177.09/45.92	YES
177.65/46.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
177.65/46.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
177.65/46.04	
177.65/46.04	
177.65/46.04	Termination w.r.t. Q of the given QTRS could be proven:
177.65/46.04	
177.65/46.04	(0) QTRS
177.65/46.04	(1) DependencyPairsProof [EQUIVALENT, 34 ms]
177.65/46.04	(2) QDP
177.65/46.04	(3) MRRProof [EQUIVALENT, 79 ms]
177.65/46.04	(4) QDP
177.65/46.04	(5) QDPOrderProof [EQUIVALENT, 712 ms]
177.65/46.04	(6) QDP
177.65/46.04	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
177.65/46.04	(8) TRUE
177.65/46.04	
177.65/46.04	
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(0)
177.65/46.04	Obligation:
177.65/46.04	Q restricted rewrite system:
177.65/46.04	The TRS R consists of the following rules:
177.65/46.04	
177.65/46.04	   b(b(a(a(x1)))) -> a(a(a(b(x1))))
177.65/46.04	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
177.65/46.04	   a(b(a(b(x1)))) -> a(a(b(b(x1))))
177.65/46.04	
177.65/46.04	Q is empty.
177.65/46.04	
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(1) DependencyPairsProof (EQUIVALENT)
177.65/46.04	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(2)
177.65/46.04	Obligation:
177.65/46.04	Q DP problem:
177.65/46.04	The TRS P consists of the following rules:
177.65/46.04	
177.65/46.04	   B(b(a(a(x1)))) -> A(a(a(b(x1))))
177.65/46.04	   B(b(a(a(x1)))) -> A(a(b(x1)))
177.65/46.04	   B(b(a(a(x1)))) -> A(b(x1))
177.65/46.04	   B(b(a(a(x1)))) -> B(x1)
177.65/46.04	   A(a(a(b(x1)))) -> B(a(b(a(x1))))
177.65/46.04	   A(a(a(b(x1)))) -> A(b(a(x1)))
177.65/46.04	   A(a(a(b(x1)))) -> B(a(x1))
177.65/46.04	   A(a(a(b(x1)))) -> A(x1)
177.65/46.04	   A(b(a(b(x1)))) -> A(a(b(b(x1))))
177.65/46.04	   A(b(a(b(x1)))) -> A(b(b(x1)))
177.65/46.04	   A(b(a(b(x1)))) -> B(b(x1))
177.65/46.04	
177.65/46.04	The TRS R consists of the following rules:
177.65/46.04	
177.65/46.04	   b(b(a(a(x1)))) -> a(a(a(b(x1))))
177.65/46.04	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
177.65/46.04	   a(b(a(b(x1)))) -> a(a(b(b(x1))))
177.65/46.04	
177.65/46.04	Q is empty.
177.65/46.04	We have to consider all minimal (P,Q,R)-chains.
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(3) MRRProof (EQUIVALENT)
177.65/46.04	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
177.65/46.04	
177.65/46.04	Strictly oriented dependency pairs:
177.65/46.04	
177.65/46.04	   B(b(a(a(x1)))) -> A(a(b(x1)))
177.65/46.04	   B(b(a(a(x1)))) -> A(b(x1))
177.65/46.04	   B(b(a(a(x1)))) -> B(x1)
177.65/46.04	   A(a(a(b(x1)))) -> A(b(a(x1)))
177.65/46.04	   A(a(a(b(x1)))) -> B(a(x1))
177.65/46.04	   A(a(a(b(x1)))) -> A(x1)
177.65/46.04	   A(b(a(b(x1)))) -> A(b(b(x1)))
177.65/46.04	   A(b(a(b(x1)))) -> B(b(x1))
177.65/46.04	
177.65/46.04	
177.65/46.04	Used ordering: Polynomial interpretation [POLO]:
177.65/46.04	
177.65/46.04	   POL(A(x_1)) = 2*x_1
177.65/46.04	   POL(B(x_1)) = 2*x_1
177.65/46.04	   POL(a(x_1)) = 2 + x_1
177.65/46.04	   POL(b(x_1)) = 2 + x_1
177.65/46.04	
177.65/46.04	
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(4)
177.65/46.04	Obligation:
177.65/46.04	Q DP problem:
177.65/46.04	The TRS P consists of the following rules:
177.65/46.04	
177.65/46.04	   B(b(a(a(x1)))) -> A(a(a(b(x1))))
177.65/46.04	   A(a(a(b(x1)))) -> B(a(b(a(x1))))
177.65/46.04	   A(b(a(b(x1)))) -> A(a(b(b(x1))))
177.65/46.04	
177.65/46.04	The TRS R consists of the following rules:
177.65/46.04	
177.65/46.04	   b(b(a(a(x1)))) -> a(a(a(b(x1))))
177.65/46.04	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
177.65/46.04	   a(b(a(b(x1)))) -> a(a(b(b(x1))))
177.65/46.04	
177.65/46.04	Q is empty.
