14.15/4.41	YES
14.57/4.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
14.57/4.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
14.57/4.52	
14.57/4.52	
14.57/4.52	Termination w.r.t. Q of the given QTRS could be proven:
14.57/4.52	
14.57/4.52	(0) QTRS
14.57/4.52	(1) QTRS Reverse [EQUIVALENT, 0 ms]
14.57/4.52	(2) QTRS
14.57/4.52	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
14.57/4.52	(4) QDP
14.57/4.52	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
14.57/4.52	(6) QDP
14.57/4.52	(7) QDPOrderProof [EQUIVALENT, 129 ms]
14.57/4.52	(8) QDP
14.57/4.52	(9) PisEmptyProof [EQUIVALENT, 0 ms]
14.57/4.52	(10) YES
14.57/4.52	
14.57/4.52	
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(0)
14.57/4.52	Obligation:
14.57/4.52	Q restricted rewrite system:
14.57/4.52	The TRS R consists of the following rules:
14.57/4.52	
14.57/4.52	   a(x1) -> b(x1)
14.57/4.52	   a(c(x1)) -> c(c(c(a(a(b(x1))))))
14.57/4.52	   b(b(x1)) -> x1
14.57/4.52	
14.57/4.52	Q is empty.
14.57/4.52	
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(1) QTRS Reverse (EQUIVALENT)
14.57/4.52	We applied the QTRS Reverse Processor [REVERSE].
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(2)
14.57/4.52	Obligation:
14.57/4.52	Q restricted rewrite system:
14.57/4.52	The TRS R consists of the following rules:
14.57/4.52	
14.57/4.52	   a(x1) -> b(x1)
14.57/4.52	   c(a(x1)) -> b(a(a(c(c(c(x1))))))
14.57/4.52	   b(b(x1)) -> x1
14.57/4.52	
14.57/4.52	Q is empty.
14.57/4.52	
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(3) DependencyPairsProof (EQUIVALENT)
14.57/4.52	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(4)
14.57/4.52	Obligation:
14.57/4.52	Q DP problem:
14.57/4.52	The TRS P consists of the following rules:
14.57/4.52	
14.57/4.52	   A(x1) -> B(x1)
14.57/4.52	   C(a(x1)) -> B(a(a(c(c(c(x1))))))
14.57/4.52	   C(a(x1)) -> A(a(c(c(c(x1)))))
14.57/4.52	   C(a(x1)) -> A(c(c(c(x1))))
14.57/4.52	   C(a(x1)) -> C(c(c(x1)))
14.57/4.52	   C(a(x1)) -> C(c(x1))
14.57/4.52	   C(a(x1)) -> C(x1)
14.57/4.52	
14.57/4.52	The TRS R consists of the following rules:
14.57/4.52	
14.57/4.52	   a(x1) -> b(x1)
14.57/4.52	   c(a(x1)) -> b(a(a(c(c(c(x1))))))
14.57/4.52	   b(b(x1)) -> x1
14.57/4.52	
14.57/4.52	Q is empty.
14.57/4.52	We have to consider all minimal (P,Q,R)-chains.
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(5) DependencyGraphProof (EQUIVALENT)
14.57/4.52	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(6)
14.57/4.52	Obligation:
14.57/4.52	Q DP problem:
14.57/4.52	The TRS P consists of the following rules:
14.57/4.52	
14.57/4.52	   C(a(x1)) -> C(c(x1))
14.57/4.52	   C(a(x1)) -> C(c(c(x1)))
14.57/4.52	   C(a(x1)) -> C(x1)
14.57/4.52	
14.57/4.52	The TRS R consists of the following rules:
14.57/4.52	
14.57/4.52	   a(x1) -> b(x1)
14.57/4.52	   c(a(x1)) -> b(a(a(c(c(c(x1))))))
14.57/4.52	   b(b(x1)) -> x1
14.57/4.52	
14.57/4.52	Q is empty.
14.57/4.52	We have to consider all minimal (P,Q,R)-chains.
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(7) QDPOrderProof (EQUIVALENT)
14.57/4.52	We use the reduction pair processor [LPAR04,JAR06].
14.57/4.52	
14.57/4.52	
14.57/4.52	The following pairs can be oriented strictly and are deleted.
14.57/4.52	
14.57/4.52	   C(a(x1)) -> C(c(x1))
14.57/4.52	   C(a(x1)) -> C(c(c(x1)))
14.57/4.52	   C(a(x1)) -> C(x1)
14.57/4.52	The remaining pairs can at least be oriented weakly.
14.57/4.52	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
14.57/4.52	
14.57/4.52	   <<<
14.57/4.52	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
14.57/4.52	>>>
14.57/4.52	
14.57/4.52	   <<<
14.57/4.52	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
14.57/4.52	>>>
14.57/4.52	
14.57/4.52	   <<<
14.57/4.52	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [1A, 1A, 0A]] 	* 	x_1
14.57/4.52	>>>
14.57/4.52	
14.57/4.52	   <<<
14.57/4.52	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
14.57/4.52	>>>
14.57/4.52	
14.57/4.52	
14.57/4.52	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
14.57/4.52	
14.57/4.52	   c(a(x1)) -> b(a(a(c(c(c(x1))))))
14.57/4.52	   a(x1) -> b(x1)
14.57/4.52	   b(b(x1)) -> x1
14.57/4.52	
14.57/4.52	
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(8)
14.57/4.52	Obligation:
14.57/4.52	Q DP problem:
14.57/4.52	P is empty.
14.57/4.52	The TRS R consists of the following rules:
14.57/4.52	
14.57/4.52	   a(x1) -> b(x1)
14.57/4.52	   c(a(x1)) -> b(a(a(c(c(c(x1))))))
14.57/4.52	   b(b(x1)) -> x1
14.57/4.52	
14.57/4.52	Q is empty.
14.57/4.52	We have to consider all minimal (P,Q,R)-chains.
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(9) PisEmptyProof (EQUIVALENT)
14.57/4.52	The TRS P is empty. Hence, there is no (P,Q,R) chain.
14.57/4.52	----------------------------------------
14.57/4.52	
14.57/4.52	(10)
14.57/4.52	YES
14.92/4.60	EOF