94.79/25.11	YES
94.79/25.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
94.79/25.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
94.79/25.13	
94.79/25.13	
94.79/25.13	Termination w.r.t. Q of the given QTRS could be proven:
94.79/25.13	
94.79/25.13	(0) QTRS
94.79/25.13	(1) DependencyPairsProof [EQUIVALENT, 27 ms]
94.79/25.13	(2) QDP
94.79/25.13	(3) QDPOrderProof [EQUIVALENT, 36 ms]
94.79/25.13	(4) QDP
94.79/25.13	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
94.79/25.13	(6) AND
94.79/25.13	    (7) QDP
94.79/25.13	        (8) QDPOrderProof [EQUIVALENT, 6621 ms]
94.79/25.13	        (9) QDP
94.79/25.13	        (10) PisEmptyProof [EQUIVALENT, 0 ms]
94.79/25.13	        (11) YES
94.79/25.13	    (12) QDP
94.79/25.13	        (13) QDPOrderProof [EQUIVALENT, 0 ms]
94.79/25.13	        (14) QDP
94.79/25.13	        (15) PisEmptyProof [EQUIVALENT, 0 ms]
94.79/25.13	        (16) YES
94.79/25.13	
94.79/25.13	
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(0)
94.79/25.13	Obligation:
94.79/25.13	Q restricted rewrite system:
94.79/25.13	The TRS R consists of the following rules:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	Q is empty.
94.79/25.13	
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(1) DependencyPairsProof (EQUIVALENT)
94.79/25.13	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(2)
94.79/25.13	Obligation:
94.79/25.13	Q DP problem:
94.79/25.13	The TRS P consists of the following rules:
94.79/25.13	
94.79/25.13	   B(b(b(a(x1)))) -> B(b(a(a(x1))))
94.79/25.13	   B(b(b(a(x1)))) -> B(a(a(x1)))
94.79/25.13	   B(a(b(a(x1)))) -> B(a(b(b(x1))))
94.79/25.13	   B(a(b(a(x1)))) -> B(b(x1))
94.79/25.13	   B(a(b(a(x1)))) -> B(x1)
94.79/25.13	   B(a(a(a(x1)))) -> B(x1)
94.79/25.13	
94.79/25.13	The TRS R consists of the following rules:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	Q is empty.
94.79/25.13	We have to consider all minimal (P,Q,R)-chains.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(3) QDPOrderProof (EQUIVALENT)
94.79/25.13	We use the reduction pair processor [LPAR04,JAR06].
94.79/25.13	
94.79/25.13	
94.79/25.13	The following pairs can be oriented strictly and are deleted.
94.79/25.13	
94.79/25.13	   B(b(b(a(x1)))) -> B(a(a(x1)))
94.79/25.13	   B(a(b(a(x1)))) -> B(b(x1))
94.79/25.13	   B(a(b(a(x1)))) -> B(x1)
94.79/25.13	   B(a(a(a(x1)))) -> B(x1)
94.79/25.13	The remaining pairs can at least be oriented weakly.
94.79/25.13	Used ordering:  Polynomial interpretation [POLO]:
94.79/25.13	
94.79/25.13	   POL(B(x_1)) = x_1
94.79/25.13	   POL(a(x_1)) = 1 + x_1
94.79/25.13	   POL(b(x_1)) = 1 + x_1
94.79/25.13	
94.79/25.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
94.79/25.13	
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	
94.79/25.13	
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(4)
94.79/25.13	Obligation:
94.79/25.13	Q DP problem:
94.79/25.13	The TRS P consists of the following rules:
94.79/25.13	
94.79/25.13	   B(b(b(a(x1)))) -> B(b(a(a(x1))))
94.79/25.13	   B(a(b(a(x1)))) -> B(a(b(b(x1))))
94.79/25.13	
94.79/25.13	The TRS R consists of the following rules:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	Q is empty.
