42.72/11.76	YES
42.72/11.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
42.72/11.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
42.72/11.78	
42.72/11.78	
42.72/11.78	Termination w.r.t. Q of the given QTRS could be proven:
42.72/11.78	
42.72/11.78	(0) QTRS
42.72/11.78	(1) QTRS Reverse [EQUIVALENT, 0 ms]
42.72/11.78	(2) QTRS
42.72/11.78	(3) FlatCCProof [EQUIVALENT, 0 ms]
42.72/11.78	(4) QTRS
42.72/11.78	(5) RootLabelingProof [EQUIVALENT, 0 ms]
42.72/11.78	(6) QTRS
42.72/11.78	(7) QTRSRRRProof [EQUIVALENT, 30 ms]
42.72/11.78	(8) QTRS
42.72/11.78	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
42.72/11.78	(10) QDP
42.72/11.78	(11) QDPOrderProof [EQUIVALENT, 42 ms]
42.72/11.78	(12) QDP
42.72/11.78	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
42.72/11.78	(14) AND
42.72/11.78	    (15) QDP
42.72/11.78	        (16) QDPOrderProof [EQUIVALENT, 10 ms]
42.72/11.78	        (17) QDP
42.72/11.78	        (18) PisEmptyProof [EQUIVALENT, 0 ms]
42.72/11.78	        (19) YES
42.72/11.78	    (20) QDP
42.72/11.78	        (21) QDPOrderProof [EQUIVALENT, 15 ms]
42.72/11.78	        (22) QDP
42.72/11.78	        (23) QDPOrderProof [EQUIVALENT, 447 ms]
42.72/11.78	        (24) QDP
42.72/11.78	        (25) PisEmptyProof [EQUIVALENT, 0 ms]
42.72/11.78	        (26) YES
42.72/11.78	
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(0)
42.72/11.78	Obligation:
42.72/11.78	Q restricted rewrite system:
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   b(b(b(a(x1)))) -> a(b(a(a(x1))))
42.72/11.78	   a(a(a(b(x1)))) -> b(a(a(b(x1))))
42.72/11.78	   a(a(a(a(x1)))) -> a(a(b(b(x1))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(1) QTRS Reverse (EQUIVALENT)
42.72/11.78	We applied the QTRS Reverse Processor [REVERSE].
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(2)
42.72/11.78	Obligation:
42.72/11.78	Q restricted rewrite system:
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
42.72/11.78	   b(a(a(a(x1)))) -> b(a(a(b(x1))))
42.72/11.78	   a(a(a(a(x1)))) -> b(b(a(a(x1))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(3) FlatCCProof (EQUIVALENT)
42.72/11.78	We used flat context closure [ROOTLAB]
42.72/11.78	As Q is empty the flat context closure was sound AND complete.
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(4)
42.72/11.78	Obligation:
42.72/11.78	Q restricted rewrite system:
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
42.72/11.78	   b(a(a(a(x1)))) -> b(a(a(b(x1))))
42.72/11.78	   a(a(a(a(a(x1))))) -> a(b(b(a(a(x1)))))
42.72/11.78	   b(a(a(a(a(x1))))) -> b(b(b(a(a(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(5) RootLabelingProof (EQUIVALENT)
42.72/11.78	We used plain root labeling [ROOTLAB] with the following heuristic:
42.72/11.78	LabelAll: All function symbols get labeled
42.72/11.78	
42.72/11.78	As Q is empty the root labeling was sound AND complete.
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(6)
42.72/11.78	Obligation:
42.72/11.78	Q restricted rewrite system:
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(7) QTRSRRRProof (EQUIVALENT)
42.72/11.78	Used ordering:
42.72/11.78	Polynomial interpretation [POLO]:
42.72/11.78	
42.72/11.78	   POL(a_{a_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(a_{b_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(b_{a_1}(x_1)) = x_1
42.72/11.78	   POL(b_{b_1}(x_1)) = 1 + x_1
42.72/11.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
42.72/11.78	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	
42.72/11.78	
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(8)
42.72/11.78	Obligation:
42.72/11.78	Q restricted rewrite system:
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(9) DependencyPairsProof (EQUIVALENT)
42.72/11.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(10)
42.72/11.78	Obligation:
42.72/11.78	Q DP problem:
42.72/11.78	The TRS P consists of the following rules:
42.72/11.78	
42.72/11.78	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
42.72/11.78	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
42.72/11.78	
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	We have to consider all minimal (P,Q,R)-chains.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(11) QDPOrderProof (EQUIVALENT)
42.72/11.78	We use the reduction pair processor [LPAR04,JAR06].
42.72/11.78	
42.72/11.78	
42.72/11.78	The following pairs can be oriented strictly and are deleted.
42.72/11.78	
42.72/11.78	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
42.72/11.78	The remaining pairs can at least be oriented weakly.
