33.49/9.49	YES
33.74/9.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.74/9.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.74/9.54	
33.74/9.54	
33.74/9.54	Termination w.r.t. Q of the given QTRS could be proven:
33.74/9.54	
33.74/9.54	(0) QTRS
33.74/9.54	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.74/9.54	(2) QTRS
33.74/9.54	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
33.74/9.54	(4) QDP
33.74/9.54	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
33.74/9.54	(6) QDP
33.74/9.54	(7) QDPOrderProof [EQUIVALENT, 874 ms]
33.74/9.54	(8) QDP
33.74/9.54	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
33.74/9.54	(10) QDP
33.74/9.54	(11) QDPOrderProof [EQUIVALENT, 5 ms]
33.74/9.54	(12) QDP
33.74/9.54	(13) PisEmptyProof [EQUIVALENT, 0 ms]
33.74/9.54	(14) YES
33.74/9.54	
33.74/9.54	
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(0)
33.74/9.54	Obligation:
33.74/9.54	Q restricted rewrite system:
33.74/9.54	The TRS R consists of the following rules:
33.74/9.54	
33.74/9.54	   a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1)))))))
33.74/9.54	   a(a(c(x1))) -> c(c(a(a(x1))))
33.74/9.54	   c(c(c(x1))) -> b(d(c(b(d(x1)))))
33.74/9.54	
33.74/9.54	Q is empty.
33.74/9.54	
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(1) QTRS Reverse (EQUIVALENT)
33.74/9.54	We applied the QTRS Reverse Processor [REVERSE].
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(2)
33.74/9.54	Obligation:
33.74/9.54	Q restricted rewrite system:
33.74/9.54	The TRS R consists of the following rules:
33.74/9.54	
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	
33.74/9.54	Q is empty.
33.74/9.54	
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(3) DependencyPairsProof (EQUIVALENT)
33.74/9.54	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(4)
33.74/9.54	Obligation:
33.74/9.54	Q DP problem:
33.74/9.54	The TRS P consists of the following rules:
33.74/9.54	
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> A(a(c(a(a(x1)))))
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> A(c(a(a(x1))))
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> C(a(a(x1)))
33.74/9.54	   C(a(a(x1))) -> A(a(c(c(x1))))
33.74/9.54	   C(a(a(x1))) -> A(c(c(x1)))
33.74/9.54	   C(a(a(x1))) -> C(c(x1))
33.74/9.54	   C(a(a(x1))) -> C(x1)
33.74/9.54	   C(c(c(x1))) -> C(d(b(x1)))
33.74/9.54	
33.74/9.54	The TRS R consists of the following rules:
33.74/9.54	
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	
33.74/9.54	Q is empty.
33.74/9.54	We have to consider all minimal (P,Q,R)-chains.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(5) DependencyGraphProof (EQUIVALENT)
33.74/9.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(6)
33.74/9.54	Obligation:
33.74/9.54	Q DP problem:
33.74/9.54	The TRS P consists of the following rules:
33.74/9.54	
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> A(c(a(a(x1))))
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> A(a(c(a(a(x1)))))
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> C(a(a(x1)))
33.74/9.54	   C(a(a(x1))) -> A(a(c(c(x1))))
33.74/9.54	   C(a(a(x1))) -> A(c(c(x1)))
33.74/9.54	   C(a(a(x1))) -> C(c(x1))
33.74/9.54	   C(a(a(x1))) -> C(x1)
33.74/9.54	
33.74/9.54	The TRS R consists of the following rules:
33.74/9.54	
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	
33.74/9.54	Q is empty.
33.74/9.54	We have to consider all minimal (P,Q,R)-chains.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(7) QDPOrderProof (EQUIVALENT)
33.74/9.54	We use the reduction pair processor [LPAR04,JAR06].
33.74/9.54	
33.74/9.54	
33.74/9.54	The following pairs can be oriented strictly and are deleted.
33.74/9.54	
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> A(c(a(a(x1))))
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> A(a(c(a(a(x1)))))
33.74/9.54	   A(d(b(d(b(a(a(x1))))))) -> C(a(a(x1)))
33.74/9.54	The remaining pairs can at least be oriented weakly.
