45/6.88	   b(b(x1)) -> c(x1)
23.45/6.88	   c(a(b(x1))) -> a(a(b(c(x1))))
23.45/6.88	   a(x1) -> b(x1)
23.45/6.88	
23.45/6.88	
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(8)
23.45/6.88	Obligation:
23.45/6.88	Q DP problem:
23.45/6.88	The TRS P consists of the following rules:
23.45/6.88	
23.45/6.88	   A(x1) -> B(x1)
23.45/6.88	   C(a(b(x1))) -> A(a(b(c(x1))))
23.45/6.88	   C(a(b(x1))) -> A(b(c(x1)))
23.45/6.88	   C(a(b(x1))) -> B(c(x1))
23.45/6.88	
23.45/6.88	The TRS R consists of the following rules:
23.45/6.88	
23.45/6.88	   a(x1) -> b(x1)
23.45/6.88	   c(a(b(x1))) -> a(a(b(c(x1))))
23.45/6.88	   b(b(x1)) -> c(x1)
23.45/6.88	
23.45/6.88	Q is empty.
23.45/6.88	We have to consider all minimal (P,Q,R)-chains.
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(9) DependencyGraphProof (EQUIVALENT)
23.45/6.88	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
23.45/6.88	----------------------------------------
23.45/6.88	
23.45/6.88	(10)
23.45/6.88	TRUE
23.56/10.18	EOF