11.39/3.67	YES
11.59/3.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.59/3.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.59/3.73	
11.59/3.73	
11.59/3.73	Termination w.r.t. Q of the given QTRS could be proven:
11.59/3.73	
11.59/3.73	(0) QTRS
11.59/3.73	(1) FlatCCProof [EQUIVALENT, 0 ms]
11.59/3.73	(2) QTRS
11.59/3.73	(3) RootLabelingProof [EQUIVALENT, 0 ms]
11.59/3.73	(4) QTRS
11.59/3.73	(5) QTRSRRRProof [EQUIVALENT, 42 ms]
11.59/3.73	(6) QTRS
11.59/3.73	(7) QTRSRRRProof [EQUIVALENT, 3 ms]
11.59/3.73	(8) QTRS
11.59/3.73	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
11.59/3.73	(10) QDP
11.59/3.73	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.59/3.73	(12) YES
11.59/3.73	
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(0)
11.59/3.73	Obligation:
11.59/3.73	Q restricted rewrite system:
11.59/3.73	The TRS R consists of the following rules:
11.59/3.73	
11.59/3.73	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
11.59/3.73	   b(a(b(b(x1)))) -> b(b(b(b(x1))))
11.59/3.73	   a(b(b(a(x1)))) -> b(a(a(a(x1))))
11.59/3.73	   b(a(b(a(x1)))) -> a(a(a(a(x1))))
11.59/3.73	
11.59/3.73	Q is empty.
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(1) FlatCCProof (EQUIVALENT)
11.59/3.73	We used flat context closure [ROOTLAB]
11.59/3.73	As Q is empty the flat context closure was sound AND complete.
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(2)
11.59/3.73	Obligation:
11.59/3.73	Q restricted rewrite system:
11.59/3.73	The TRS R consists of the following rules:
11.59/3.73	
11.59/3.73	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
11.59/3.73	   b(a(b(b(x1)))) -> b(b(b(b(x1))))
11.59/3.73	   b(a(b(b(a(x1))))) -> b(b(a(a(a(x1)))))
11.59/3.73	   a(a(b(b(a(x1))))) -> a(b(a(a(a(x1)))))
11.59/3.73	   b(b(a(b(a(x1))))) -> b(a(a(a(a(x1)))))
11.59/3.73	   a(b(a(b(a(x1))))) -> a(a(a(a(a(x1)))))
11.59/3.73	
11.59/3.73	Q is empty.
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(3) RootLabelingProof (EQUIVALENT)
11.59/3.73	We used plain root labeling [ROOTLAB] with the following heuristic:
11.59/3.73	LabelAll: All function symbols get labeled
11.59/3.73	
11.59/3.73	As Q is empty the root labeling was sound AND complete.
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(4)
11.59/3.73	Obligation:
11.59/3.73	Q restricted rewrite system:
11.59/3.73	The TRS R consists of the following rules:
11.59/3.73	
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	
11.59/3.73	Q is empty.
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(5) QTRSRRRProof (EQUIVALENT)
11.59/3.73	Used ordering:
11.59/3.73	Polynomial interpretation [POLO]:
11.59/3.73	
11.59/3.73	   POL(a_{a_1}(x_1)) = x_1
11.59/3.73	   POL(a_{b_1}(x_1)) = 1 + x_1
11.59/3.73	   POL(b_{a_1}(x_1)) = x_1
11.59/3.73	   POL(b_{b_1}(x_1)) = x_1
11.59/3.73	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.59/3.73	
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	
11.59/3.73	
11.59/3.73	
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(6)
11.59/3.73	Obligation:
11.59/3.73	Q restricted rewrite system:
11.59/3.73	The TRS R consists of the following rules:
11.59/3.73	
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
11.59/3.73	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	
11.59/3.73	Q is empty.
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(7) QTRSRRRProof (EQUIVALENT)
11.59/3.73	Used ordering:
11.59/3.73	Polynomial interpretation [POLO]:
11.59/3.73	
11.59/3.73	   POL(a_{a_1}(x_1)) = x_1
11.59/3.73	   POL(a_{b_1}(x_1)) = x_1
11.59/3.73	   POL(b_{a_1}(x_1)) = x_1
11.59/3.73	   POL(b_{b_1}(x_1)) = 1 + x_1
11.59/3.73	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.59/3.73	
11.59/3.73	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
11.59/3.73	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
11.59/3.73	
11.59/3.73	
11.59/3.73	
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(8)
11.59/3.73	Obligation:
11.59/3.73	Q restricted rewrite system:
11.59/3.73	The TRS R consists of the following rules:
11.59/3.73	
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
11.59/3.73	
11.59/3.73	Q is empty.
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(9) DependencyPairsProof (EQUIVALENT)
11.59/3.73	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(10)
11.59/3.73	Obligation:
11.59/3.73	Q DP problem:
11.59/3.73	The TRS P consists of the following rules:
11.59/3.73	
11.59/3.73	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
11.59/3.73	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
11.59/3.73	
11.59/3.73	The TRS R consists of the following rules:
11.59/3.73	
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
11.59/3.73	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
11.59/3.73	
11.59/3.73	Q is empty.
11.59/3.73	We have to consider all minimal (P,Q,R)-chains.
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(11) QDPSizeChangeProof (EQUIVALENT)
11.59/3.73	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
11.59/3.73	
11.59/3.73	From the DPs we obtained the following set of size-change graphs:
11.59/3.73	*B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
11.59/3.73	The graph contains the following edges 1 > 1
11.59/3.73	
11.59/3.73	
11.59/3.73	*B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
11.59/3.73	The graph contains the following edges 1 > 1
11.59/3.73	
11.59/3.73	
11.59/3.73	----------------------------------------
11.59/3.73	
11.59/3.73	(12)
11.59/3.73	YES
11.86/3.86	EOF