44.19/12.14	YES
44.19/12.14	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
44.19/12.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
44.19/12.14	
44.19/12.14	
44.19/12.14	Termination w.r.t. Q of the given QTRS could be proven:
44.19/12.14	
44.19/12.14	(0) QTRS
44.19/12.14	(1) DependencyPairsProof [EQUIVALENT, 4 ms]
44.19/12.14	(2) QDP
44.19/12.14	(3) QDPOrderProof [EQUIVALENT, 35 ms]
44.19/12.14	(4) QDP
44.19/12.14	(5) QDPOrderProof [EQUIVALENT, 54 ms]
44.19/12.14	(6) QDP
44.19/12.14	(7) QDPOrderProof [EQUIVALENT, 29 ms]
44.19/12.14	(8) QDP
44.19/12.14	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
44.19/12.14	(10) TRUE
44.19/12.14	
44.19/12.14	
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(0)
44.19/12.14	Obligation:
44.19/12.14	Q restricted rewrite system:
44.19/12.14	The TRS R consists of the following rules:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	Q is empty.
44.19/12.14	
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(1) DependencyPairsProof (EQUIVALENT)
44.19/12.14	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(2)
44.19/12.14	Obligation:
44.19/12.14	Q DP problem:
44.19/12.14	The TRS P consists of the following rules:
44.19/12.14	
44.19/12.14	   A(c(x1)) -> C(a(a(x1)))
44.19/12.14	   A(c(x1)) -> A(a(x1))
44.19/12.14	   A(c(x1)) -> A(x1)
44.19/12.14	   C(b(x1)) -> A(c(x1))
44.19/12.14	   C(b(x1)) -> C(x1)
44.19/12.14	
44.19/12.14	The TRS R consists of the following rules:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	Q is empty.
44.19/12.14	We have to consider all minimal (P,Q,R)-chains.
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(3) QDPOrderProof (EQUIVALENT)
44.19/12.14	We use the reduction pair processor [LPAR04,JAR06].
44.19/12.14	
44.19/12.14	
44.19/12.14	The following pairs can be oriented strictly and are deleted.
44.19/12.14	
44.19/12.14	   A(c(x1)) -> A(a(x1))
44.19/12.14	   A(c(x1)) -> A(x1)
44.19/12.14	The remaining pairs can at least be oriented weakly.
44.19/12.14	Used ordering:  Polynomial interpretation [POLO]:
44.19/12.14	
44.19/12.14	   POL(A(x_1)) = x_1
44.19/12.14	   POL(C(x_1)) = 1 + x_1
44.19/12.14	   POL(a(x_1)) = x_1
44.19/12.14	   POL(b(x_1)) = x_1
44.19/12.14	   POL(c(x_1)) = 1 + x_1
44.19/12.14	
44.19/12.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(4)
44.19/12.14	Obligation:
44.19/12.14	Q DP problem:
44.19/12.14	The TRS P consists of the following rules:
44.19/12.14	
44.19/12.14	   A(c(x1)) -> C(a(a(x1)))
44.19/12.14	   C(b(x1)) -> A(c(x1))
44.19/12.14	   C(b(x1)) -> C(x1)
44.19/12.14	
44.19/12.14	The TRS R consists of the following rules:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	Q is empty.
44.19/12.14	We have to consider all minimal (P,Q,R)-chains.
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(5) QDPOrderProof (EQUIVALENT)
44.19/12.14	We use the reduction pair processor [LPAR04,JAR06].
44.19/12.14	
44.19/12.14	
44.19/12.14	The following pairs can be oriented strictly and are deleted.
44.19/12.14	
44.19/12.14	   C(b(x1)) -> C(x1)
44.19/12.14	The remaining pairs can at least be oriented weakly.
44.19/12.14	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 1A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	
44.19/12.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(6)
44.19/12.14	Obligation:
44.19/12.14	Q DP problem:
44.19/12.14	The TRS P consists of the following rules:
44.19/12.14	
44.19/12.14	   A(c(x1)) -> C(a(a(x1)))
44.19/12.14	   C(b(x1)) -> A(c(x1))
44.19/12.14	
44.19/12.14	The TRS R consists of the following rules:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	Q is empty.
44.19/12.14	We have to consider all minimal (P,Q,R)-chains.
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(7) QDPOrderProof (EQUIVALENT)
44.19/12.14	We use the reduction pair processor [LPAR04,JAR06].
44.19/12.14	
44.19/12.14	
44.19/12.14	The following pairs can be oriented strictly and are deleted.
44.19/12.14	
44.19/12.14	   A(c(x1)) -> C(a(a(x1)))
44.19/12.14	The remaining pairs can at least be oriented weakly.
44.19/12.14	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(c(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[0A, 1A, 1A], [-I, -I, -I], [-I, 0A, 0A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	   <<<
44.19/12.14	 POL(b(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, 0A, 0A], [-I, 1A, -I], [0A, 0A, 1A]] 	* 	x_1
44.19/12.14	>>>
44.19/12.14	
44.19/12.14	
44.19/12.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(8)
44.19/12.14	Obligation:
44.19/12.14	Q DP problem:
44.19/12.14	The TRS P consists of the following rules:
44.19/12.14	
44.19/12.14	   C(b(x1)) -> A(c(x1))
44.19/12.14	
44.19/12.14	The TRS R consists of the following rules:
44.19/12.14	
44.19/12.14	   a(b(x1)) -> x1
44.19/12.14	   a(c(x1)) -> b(c(a(a(x1))))
44.19/12.14	   c(b(x1)) -> a(c(x1))
44.19/12.14	
44.19/12.14	Q is empty.
44.19/12.14	We have to consider all minimal (P,Q,R)-chains.
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(9) DependencyGraphProof (EQUIVALENT)
44.19/12.14	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
44.19/12.14	----------------------------------------
44.19/12.14	
44.19/12.14	(10)
44.19/12.14	TRUE
44.61/12.33	EOF