58.33/15.82	YES
58.33/15.86	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
58.33/15.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
58.33/15.86	
58.33/15.86	
58.33/15.86	Termination w.r.t. Q of the given QTRS could be proven:
58.33/15.86	
58.33/15.86	(0) QTRS
58.33/15.86	(1) QTRS Reverse [EQUIVALENT, 0 ms]
58.33/15.86	(2) QTRS
58.33/15.86	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
58.33/15.86	(4) QDP
58.33/15.86	(5) QDPOrderProof [EQUIVALENT, 251 ms]
58.33/15.86	(6) QDP
58.33/15.86	(7) QDPOrderProof [EQUIVALENT, 11 ms]
58.33/15.86	(8) QDP
58.33/15.86	(9) QDPOrderProof [EQUIVALENT, 24 ms]
58.33/15.86	(10) QDP
58.33/15.86	(11) QDPOrderProof [EQUIVALENT, 28 ms]
58.33/15.86	(12) QDP
58.33/15.86	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
58.33/15.86	(14) TRUE
58.33/15.86	
58.33/15.86	
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(0)
58.33/15.86	Obligation:
58.33/15.86	Q restricted rewrite system:
58.33/15.86	The TRS R consists of the following rules:
58.33/15.86	
58.33/15.86	   a(x1) -> x1
58.33/15.86	   a(a(b(x1))) -> b(c(a(a(a(x1)))))
58.33/15.86	   c(a(x1)) -> b(x1)
58.33/15.86	
58.33/15.86	Q is empty.
58.33/15.86	
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(1) QTRS Reverse (EQUIVALENT)
58.33/15.86	We applied the QTRS Reverse Processor [REVERSE].
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(2)
58.33/15.86	Obligation:
58.33/15.86	Q restricted rewrite system:
58.33/15.86	The TRS R consists of the following rules:
58.33/15.86	
58.33/15.86	   a(x1) -> x1
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	
58.33/15.86	Q is empty.
58.33/15.86	
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(3) DependencyPairsProof (EQUIVALENT)
58.33/15.86	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(4)
58.33/15.86	Obligation:
58.33/15.86	Q DP problem:
58.33/15.86	The TRS P consists of the following rules:
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> A(a(a(c(b(x1)))))
58.33/15.86	   B(a(a(x1))) -> A(a(c(b(x1))))
58.33/15.86	   B(a(a(x1))) -> A(c(b(x1)))
58.33/15.86	   B(a(a(x1))) -> B(x1)
58.33/15.86	   A(c(x1)) -> B(x1)
58.33/15.86	
58.33/15.86	The TRS R consists of the following rules:
58.33/15.86	
58.33/15.86	   a(x1) -> x1
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	
58.33/15.86	Q is empty.
58.33/15.86	We have to consider all minimal (P,Q,R)-chains.
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(5) QDPOrderProof (EQUIVALENT)
58.33/15.86	We use the reduction pair processor [LPAR04,JAR06].
58.33/15.86	
58.33/15.86	
58.33/15.86	The following pairs can be oriented strictly and are deleted.
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> B(x1)
58.33/15.86	The remaining pairs can at least be oriented weakly.
58.33/15.86	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [-I, 1A, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, -I], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 1A, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	
58.33/15.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
58.33/15.86	
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(x1) -> x1
58.33/15.86	
58.33/15.86	
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(6)
58.33/15.86	Obligation:
58.33/15.86	Q DP problem:
58.33/15.86	The TRS P consists of the following rules:
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> A(a(a(c(b(x1)))))
58.33/15.86	   B(a(a(x1))) -> A(a(c(b(x1))))
58.33/15.86	   B(a(a(x1))) -> A(c(b(x1)))
58.33/15.86	   A(c(x1)) -> B(x1)
58.33/15.86	
58.33/15.86	The TRS R consists of the following rules:
58.33/15.86	
58.33/15.86	   a(x1) -> x1
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	
58.33/15.86	Q is empty.
