32.90/9.35	YES
33.26/9.40	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
33.26/9.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.26/9.40	
33.26/9.40	
33.26/9.40	Termination w.r.t. Q of the given QTRS could be proven:
33.26/9.40	
33.26/9.40	(0) QTRS
33.26/9.40	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.26/9.40	(2) QTRS
33.26/9.40	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
33.26/9.40	(4) QDP
33.26/9.40	(5) QDPOrderProof [EQUIVALENT, 120 ms]
33.26/9.40	(6) QDP
33.26/9.40	(7) QDPOrderProof [EQUIVALENT, 172 ms]
33.26/9.40	(8) QDP
33.26/9.40	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
33.26/9.40	(10) TRUE
33.26/9.40	
33.26/9.40	
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(0)
33.26/9.40	Obligation:
33.26/9.40	Q restricted rewrite system:
33.26/9.40	The TRS R consists of the following rules:
33.26/9.40	
33.26/9.40	   a(a(x1)) -> a(b(b(c(x1))))
33.26/9.40	   c(a(x1)) -> x1
33.26/9.40	   c(b(x1)) -> a(c(x1))
33.26/9.40	
33.26/9.40	Q is empty.
33.26/9.40	
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(1) QTRS Reverse (EQUIVALENT)
33.26/9.40	We applied the QTRS Reverse Processor [REVERSE].
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(2)
33.26/9.40	Obligation:
33.26/9.40	Q restricted rewrite system:
33.26/9.40	The TRS R consists of the following rules:
33.26/9.40	
33.26/9.40	   a(a(x1)) -> c(b(b(a(x1))))
33.26/9.40	   a(c(x1)) -> x1
33.26/9.40	   b(c(x1)) -> c(a(x1))
33.26/9.40	
33.26/9.40	Q is empty.
33.26/9.40	
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(3) DependencyPairsProof (EQUIVALENT)
33.26/9.40	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(4)
33.26/9.40	Obligation:
33.26/9.40	Q DP problem:
33.26/9.40	The TRS P consists of the following rules:
33.26/9.40	
33.26/9.40	   A(a(x1)) -> B(b(a(x1)))
33.26/9.40	   A(a(x1)) -> B(a(x1))
33.26/9.40	   B(c(x1)) -> A(x1)
33.26/9.40	
33.26/9.40	The TRS R consists of the following rules:
33.26/9.40	
33.26/9.40	   a(a(x1)) -> c(b(b(a(x1))))
33.26/9.40	   a(c(x1)) -> x1
33.26/9.40	   b(c(x1)) -> c(a(x1))
33.26/9.40	
33.26/9.40	Q is empty.
33.26/9.40	We have to consider all minimal (P,Q,R)-chains.
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(5) QDPOrderProof (EQUIVALENT)
33.26/9.40	We use the reduction pair processor [LPAR04,JAR06].
33.26/9.40	
33.26/9.40	
33.26/9.40	The following pairs can be oriented strictly and are deleted.
33.26/9.40	
33.26/9.40	   A(a(x1)) -> B(a(x1))
33.26/9.40	The remaining pairs can at least be oriented weakly.
33.26/9.40	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [1A, 0A, 1A]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(c(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	
33.26/9.40	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.26/9.40	
33.26/9.40	   a(a(x1)) -> c(b(b(a(x1))))
33.26/9.40	   a(c(x1)) -> x1
33.26/9.40	   b(c(x1)) -> c(a(x1))
33.26/9.40	
33.26/9.40	
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(6)
33.26/9.40	Obligation:
33.26/9.40	Q DP problem:
33.26/9.40	The TRS P consists of the following rules:
33.26/9.40	
33.26/9.40	   A(a(x1)) -> B(b(a(x1)))
33.26/9.40	   B(c(x1)) -> A(x1)
33.26/9.40	
33.26/9.40	The TRS R consists of the following rules:
33.26/9.40	
33.26/9.40	   a(a(x1)) -> c(b(b(a(x1))))
33.26/9.40	   a(c(x1)) -> x1
33.26/9.40	   b(c(x1)) -> c(a(x1))
33.26/9.40	
33.26/9.40	Q is empty.
33.26/9.40	We have to consider all minimal (P,Q,R)-chains.
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(7) QDPOrderProof (EQUIVALENT)
33.26/9.40	We use the reduction pair processor [LPAR04,JAR06].
33.26/9.40	
33.26/9.40	
33.26/9.40	The following pairs can be oriented strictly and are deleted.
33.26/9.40	
33.26/9.40	   B(c(x1)) -> A(x1)
33.26/9.40	The remaining pairs can at least be oriented weakly.
33.26/9.40	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 1A]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [-I, 0A, 0A], [-I, 0A, 1A]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(B(x_1)) =  	[[1A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [-I, 0A, -I]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	   <<<
33.26/9.40	 POL(c(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[-I, 0A, 1A], [-I, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
33.26/9.40	>>>
33.26/9.40	
33.26/9.40	
33.26/9.40	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.26/9.40	
33.26/9.40	   a(a(x1)) -> c(b(b(a(x1))))
33.26/9.40	   a(c(x1)) -> x1
33.26/9.40	   b(c(x1)) -> c(a(x1))
33.26/9.40	
33.26/9.40	
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(8)
33.26/9.40	Obligation:
33.26/9.40	Q DP problem:
33.26/9.40	The TRS P consists of the following rules:
33.26/9.40	
33.26/9.40	   A(a(x1)) -> B(b(a(x1)))
33.26/9.40	
33.26/9.40	The TRS R consists of the following rules:
33.26/9.40	
33.26/9.40	   a(a(x1)) -> c(b(b(a(x1))))
33.26/9.40	   a(c(x1)) -> x1
33.26/9.40	   b(c(x1)) -> c(a(x1))
33.26/9.40	
33.26/9.40	Q is empty.
33.26/9.40	We have to consider all minimal (P,Q,R)-chains.
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(9) DependencyGraphProof (EQUIVALENT)
33.26/9.40	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
33.26/9.40	----------------------------------------
33.26/9.40	
33.26/9.40	(10)
33.26/9.40	TRUE
33.49/9.48	EOF