33.74/9.54	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.74/9.54	
33.74/9.54	   <<<
33.74/9.54	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
33.74/9.54	>>>
33.74/9.54	
33.74/9.54	   <<<
33.74/9.54	 POL(d(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
33.74/9.54	>>>
33.74/9.54	
33.74/9.54	   <<<
33.74/9.54	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [-I, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
33.74/9.54	>>>
33.74/9.54	
33.74/9.54	   <<<
33.74/9.54	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
33.74/9.54	>>>
33.74/9.54	
33.74/9.54	   <<<
33.74/9.54	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
33.74/9.54	>>>
33.74/9.54	
33.74/9.54	   <<<
33.74/9.54	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
33.74/9.54	>>>
33.74/9.54	
33.74/9.54	
33.74/9.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.74/9.54	
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	
33.74/9.54	
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(8)
33.74/9.54	Obligation:
33.74/9.54	Q DP problem:
33.74/9.54	The TRS P consists of the following rules:
33.74/9.54	
33.74/9.54	   C(a(a(x1))) -> A(a(c(c(x1))))
33.74/9.54	   C(a(a(x1))) -> A(c(c(x1)))
33.74/9.54	   C(a(a(x1))) -> C(c(x1))
33.74/9.54	   C(a(a(x1))) -> C(x1)
33.74/9.54	
33.74/9.54	The TRS R consists of the following rules:
33.74/9.54	
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	
33.74/9.54	Q is empty.
33.74/9.54	We have to consider all minimal (P,Q,R)-chains.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(9) DependencyGraphProof (EQUIVALENT)
33.74/9.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(10)
33.74/9.54	Obligation:
33.74/9.54	Q DP problem:
33.74/9.54	The TRS P consists of the following rules:
33.74/9.54	
33.74/9.54	   C(a(a(x1))) -> C(x1)
33.74/9.54	   C(a(a(x1))) -> C(c(x1))
33.74/9.54	
33.74/9.54	The TRS R consists of the following rules:
33.74/9.54	
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	
33.74/9.54	Q is empty.
33.74/9.54	We have to consider all minimal (P,Q,R)-chains.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(11) QDPOrderProof (EQUIVALENT)
33.74/9.54	We use the reduction pair processor [LPAR04,JAR06].
33.74/9.54	
33.74/9.54	
33.74/9.54	The following pairs can be oriented strictly and are deleted.
33.74/9.54	
33.74/9.54	   C(a(a(x1))) -> C(x1)
33.74/9.54	   C(a(a(x1))) -> C(c(x1))
33.74/9.54	The remaining pairs can at least be oriented weakly.
33.74/9.54	Used ordering:  Polynomial interpretation [POLO]:
33.74/9.54	
33.74/9.54	   POL(C(x_1)) = x_1
33.74/9.54	   POL(a(x_1)) = 1 + x_1
33.74/9.54	   POL(b(x_1)) = 1 + x_1
33.74/9.54	   POL(c(x_1)) = x_1
33.74/9.54	   POL(d(x_1)) = 0
33.74/9.54	
33.74/9.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.74/9.54	
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	
33.74/9.54	
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(12)
33.74/9.54	Obligation:
33.74/9.54	Q DP problem:
33.74/9.54	P is empty.
33.74/9.54	The TRS R consists of the following rules:
33.74/9.54	
33.74/9.54	   a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1)))))))
33.74/9.54	   c(a(a(x1))) -> a(a(c(c(x1))))
33.74/9.54	   c(c(c(x1))) -> d(b(c(d(b(x1)))))
33.74/9.54	
33.74/9.54	Q is empty.
33.74/9.54	We have to consider all minimal (P,Q,R)-chains.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(13) PisEmptyProof (EQUIVALENT)
33.74/9.54	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.74/9.54	----------------------------------------
33.74/9.54	
33.74/9.54	(14)
33.74/9.54	YES
34.02/9.64	EOF