32.23/9.15	YES
32.23/9.15	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
32.23/9.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.23/9.15	
32.23/9.15	
32.23/9.15	Termination w.r.t. Q of the given QTRS could be proven:
32.23/9.15	
32.23/9.15	(0) QTRS
32.23/9.15	(1) QTRS Reverse [EQUIVALENT, 0 ms]
32.23/9.15	(2) QTRS
32.23/9.15	(3) DependencyPairsProof [EQUIVALENT, 20 ms]
32.23/9.15	(4) QDP
32.23/9.15	(5) QDPOrderProof [EQUIVALENT, 24 ms]
32.23/9.15	(6) QDP
32.23/9.15	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
32.23/9.15	(8) AND
32.23/9.15	    (9) QDP
32.23/9.15	        (10) QDPOrderProof [EQUIVALENT, 1206 ms]
32.23/9.15	        (11) QDP
32.23/9.15	        (12) PisEmptyProof [EQUIVALENT, 0 ms]
32.23/9.15	        (13) YES
32.23/9.15	    (14) QDP
32.23/9.15	        (15) QDPOrderProof [EQUIVALENT, 0 ms]
32.23/9.15	        (16) QDP
32.23/9.15	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
32.23/9.15	        (18) YES
32.23/9.15	
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(0)
32.23/9.15	Obligation:
32.23/9.15	Q restricted rewrite system:
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> a(a(b(b(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> b(a(a(a(x1))))
32.23/9.15	   a(b(a(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(1) QTRS Reverse (EQUIVALENT)
32.23/9.15	We applied the QTRS Reverse Processor [REVERSE].
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(2)
32.23/9.15	Obligation:
32.23/9.15	Q restricted rewrite system:
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(3) DependencyPairsProof (EQUIVALENT)
32.23/9.15	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(4)
32.23/9.15	Obligation:
32.23/9.15	Q DP problem:
32.23/9.15	The TRS P consists of the following rules:
32.23/9.15	
32.23/9.15	   B(b(b(b(x1)))) -> B(b(a(a(x1))))
32.23/9.15	   B(b(b(b(x1)))) -> B(a(a(x1)))
32.23/9.15	   B(b(b(b(x1)))) -> A(a(x1))
32.23/9.15	   B(b(b(b(x1)))) -> A(x1)
32.23/9.15	   A(a(a(a(x1)))) -> A(a(a(b(x1))))
32.23/9.15	   A(a(a(a(x1)))) -> A(a(b(x1)))
32.23/9.15	   A(a(a(a(x1)))) -> A(b(x1))
32.23/9.15	   A(a(a(a(x1)))) -> B(x1)
32.23/9.15	   B(a(b(a(x1)))) -> A(a(b(b(x1))))
32.23/9.15	   B(a(b(a(x1)))) -> A(b(b(x1)))
32.23/9.15	   B(a(b(a(x1)))) -> B(b(x1))
32.23/9.15	   B(a(b(a(x1)))) -> B(x1)
32.23/9.15	
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	We have to consider all minimal (P,Q,R)-chains.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(5) QDPOrderProof (EQUIVALENT)
32.23/9.15	We use the reduction pair processor [LPAR04,JAR06].
32.23/9.15	
32.23/9.15	
32.23/9.15	The following pairs can be oriented strictly and are deleted.
32.23/9.15	
32.23/9.15	   B(b(b(b(x1)))) -> B(a(a(x1)))
32.23/9.15	   B(b(b(b(x1)))) -> A(a(x1))
32.23/9.15	   B(b(b(b(x1)))) -> A(x1)
32.23/9.15	   A(a(a(a(x1)))) -> A(a(b(x1)))
32.23/9.15	   A(a(a(a(x1)))) -> A(b(x1))
32.23/9.15	   A(a(a(a(x1)))) -> B(x1)
32.23/9.15	   B(a(b(a(x1)))) -> A(b(b(x1)))
32.23/9.15	   B(a(b(a(x1)))) -> B(b(x1))
32.23/9.15	   B(a(b(a(x1)))) -> B(x1)
32.23/9.15	The remaining pairs can at least be oriented weakly.
