23.37/6.99	YES
23.37/7.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.37/7.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.37/7.00	
23.37/7.00	
23.37/7.00	Termination w.r.t. Q of the given QTRS could be proven:
23.37/7.00	
23.37/7.00	(0) QTRS
23.37/7.00	(1) QTRS Reverse [EQUIVALENT, 0 ms]
23.37/7.00	(2) QTRS
23.37/7.00	(3) DependencyPairsProof [EQUIVALENT, 21 ms]
23.37/7.00	(4) QDP
23.37/7.00	(5) QDPOrderProof [EQUIVALENT, 125 ms]
23.37/7.00	(6) QDP
23.37/7.00	(7) QDPOrderProof [EQUIVALENT, 0 ms]
23.37/7.00	(8) QDP
23.37/7.00	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
23.37/7.00	(10) TRUE
23.37/7.00	
23.37/7.00	
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(0)
23.37/7.00	Obligation:
23.37/7.00	Q restricted rewrite system:
23.37/7.00	The TRS R consists of the following rules:
23.37/7.00	
23.37/7.00	   a(a(x1)) -> b(b(c(x1)))
23.37/7.00	   b(a(x1)) -> c(x1)
23.37/7.00	   c(b(x1)) -> a(a(x1))
23.37/7.00	
23.37/7.00	Q is empty.
23.37/7.00	
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(1) QTRS Reverse (EQUIVALENT)
23.37/7.00	We applied the QTRS Reverse Processor [REVERSE].
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(2)
23.37/7.00	Obligation:
23.37/7.00	Q restricted rewrite system:
23.37/7.00	The TRS R consists of the following rules:
23.37/7.00	
23.37/7.00	   a(a(x1)) -> c(b(b(x1)))
23.37/7.00	   a(b(x1)) -> c(x1)
23.37/7.00	   b(c(x1)) -> a(a(x1))
23.37/7.00	
23.37/7.00	Q is empty.
23.37/7.00	
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(3) DependencyPairsProof (EQUIVALENT)
23.37/7.00	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(4)
23.37/7.00	Obligation:
23.37/7.00	Q DP problem:
23.37/7.00	The TRS P consists of the following rules:
23.37/7.00	
23.37/7.00	   A(a(x1)) -> B(b(x1))
23.37/7.00	   A(a(x1)) -> B(x1)
23.37/7.00	   B(c(x1)) -> A(a(x1))
23.37/7.00	   B(c(x1)) -> A(x1)
23.37/7.00	
23.37/7.00	The TRS R consists of the following rules:
23.37/7.00	
23.37/7.00	   a(a(x1)) -> c(b(b(x1)))
23.37/7.00	   a(b(x1)) -> c(x1)
23.37/7.00	   b(c(x1)) -> a(a(x1))
23.37/7.00	
23.37/7.00	Q is empty.
23.37/7.00	We have to consider all minimal (P,Q,R)-chains.
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(5) QDPOrderProof (EQUIVALENT)
23.37/7.00	We use the reduction pair processor [LPAR04,JAR06].
23.37/7.00	
23.37/7.00	
23.37/7.00	The following pairs can be oriented strictly and are deleted.
23.37/7.00	
23.37/7.00	   B(c(x1)) -> A(x1)
23.37/7.00	The remaining pairs can at least be oriented weakly.
23.37/7.00	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 1A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[1A, 1A, 1A], [-I, -I, -I], [-I, -I, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	
23.37/7.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.37/7.00	
23.37/7.00	   b(c(x1)) -> a(a(x1))
23.37/7.00	   a(a(x1)) -> c(b(b(x1)))
23.37/7.00	   a(b(x1)) -> c(x1)
23.37/7.00	
23.37/7.00	
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(6)
23.37/7.00	Obligation:
23.37/7.00	Q DP problem:
23.37/7.00	The TRS P consists of the following rules:
23.37/7.00	
23.37/7.00	   A(a(x1)) -> B(b(x1))
23.37/7.00	   A(a(x1)) -> B(x1)
23.37/7.00	   B(c(x1)) -> A(a(x1))
23.37/7.00	
23.37/7.00	The TRS R consists of the following rules:
23.37/7.00	
23.37/7.00	   a(a(x1)) -> c(b(b(x1)))
23.37/7.00	   a(b(x1)) -> c(x1)
23.37/7.00	   b(c(x1)) -> a(a(x1))
23.37/7.00	
23.37/7.00	Q is empty.
23.37/7.00	We have to consider all minimal (P,Q,R)-chains.
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(7) QDPOrderProof (EQUIVALENT)
23.37/7.00	We use the reduction pair processor [LPAR04,JAR06].
23.37/7.00	
23.37/7.00	
23.37/7.00	The following pairs can be oriented strictly and are deleted.
23.37/7.00	
23.37/7.00	   B(c(x1)) -> A(a(x1))
23.37/7.00	The remaining pairs can at least be oriented weakly.
23.37/7.00	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 1A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	   <<<
23.37/7.00	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 0A], [-I, 1A, 0A]] 	* 	x_1
23.37/7.00	>>>
23.37/7.00	
23.37/7.00	
23.37/7.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.37/7.00	
23.37/7.00	   b(c(x1)) -> a(a(x1))
23.37/7.00	   a(a(x1)) -> c(b(b(x1)))
23.37/7.00	   a(b(x1)) -> c(x1)
23.37/7.00	
23.37/7.00	
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(8)
23.37/7.00	Obligation:
23.37/7.00	Q DP problem:
23.37/7.00	The TRS P consists of the following rules:
23.37/7.00	
23.37/7.00	   A(a(x1)) -> B(b(x1))
23.37/7.00	   A(a(x1)) -> B(x1)
23.37/7.00	
23.37/7.00	The TRS R consists of the following rules:
23.37/7.00	
23.37/7.00	   a(a(x1)) -> c(b(b(x1)))
23.37/7.00	   a(b(x1)) -> c(x1)
23.37/7.00	   b(c(x1)) -> a(a(x1))
23.37/7.00	
23.37/7.00	Q is empty.
23.37/7.00	We have to consider all minimal (P,Q,R)-chains.
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(9) DependencyGraphProof (EQUIVALENT)
23.37/7.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
23.37/7.00	----------------------------------------
23.37/7.00	
23.37/7.00	(10)
23.37/7.00	TRUE
23.79/7.07	EOF