21.77/6.49	YES
21.88/6.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
21.88/6.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.88/6.53	
21.88/6.53	
21.88/6.53	Termination w.r.t. Q of the given QTRS could be proven:
21.88/6.53	
21.88/6.53	(0) QTRS
21.88/6.53	(1) FlatCCProof [EQUIVALENT, 0 ms]
21.88/6.53	(2) QTRS
21.88/6.53	(3) RootLabelingProof [EQUIVALENT, 0 ms]
21.88/6.53	(4) QTRS
21.88/6.53	(5) QTRSRRRProof [EQUIVALENT, 15 ms]
21.88/6.53	(6) QTRS
21.88/6.53	(7) DependencyPairsProof [EQUIVALENT, 19 ms]
21.88/6.53	(8) QDP
21.88/6.53	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
21.88/6.53	(10) AND
21.88/6.53	    (11) QDP
21.88/6.53	        (12) QDPOrderProof [EQUIVALENT, 85 ms]
21.88/6.53	        (13) QDP
21.88/6.53	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
21.88/6.53	        (15) YES
21.88/6.53	    (16) QDP
21.88/6.53	        (17) UsableRulesProof [EQUIVALENT, 2 ms]
21.88/6.53	        (18) QDP
21.88/6.53	        (19) MNOCProof [EQUIVALENT, 0 ms]
21.88/6.53	        (20) QDP
21.88/6.53	        (21) QDPOrderProof [EQUIVALENT, 5 ms]
21.88/6.53	        (22) QDP
21.88/6.53	        (23) PisEmptyProof [EQUIVALENT, 0 ms]
21.88/6.53	        (24) YES
21.88/6.53	    (25) QDP
21.88/6.53	        (26) QDPOrderProof [EQUIVALENT, 95 ms]
21.88/6.53	        (27) QDP
21.88/6.53	        (28) PisEmptyProof [EQUIVALENT, 0 ms]
21.88/6.53	        (29) YES
21.88/6.53	
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(0)
21.88/6.53	Obligation:
21.88/6.53	Q restricted rewrite system:
21.88/6.53	The TRS R consists of the following rules:
21.88/6.53	
21.88/6.53	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
21.88/6.53	   a(a(b(b(x1)))) -> a(a(a(b(x1))))
21.88/6.53	   a(a(a(b(x1)))) -> b(b(a(b(x1))))
21.88/6.53	   b(b(b(a(x1)))) -> b(a(b(b(x1))))
21.88/6.53	
21.88/6.53	Q is empty.
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(1) FlatCCProof (EQUIVALENT)
21.88/6.53	We used flat context closure [ROOTLAB]
21.88/6.53	As Q is empty the flat context closure was sound AND complete.
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(2)
21.88/6.53	Obligation:
21.88/6.53	Q restricted rewrite system:
21.88/6.53	The TRS R consists of the following rules:
21.88/6.53	
21.88/6.53	   a(a(b(b(x1)))) -> a(a(a(b(x1))))
21.88/6.53	   b(b(b(a(x1)))) -> b(a(b(b(x1))))
21.88/6.53	   a(a(a(a(b(x1))))) -> a(b(a(b(a(x1)))))
21.88/6.53	   b(a(a(a(b(x1))))) -> b(b(a(b(a(x1)))))
21.88/6.53	   a(a(a(a(b(x1))))) -> a(b(b(a(b(x1)))))
21.88/6.53	   b(a(a(a(b(x1))))) -> b(b(b(a(b(x1)))))
21.88/6.53	
21.88/6.53	Q is empty.
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(3) RootLabelingProof (EQUIVALENT)
21.88/6.53	We used plain root labeling [ROOTLAB] with the following heuristic:
21.88/6.53	LabelAll: All function symbols get labeled
21.88/6.53	
21.88/6.53	As Q is empty the root labeling was sound AND complete.
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(4)
21.88/6.53	Obligation:
21.88/6.53	Q restricted rewrite system:
21.88/6.53	The TRS R consists of the following rules:
21.88/6.53	
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.88/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
21.88/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.88/6.53	
21.88/6.53	Q is empty.
