23.51/6.91	YES
23.78/7.01	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.78/7.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.78/7.01	
23.78/7.01	
23.78/7.01	Termination w.r.t. Q of the given QTRS could be proven:
23.78/7.01	
23.78/7.01	(0) QTRS
23.78/7.01	(1) QTRS Reverse [EQUIVALENT, 0 ms]
23.78/7.01	(2) QTRS
23.78/7.01	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
23.78/7.01	(4) QDP
23.78/7.01	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
23.78/7.01	(6) QDP
23.78/7.01	(7) QDPOrderProof [EQUIVALENT, 126 ms]
23.78/7.01	(8) QDP
23.78/7.01	(9) QDPOrderProof [EQUIVALENT, 0 ms]
23.78/7.01	(10) QDP
23.78/7.01	(11) PisEmptyProof [EQUIVALENT, 0 ms]
23.78/7.01	(12) YES
23.78/7.01	
23.78/7.01	
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(0)
23.78/7.01	Obligation:
23.78/7.01	Q restricted rewrite system:
23.78/7.01	The TRS R consists of the following rules:
23.78/7.01	
23.78/7.01	   a(x1) -> x1
23.78/7.01	   a(b(x1)) -> b(b(a(a(c(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	Q is empty.
23.78/7.01	
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(1) QTRS Reverse (EQUIVALENT)
23.78/7.01	We applied the QTRS Reverse Processor [REVERSE].
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(2)
23.78/7.01	Obligation:
23.78/7.01	Q restricted rewrite system:
23.78/7.01	The TRS R consists of the following rules:
23.78/7.01	
23.78/7.01	   a(x1) -> x1
23.78/7.01	   b(a(x1)) -> c(a(a(b(b(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	Q is empty.
23.78/7.01	
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(3) DependencyPairsProof (EQUIVALENT)
23.78/7.01	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(4)
23.78/7.01	Obligation:
23.78/7.01	Q DP problem:
23.78/7.01	The TRS P consists of the following rules:
23.78/7.01	
23.78/7.01	   B(a(x1)) -> C(a(a(b(b(x1)))))
23.78/7.01	   B(a(x1)) -> A(a(b(b(x1))))
23.78/7.01	   B(a(x1)) -> A(b(b(x1)))
23.78/7.01	   B(a(x1)) -> B(b(x1))
23.78/7.01	   B(a(x1)) -> B(x1)
23.78/7.01	
23.78/7.01	The TRS R consists of the following rules:
23.78/7.01	
23.78/7.01	   a(x1) -> x1
23.78/7.01	   b(a(x1)) -> c(a(a(b(b(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	Q is empty.
23.78/7.01	We have to consider all minimal (P,Q,R)-chains.
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(5) DependencyGraphProof (EQUIVALENT)
23.78/7.01	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(6)
23.78/7.01	Obligation:
23.78/7.01	Q DP problem:
23.78/7.01	The TRS P consists of the following rules:
23.78/7.01	
23.78/7.01	   B(a(x1)) -> B(x1)
23.78/7.01	   B(a(x1)) -> B(b(x1))
23.78/7.01	
23.78/7.01	The TRS R consists of the following rules:
23.78/7.01	
23.78/7.01	   a(x1) -> x1
23.78/7.01	   b(a(x1)) -> c(a(a(b(b(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	Q is empty.
23.78/7.01	We have to consider all minimal (P,Q,R)-chains.
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(7) QDPOrderProof (EQUIVALENT)
23.78/7.01	We use the reduction pair processor [LPAR04,JAR06].
23.78/7.01	
23.78/7.01	
23.78/7.01	The following pairs can be oriented strictly and are deleted.
23.78/7.01	
23.78/7.01	   B(a(x1)) -> B(x1)
23.78/7.01	The remaining pairs can at least be oriented weakly.
23.78/7.01	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 1A]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[1A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 1A]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, -I], [0A, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	
23.78/7.01	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.78/7.01	
23.78/7.01	   b(a(x1)) -> c(a(a(b(b(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   a(x1) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(8)
23.78/7.01	Obligation:
23.78/7.01	Q DP problem:
23.78/7.01	The TRS P consists of the following rules:
23.78/7.01	
23.78/7.01	   B(a(x1)) -> B(b(x1))
23.78/7.01	
23.78/7.01	The TRS R consists of the following rules:
23.78/7.01	
23.78/7.01	   a(x1) -> x1
23.78/7.01	   b(a(x1)) -> c(a(a(b(b(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	Q is empty.
23.78/7.01	We have to consider all minimal (P,Q,R)-chains.
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(9) QDPOrderProof (EQUIVALENT)
23.78/7.01	We use the reduction pair processor [LPAR04,JAR06].
23.78/7.01	
23.78/7.01	
23.78/7.01	The following pairs can be oriented strictly and are deleted.
23.78/7.01	
23.78/7.01	   B(a(x1)) -> B(b(x1))
23.78/7.01	The remaining pairs can at least be oriented weakly.
23.78/7.01	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [-I, -I, 0A]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	   <<<
23.78/7.01	 POL(c(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
23.78/7.01	>>>
23.78/7.01	
23.78/7.01	
23.78/7.01	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.78/7.01	
23.78/7.01	   b(a(x1)) -> c(a(a(b(b(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   a(x1) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(10)
23.78/7.01	Obligation:
23.78/7.01	Q DP problem:
23.78/7.01	P is empty.
23.78/7.01	The TRS R consists of the following rules:
23.78/7.01	
23.78/7.01	   a(x1) -> x1
23.78/7.01	   b(a(x1)) -> c(a(a(b(b(x1)))))
23.78/7.01	   b(b(x1)) -> x1
23.78/7.01	   c(c(x1)) -> x1
23.78/7.01	
23.78/7.01	Q is empty.
23.78/7.01	We have to consider all minimal (P,Q,R)-chains.
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(11) PisEmptyProof (EQUIVALENT)
23.78/7.01	The TRS P is empty. Hence, there is no (P,Q,R) chain.
23.78/7.01	----------------------------------------
23.78/7.01	
23.78/7.01	(12)
23.78/7.01	YES
24.09/7.12	EOF