94.79/25.13	We have to consider all minimal (P,Q,R)-chains.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(5) DependencyGraphProof (EQUIVALENT)
94.79/25.13	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(6)
94.79/25.13	Complex Obligation (AND)
94.79/25.13	
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(7)
94.79/25.13	Obligation:
94.79/25.13	Q DP problem:
94.79/25.13	The TRS P consists of the following rules:
94.79/25.13	
94.79/25.13	   B(a(b(a(x1)))) -> B(a(b(b(x1))))
94.79/25.13	
94.79/25.13	The TRS R consists of the following rules:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	Q is empty.
94.79/25.13	We have to consider all minimal (P,Q,R)-chains.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(8) QDPOrderProof (EQUIVALENT)
94.79/25.13	We use the reduction pair processor [LPAR04,JAR06].
94.79/25.13	
94.79/25.13	
94.79/25.13	The following pairs can be oriented strictly and are deleted.
94.79/25.13	
94.79/25.13	   B(a(b(a(x1)))) -> B(a(b(b(x1))))
94.79/25.13	The remaining pairs can at least be oriented weakly.
94.79/25.13	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
94.79/25.13	
94.79/25.13	   <<<
94.79/25.13	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
94.79/25.13	>>>
94.79/25.13	
94.79/25.13	   <<<
94.79/25.13	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 1A, -I], [-I, -I, 1A], [1A, -I, -I]] 	* 	x_1
94.79/25.13	>>>
94.79/25.13	
94.79/25.13	   <<<
94.79/25.13	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, 1A, 1A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
94.79/25.13	>>>
94.79/25.13	
94.79/25.13	
94.79/25.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(9)
94.79/25.13	Obligation:
94.79/25.13	Q DP problem:
94.79/25.13	P is empty.
94.79/25.13	The TRS R consists of the following rules:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	Q is empty.
94.79/25.13	We have to consider all minimal (P,Q,R)-chains.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(10) PisEmptyProof (EQUIVALENT)
94.79/25.13	The TRS P is empty. Hence, there is no (P,Q,R) chain.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(11)
94.79/25.13	YES
94.79/25.13	
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(12)
94.79/25.13	Obligation:
94.79/25.13	Q DP problem:
94.79/25.13	The TRS P consists of the following rules:
94.79/25.13	
94.79/25.13	   B(b(b(a(x1)))) -> B(b(a(a(x1))))
94.79/25.13	
94.79/25.13	The TRS R consists of the following rules:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	Q is empty.
94.79/25.13	We have to consider all minimal (P,Q,R)-chains.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(13) QDPOrderProof (EQUIVALENT)
94.79/25.13	We use the reduction pair processor [LPAR04,JAR06].
94.79/25.13	
94.79/25.13	
94.79/25.13	The following pairs can be oriented strictly and are deleted.
94.79/25.13	
94.79/25.13	   B(b(b(a(x1)))) -> B(b(a(a(x1))))
94.79/25.13	The remaining pairs can at least be oriented weakly.
94.79/25.13	Used ordering:  Polynomial interpretation [POLO]:
94.79/25.13	
94.79/25.13	   POL(B(x_1)) = x_1
94.79/25.13	   POL(a(x_1)) = 1
94.79/25.13	   POL(b(x_1)) = 1 + x_1
94.79/25.13	
94.79/25.13	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
94.79/25.13	
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(14)
94.79/25.13	Obligation:
94.79/25.13	Q DP problem:
94.79/25.13	P is empty.
94.79/25.13	The TRS R consists of the following rules:
94.79/25.13	
94.79/25.13	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
94.79/25.13	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
94.79/25.13	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
94.79/25.13	
94.79/25.13	Q is empty.
94.79/25.13	We have to consider all minimal (P,Q,R)-chains.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(15) PisEmptyProof (EQUIVALENT)
94.79/25.13	The TRS P is empty. Hence, there is no (P,Q,R) chain.
94.79/25.13	----------------------------------------
94.79/25.13	
94.79/25.13	(16)
94.79/25.13	YES
95.14/28.48	EOF