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(5) QTRSRRRProof (EQUIVALENT)
21.88/6.53	Used ordering:
21.88/6.53	Polynomial interpretation [POLO]:
21.88/6.53	
21.88/6.53	   POL(a_{a_1}(x_1)) = 1 + x_1
21.88/6.53	   POL(a_{b_1}(x_1)) = x_1
21.88/6.53	   POL(b_{a_1}(x_1)) = 1 + x_1
21.88/6.53	   POL(b_{b_1}(x_1)) = 1 + x_1
21.88/6.53	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
21.88/6.53	
21.88/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.88/6.53	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.88/6.53	
21.88/6.53	
21.88/6.53	
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(6)
21.88/6.53	Obligation:
21.88/6.53	Q restricted rewrite system:
21.88/6.53	The TRS R consists of the following rules:
21.88/6.53	
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.88/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.88/6.53	
21.88/6.53	Q is empty.
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(7) DependencyPairsProof (EQUIVALENT)
21.88/6.53	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(8)
21.88/6.53	Obligation:
21.88/6.53	Q DP problem:
21.88/6.53	The TRS P consists of the following rules:
21.88/6.53	
21.88/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.88/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
21.88/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.88/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
21.88/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.88/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
21.88/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.88/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
21.88/6.53	
21.88/6.53	The TRS R consists of the following rules:
21.88/6.53	
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.88/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.88/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.88/6.53	
21.88/6.53	Q is empty.
21.88/6.53	We have to consider all minimal (P,Q,R)-chains.
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(9) DependencyGraphProof (EQUIVALENT)
21.88/6.53	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 12 less nodes.
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(10)
21.88/6.53	Complex Obligation (AND)
21.88/6.53	
21.88/6.53	----------------------------------------
21.88/6.53	
21.88/6.53	(11)
21.88/6.53	Obligation:
21.88/6.53	Q DP problem:
21.88/6.53	The TRS P consists of the following rules:
21.88/6.53	
21.88/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
21.88/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
21.88/6.53	
21.88/6.53	The TRS R consists of the following rules:
21.88/6.53	
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.88/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.88/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.92/6.53	
21.92/6.53	Q is empty.
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(12) QDPOrderProof (EQUIVALENT)
21.92/6.53	We use the reduction pair processor [LPAR04,JAR06].
21.92/6.53	
21.92/6.53	
21.92/6.53	The following pairs can be oriented strictly and are deleted.
21.92/6.53	
21.92/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
21.92/6.53	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
21.92/6.53	The remaining pairs can at least be oriented weakly.
21.92/6.53	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
21.92/6.53	
21.92/6.53	POL( A_{A_1}_1(x_1) ) = 2x_1
21.92/6.53	POL( a_{b_1}_1(x_1) ) = 2x_1 + 2
21.92/6.53	POL( b_{b_1}_1(x_1) ) = 2x_1 + 2
21.92/6.53	POL( b_{a_1}_1(x_1) ) = 2x_1 + 2
21.92/6.53	POL( a_{a_1}_1(x_1) ) = 2x_1 + 2
21.92/6.53	
21.92/6.53	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.92/6.53	
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.92/6.53	
21.92/6.53	
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(13)
21.92/6.53	Obligation:
21.92/6.53	Q DP problem:
21.92/6.53	P is empty.
21.92/6.53	The TRS R consists of the following rules:
21.92/6.53	
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.92/6.53	
21.92/6.53	Q is empty.
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(14) PisEmptyProof (EQUIVALENT)
21.92/6.53	The TRS P is empty. Hence, there is no (P,Q,R) chain.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(15)
21.92/6.53	YES
21.92/6.53	
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(16)
21.92/6.53	Obligation:
21.92/6.53	Q DP problem:
21.92/6.53	The TRS P consists of the following rules:
21.92/6.53	
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
21.92/6.53	
21.92/6.53	The TRS R consists of the following rules:
21.92/6.53	
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.92/6.53	
21.92/6.53	Q is empty.