32.23/9.15	Used ordering:  Polynomial interpretation [POLO]:
32.23/9.15	
32.23/9.15	   POL(A(x_1)) = 1 + x_1
32.23/9.15	   POL(B(x_1)) = 1 + x_1
32.23/9.15	   POL(a(x_1)) = 1 + x_1
32.23/9.15	   POL(b(x_1)) = 1 + x_1
32.23/9.15	
32.23/9.15	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.23/9.15	
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(6)
32.23/9.15	Obligation:
32.23/9.15	Q DP problem:
32.23/9.15	The TRS P consists of the following rules:
32.23/9.15	
32.23/9.15	   B(b(b(b(x1)))) -> B(b(a(a(x1))))
32.23/9.15	   A(a(a(a(x1)))) -> A(a(a(b(x1))))
32.23/9.15	   B(a(b(a(x1)))) -> A(a(b(b(x1))))
32.23/9.15	
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	We have to consider all minimal (P,Q,R)-chains.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(7) DependencyGraphProof (EQUIVALENT)
32.23/9.15	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(8)
32.23/9.15	Complex Obligation (AND)
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(9)
32.23/9.15	Obligation:
32.23/9.15	Q DP problem:
32.23/9.15	The TRS P consists of the following rules:
32.23/9.15	
32.23/9.15	   A(a(a(a(x1)))) -> A(a(a(b(x1))))
32.23/9.15	
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	We have to consider all minimal (P,Q,R)-chains.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(10) QDPOrderProof (EQUIVALENT)
32.23/9.15	We use the reduction pair processor [LPAR04,JAR06].
32.23/9.15	
32.23/9.15	
32.23/9.15	The following pairs can be oriented strictly and are deleted.
32.23/9.15	
32.23/9.15	   A(a(a(a(x1)))) -> A(a(a(b(x1))))
32.23/9.15	The remaining pairs can at least be oriented weakly.
32.23/9.15	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.23/9.15	
32.23/9.15	   <<<
32.23/9.15	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
32.23/9.15	>>>
32.23/9.15	
32.23/9.15	   <<<
32.23/9.15	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 1A, -I], [-I, -I, 0A], [-I, -I, -I]] 	* 	x_1
32.23/9.15	>>>
32.23/9.15	
32.23/9.15	   <<<
32.23/9.15	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [1A, 0A, -I]] 	* 	x_1
32.23/9.15	>>>
32.23/9.15	
32.23/9.15	
32.23/9.15	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(11)
32.23/9.15	Obligation:
32.23/9.15	Q DP problem:
32.23/9.15	P is empty.
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	We have to consider all minimal (P,Q,R)-chains.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(12) PisEmptyProof (EQUIVALENT)
32.23/9.15	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(13)
32.23/9.15	YES
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(14)
32.23/9.15	Obligation:
32.23/9.15	Q DP problem:
32.23/9.15	The TRS P consists of the following rules:
32.23/9.15	
32.23/9.15	   B(b(b(b(x1)))) -> B(b(a(a(x1))))
32.23/9.15	
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	We have to consider all minimal (P,Q,R)-chains.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(15) QDPOrderProof (EQUIVALENT)
32.23/9.15	We use the reduction pair processor [LPAR04,JAR06].
32.23/9.15	
32.23/9.15	
32.23/9.15	The following pairs can be oriented strictly and are deleted.
32.23/9.15	
32.23/9.15	   B(b(b(b(x1)))) -> B(b(a(a(x1))))
32.23/9.15	The remaining pairs can at least be oriented weakly.
32.23/9.15	Used ordering:  Polynomial interpretation [POLO]:
32.23/9.15	
32.23/9.15	   POL(B(x_1)) = x_1
32.23/9.15	   POL(a(x_1)) = 0
32.23/9.15	   POL(b(x_1)) = 1 + x_1
32.23/9.15	
32.23/9.15	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.23/9.15	
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(16)
32.23/9.15	Obligation:
32.23/9.15	Q DP problem:
32.23/9.15	P is empty.
32.23/9.15	The TRS R consists of the following rules:
32.23/9.15	
32.23/9.15	   b(b(b(b(x1)))) -> b(b(a(a(x1))))
32.23/9.15	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
32.23/9.15	   b(a(b(a(x1)))) -> a(a(b(b(x1))))
32.23/9.15	
32.23/9.15	Q is empty.
32.23/9.15	We have to consider all minimal (P,Q,R)-chains.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(17) PisEmptyProof (EQUIVALENT)
32.23/9.15	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.23/9.15	----------------------------------------
32.23/9.15	
32.23/9.15	(18)
32.23/9.15	YES
32.54/9.28	EOF