26.96/7.77	YES
28.40/8.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
28.40/8.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
28.40/8.22	
28.40/8.22	
28.40/8.22	Termination w.r.t. Q of the given QTRS could be proven:
28.40/8.22	
28.40/8.22	(0) QTRS
28.40/8.22	(1) QTRS Reverse [EQUIVALENT, 0 ms]
28.40/8.22	(2) QTRS
28.40/8.22	(3) DependencyPairsProof [EQUIVALENT, 30 ms]
28.40/8.22	(4) QDP
28.40/8.22	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
28.40/8.22	(6) QDP
28.40/8.22	(7) QDPOrderProof [EQUIVALENT, 132 ms]
28.40/8.22	(8) QDP
28.40/8.22	(9) PisEmptyProof [EQUIVALENT, 0 ms]
28.40/8.22	(10) YES
28.40/8.22	
28.40/8.22	
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(0)
28.40/8.22	Obligation:
28.40/8.22	Q restricted rewrite system:
28.40/8.22	The TRS R consists of the following rules:
28.40/8.22	
28.40/8.22	   a(x1) -> x1
28.40/8.22	   a(b(x1)) -> b(b(a(a(c(x1)))))
28.40/8.22	   b(x1) -> x1
28.40/8.22	   c(b(c(x1))) -> x1
28.40/8.22	
28.40/8.22	Q is empty.
28.40/8.22	
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(1) QTRS Reverse (EQUIVALENT)
28.40/8.22	We applied the QTRS Reverse Processor [REVERSE].
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(2)
28.40/8.22	Obligation:
28.40/8.22	Q restricted rewrite system:
28.40/8.22	The TRS R consists of the following rules:
28.40/8.22	
28.40/8.22	   a(x1) -> x1
28.40/8.22	   b(a(x1)) -> c(a(a(b(b(x1)))))
28.40/8.22	   b(x1) -> x1
28.40/8.22	   c(b(c(x1))) -> x1
28.40/8.22	
28.40/8.22	Q is empty.
28.40/8.22	
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(3) DependencyPairsProof (EQUIVALENT)
28.40/8.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(4)
28.40/8.22	Obligation:
28.40/8.22	Q DP problem:
28.40/8.22	The TRS P consists of the following rules:
28.40/8.22	
28.40/8.22	   B(a(x1)) -> C(a(a(b(b(x1)))))
28.40/8.22	   B(a(x1)) -> A(a(b(b(x1))))
28.40/8.22	   B(a(x1)) -> A(b(b(x1)))
28.40/8.22	   B(a(x1)) -> B(b(x1))
28.40/8.22	   B(a(x1)) -> B(x1)
28.40/8.22	
28.40/8.22	The TRS R consists of the following rules:
28.40/8.22	
28.40/8.22	   a(x1) -> x1
28.40/8.22	   b(a(x1)) -> c(a(a(b(b(x1)))))
28.40/8.22	   b(x1) -> x1
28.40/8.22	   c(b(c(x1))) -> x1
28.40/8.22	
28.40/8.22	Q is empty.
28.40/8.22	We have to consider all minimal (P,Q,R)-chains.
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(5) DependencyGraphProof (EQUIVALENT)
28.40/8.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(6)
28.40/8.22	Obligation:
28.40/8.22	Q DP problem:
28.40/8.22	The TRS P consists of the following rules:
28.40/8.22	
28.40/8.22	   B(a(x1)) -> B(x1)
28.40/8.22	   B(a(x1)) -> B(b(x1))
28.40/8.22	
28.40/8.22	The TRS R consists of the following rules:
28.40/8.22	
28.40/8.22	   a(x1) -> x1
28.40/8.22	   b(a(x1)) -> c(a(a(b(b(x1)))))
28.40/8.22	   b(x1) -> x1
28.40/8.22	   c(b(c(x1))) -> x1
28.40/8.22	
28.40/8.22	Q is empty.
28.40/8.22	We have to consider all minimal (P,Q,R)-chains.
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(7) QDPOrderProof (EQUIVALENT)
28.40/8.22	We use the reduction pair processor [LPAR04,JAR06].
28.40/8.22	
28.40/8.22	
28.40/8.22	The following pairs can be oriented strictly and are deleted.
28.40/8.22	
28.40/8.22	   B(a(x1)) -> B(x1)
28.40/8.22	   B(a(x1)) -> B(b(x1))
28.40/8.22	The remaining pairs can at least be oriented weakly.
28.40/8.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
28.40/8.22	
28.40/8.22	   <<<
28.40/8.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
28.40/8.22	>>>
28.40/8.22	
28.40/8.22	   <<<
28.40/8.22	 POL(a(x_1)) =  	[[-I], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
28.40/8.22	>>>
28.40/8.22	
28.40/8.22	   <<<
28.40/8.22	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
28.40/8.22	>>>
28.40/8.22	
28.40/8.22	   <<<
28.40/8.22	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
28.40/8.22	>>>
28.40/8.22	
28.40/8.22	
28.40/8.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.40/8.22	
28.40/8.22	   b(a(x1)) -> c(a(a(b(b(x1)))))
28.40/8.22	   b(x1) -> x1
28.40/8.22	   a(x1) -> x1
28.40/8.22	   c(b(c(x1))) -> x1
28.40/8.22	
28.40/8.22	
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(8)
28.40/8.22	Obligation:
28.40/8.22	Q DP problem:
28.40/8.22	P is empty.
28.40/8.22	The TRS R consists of the following rules:
28.40/8.22	
28.40/8.22	   a(x1) -> x1
28.40/8.22	   b(a(x1)) -> c(a(a(b(b(x1)))))
28.40/8.22	   b(x1) -> x1
28.40/8.22	   c(b(c(x1))) -> x1
28.40/8.22	
28.40/8.22	Q is empty.
28.40/8.22	We have to consider all minimal (P,Q,R)-chains.
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(9) PisEmptyProof (EQUIVALENT)
28.40/8.22	The TRS P is empty. Hence, there is no (P,Q,R) chain.
28.40/8.22	----------------------------------------
28.40/8.22	
28.40/8.22	(10)
28.40/8.22	YES
28.54/8.40	EOF