38.19/10.66	YES
38.54/10.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
38.54/10.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
38.54/10.77	
38.54/10.77	
38.54/10.77	Termination w.r.t. Q of the given QTRS could be proven:
38.54/10.77	
38.54/10.77	(0) QTRS
38.54/10.77	(1) QTRS Reverse [EQUIVALENT, 0 ms]
38.54/10.77	(2) QTRS
38.54/10.77	(3) DependencyPairsProof [EQUIVALENT, 23 ms]
38.54/10.77	(4) QDP
38.54/10.77	(5) DependencyGraphProof [EQUIVALENT, 3 ms]
38.54/10.77	(6) QDP
38.54/10.77	(7) QDPOrderProof [EQUIVALENT, 132 ms]
38.54/10.77	(8) QDP
38.54/10.77	(9) QDPOrderProof [EQUIVALENT, 0 ms]
38.54/10.77	(10) QDP
38.54/10.77	(11) QDPOrderProof [EQUIVALENT, 48 ms]
38.54/10.77	(12) QDP
38.54/10.77	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
38.54/10.77	(14) TRUE
38.54/10.77	
38.54/10.77	
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(0)
38.54/10.77	Obligation:
38.54/10.77	Q restricted rewrite system:
38.54/10.77	The TRS R consists of the following rules:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> b(b(c(x1)))
38.54/10.77	   a(c(x1)) -> x1
38.54/10.77	   c(b(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	Q is empty.
38.54/10.77	
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(1) QTRS Reverse (EQUIVALENT)
38.54/10.77	We applied the QTRS Reverse Processor [REVERSE].
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(2)
38.54/10.77	Obligation:
38.54/10.77	Q restricted rewrite system:
38.54/10.77	The TRS R consists of the following rules:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	Q is empty.
38.54/10.77	
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(3) DependencyPairsProof (EQUIVALENT)
38.54/10.77	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(4)
38.54/10.77	Obligation:
38.54/10.77	Q DP problem:
38.54/10.77	The TRS P consists of the following rules:
38.54/10.77	
38.54/10.77	   A(a(x1)) -> C(b(b(x1)))
38.54/10.77	   A(a(x1)) -> B(b(x1))
38.54/10.77	   A(a(x1)) -> B(x1)
38.54/10.77	   B(c(x1)) -> A(c(a(x1)))
38.54/10.77	   B(c(x1)) -> C(a(x1))
38.54/10.77	   B(c(x1)) -> A(x1)
38.54/10.77	
38.54/10.77	The TRS R consists of the following rules:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	Q is empty.
38.54/10.77	We have to consider all minimal (P,Q,R)-chains.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(5) DependencyGraphProof (EQUIVALENT)
38.54/10.77	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(6)
38.54/10.77	Obligation:
38.54/10.77	Q DP problem:
38.54/10.77	The TRS P consists of the following rules:
38.54/10.77	
38.54/10.77	   A(a(x1)) -> B(b(x1))
38.54/10.77	   B(c(x1)) -> A(c(a(x1)))
38.54/10.77	   A(a(x1)) -> B(x1)
38.54/10.77	   B(c(x1)) -> A(x1)
38.54/10.77	
38.54/10.77	The TRS R consists of the following rules:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	Q is empty.
38.54/10.77	We have to consider all minimal (P,Q,R)-chains.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(7) QDPOrderProof (EQUIVALENT)
38.54/10.77	We use the reduction pair processor [LPAR04,JAR06].
38.54/10.77	
38.54/10.77	
38.54/10.77	The following pairs can be oriented strictly and are deleted.
38.54/10.77	
38.54/10.77	   A(a(x1)) -> B(b(x1))
38.54/10.77	The remaining pairs can at least be oriented weakly.
38.54/10.77	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [0A, 0A, -I], [0A, -I, -I]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 1A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	
38.54/10.77	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.54/10.77	
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	
38.54/10.77	
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(8)
38.54/10.77	Obligation:
38.54/10.77	Q DP problem:
38.54/10.77	The TRS P consists of the following rules:
38.54/10.77	
38.54/10.77	   B(c(x1)) -> A(c(a(x1)))
38.54/10.77	   A(a(x1)) -> B(x1)
38.54/10.77	   B(c(x1)) -> A(x1)
38.54/10.77	
38.54/10.77	The TRS R consists of the following rules:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	Q is empty.
38.54/10.77	We have to consider all minimal (P,Q,R)-chains.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(9) QDPOrderProof (EQUIVALENT)
38.54/10.77	We use the reduction pair processor [LPAR04,JAR06].
38.54/10.77	
38.54/10.77	
38.54/10.77	The following pairs can be oriented strictly and are deleted.
38.54/10.77	
38.54/10.77	   B(c(x1)) -> A(x1)
38.54/10.77	The remaining pairs can at least be oriented weakly.
38.54/10.77	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(B(x_1)) =  	[[1A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(a(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [1A, 0A, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [0A, 1A, -I]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	
38.54/10.77	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(10)
38.54/10.77	Obligation:
38.54/10.77	Q DP problem:
38.54/10.77	The TRS P consists of the following rules:
38.54/10.77	
38.54/10.77	   B(c(x1)) -> A(c(a(x1)))
38.54/10.77	   A(a(x1)) -> B(x1)
38.54/10.77	
38.54/10.77	The TRS R consists of the following rules:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	Q is empty.
38.54/10.77	We have to consider all minimal (P,Q,R)-chains.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(11) QDPOrderProof (EQUIVALENT)
38.54/10.77	We use the reduction pair processor [LPAR04,JAR06].
38.54/10.77	
38.54/10.77	
38.54/10.77	The following pairs can be oriented strictly and are deleted.
38.54/10.77	
38.54/10.77	   A(a(x1)) -> B(x1)
38.54/10.77	The remaining pairs can at least be oriented weakly.
38.54/10.77	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 0A, 1A], [-I, 0A, -I]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	   <<<
38.54/10.77	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [1A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
38.54/10.77	>>>
38.54/10.77	
38.54/10.77	
38.54/10.77	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(12)
38.54/10.77	Obligation:
38.54/10.77	Q DP problem:
38.54/10.77	The TRS P consists of the following rules:
38.54/10.77	
38.54/10.77	   B(c(x1)) -> A(c(a(x1)))
38.54/10.77	
38.54/10.77	The TRS R consists of the following rules:
38.54/10.77	
38.54/10.77	   a(a(x1)) -> c(b(b(x1)))
38.54/10.77	   c(a(x1)) -> x1
38.54/10.77	   b(c(x1)) -> a(c(a(x1)))
38.54/10.77	
38.54/10.77	Q is empty.
38.54/10.77	We have to consider all minimal (P,Q,R)-chains.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(13) DependencyGraphProof (EQUIVALENT)
38.54/10.77	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
38.54/10.77	----------------------------------------
38.54/10.77	
38.54/10.77	(14)
38.54/10.77	TRUE
38.88/10.94	EOF