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(17) UsableRulesProof (EQUIVALENT)
21.92/6.53	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(18)
21.92/6.53	Obligation:
21.92/6.53	Q DP problem:
21.92/6.53	The TRS P consists of the following rules:
21.92/6.53	
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
21.92/6.53	
21.92/6.53	The TRS R consists of the following rules:
21.92/6.53	
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	
21.92/6.53	Q is empty.
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(19) MNOCProof (EQUIVALENT)
21.92/6.53	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(20)
21.92/6.53	Obligation:
21.92/6.53	Q DP problem:
21.92/6.53	The TRS P consists of the following rules:
21.92/6.53	
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
21.92/6.53	
21.92/6.53	The TRS R consists of the following rules:
21.92/6.53	
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	
21.92/6.53	The set Q consists of the following terms:
21.92/6.53	
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x0))))
21.92/6.53	
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(21) QDPOrderProof (EQUIVALENT)
21.92/6.53	We use the reduction pair processor [LPAR04,JAR06].
21.92/6.53	
21.92/6.53	
21.92/6.53	The following pairs can be oriented strictly and are deleted.
21.92/6.53	
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
21.92/6.53	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
21.92/6.53	The remaining pairs can at least be oriented weakly.
21.92/6.53	Used ordering:  Polynomial interpretation [POLO]:
21.92/6.53	
21.92/6.53	   POL(B_{B_1}(x_1)) = x_1
21.92/6.53	   POL(a_{b_1}(x_1)) = x_1
21.92/6.53	   POL(b_{a_1}(x_1)) = 1 + x_1
21.92/6.53	   POL(b_{b_1}(x_1)) = x_1
21.92/6.53	
21.92/6.53	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.92/6.53	
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	
21.92/6.53	
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(22)
21.92/6.53	Obligation:
21.92/6.53	Q DP problem:
21.92/6.53	P is empty.
21.92/6.53	The TRS R consists of the following rules:
21.92/6.53	
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	
21.92/6.53	The set Q consists of the following terms:
21.92/6.53	
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x0))))
21.92/6.53	
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(23) PisEmptyProof (EQUIVALENT)
21.92/6.53	The TRS P is empty. Hence, there is no (P,Q,R) chain.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(24)
21.92/6.53	YES
21.92/6.53	
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(25)
21.92/6.53	Obligation:
21.92/6.53	Q DP problem:
21.92/6.53	The TRS P consists of the following rules:
21.92/6.53	
21.92/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
21.92/6.53	
21.92/6.53	The TRS R consists of the following rules:
21.92/6.53	
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.92/6.53	
21.92/6.53	Q is empty.
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(26) QDPOrderProof (EQUIVALENT)
21.92/6.53	We use the reduction pair processor [LPAR04,JAR06].
21.92/6.53	
21.92/6.53	
21.92/6.53	The following pairs can be oriented strictly and are deleted.
21.92/6.53	
21.92/6.53	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
21.92/6.53	The remaining pairs can at least be oriented weakly.
21.92/6.53	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
21.92/6.53	
21.92/6.53	POL( B_{A_1}_1(x_1) ) = 2x_1 + 2
21.92/6.53	POL( a_{a_1}_1(x_1) ) = x_1 + 2
21.92/6.53	POL( a_{b_1}_1(x_1) ) = max{0, x_1 - 1}
21.92/6.53	POL( b_{b_1}_1(x_1) ) = x_1 + 2
21.92/6.53	POL( b_{a_1}_1(x_1) ) = x_1 + 1
21.92/6.53	
21.92/6.53	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.92/6.53	
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	
21.92/6.53	
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(27)
21.92/6.53	Obligation:
21.92/6.53	Q DP problem:
21.92/6.53	P is empty.
21.92/6.53	The TRS R consists of the following rules:
21.92/6.53	
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
21.92/6.53	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
21.92/6.53	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
21.92/6.53	
21.92/6.53	Q is empty.
21.92/6.53	We have to consider all minimal (P,Q,R)-chains.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(28) PisEmptyProof (EQUIVALENT)
21.92/6.53	The TRS P is empty. Hence, there is no (P,Q,R) chain.
21.92/6.53	----------------------------------------
21.92/6.53	
21.92/6.53	(29)
21.92/6.53	YES
21.92/6.58	EOF