41.75/11.58	YES
42.15/11.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
42.15/11.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
42.15/11.71	
42.15/11.71	
42.15/11.71	Termination w.r.t. Q of the given QTRS could be proven:
42.15/11.71	
42.15/11.71	(0) QTRS
42.15/11.71	(1) QTRS Reverse [EQUIVALENT, 0 ms]
42.15/11.71	(2) QTRS
42.15/11.71	(3) DependencyPairsProof [EQUIVALENT, 12 ms]
42.15/11.71	(4) QDP
42.15/11.71	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
42.15/11.71	(6) QDP
42.15/11.71	(7) QDPOrderProof [EQUIVALENT, 142 ms]
42.15/11.71	(8) QDP
42.15/11.71	(9) QDPOrderProof [EQUIVALENT, 48 ms]
42.15/11.71	(10) QDP
42.15/11.71	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
42.15/11.71	(12) TRUE
42.15/11.71	
42.15/11.71	
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(0)
42.15/11.71	Obligation:
42.15/11.71	Q restricted rewrite system:
42.15/11.71	The TRS R consists of the following rules:
42.15/11.71	
42.15/11.71	   a(x1) -> x1
42.15/11.71	   a(a(b(b(x1)))) -> b(c(a(c(x1))))
42.15/11.71	   c(x1) -> b(a(x1))
42.15/11.71	
42.15/11.71	Q is empty.
42.15/11.71	
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(1) QTRS Reverse (EQUIVALENT)
42.15/11.71	We applied the QTRS Reverse Processor [REVERSE].
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(2)
42.15/11.71	Obligation:
42.15/11.71	Q restricted rewrite system:
42.15/11.71	The TRS R consists of the following rules:
42.15/11.71	
42.15/11.71	   a(x1) -> x1
42.15/11.71	   b(b(a(a(x1)))) -> c(a(c(b(x1))))
42.15/11.71	   c(x1) -> a(b(x1))
42.15/11.71	
42.15/11.71	Q is empty.
42.15/11.71	
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(3) DependencyPairsProof (EQUIVALENT)
42.15/11.71	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(4)
42.15/11.71	Obligation:
42.15/11.71	Q DP problem:
42.15/11.71	The TRS P consists of the following rules:
42.15/11.71	
42.15/11.71	   B(b(a(a(x1)))) -> C(a(c(b(x1))))
42.15/11.71	   B(b(a(a(x1)))) -> A(c(b(x1)))
42.15/11.71	   B(b(a(a(x1)))) -> C(b(x1))
42.15/11.71	   B(b(a(a(x1)))) -> B(x1)
42.15/11.71	   C(x1) -> A(b(x1))
42.15/11.71	   C(x1) -> B(x1)
42.15/11.71	
42.15/11.71	The TRS R consists of the following rules:
42.15/11.71	
42.15/11.71	   a(x1) -> x1
42.15/11.71	   b(b(a(a(x1)))) -> c(a(c(b(x1))))
42.15/11.71	   c(x1) -> a(b(x1))
42.15/11.71	
42.15/11.71	Q is empty.
42.15/11.71	We have to consider all minimal (P,Q,R)-chains.
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(5) DependencyGraphProof (EQUIVALENT)
42.15/11.71	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(6)
42.15/11.71	Obligation:
42.15/11.71	Q DP problem:
42.15/11.71	The TRS P consists of the following rules:
42.15/11.71	
42.15/11.71	   C(x1) -> B(x1)
42.15/11.71	   B(b(a(a(x1)))) -> C(a(c(b(x1))))
42.15/11.71	   B(b(a(a(x1)))) -> C(b(x1))
42.15/11.71	   B(b(a(a(x1)))) -> B(x1)
42.15/11.71	
42.15/11.71	The TRS R consists of the following rules:
42.15/11.71	
42.15/11.71	   a(x1) -> x1
42.15/11.71	   b(b(a(a(x1)))) -> c(a(c(b(x1))))
42.15/11.71	   c(x1) -> a(b(x1))
42.15/11.71	
42.15/11.71	Q is empty.
42.15/11.71	We have to consider all minimal (P,Q,R)-chains.
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(7) QDPOrderProof (EQUIVALENT)
42.15/11.71	We use the reduction pair processor [LPAR04,JAR06].
42.15/11.71	
42.15/11.71	
42.15/11.71	The following pairs can be oriented strictly and are deleted.
42.15/11.71	
42.15/11.71	   B(b(a(a(x1)))) -> B(x1)
42.15/11.71	The remaining pairs can at least be oriented weakly.
42.15/11.71	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(C(x_1)) =  	[[1A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [0A, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	
42.15/11.71	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.15/11.71	
42.15/11.71	   b(b(a(a(x1)))) -> c(a(c(b(x1))))
42.15/11.71	   c(x1) -> a(b(x1))
42.15/11.71	   a(x1) -> x1
42.15/11.71	
42.15/11.71	
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(8)
42.15/11.71	Obligation:
42.15/11.71	Q DP problem:
42.15/11.71	The TRS P consists of the following rules:
42.15/11.71	
42.15/11.71	   C(x1) -> B(x1)
42.15/11.71	   B(b(a(a(x1)))) -> C(a(c(b(x1))))
42.15/11.71	   B(b(a(a(x1)))) -> C(b(x1))
42.15/11.71	
42.15/11.71	The TRS R consists of the following rules:
42.15/11.71	
42.15/11.71	   a(x1) -> x1
42.15/11.71	   b(b(a(a(x1)))) -> c(a(c(b(x1))))
42.15/11.71	   c(x1) -> a(b(x1))
42.15/11.71	
42.15/11.71	Q is empty.
42.15/11.71	We have to consider all minimal (P,Q,R)-chains.
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(9) QDPOrderProof (EQUIVALENT)
42.15/11.71	We use the reduction pair processor [LPAR04,JAR06].
42.15/11.71	
42.15/11.71	
42.15/11.71	The following pairs can be oriented strictly and are deleted.
42.15/11.71	
42.15/11.71	   C(x1) -> B(x1)
42.15/11.71	The remaining pairs can at least be oriented weakly.
42.15/11.71	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(C(x_1)) =  	[[1A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, -I, -I], [0A, 0A, -I]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	   <<<
42.15/11.71	 POL(c(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
42.15/11.71	>>>
42.15/11.71	
42.15/11.71	
42.15/11.71	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.15/11.71	
42.15/11.71	   b(b(a(a(x1)))) -> c(a(c(b(x1))))
42.15/11.71	   c(x1) -> a(b(x1))
42.15/11.71	   a(x1) -> x1
42.15/11.71	
42.15/11.71	
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(10)
42.15/11.71	Obligation:
42.15/11.71	Q DP problem:
42.15/11.71	The TRS P consists of the following rules:
42.15/11.71	
42.15/11.71	   B(b(a(a(x1)))) -> C(a(c(b(x1))))
42.15/11.71	   B(b(a(a(x1)))) -> C(b(x1))
42.15/11.71	
42.15/11.71	The TRS R consists of the following rules:
42.15/11.71	
42.15/11.71	   a(x1) -> x1
42.15/11.71	   b(b(a(a(x1)))) -> c(a(c(b(x1))))
42.15/11.71	   c(x1) -> a(b(x1))
42.15/11.71	
42.15/11.71	Q is empty.
42.15/11.71	We have to consider all minimal (P,Q,R)-chains.
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(11) DependencyGraphProof (EQUIVALENT)
42.15/11.71	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
42.15/11.71	----------------------------------------
42.15/11.71	
42.15/11.71	(12)
42.15/11.71	TRUE
42.52/11.80	EOF