177.65/46.04	We have to consider all minimal (P,Q,R)-chains.
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(5) QDPOrderProof (EQUIVALENT)
177.65/46.04	We use the reduction pair processor [LPAR04,JAR06].
177.65/46.04	
177.65/46.04	
177.65/46.04	The following pairs can be oriented strictly and are deleted.
177.65/46.04	
177.65/46.04	   B(b(a(a(x1)))) -> A(a(a(b(x1))))
177.65/46.04	   A(b(a(b(x1)))) -> A(a(b(b(x1))))
177.65/46.04	The remaining pairs can at least be oriented weakly.
177.65/46.04	Used ordering:  Matrix interpretation [MATRO] to (N^5, +, *, >=, >) :
177.65/46.04	
177.65/46.04	   <<<
177.65/46.04	 POL(B(x_1)) =  	[[0]] 	 +  	[[0, 0, 1, 0, 0]] 	* 	x_1
177.65/46.04	>>>
177.65/46.04	
177.65/46.04	   <<<
177.65/46.04	 POL(b(x_1)) =  	[[0], [0], [0], [0], [0]] 	 +  	[[0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 1, 1, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 0]] 	* 	x_1
177.65/46.04	>>>
177.65/46.04	
177.65/46.04	   <<<
177.65/46.04	 POL(a(x_1)) =  	[[0], [0], [0], [1], [0]] 	 +  	[[0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 1, 1, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 0]] 	* 	x_1
177.65/46.04	>>>
177.65/46.04	
177.65/46.04	   <<<
177.65/46.04	 POL(A(x_1)) =  	[[0]] 	 +  	[[0, 0, 1, 0, 0]] 	* 	x_1
177.65/46.04	>>>
177.65/46.04	
177.65/46.04	
177.65/46.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
177.65/46.04	
177.65/46.04	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
177.65/46.04	   b(b(a(a(x1)))) -> a(a(a(b(x1))))
177.65/46.04	   a(b(a(b(x1)))) -> a(a(b(b(x1))))
177.65/46.04	
177.65/46.04	
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(6)
177.65/46.04	Obligation:
177.65/46.04	Q DP problem:
177.65/46.04	The TRS P consists of the following rules:
177.65/46.04	
177.65/46.04	   A(a(a(b(x1)))) -> B(a(b(a(x1))))
177.65/46.04	
177.65/46.04	The TRS R consists of the following rules:
177.65/46.04	
177.65/46.04	   b(b(a(a(x1)))) -> a(a(a(b(x1))))
177.65/46.04	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
177.65/46.04	   a(b(a(b(x1)))) -> a(a(b(b(x1))))
177.65/46.04	
177.65/46.04	Q is empty.
177.65/46.04	We have to consider all minimal (P,Q,R)-chains.
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(7) DependencyGraphProof (EQUIVALENT)
177.65/46.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
177.65/46.04	----------------------------------------
177.65/46.04	
177.65/46.04	(8)
177.65/46.04	TRUE
177.92/46.15	EOF