33.63/9.52	YES
34.02/9.59	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
34.02/9.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.02/9.59	
34.02/9.59	
34.02/9.59	Termination w.r.t. Q of the given QTRS could be proven:
34.02/9.59	
34.02/9.59	(0) QTRS
34.02/9.59	(1) QTRS Reverse [EQUIVALENT, 0 ms]
34.02/9.59	(2) QTRS
34.02/9.59	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
34.02/9.59	(4) QDP
34.02/9.59	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
34.02/9.59	(6) QDP
34.02/9.59	(7) QDPOrderProof [EQUIVALENT, 58 ms]
34.02/9.59	(8) QDP
34.02/9.59	(9) QDPOrderProof [EQUIVALENT, 40 ms]
34.02/9.59	(10) QDP
34.02/9.59	(11) PisEmptyProof [EQUIVALENT, 0 ms]
34.02/9.59	(12) YES
34.02/9.59	
34.02/9.59	
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(0)
34.02/9.59	Obligation:
34.02/9.59	Q restricted rewrite system:
34.02/9.59	The TRS R consists of the following rules:
34.02/9.59	
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	   b(x1) -> x1
34.02/9.59	   b(c(x1)) -> c(c(b(a(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	
34.02/9.59	Q is empty.
34.02/9.59	
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(1) QTRS Reverse (EQUIVALENT)
34.02/9.59	We applied the QTRS Reverse Processor [REVERSE].
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(2)
34.02/9.59	Obligation:
34.02/9.59	Q restricted rewrite system:
34.02/9.59	The TRS R consists of the following rules:
34.02/9.59	
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	   b(x1) -> x1
34.02/9.59	   c(b(x1)) -> a(b(c(c(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	
34.02/9.59	Q is empty.
34.02/9.59	
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(3) DependencyPairsProof (EQUIVALENT)
34.02/9.59	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(4)
34.02/9.59	Obligation:
34.02/9.59	Q DP problem:
34.02/9.59	The TRS P consists of the following rules:
34.02/9.59	
34.02/9.59	   A(a(x1)) -> B(x1)
34.02/9.59	   C(b(x1)) -> A(b(c(c(x1))))
34.02/9.59	   C(b(x1)) -> B(c(c(x1)))
34.02/9.59	   C(b(x1)) -> C(c(x1))
34.02/9.59	   C(b(x1)) -> C(x1)
34.02/9.59	
34.02/9.59	The TRS R consists of the following rules:
34.02/9.59	
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	   b(x1) -> x1
34.02/9.59	   c(b(x1)) -> a(b(c(c(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	
34.02/9.59	Q is empty.
34.02/9.59	We have to consider all minimal (P,Q,R)-chains.
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(5) DependencyGraphProof (EQUIVALENT)
34.02/9.59	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(6)
34.02/9.59	Obligation:
34.02/9.59	Q DP problem:
34.02/9.59	The TRS P consists of the following rules:
34.02/9.59	
34.02/9.59	   C(b(x1)) -> C(x1)
34.02/9.59	   C(b(x1)) -> C(c(x1))
34.02/9.59	
34.02/9.59	The TRS R consists of the following rules:
34.02/9.59	
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	   b(x1) -> x1
34.02/9.59	   c(b(x1)) -> a(b(c(c(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	
34.02/9.59	Q is empty.
34.02/9.59	We have to consider all minimal (P,Q,R)-chains.
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(7) QDPOrderProof (EQUIVALENT)
34.02/9.59	We use the reduction pair processor [LPAR04,JAR06].
34.02/9.59	
34.02/9.59	
34.02/9.59	The following pairs can be oriented strictly and are deleted.
34.02/9.59	
34.02/9.59	   C(b(x1)) -> C(x1)
34.02/9.59	The remaining pairs can at least be oriented weakly.
34.02/9.59	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(b(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[1A, 1A, 1A], [-I, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(c(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	
34.02/9.59	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.02/9.59	
34.02/9.59	   c(b(x1)) -> a(b(c(c(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	   b(x1) -> x1
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	
34.02/9.59	
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(8)
34.02/9.59	Obligation:
34.02/9.59	Q DP problem:
34.02/9.59	The TRS P consists of the following rules:
34.02/9.59	
34.02/9.59	   C(b(x1)) -> C(c(x1))
34.02/9.59	
34.02/9.59	The TRS R consists of the following rules:
34.02/9.59	
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	   b(x1) -> x1
34.02/9.59	   c(b(x1)) -> a(b(c(c(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	
34.02/9.59	Q is empty.
34.02/9.59	We have to consider all minimal (P,Q,R)-chains.
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(9) QDPOrderProof (EQUIVALENT)
34.02/9.59	We use the reduction pair processor [LPAR04,JAR06].
34.02/9.59	
34.02/9.59	
34.02/9.59	The following pairs can be oriented strictly and are deleted.
34.02/9.59	
34.02/9.59	   C(b(x1)) -> C(c(x1))
34.02/9.59	The remaining pairs can at least be oriented weakly.
34.02/9.59	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[1A, -I, 1A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	   <<<
34.02/9.59	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
34.02/9.59	>>>
34.02/9.59	
34.02/9.59	
34.02/9.59	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.02/9.59	
34.02/9.59	   c(b(x1)) -> a(b(c(c(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	   b(x1) -> x1
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	
34.02/9.59	
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(10)
34.02/9.59	Obligation:
34.02/9.59	Q DP problem:
34.02/9.59	P is empty.
34.02/9.59	The TRS R consists of the following rules:
34.02/9.59	
34.02/9.59	   a(x1) -> x1
34.02/9.59	   a(a(x1)) -> b(x1)
34.02/9.59	   b(x1) -> x1
34.02/9.59	   c(b(x1)) -> a(b(c(c(x1))))
34.02/9.59	   c(x1) -> x1
34.02/9.59	
34.02/9.59	Q is empty.
34.02/9.59	We have to consider all minimal (P,Q,R)-chains.
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(11) PisEmptyProof (EQUIVALENT)
34.02/9.59	The TRS P is empty. Hence, there is no (P,Q,R) chain.
34.02/9.59	----------------------------------------
34.02/9.59	
34.02/9.59	(12)
34.02/9.59	YES
34.32/9.67	EOF