58.33/15.86	We have to consider all minimal (P,Q,R)-chains.
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(7) QDPOrderProof (EQUIVALENT)
58.33/15.86	We use the reduction pair processor [LPAR04,JAR06].
58.33/15.86	
58.33/15.86	
58.33/15.86	The following pairs can be oriented strictly and are deleted.
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> A(c(b(x1)))
58.33/15.86	The remaining pairs can at least be oriented weakly.
58.33/15.86	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [-I, -I, 1A], [0A, -I, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	
58.33/15.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
58.33/15.86	
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(x1) -> x1
58.33/15.86	
58.33/15.86	
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(8)
58.33/15.86	Obligation:
58.33/15.86	Q DP problem:
58.33/15.86	The TRS P consists of the following rules:
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> A(a(a(c(b(x1)))))
58.33/15.86	   B(a(a(x1))) -> A(a(c(b(x1))))
58.33/15.86	   A(c(x1)) -> B(x1)
58.33/15.86	
58.33/15.86	The TRS R consists of the following rules:
58.33/15.86	
58.33/15.86	   a(x1) -> x1
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	
58.33/15.86	Q is empty.
58.33/15.86	We have to consider all minimal (P,Q,R)-chains.
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(9) QDPOrderProof (EQUIVALENT)
58.33/15.86	We use the reduction pair processor [LPAR04,JAR06].
58.33/15.86	
58.33/15.86	
58.33/15.86	The following pairs can be oriented strictly and are deleted.
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> A(a(c(b(x1))))
58.33/15.86	The remaining pairs can at least be oriented weakly.
58.33/15.86	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, 0A, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 1A, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	
58.33/15.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
58.33/15.86	
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(x1) -> x1
58.33/15.86	
58.33/15.86	
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(10)
58.33/15.86	Obligation:
58.33/15.86	Q DP problem:
58.33/15.86	The TRS P consists of the following rules:
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> A(a(a(c(b(x1)))))
58.33/15.86	   A(c(x1)) -> B(x1)
58.33/15.86	
58.33/15.86	The TRS R consists of the following rules:
58.33/15.86	
58.33/15.86	   a(x1) -> x1
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	
58.33/15.86	Q is empty.
58.33/15.86	We have to consider all minimal (P,Q,R)-chains.
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(11) QDPOrderProof (EQUIVALENT)
58.33/15.86	We use the reduction pair processor [LPAR04,JAR06].
58.33/15.86	
58.33/15.86	
58.33/15.86	The following pairs can be oriented strictly and are deleted.
58.33/15.86	
58.33/15.86	   A(c(x1)) -> B(x1)
58.33/15.86	The remaining pairs can at least be oriented weakly.
58.33/15.86	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 1A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [-I, -I, -I]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	   <<<
58.33/15.86	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
58.33/15.86	>>>
58.33/15.86	
58.33/15.86	
58.33/15.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
58.33/15.86	
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(x1) -> x1
58.33/15.86	
58.33/15.86	
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(12)
58.33/15.86	Obligation:
58.33/15.86	Q DP problem:
58.33/15.86	The TRS P consists of the following rules:
58.33/15.86	
58.33/15.86	   B(a(a(x1))) -> A(a(a(c(b(x1)))))
58.33/15.86	
58.33/15.86	The TRS R consists of the following rules:
58.33/15.86	
58.33/15.86	   a(x1) -> x1
58.33/15.86	   b(a(a(x1))) -> a(a(a(c(b(x1)))))
58.33/15.86	   a(c(x1)) -> b(x1)
58.33/15.86	
58.33/15.86	Q is empty.
58.33/15.86	We have to consider all minimal (P,Q,R)-chains.
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(13) DependencyGraphProof (EQUIVALENT)
58.33/15.86	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
58.33/15.86	----------------------------------------
58.33/15.86	
58.33/15.86	(14)
58.33/15.86	TRUE
58.92/15.95	EOF