28.85/8.26	YES
31.38/9.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.38/9.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.38/9.04	
31.38/9.04	
31.38/9.04	Termination w.r.t. Q of the given QTRS could be proven:
31.38/9.04	
31.38/9.04	(0) QTRS
31.38/9.04	(1) QTRS Reverse [EQUIVALENT, 0 ms]
31.38/9.04	(2) QTRS
31.38/9.04	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
31.38/9.04	(4) QDP
31.38/9.04	(5) QDPOrderProof [EQUIVALENT, 187 ms]
31.38/9.04	(6) QDP
31.38/9.04	(7) QDPOrderProof [EQUIVALENT, 0 ms]
31.38/9.04	(8) QDP
31.38/9.04	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
31.38/9.04	(10) TRUE
31.38/9.04	
31.38/9.04	
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(0)
31.38/9.04	Obligation:
31.38/9.04	Q restricted rewrite system:
31.38/9.04	The TRS R consists of the following rules:
31.38/9.04	
31.38/9.04	   a(x1) -> b(x1)
31.38/9.04	   a(b(b(x1))) -> c(x1)
31.38/9.04	   a(c(x1)) -> c(b(a(a(x1))))
31.38/9.04	
31.38/9.04	Q is empty.
31.38/9.04	
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(1) QTRS Reverse (EQUIVALENT)
31.38/9.04	We applied the QTRS Reverse Processor [REVERSE].
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(2)
31.38/9.04	Obligation:
31.38/9.04	Q restricted rewrite system:
31.38/9.04	The TRS R consists of the following rules:
31.38/9.04	
31.38/9.04	   a(x1) -> b(x1)
31.38/9.04	   b(b(a(x1))) -> c(x1)
31.38/9.04	   c(a(x1)) -> a(a(b(c(x1))))
31.38/9.04	
31.38/9.04	Q is empty.
31.38/9.04	
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(3) DependencyPairsProof (EQUIVALENT)
31.38/9.04	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(4)
31.38/9.04	Obligation:
31.38/9.04	Q DP problem:
31.38/9.04	The TRS P consists of the following rules:
31.38/9.04	
31.38/9.04	   A(x1) -> B(x1)
31.38/9.04	   B(b(a(x1))) -> C(x1)
31.38/9.04	   C(a(x1)) -> A(a(b(c(x1))))
31.38/9.04	   C(a(x1)) -> A(b(c(x1)))
31.38/9.04	   C(a(x1)) -> B(c(x1))
31.38/9.04	   C(a(x1)) -> C(x1)
31.38/9.04	
31.38/9.04	The TRS R consists of the following rules:
31.38/9.04	
31.38/9.04	   a(x1) -> b(x1)
31.38/9.04	   b(b(a(x1))) -> c(x1)
31.38/9.04	   c(a(x1)) -> a(a(b(c(x1))))
31.38/9.04	
31.38/9.04	Q is empty.
31.38/9.04	We have to consider all minimal (P,Q,R)-chains.
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(5) QDPOrderProof (EQUIVALENT)
31.38/9.04	We use the reduction pair processor [LPAR04,JAR06].
31.38/9.04	
31.38/9.04	
31.38/9.04	The following pairs can be oriented strictly and are deleted.
31.38/9.04	
31.38/9.04	   C(a(x1)) -> A(b(c(x1)))
31.38/9.04	The remaining pairs can at least be oriented weakly.
31.38/9.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	
31.38/9.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.38/9.04	
31.38/9.04	   b(b(a(x1))) -> c(x1)
31.38/9.04	   c(a(x1)) -> a(a(b(c(x1))))
31.38/9.04	   a(x1) -> b(x1)
31.38/9.04	
31.38/9.04	
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(6)
31.38/9.04	Obligation:
31.38/9.04	Q DP problem:
31.38/9.04	The TRS P consists of the following rules:
31.38/9.04	
31.38/9.04	   A(x1) -> B(x1)
31.38/9.04	   B(b(a(x1))) -> C(x1)
31.38/9.04	   C(a(x1)) -> A(a(b(c(x1))))
31.38/9.04	   C(a(x1)) -> B(c(x1))
31.38/9.04	   C(a(x1)) -> C(x1)
31.38/9.04	
31.38/9.04	The TRS R consists of the following rules:
31.38/9.04	
31.38/9.04	   a(x1) -> b(x1)
31.38/9.04	   b(b(a(x1))) -> c(x1)
31.38/9.04	   c(a(x1)) -> a(a(b(c(x1))))
31.38/9.04	
31.38/9.04	Q is empty.
31.38/9.04	We have to consider all minimal (P,Q,R)-chains.
31.38/9.04	----------------------------------------
31.38/9.04	
31.38/9.04	(7) QDPOrderProof (EQUIVALENT)
31.38/9.04	We use the reduction pair processor [LPAR04,JAR06].
31.38/9.04	
31.38/9.04	
31.38/9.04	The following pairs can be oriented strictly and are deleted.
31.38/9.04	
31.38/9.04	   B(b(a(x1))) -> C(x1)
31.38/9.04	   C(a(x1)) -> C(x1)
31.38/9.04	The remaining pairs can at least be oriented weakly.
31.38/9.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-I, 0A, -I], [0A, -I, 0A]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(a(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[1A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	   <<<
31.38/9.04	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[1A, 0A, 0A], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
31.38/9.04	>>>
31.38/9.04	
31.38/9.04	
31.38/9.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
31.38/9.04	
31.38/9.04	   b(b(a(x1))) -> c(x1)
31.38/9.05	   c(a(x1)) -> a(a(b(c(x1))))
31.38/9.05	   a(x1) -> b(x1)
31.38/9.05	
31.38/9.05	
31.38/9.05	----------------------------------------
31.38/9.05	
31.38/9.05	(8)
31.38/9.05	Obligation:
31.38/9.05	Q DP problem:
31.38/9.05	The TRS P consists of the following rules:
31.38/9.05	
31.38/9.05	   A(x1) -> B(x1)
31.38/9.05	   C(a(x1)) -> A(a(b(c(x1))))
31.38/9.05	   C(a(x1)) -> B(c(x1))
31.38/9.05	
31.38/9.05	The TRS R consists of the following rules:
31.38/9.05	
31.38/9.05	   a(x1) -> b(x1)
31.38/9.05	   b(b(a(x1))) -> c(x1)
31.38/9.05	   c(a(x1)) -> a(a(b(c(x1))))
31.38/9.05	
31.38/9.05	Q is empty.
31.38/9.05	We have to consider all minimal (P,Q,R)-chains.
31.38/9.05	----------------------------------------
31.38/9.05	
31.38/9.05	(9) DependencyGraphProof (EQUIVALENT)
31.38/9.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
31.38/9.05	----------------------------------------
31.38/9.05	
31.38/9.05	(10)
31.38/9.05	TRUE
31.50/9.17	EOF