32.98/9.29	YES
32.98/9.31	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.98/9.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.98/9.31	
32.98/9.31	
32.98/9.31	Termination w.r.t. Q of the given QTRS could be proven:
32.98/9.31	
32.98/9.31	(0) QTRS
32.98/9.31	(1) QTRS Reverse [EQUIVALENT, 0 ms]
32.98/9.31	(2) QTRS
32.98/9.31	(3) DependencyPairsProof [EQUIVALENT, 24 ms]
32.98/9.31	(4) QDP
32.98/9.31	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
32.98/9.31	(6) QDP
32.98/9.31	(7) QDPOrderProof [EQUIVALENT, 110 ms]
32.98/9.31	(8) QDP
32.98/9.31	(9) QDPOrderProof [EQUIVALENT, 69 ms]
32.98/9.31	(10) QDP
32.98/9.31	(11) QDPOrderProof [EQUIVALENT, 25 ms]
32.98/9.31	(12) QDP
32.98/9.31	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
32.98/9.31	(14) TRUE
32.98/9.31	
32.98/9.31	
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(0)
32.98/9.31	Obligation:
32.98/9.31	Q restricted rewrite system:
32.98/9.31	The TRS R consists of the following rules:
32.98/9.31	
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> b(b(c(x1)))
32.98/9.31	   a(c(b(x1))) -> c(a(a(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	
32.98/9.31	Q is empty.
32.98/9.31	
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(1) QTRS Reverse (EQUIVALENT)
32.98/9.31	We applied the QTRS Reverse Processor [REVERSE].
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(2)
32.98/9.31	Obligation:
32.98/9.31	Q restricted rewrite system:
32.98/9.31	The TRS R consists of the following rules:
32.98/9.31	
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	
32.98/9.31	Q is empty.
32.98/9.31	
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(3) DependencyPairsProof (EQUIVALENT)
32.98/9.31	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(4)
32.98/9.31	Obligation:
32.98/9.31	Q DP problem:
32.98/9.31	The TRS P consists of the following rules:
32.98/9.31	
32.98/9.31	   A(x1) -> C(b(b(x1)))
32.98/9.31	   A(x1) -> B(b(x1))
32.98/9.31	   A(x1) -> B(x1)
32.98/9.31	   B(c(a(x1))) -> A(a(c(x1)))
32.98/9.31	   B(c(a(x1))) -> A(c(x1))
32.98/9.31	   B(c(a(x1))) -> C(x1)
32.98/9.31	
32.98/9.31	The TRS R consists of the following rules:
32.98/9.31	
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	
32.98/9.31	Q is empty.
32.98/9.31	We have to consider all minimal (P,Q,R)-chains.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(5) DependencyGraphProof (EQUIVALENT)
32.98/9.31	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(6)
32.98/9.31	Obligation:
32.98/9.31	Q DP problem:
32.98/9.31	The TRS P consists of the following rules:
32.98/9.31	
32.98/9.31	   A(x1) -> B(b(x1))
32.98/9.31	   B(c(a(x1))) -> A(a(c(x1)))
32.98/9.31	   A(x1) -> B(x1)
32.98/9.31	   B(c(a(x1))) -> A(c(x1))
32.98/9.31	
32.98/9.31	The TRS R consists of the following rules:
32.98/9.31	
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	
32.98/9.31	Q is empty.
32.98/9.31	We have to consider all minimal (P,Q,R)-chains.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(7) QDPOrderProof (EQUIVALENT)
32.98/9.31	We use the reduction pair processor [LPAR04,JAR06].
32.98/9.31	
32.98/9.31	
32.98/9.31	The following pairs can be oriented strictly and are deleted.
32.98/9.31	
32.98/9.31	   B(c(a(x1))) -> A(c(x1))
32.98/9.31	The remaining pairs can at least be oriented weakly.
32.98/9.31	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(c(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [1A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(a(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[1A, 0A, 1A], [0A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	
32.98/9.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.98/9.31	
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	
32.98/9.31	
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(8)
32.98/9.31	Obligation:
32.98/9.31	Q DP problem:
32.98/9.31	The TRS P consists of the following rules:
32.98/9.31	
32.98/9.31	   A(x1) -> B(b(x1))
32.98/9.31	   B(c(a(x1))) -> A(a(c(x1)))
32.98/9.31	   A(x1) -> B(x1)
32.98/9.31	
32.98/9.31	The TRS R consists of the following rules:
32.98/9.31	
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	
32.98/9.31	Q is empty.
32.98/9.31	We have to consider all minimal (P,Q,R)-chains.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(9) QDPOrderProof (EQUIVALENT)
32.98/9.31	We use the reduction pair processor [LPAR04,JAR06].
32.98/9.31	
32.98/9.31	
32.98/9.31	The following pairs can be oriented strictly and are deleted.
32.98/9.31	
32.98/9.31	   A(x1) -> B(x1)
32.98/9.31	The remaining pairs can at least be oriented weakly.
32.98/9.31	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(a(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [0A, 1A, 1A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	
32.98/9.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.98/9.31	
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	
32.98/9.31	
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(10)
32.98/9.31	Obligation:
32.98/9.31	Q DP problem:
32.98/9.31	The TRS P consists of the following rules:
32.98/9.31	
32.98/9.31	   A(x1) -> B(b(x1))
32.98/9.31	   B(c(a(x1))) -> A(a(c(x1)))
32.98/9.31	
32.98/9.31	The TRS R consists of the following rules:
32.98/9.31	
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	
32.98/9.31	Q is empty.
32.98/9.31	We have to consider all minimal (P,Q,R)-chains.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(11) QDPOrderProof (EQUIVALENT)
32.98/9.31	We use the reduction pair processor [LPAR04,JAR06].
32.98/9.31	
32.98/9.31	
32.98/9.31	The following pairs can be oriented strictly and are deleted.
32.98/9.31	
32.98/9.31	   A(x1) -> B(b(x1))
32.98/9.31	The remaining pairs can at least be oriented weakly.
32.98/9.31	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(A(x_1)) =  	[[1A]] 	 +  	[[1A, 1A, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	   <<<
32.98/9.31	 POL(a(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, 0A], [0A, 1A, 1A]] 	* 	x_1
32.98/9.31	>>>
32.98/9.31	
32.98/9.31	
32.98/9.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.98/9.31	
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	
32.98/9.31	
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(12)
32.98/9.31	Obligation:
32.98/9.31	Q DP problem:
32.98/9.31	The TRS P consists of the following rules:
32.98/9.31	
32.98/9.31	   B(c(a(x1))) -> A(a(c(x1)))
32.98/9.31	
32.98/9.31	The TRS R consists of the following rules:
32.98/9.31	
32.98/9.31	   a(x1) -> x1
32.98/9.31	   a(x1) -> c(b(b(x1)))
32.98/9.31	   b(c(a(x1))) -> a(a(c(x1)))
32.98/9.31	   c(x1) -> x1
32.98/9.31	
32.98/9.31	Q is empty.
32.98/9.31	We have to consider all minimal (P,Q,R)-chains.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(13) DependencyGraphProof (EQUIVALENT)
32.98/9.31	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
32.98/9.31	----------------------------------------
32.98/9.31	
32.98/9.31	(14)
32.98/9.31	TRUE
33.23/9.35	EOF