18.63/5.83	YES
19.37/5.92	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.37/5.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.37/5.92	
19.37/5.92	
19.37/5.92	Termination w.r.t. Q of the given QTRS could be proven:
19.37/5.92	
19.37/5.92	(0) QTRS
19.37/5.92	(1) QTRS Reverse [EQUIVALENT, 0 ms]
19.37/5.92	(2) QTRS
19.37/5.92	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
19.37/5.92	(4) QDP
19.37/5.92	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
19.37/5.92	(6) QDP
19.37/5.92	(7) QDPOrderProof [EQUIVALENT, 86 ms]
19.37/5.92	(8) QDP
19.37/5.92	(9) PisEmptyProof [EQUIVALENT, 0 ms]
19.37/5.92	(10) YES
19.37/5.92	
19.37/5.92	
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(0)
19.37/5.92	Obligation:
19.37/5.92	Q restricted rewrite system:
19.37/5.92	The TRS R consists of the following rules:
19.37/5.92	
19.37/5.92	   a(x1) -> x1
19.37/5.92	   a(x1) -> b(x1)
19.37/5.92	   a(b(c(x1))) -> c(c(a(b(a(x1)))))
19.37/5.92	   b(x1) -> x1
19.37/5.92	
19.37/5.92	Q is empty.
19.37/5.92	
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(1) QTRS Reverse (EQUIVALENT)
19.37/5.92	We applied the QTRS Reverse Processor [REVERSE].
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(2)
19.37/5.92	Obligation:
19.37/5.92	Q restricted rewrite system:
19.37/5.92	The TRS R consists of the following rules:
19.37/5.92	
19.37/5.92	   a(x1) -> x1
19.37/5.92	   a(x1) -> b(x1)
19.37/5.92	   c(b(a(x1))) -> a(b(a(c(c(x1)))))
19.37/5.92	   b(x1) -> x1
19.37/5.92	
19.37/5.92	Q is empty.
19.37/5.92	
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(3) DependencyPairsProof (EQUIVALENT)
19.37/5.92	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(4)
19.37/5.92	Obligation:
19.37/5.92	Q DP problem:
19.37/5.92	The TRS P consists of the following rules:
19.37/5.92	
19.37/5.92	   A(x1) -> B(x1)
19.37/5.92	   C(b(a(x1))) -> A(b(a(c(c(x1)))))
19.37/5.92	   C(b(a(x1))) -> B(a(c(c(x1))))
19.37/5.92	   C(b(a(x1))) -> A(c(c(x1)))
19.37/5.92	   C(b(a(x1))) -> C(c(x1))
19.37/5.92	   C(b(a(x1))) -> C(x1)
19.37/5.92	
19.37/5.92	The TRS R consists of the following rules:
19.37/5.92	
19.37/5.92	   a(x1) -> x1
19.37/5.92	   a(x1) -> b(x1)
19.37/5.92	   c(b(a(x1))) -> a(b(a(c(c(x1)))))
19.37/5.92	   b(x1) -> x1
19.37/5.92	
19.37/5.92	Q is empty.
19.37/5.92	We have to consider all minimal (P,Q,R)-chains.
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(5) DependencyGraphProof (EQUIVALENT)
19.37/5.92	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(6)
19.37/5.92	Obligation:
19.37/5.92	Q DP problem:
19.37/5.92	The TRS P consists of the following rules:
19.37/5.92	
19.37/5.92	   C(b(a(x1))) -> C(x1)
19.37/5.92	   C(b(a(x1))) -> C(c(x1))
19.37/5.92	
19.37/5.92	The TRS R consists of the following rules:
19.37/5.92	
19.37/5.92	   a(x1) -> x1
19.37/5.92	   a(x1) -> b(x1)
19.37/5.92	   c(b(a(x1))) -> a(b(a(c(c(x1)))))
19.37/5.92	   b(x1) -> x1
19.37/5.92	
19.37/5.92	Q is empty.
19.37/5.92	We have to consider all minimal (P,Q,R)-chains.
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(7) QDPOrderProof (EQUIVALENT)
19.37/5.92	We use the reduction pair processor [LPAR04,JAR06].
19.37/5.92	
19.37/5.92	
19.37/5.92	The following pairs can be oriented strictly and are deleted.
19.37/5.92	
19.37/5.92	   C(b(a(x1))) -> C(x1)
19.37/5.92	   C(b(a(x1))) -> C(c(x1))
19.37/5.92	The remaining pairs can at least be oriented weakly.
19.37/5.92	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
19.37/5.92	
19.37/5.92	   <<<
19.37/5.92	 POL(C(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
19.37/5.92	>>>
19.37/5.92	
19.37/5.92	   <<<
19.37/5.92	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
19.37/5.92	>>>
19.37/5.92	
19.37/5.92	   <<<
19.37/5.92	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
19.37/5.92	>>>
19.37/5.92	
19.37/5.92	   <<<
19.37/5.92	 POL(c(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, -I], [-I, 0A, -I]] 	* 	x_1
19.37/5.92	>>>
19.37/5.92	
19.37/5.92	
19.37/5.92	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.37/5.92	
19.37/5.92	   c(b(a(x1))) -> a(b(a(c(c(x1)))))
19.37/5.92	   a(x1) -> x1
19.37/5.92	   a(x1) -> b(x1)
19.37/5.92	   b(x1) -> x1
19.37/5.92	
19.37/5.92	
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(8)
19.37/5.92	Obligation:
19.37/5.92	Q DP problem:
19.37/5.92	P is empty.
19.37/5.92	The TRS R consists of the following rules:
19.37/5.92	
19.37/5.92	   a(x1) -> x1
19.37/5.92	   a(x1) -> b(x1)
19.37/5.92	   c(b(a(x1))) -> a(b(a(c(c(x1)))))
19.37/5.92	   b(x1) -> x1
19.37/5.92	
19.37/5.92	Q is empty.
19.37/5.92	We have to consider all minimal (P,Q,R)-chains.
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(9) PisEmptyProof (EQUIVALENT)
19.37/5.92	The TRS P is empty. Hence, there is no (P,Q,R) chain.
19.37/5.92	----------------------------------------
19.37/5.92	
19.37/5.92	(10)
19.37/5.92	YES
19.60/6.07	EOF