43.12/11.93	YES
43.79/12.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
43.79/12.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
43.79/12.06	
43.79/12.06	
43.79/12.06	Termination w.r.t. Q of the given QTRS could be proven:
43.79/12.06	
43.79/12.06	(0) QTRS
43.79/12.06	(1) DependencyPairsProof [EQUIVALENT, 6 ms]
43.79/12.06	(2) QDP
43.79/12.06	(3) QDPOrderProof [EQUIVALENT, 29 ms]
43.79/12.06	(4) QDP
43.79/12.06	(5) QDPOrderProof [EQUIVALENT, 77 ms]
43.79/12.06	(6) QDP
43.79/12.06	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
43.79/12.06	(8) QDP
43.79/12.06	(9) QDPOrderProof [EQUIVALENT, 26 ms]
43.79/12.06	(10) QDP
43.79/12.06	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
43.79/12.06	(12) TRUE
43.79/12.06	
43.79/12.06	
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(0)
43.79/12.06	Obligation:
43.79/12.06	Q restricted rewrite system:
43.79/12.06	The TRS R consists of the following rules:
43.79/12.06	
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	
43.79/12.06	Q is empty.
43.79/12.06	
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(1) DependencyPairsProof (EQUIVALENT)
43.79/12.06	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(2)
43.79/12.06	Obligation:
43.79/12.06	Q DP problem:
43.79/12.06	The TRS P consists of the following rules:
43.79/12.06	
43.79/12.06	   A(x1) -> B(c(x1))
43.79/12.06	   A(x1) -> C(x1)
43.79/12.06	   B(a(b(x1))) -> C(x1)
43.79/12.06	   C(c(x1)) -> A(a(b(x1)))
43.79/12.06	   C(c(x1)) -> A(b(x1))
43.79/12.06	   C(c(x1)) -> B(x1)
43.79/12.06	
43.79/12.06	The TRS R consists of the following rules:
43.79/12.06	
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	
43.79/12.06	Q is empty.
43.79/12.06	We have to consider all minimal (P,Q,R)-chains.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(3) QDPOrderProof (EQUIVALENT)
43.79/12.06	We use the reduction pair processor [LPAR04,JAR06].
43.79/12.06	
43.79/12.06	
43.79/12.06	The following pairs can be oriented strictly and are deleted.
43.79/12.06	
43.79/12.06	   C(c(x1)) -> A(b(x1))
43.79/12.06	   C(c(x1)) -> B(x1)
43.79/12.06	The remaining pairs can at least be oriented weakly.
43.79/12.06	Used ordering:  Polynomial interpretation [POLO]:
43.79/12.06	
43.79/12.06	   POL(A(x_1)) = 1 + x_1
43.79/12.06	   POL(B(x_1)) = x_1
43.79/12.06	   POL(C(x_1)) = 1 + x_1
43.79/12.06	   POL(a(x_1)) = 1 + x_1
43.79/12.06	   POL(b(x_1)) = x_1
43.79/12.06	   POL(c(x_1)) = 1 + x_1
43.79/12.06	
43.79/12.06	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.79/12.06	
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	
43.79/12.06	
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(4)
43.79/12.06	Obligation:
43.79/12.06	Q DP problem:
43.79/12.06	The TRS P consists of the following rules:
43.79/12.06	
43.79/12.06	   A(x1) -> B(c(x1))
43.79/12.06	   A(x1) -> C(x1)
43.79/12.06	   B(a(b(x1))) -> C(x1)
43.79/12.06	   C(c(x1)) -> A(a(b(x1)))
43.79/12.06	
43.79/12.06	The TRS R consists of the following rules:
43.79/12.06	
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	
43.79/12.06	Q is empty.
43.79/12.06	We have to consider all minimal (P,Q,R)-chains.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(5) QDPOrderProof (EQUIVALENT)
43.79/12.06	We use the reduction pair processor [LPAR04,JAR06].
43.79/12.06	
43.79/12.06	
43.79/12.06	The following pairs can be oriented strictly and are deleted.
43.79/12.06	
43.79/12.06	   A(x1) -> B(c(x1))
43.79/12.06	The remaining pairs can at least be oriented weakly.
43.79/12.06	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 1A, 1A], [-I, -I, 0A], [0A, 1A, 0A]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 1A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	
43.79/12.06	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.79/12.06	
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	
43.79/12.06	
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(6)
43.79/12.06	Obligation:
43.79/12.06	Q DP problem:
43.79/12.06	The TRS P consists of the following rules:
43.79/12.06	
43.79/12.06	   A(x1) -> C(x1)
43.79/12.06	   B(a(b(x1))) -> C(x1)
43.79/12.06	   C(c(x1)) -> A(a(b(x1)))
43.79/12.06	
43.79/12.06	The TRS R consists of the following rules:
43.79/12.06	
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	
43.79/12.06	Q is empty.
43.79/12.06	We have to consider all minimal (P,Q,R)-chains.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(7) DependencyGraphProof (EQUIVALENT)
43.79/12.06	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(8)
43.79/12.06	Obligation:
43.79/12.06	Q DP problem:
43.79/12.06	The TRS P consists of the following rules:
43.79/12.06	
43.79/12.06	   C(c(x1)) -> A(a(b(x1)))
43.79/12.06	   A(x1) -> C(x1)
43.79/12.06	
43.79/12.06	The TRS R consists of the following rules:
43.79/12.06	
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	
43.79/12.06	Q is empty.
43.79/12.06	We have to consider all minimal (P,Q,R)-chains.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(9) QDPOrderProof (EQUIVALENT)
43.79/12.06	We use the reduction pair processor [LPAR04,JAR06].
43.79/12.06	
43.79/12.06	
43.79/12.06	The following pairs can be oriented strictly and are deleted.
43.79/12.06	
43.79/12.06	   A(x1) -> C(x1)
43.79/12.06	The remaining pairs can at least be oriented weakly.
43.79/12.06	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 1A], [1A, 0A, 0A]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(a(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	   <<<
43.79/12.06	 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
43.79/12.06	>>>
43.79/12.06	
43.79/12.06	
43.79/12.06	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.79/12.06	
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	
43.79/12.06	
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(10)
43.79/12.06	Obligation:
43.79/12.06	Q DP problem:
43.79/12.06	The TRS P consists of the following rules:
43.79/12.06	
43.79/12.06	   C(c(x1)) -> A(a(b(x1)))
43.79/12.06	
43.79/12.06	The TRS R consists of the following rules:
43.79/12.06	
43.79/12.06	   a(x1) -> b(c(x1))
43.79/12.06	   b(a(b(x1))) -> c(x1)
43.79/12.06	   c(c(x1)) -> a(a(b(x1)))
43.79/12.06	
43.79/12.06	Q is empty.
43.79/12.06	We have to consider all minimal (P,Q,R)-chains.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(11) DependencyGraphProof (EQUIVALENT)
43.79/12.06	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
43.79/12.06	----------------------------------------
43.79/12.06	
43.79/12.06	(12)
43.79/12.06	TRUE
44.07/12.16	EOF