31.95/9.06	YES
32.26/9.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.26/9.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.26/9.09	
32.26/9.09	
32.26/9.09	Termination w.r.t. Q of the given QTRS could be proven:
32.26/9.09	
32.26/9.09	(0) QTRS
32.26/9.09	(1) DependencyPairsProof [EQUIVALENT, 22 ms]
32.26/9.09	(2) QDP
32.26/9.09	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
32.26/9.09	(4) QDP
32.26/9.09	(5) QDPOrderProof [EQUIVALENT, 50 ms]
32.26/9.09	(6) QDP
32.26/9.09	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
32.26/9.09	(8) TRUE
32.26/9.09	
32.26/9.09	
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(0)
32.26/9.09	Obligation:
32.26/9.09	Q restricted rewrite system:
32.26/9.09	The TRS R consists of the following rules:
32.26/9.09	
32.26/9.09	   a(x1) -> x1
32.26/9.09	   a(b(x1)) -> c(b(c(b(x1))))
32.26/9.09	   b(x1) -> a(a(x1))
32.26/9.09	   c(c(x1)) -> x1
32.26/9.09	
32.26/9.09	Q is empty.
32.26/9.09	
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(1) DependencyPairsProof (EQUIVALENT)
32.26/9.09	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(2)
32.26/9.09	Obligation:
32.26/9.09	Q DP problem:
32.26/9.09	The TRS P consists of the following rules:
32.26/9.09	
32.26/9.09	   A(b(x1)) -> C(b(c(b(x1))))
32.26/9.09	   A(b(x1)) -> B(c(b(x1)))
32.26/9.09	   A(b(x1)) -> C(b(x1))
32.26/9.09	   B(x1) -> A(a(x1))
32.26/9.09	   B(x1) -> A(x1)
32.26/9.09	
32.26/9.09	The TRS R consists of the following rules:
32.26/9.09	
32.26/9.09	   a(x1) -> x1
32.26/9.09	   a(b(x1)) -> c(b(c(b(x1))))
32.26/9.09	   b(x1) -> a(a(x1))
32.26/9.09	   c(c(x1)) -> x1
32.26/9.09	
32.26/9.09	Q is empty.
32.26/9.09	We have to consider all minimal (P,Q,R)-chains.
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(3) DependencyGraphProof (EQUIVALENT)
32.26/9.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(4)
32.26/9.09	Obligation:
32.26/9.09	Q DP problem:
32.26/9.09	The TRS P consists of the following rules:
32.26/9.09	
32.26/9.09	   A(b(x1)) -> B(c(b(x1)))
32.26/9.09	   B(x1) -> A(a(x1))
32.26/9.09	   B(x1) -> A(x1)
32.26/9.09	
32.26/9.09	The TRS R consists of the following rules:
32.26/9.09	
32.26/9.09	   a(x1) -> x1
32.26/9.09	   a(b(x1)) -> c(b(c(b(x1))))
32.26/9.09	   b(x1) -> a(a(x1))
32.26/9.09	   c(c(x1)) -> x1
32.26/9.09	
32.26/9.09	Q is empty.
32.26/9.09	We have to consider all minimal (P,Q,R)-chains.
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(5) QDPOrderProof (EQUIVALENT)
32.26/9.09	We use the reduction pair processor [LPAR04,JAR06].
32.26/9.09	
32.26/9.09	
32.26/9.09	The following pairs can be oriented strictly and are deleted.
32.26/9.09	
32.26/9.09	   B(x1) -> A(a(x1))
32.26/9.09	   B(x1) -> A(x1)
32.26/9.09	The remaining pairs can at least be oriented weakly.
32.26/9.09	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
32.26/9.09	
32.26/9.09	   <<<
32.26/9.09	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 1A]] 	* 	x_1
32.26/9.09	>>>
32.26/9.09	
32.26/9.09	   <<<
32.26/9.09	 POL(b(x_1)) =  	[[-1A], [-I], [0A]] 	 +  	[[0A, -1A, 2A], [-1A, 0A, 2A], [-1A, 0A, 2A]] 	* 	x_1
32.26/9.09	>>>
32.26/9.09	
32.26/9.09	   <<<
32.26/9.09	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, -I, 2A]] 	* 	x_1
32.26/9.09	>>>
32.26/9.09	
32.26/9.09	   <<<
32.26/9.09	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-1A, 1A, -1A], [-1A, -1A, 1A], [-I, -1A, -I]] 	* 	x_1
32.26/9.09	>>>
32.26/9.09	
32.26/9.09	   <<<
32.26/9.09	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 2A], [-1A, 0A, 2A], [-I, -I, 0A]] 	* 	x_1
32.26/9.09	>>>
32.26/9.09	
32.26/9.09	
32.26/9.09	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.26/9.09	
32.26/9.09	   b(x1) -> a(a(x1))
32.26/9.09	   c(c(x1)) -> x1
32.26/9.09	   a(x1) -> x1
32.26/9.09	   a(b(x1)) -> c(b(c(b(x1))))
32.26/9.09	
32.26/9.09	
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(6)
32.26/9.09	Obligation:
32.26/9.09	Q DP problem:
32.26/9.09	The TRS P consists of the following rules:
32.26/9.09	
32.26/9.09	   A(b(x1)) -> B(c(b(x1)))
32.26/9.09	
32.26/9.09	The TRS R consists of the following rules:
32.26/9.09	
32.26/9.09	   a(x1) -> x1
32.26/9.09	   a(b(x1)) -> c(b(c(b(x1))))
32.26/9.09	   b(x1) -> a(a(x1))
32.26/9.09	   c(c(x1)) -> x1
32.26/9.09	
32.26/9.09	Q is empty.
32.26/9.09	We have to consider all minimal (P,Q,R)-chains.
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(7) DependencyGraphProof (EQUIVALENT)
32.26/9.09	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
32.26/9.09	----------------------------------------
32.26/9.09	
32.26/9.09	(8)
32.26/9.09	TRUE
32.58/9.24	EOF