36.00/10.09	YES
36.00/10.09	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
36.00/10.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.00/10.09	
36.00/10.09	
36.00/10.09	Termination w.r.t. Q of the given QTRS could be proven:
36.00/10.09	
36.00/10.09	(0) QTRS
36.00/10.09	(1) QTRS Reverse [EQUIVALENT, 0 ms]
36.00/10.09	(2) QTRS
36.00/10.09	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
36.00/10.09	(4) QDP
36.00/10.09	(5) QDPOrderProof [EQUIVALENT, 124 ms]
36.00/10.09	(6) QDP
36.00/10.09	(7) QDPOrderProof [EQUIVALENT, 57 ms]
36.00/10.09	(8) QDP
36.00/10.09	(9) QDPOrderProof [EQUIVALENT, 0 ms]
36.00/10.09	(10) QDP
36.00/10.09	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
36.00/10.09	(12) TRUE
36.00/10.09	
36.00/10.09	
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(0)
36.00/10.09	Obligation:
36.00/10.09	Q restricted rewrite system:
36.00/10.09	The TRS R consists of the following rules:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   a(a(b(x1))) -> c(a(a(a(x1))))
36.00/10.09	   c(a(x1)) -> b(b(x1))
36.00/10.09	
36.00/10.09	Q is empty.
36.00/10.09	
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(1) QTRS Reverse (EQUIVALENT)
36.00/10.09	We applied the QTRS Reverse Processor [REVERSE].
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(2)
36.00/10.09	Obligation:
36.00/10.09	Q restricted rewrite system:
36.00/10.09	The TRS R consists of the following rules:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	
36.00/10.09	Q is empty.
36.00/10.09	
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(3) DependencyPairsProof (EQUIVALENT)
36.00/10.09	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(4)
36.00/10.09	Obligation:
36.00/10.09	Q DP problem:
36.00/10.09	The TRS P consists of the following rules:
36.00/10.09	
36.00/10.09	   B(a(a(x1))) -> A(a(a(c(x1))))
36.00/10.09	   B(a(a(x1))) -> A(a(c(x1)))
36.00/10.09	   B(a(a(x1))) -> A(c(x1))
36.00/10.09	   A(c(x1)) -> B(b(x1))
36.00/10.09	   A(c(x1)) -> B(x1)
36.00/10.09	
36.00/10.09	The TRS R consists of the following rules:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	
36.00/10.09	Q is empty.
36.00/10.09	We have to consider all minimal (P,Q,R)-chains.
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(5) QDPOrderProof (EQUIVALENT)
36.00/10.09	We use the reduction pair processor [LPAR04,JAR06].
36.00/10.09	
36.00/10.09	
36.00/10.09	The following pairs can be oriented strictly and are deleted.
36.00/10.09	
36.00/10.09	   A(c(x1)) -> B(x1)
36.00/10.09	The remaining pairs can at least be oriented weakly.
36.00/10.09	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(A(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(c(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, 0A], [-I, -I, -I], [0A, -I, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	
36.00/10.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	
36.00/10.09	
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(6)
36.00/10.09	Obligation:
36.00/10.09	Q DP problem:
36.00/10.09	The TRS P consists of the following rules:
36.00/10.09	
36.00/10.09	   B(a(a(x1))) -> A(a(a(c(x1))))
36.00/10.09	   B(a(a(x1))) -> A(a(c(x1)))
36.00/10.09	   B(a(a(x1))) -> A(c(x1))
36.00/10.09	   A(c(x1)) -> B(b(x1))
36.00/10.09	
36.00/10.09	The TRS R consists of the following rules:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	
36.00/10.09	Q is empty.
36.00/10.09	We have to consider all minimal (P,Q,R)-chains.
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(7) QDPOrderProof (EQUIVALENT)
36.00/10.09	We use the reduction pair processor [LPAR04,JAR06].
36.00/10.09	
36.00/10.09	
36.00/10.09	The following pairs can be oriented strictly and are deleted.
36.00/10.09	
36.00/10.09	   B(a(a(x1))) -> A(c(x1))
36.00/10.09	The remaining pairs can at least be oriented weakly.
36.00/10.09	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, -I], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	
36.00/10.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	
36.00/10.09	
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(8)
36.00/10.09	Obligation:
36.00/10.09	Q DP problem:
36.00/10.09	The TRS P consists of the following rules:
36.00/10.09	
36.00/10.09	   B(a(a(x1))) -> A(a(a(c(x1))))
36.00/10.09	   B(a(a(x1))) -> A(a(c(x1)))
36.00/10.09	   A(c(x1)) -> B(b(x1))
36.00/10.09	
36.00/10.09	The TRS R consists of the following rules:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	
36.00/10.09	Q is empty.
36.00/10.09	We have to consider all minimal (P,Q,R)-chains.
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(9) QDPOrderProof (EQUIVALENT)
36.00/10.09	We use the reduction pair processor [LPAR04,JAR06].
36.00/10.09	
36.00/10.09	
36.00/10.09	The following pairs can be oriented strictly and are deleted.
36.00/10.09	
36.00/10.09	   A(c(x1)) -> B(b(x1))
36.00/10.09	The remaining pairs can at least be oriented weakly.
36.00/10.09	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 1A, -I]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	   <<<
36.00/10.09	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 1A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
36.00/10.09	>>>
36.00/10.09	
36.00/10.09	
36.00/10.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	
36.00/10.09	
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(10)
36.00/10.09	Obligation:
36.00/10.09	Q DP problem:
36.00/10.09	The TRS P consists of the following rules:
36.00/10.09	
36.00/10.09	   B(a(a(x1))) -> A(a(a(c(x1))))
36.00/10.09	   B(a(a(x1))) -> A(a(c(x1)))
36.00/10.09	
36.00/10.09	The TRS R consists of the following rules:
36.00/10.09	
36.00/10.09	   a(x1) -> x1
36.00/10.09	   b(a(a(x1))) -> a(a(a(c(x1))))
36.00/10.09	   a(c(x1)) -> b(b(x1))
36.00/10.09	
36.00/10.09	Q is empty.
36.00/10.09	We have to consider all minimal (P,Q,R)-chains.
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(11) DependencyGraphProof (EQUIVALENT)
36.00/10.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
36.00/10.09	----------------------------------------
36.00/10.09	
36.00/10.09	(12)
36.00/10.09	TRUE
36.46/10.19	EOF