36.68/10.20	YES
37.03/10.30	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
37.03/10.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
37.03/10.30	
37.03/10.30	
37.03/10.30	Termination w.r.t. Q of the given QTRS could be proven:
37.03/10.30	
37.03/10.30	(0) QTRS
37.03/10.30	(1) QTRS Reverse [EQUIVALENT, 0 ms]
37.03/10.30	(2) QTRS
37.03/10.30	(3) DependencyPairsProof [EQUIVALENT, 15 ms]
37.03/10.30	(4) QDP
37.03/10.30	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
37.03/10.30	(6) QDP
37.03/10.30	(7) QDPOrderProof [EQUIVALENT, 137 ms]
37.03/10.30	(8) QDP
37.03/10.30	(9) QDPOrderProof [EQUIVALENT, 57 ms]
37.03/10.30	(10) QDP
37.03/10.30	(11) PisEmptyProof [EQUIVALENT, 0 ms]
37.03/10.30	(12) YES
37.03/10.30	
37.03/10.30	
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(0)
37.03/10.30	Obligation:
37.03/10.30	Q restricted rewrite system:
37.03/10.30	The TRS R consists of the following rules:
37.03/10.30	
37.03/10.30	   a(x1) -> x1
37.03/10.30	   a(b(c(x1))) -> c(c(b(b(a(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	
37.03/10.30	Q is empty.
37.03/10.30	
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(1) QTRS Reverse (EQUIVALENT)
37.03/10.30	We applied the QTRS Reverse Processor [REVERSE].
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(2)
37.03/10.30	Obligation:
37.03/10.30	Q restricted rewrite system:
37.03/10.30	The TRS R consists of the following rules:
37.03/10.30	
37.03/10.30	   a(x1) -> x1
37.03/10.30	   c(b(a(x1))) -> a(b(b(c(c(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	
37.03/10.30	Q is empty.
37.03/10.30	
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(3) DependencyPairsProof (EQUIVALENT)
37.03/10.30	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(4)
37.03/10.30	Obligation:
37.03/10.30	Q DP problem:
37.03/10.30	The TRS P consists of the following rules:
37.03/10.30	
37.03/10.30	   C(b(a(x1))) -> A(b(b(c(c(x1)))))
37.03/10.30	   C(b(a(x1))) -> B(b(c(c(x1))))
37.03/10.30	   C(b(a(x1))) -> B(c(c(x1)))
37.03/10.30	   C(b(a(x1))) -> C(c(x1))
37.03/10.30	   C(b(a(x1))) -> C(x1)
37.03/10.30	   B(x1) -> A(x1)
37.03/10.30	
37.03/10.30	The TRS R consists of the following rules:
37.03/10.30	
37.03/10.30	   a(x1) -> x1
37.03/10.30	   c(b(a(x1))) -> a(b(b(c(c(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	
37.03/10.30	Q is empty.
37.03/10.30	We have to consider all minimal (P,Q,R)-chains.
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(5) DependencyGraphProof (EQUIVALENT)
37.03/10.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(6)
37.03/10.30	Obligation:
37.03/10.30	Q DP problem:
37.03/10.30	The TRS P consists of the following rules:
37.03/10.30	
37.03/10.30	   C(b(a(x1))) -> C(x1)
37.03/10.30	   C(b(a(x1))) -> C(c(x1))
37.03/10.30	
37.03/10.30	The TRS R consists of the following rules:
37.03/10.30	
37.03/10.30	   a(x1) -> x1
37.03/10.30	   c(b(a(x1))) -> a(b(b(c(c(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	
37.03/10.30	Q is empty.
37.03/10.30	We have to consider all minimal (P,Q,R)-chains.
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(7) QDPOrderProof (EQUIVALENT)
37.03/10.30	We use the reduction pair processor [LPAR04,JAR06].
37.03/10.30	
37.03/10.30	
37.03/10.30	The following pairs can be oriented strictly and are deleted.
37.03/10.30	
37.03/10.30	   C(b(a(x1))) -> C(x1)
37.03/10.30	The remaining pairs can at least be oriented weakly.
37.03/10.30	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 1A]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 1A], [1A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, 1A], [1A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(c(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	
37.03/10.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.03/10.30	
37.03/10.30	   c(b(a(x1))) -> a(b(b(c(c(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	   a(x1) -> x1
37.03/10.30	
37.03/10.30	
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(8)
37.03/10.30	Obligation:
37.03/10.30	Q DP problem:
37.03/10.30	The TRS P consists of the following rules:
37.03/10.30	
37.03/10.30	   C(b(a(x1))) -> C(c(x1))
37.03/10.30	
37.03/10.30	The TRS R consists of the following rules:
37.03/10.30	
37.03/10.30	   a(x1) -> x1
37.03/10.30	   c(b(a(x1))) -> a(b(b(c(c(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	
37.03/10.30	Q is empty.
37.03/10.30	We have to consider all minimal (P,Q,R)-chains.
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(9) QDPOrderProof (EQUIVALENT)
37.03/10.30	We use the reduction pair processor [LPAR04,JAR06].
37.03/10.30	
37.03/10.30	
37.03/10.30	The following pairs can be oriented strictly and are deleted.
37.03/10.30	
37.03/10.30	   C(b(a(x1))) -> C(c(x1))
37.03/10.30	The remaining pairs can at least be oriented weakly.
37.03/10.30	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(b(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	   <<<
37.03/10.30	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
37.03/10.30	>>>
37.03/10.30	
37.03/10.30	
37.03/10.30	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.03/10.30	
37.03/10.30	   c(b(a(x1))) -> a(b(b(c(c(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	   a(x1) -> x1
37.03/10.30	
37.03/10.30	
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(10)
37.03/10.30	Obligation:
37.03/10.30	Q DP problem:
37.03/10.30	P is empty.
37.03/10.30	The TRS R consists of the following rules:
37.03/10.30	
37.03/10.30	   a(x1) -> x1
37.03/10.30	   c(b(a(x1))) -> a(b(b(c(c(x1)))))
37.03/10.30	   b(x1) -> a(x1)
37.03/10.30	
37.03/10.30	Q is empty.
37.03/10.30	We have to consider all minimal (P,Q,R)-chains.
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(11) PisEmptyProof (EQUIVALENT)
37.03/10.30	The TRS P is empty. Hence, there is no (P,Q,R) chain.
37.03/10.30	----------------------------------------
37.03/10.30	
37.03/10.30	(12)
37.03/10.30	YES
37.27/11.79	EOF