32.18/9.12	YES
32.18/9.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.18/9.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.18/9.12	
32.18/9.12	
32.18/9.12	Termination w.r.t. Q of the given QTRS could be proven:
32.18/9.12	
32.18/9.12	(0) QTRS
32.18/9.12	(1) QTRS Reverse [EQUIVALENT, 0 ms]
32.18/9.12	(2) QTRS
32.18/9.12	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
32.18/9.12	(4) QDP
32.18/9.12	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
32.18/9.12	(6) QDP
32.18/9.12	(7) QDPOrderProof [EQUIVALENT, 99 ms]
32.18/9.12	(8) QDP
32.18/9.12	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
32.18/9.12	(10) TRUE
32.18/9.12	
32.18/9.12	
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(0)
32.18/9.12	Obligation:
32.18/9.12	Q restricted rewrite system:
32.18/9.12	The TRS R consists of the following rules:
32.18/9.12	
32.18/9.12	   a(a(x1)) -> a(b(c(b(x1))))
32.18/9.12	   b(c(x1)) -> x1
32.18/9.12	   c(b(x1)) -> a(c(x1))
32.18/9.12	
32.18/9.12	Q is empty.
32.18/9.12	
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(1) QTRS Reverse (EQUIVALENT)
32.18/9.12	We applied the QTRS Reverse Processor [REVERSE].
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(2)
32.18/9.12	Obligation:
32.18/9.12	Q restricted rewrite system:
32.18/9.12	The TRS R consists of the following rules:
32.18/9.12	
32.18/9.12	   a(a(x1)) -> b(c(b(a(x1))))
32.18/9.12	   c(b(x1)) -> x1
32.18/9.12	   b(c(x1)) -> c(a(x1))
32.18/9.12	
32.18/9.12	Q is empty.
32.18/9.12	
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(3) DependencyPairsProof (EQUIVALENT)
32.18/9.12	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(4)
32.18/9.12	Obligation:
32.18/9.12	Q DP problem:
32.18/9.12	The TRS P consists of the following rules:
32.18/9.12	
32.18/9.12	   A(a(x1)) -> B(c(b(a(x1))))
32.18/9.12	   A(a(x1)) -> C(b(a(x1)))
32.18/9.12	   A(a(x1)) -> B(a(x1))
32.18/9.12	   B(c(x1)) -> C(a(x1))
32.18/9.12	   B(c(x1)) -> A(x1)
32.18/9.12	
32.18/9.12	The TRS R consists of the following rules:
32.18/9.12	
32.18/9.12	   a(a(x1)) -> b(c(b(a(x1))))
32.18/9.12	   c(b(x1)) -> x1
32.18/9.12	   b(c(x1)) -> c(a(x1))
32.18/9.12	
32.18/9.12	Q is empty.
32.18/9.12	We have to consider all minimal (P,Q,R)-chains.
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(5) DependencyGraphProof (EQUIVALENT)
32.18/9.12	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(6)
32.18/9.12	Obligation:
32.18/9.12	Q DP problem:
32.18/9.12	The TRS P consists of the following rules:
32.18/9.12	
32.18/9.12	   B(c(x1)) -> A(x1)
32.18/9.12	   A(a(x1)) -> B(c(b(a(x1))))
32.18/9.12	   A(a(x1)) -> B(a(x1))
32.18/9.12	
32.18/9.12	The TRS R consists of the following rules:
32.18/9.12	
32.18/9.12	   a(a(x1)) -> b(c(b(a(x1))))
32.18/9.12	   c(b(x1)) -> x1
32.18/9.12	   b(c(x1)) -> c(a(x1))
32.18/9.12	
32.18/9.12	Q is empty.
32.18/9.12	We have to consider all minimal (P,Q,R)-chains.
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(7) QDPOrderProof (EQUIVALENT)
32.18/9.12	We use the reduction pair processor [LPAR04,JAR06].
32.18/9.12	
32.18/9.12	
32.18/9.12	The following pairs can be oriented strictly and are deleted.
32.18/9.12	
32.18/9.12	   A(a(x1)) -> B(c(b(a(x1))))
32.18/9.12	   A(a(x1)) -> B(a(x1))
32.18/9.12	The remaining pairs can at least be oriented weakly.
32.18/9.12	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.18/9.12	
32.18/9.12	   <<<
32.18/9.12	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
32.18/9.12	>>>
32.18/9.12	
32.18/9.12	   <<<
32.18/9.12	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
32.18/9.12	>>>
32.18/9.12	
32.18/9.12	   <<<
32.18/9.12	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
32.18/9.12	>>>
32.18/9.12	
32.18/9.12	   <<<
32.18/9.12	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 1A, 0A], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
32.18/9.12	>>>
32.18/9.12	
32.18/9.12	   <<<
32.18/9.12	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
32.18/9.12	>>>
32.18/9.12	
32.18/9.12	
32.18/9.12	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.18/9.12	
32.18/9.12	   a(a(x1)) -> b(c(b(a(x1))))
32.18/9.12	   b(c(x1)) -> c(a(x1))
32.18/9.12	   c(b(x1)) -> x1
32.18/9.12	
32.18/9.12	
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(8)
32.18/9.12	Obligation:
32.18/9.12	Q DP problem:
32.18/9.12	The TRS P consists of the following rules:
32.18/9.12	
32.18/9.12	   B(c(x1)) -> A(x1)
32.18/9.12	
32.18/9.12	The TRS R consists of the following rules:
32.18/9.12	
32.18/9.12	   a(a(x1)) -> b(c(b(a(x1))))
32.18/9.12	   c(b(x1)) -> x1
32.18/9.12	   b(c(x1)) -> c(a(x1))
32.18/9.12	
32.18/9.12	Q is empty.
32.18/9.12	We have to consider all minimal (P,Q,R)-chains.
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(9) DependencyGraphProof (EQUIVALENT)
32.18/9.12	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
32.18/9.12	----------------------------------------
32.18/9.12	
32.18/9.12	(10)
32.18/9.12	TRUE
32.69/9.19	EOF