30.56/8.67	YES
30.91/8.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
30.91/8.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.91/8.81	
30.91/8.81	
30.91/8.81	Termination w.r.t. Q of the given QTRS could be proven:
30.91/8.81	
30.91/8.81	(0) QTRS
30.91/8.81	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
30.91/8.81	(2) QDP
30.91/8.81	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
30.91/8.81	(4) QDP
30.91/8.81	(5) MRRProof [EQUIVALENT, 10 ms]
30.91/8.81	(6) QDP
30.91/8.81	(7) QDPOrderProof [EQUIVALENT, 864 ms]
30.91/8.81	(8) QDP
30.91/8.81	(9) PisEmptyProof [EQUIVALENT, 0 ms]
30.91/8.81	(10) YES
30.91/8.81	
30.91/8.81	
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(0)
30.91/8.81	Obligation:
30.91/8.81	Q restricted rewrite system:
30.91/8.81	The TRS R consists of the following rules:
30.91/8.81	
30.91/8.81	   a(x1) -> b(x1)
30.91/8.81	   a(b(x1)) -> c(a(c(a(x1))))
30.91/8.81	   c(b(c(x1))) -> b(x1)
30.91/8.81	
30.91/8.81	Q is empty.
30.91/8.81	
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(1) DependencyPairsProof (EQUIVALENT)
30.91/8.81	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(2)
30.91/8.81	Obligation:
30.91/8.81	Q DP problem:
30.91/8.81	The TRS P consists of the following rules:
30.91/8.81	
30.91/8.81	   A(b(x1)) -> C(a(c(a(x1))))
30.91/8.81	   A(b(x1)) -> A(c(a(x1)))
30.91/8.81	   A(b(x1)) -> C(a(x1))
30.91/8.81	   A(b(x1)) -> A(x1)
30.91/8.81	
30.91/8.81	The TRS R consists of the following rules:
30.91/8.81	
30.91/8.81	   a(x1) -> b(x1)
30.91/8.81	   a(b(x1)) -> c(a(c(a(x1))))
30.91/8.81	   c(b(c(x1))) -> b(x1)
30.91/8.81	
30.91/8.81	Q is empty.
30.91/8.81	We have to consider all minimal (P,Q,R)-chains.
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(3) DependencyGraphProof (EQUIVALENT)
30.91/8.81	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(4)
30.91/8.81	Obligation:
30.91/8.81	Q DP problem:
30.91/8.81	The TRS P consists of the following rules:
30.91/8.81	
30.91/8.81	   A(b(x1)) -> A(x1)
30.91/8.81	   A(b(x1)) -> A(c(a(x1)))
30.91/8.81	
30.91/8.81	The TRS R consists of the following rules:
30.91/8.81	
30.91/8.81	   a(x1) -> b(x1)
30.91/8.81	   a(b(x1)) -> c(a(c(a(x1))))
30.91/8.81	   c(b(c(x1))) -> b(x1)
30.91/8.81	
30.91/8.81	Q is empty.
30.91/8.81	We have to consider all minimal (P,Q,R)-chains.
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(5) MRRProof (EQUIVALENT)
30.91/8.81	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
30.91/8.81	
30.91/8.81	Strictly oriented dependency pairs:
30.91/8.81	
30.91/8.81	   A(b(x1)) -> A(x1)
30.91/8.81	
30.91/8.81	
30.91/8.81	Used ordering: Polynomial interpretation [POLO]:
30.91/8.81	
30.91/8.81	   POL(A(x_1)) = 2*x_1
30.91/8.81	   POL(a(x_1)) = 2 + x_1
30.91/8.81	   POL(b(x_1)) = 2 + x_1
30.91/8.81	   POL(c(x_1)) = x_1
30.91/8.81	
30.91/8.81	
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(6)
30.91/8.81	Obligation:
30.91/8.81	Q DP problem:
30.91/8.81	The TRS P consists of the following rules:
30.91/8.81	
30.91/8.81	   A(b(x1)) -> A(c(a(x1)))
30.91/8.81	
30.91/8.81	The TRS R consists of the following rules:
30.91/8.81	
30.91/8.81	   a(x1) -> b(x1)
30.91/8.81	   a(b(x1)) -> c(a(c(a(x1))))
30.91/8.81	   c(b(c(x1))) -> b(x1)
30.91/8.81	
30.91/8.81	Q is empty.
30.91/8.81	We have to consider all minimal (P,Q,R)-chains.
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(7) QDPOrderProof (EQUIVALENT)
30.91/8.81	We use the reduction pair processor [LPAR04,JAR06].
30.91/8.81	
30.91/8.81	
30.91/8.81	The following pairs can be oriented strictly and are deleted.
30.91/8.81	
30.91/8.81	   A(b(x1)) -> A(c(a(x1)))
30.91/8.81	The remaining pairs can at least be oriented weakly.
30.91/8.81	Used ordering:  Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]:
30.91/8.81	
30.91/8.81	   <<<
30.91/8.81	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-1A, 2A, -I]] 	* 	x_1
30.91/8.81	>>>
30.91/8.81	
30.91/8.81	   <<<
30.91/8.81	 POL(b(x_1)) =  	[[0A], [-1A], [-1A]] 	 +  	[[-I, -I, -1A], [-I, -I, -1A], [-1A, 2A, -I]] 	* 	x_1
30.91/8.81	>>>
30.91/8.81	
30.91/8.81	   <<<
30.91/8.81	 POL(c(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-1A, -I, -I], [-I, -1A, -I], [-I, -I, 1A]] 	* 	x_1
30.91/8.81	>>>
30.91/8.81	
30.91/8.81	   <<<
30.91/8.81	 POL(a(x_1)) =  	[[0A], [-1A], [-1A]] 	 +  	[[-I, -I, -1A], [-I, -I, -1A], [-1A, 2A, -I]] 	* 	x_1
30.91/8.81	>>>
30.91/8.81	
30.91/8.81	
30.91/8.81	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.91/8.81	
30.91/8.81	   a(x1) -> b(x1)
30.91/8.81	   a(b(x1)) -> c(a(c(a(x1))))
30.91/8.81	   c(b(c(x1))) -> b(x1)
30.91/8.81	
30.91/8.81	
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(8)
30.91/8.81	Obligation:
30.91/8.81	Q DP problem:
30.91/8.81	P is empty.
30.91/8.81	The TRS R consists of the following rules:
30.91/8.81	
30.91/8.81	   a(x1) -> b(x1)
30.91/8.81	   a(b(x1)) -> c(a(c(a(x1))))
30.91/8.81	   c(b(c(x1))) -> b(x1)
30.91/8.81	
30.91/8.81	Q is empty.
30.91/8.81	We have to consider all minimal (P,Q,R)-chains.
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(9) PisEmptyProof (EQUIVALENT)
30.91/8.81	The TRS P is empty. Hence, there is no (P,Q,R) chain.
30.91/8.81	----------------------------------------
30.91/8.81	
30.91/8.81	(10)
30.91/8.81	YES
31.36/8.90	EOF