42.72/11.78	Used ordering:  Polynomial interpretation [POLO]:
42.72/11.78	
42.72/11.78	   POL(A_{B_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(B_{A_1}(x_1)) = x_1
42.72/11.78	   POL(a_{a_1}(x_1)) = x_1
42.72/11.78	   POL(a_{b_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(b_{a_1}(x_1)) = x_1
42.72/11.78	   POL(b_{b_1}(x_1)) = x_1
42.72/11.78	
42.72/11.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.72/11.78	
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(12)
42.72/11.78	Obligation:
42.72/11.78	Q DP problem:
42.72/11.78	The TRS P consists of the following rules:
42.72/11.78	
42.72/11.78	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
42.72/11.78	
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	We have to consider all minimal (P,Q,R)-chains.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(13) DependencyGraphProof (EQUIVALENT)
42.72/11.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(14)
42.72/11.78	Complex Obligation (AND)
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(15)
42.72/11.78	Obligation:
42.72/11.78	Q DP problem:
42.72/11.78	The TRS P consists of the following rules:
42.72/11.78	
42.72/11.78	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
42.72/11.78	
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	We have to consider all minimal (P,Q,R)-chains.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(16) QDPOrderProof (EQUIVALENT)
42.72/11.78	We use the reduction pair processor [LPAR04,JAR06].
42.72/11.78	
42.72/11.78	
42.72/11.78	The following pairs can be oriented strictly and are deleted.
42.72/11.78	
42.72/11.78	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
42.72/11.78	The remaining pairs can at least be oriented weakly.
42.72/11.78	Used ordering:  Polynomial interpretation [POLO]:
42.72/11.78	
42.72/11.78	   POL(A_{B_1}(x_1)) = x_1
42.72/11.78	   POL(a_{a_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(a_{b_1}(x_1)) = x_1
42.72/11.78	   POL(b_{a_1}(x_1)) = x_1
42.72/11.78	   POL(b_{b_1}(x_1)) = 1 + x_1
42.72/11.78	
42.72/11.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.72/11.78	
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(17)
42.72/11.78	Obligation:
42.72/11.78	Q DP problem:
42.72/11.78	P is empty.
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	We have to consider all minimal (P,Q,R)-chains.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(18) PisEmptyProof (EQUIVALENT)
42.72/11.78	The TRS P is empty. Hence, there is no (P,Q,R) chain.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(19)
42.72/11.78	YES
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(20)
42.72/11.78	Obligation:
42.72/11.78	Q DP problem:
42.72/11.78	The TRS P consists of the following rules:
42.72/11.78	
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
42.72/11.78	
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	We have to consider all minimal (P,Q,R)-chains.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(21) QDPOrderProof (EQUIVALENT)
42.72/11.78	We use the reduction pair processor [LPAR04,JAR06].
42.72/11.78	
42.72/11.78	
42.72/11.78	The following pairs can be oriented strictly and are deleted.
42.72/11.78	
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
42.72/11.78	The remaining pairs can at least be oriented weakly.
42.72/11.78	Used ordering:  Polynomial interpretation [POLO]:
42.72/11.78	
42.72/11.78	   POL(B_{A_1}(x_1)) = x_1
42.72/11.78	   POL(a_{a_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(a_{b_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(b_{a_1}(x_1)) = 1 + x_1
42.72/11.78	   POL(b_{b_1}(x_1)) = 1 + x_1
42.72/11.78	
42.72/11.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(22)
42.72/11.78	Obligation:
42.72/11.78	Q DP problem:
42.72/11.78	The TRS P consists of the following rules:
42.72/11.78	
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	We have to consider all minimal (P,Q,R)-chains.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(23) QDPOrderProof (EQUIVALENT)
42.72/11.78	We use the reduction pair processor [LPAR04,JAR06].
42.72/11.78	
42.72/11.78	
42.72/11.78	The following pairs can be oriented strictly and are deleted.
42.72/11.78	
42.72/11.78	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	The remaining pairs can at least be oriented weakly.
42.72/11.78	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
42.72/11.78	
42.72/11.78	   <<<
42.72/11.78	 POL(B_{A_1}(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
42.72/11.78	>>>
42.72/11.78	
42.72/11.78	   <<<
42.72/11.78	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, -I, 0A], [0A, -I, -I]] 	* 	x_1
42.72/11.78	>>>
42.72/11.78	
42.72/11.78	   <<<
42.72/11.78	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [1A, 0A, 1A]] 	* 	x_1
42.72/11.78	>>>
42.72/11.78	
42.72/11.78	   <<<
42.72/11.78	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, 0A], [-I, 0A, -I]] 	* 	x_1
42.72/11.78	>>>
42.72/11.78	
42.72/11.78	   <<<
42.72/11.78	 POL(b_{a_1}(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
42.72/11.78	>>>
42.72/11.78	
42.72/11.78	
42.72/11.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(24)
42.72/11.78	Obligation:
42.72/11.78	Q DP problem:
42.72/11.78	P is empty.
42.72/11.78	The TRS R consists of the following rules:
42.72/11.78	
42.72/11.78	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
42.72/11.78	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
42.72/11.78	
42.72/11.78	Q is empty.
42.72/11.78	We have to consider all minimal (P,Q,R)-chains.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(25) PisEmptyProof (EQUIVALENT)
42.72/11.78	The TRS P is empty. Hence, there is no (P,Q,R) chain.
42.72/11.78	----------------------------------------
42.72/11.78	
42.72/11.78	(26)
42.72/11.78	YES
43.00/11.91	EOF