13.54/4.41	YES
16.02/6.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
16.02/6.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.02/6.12	
16.02/6.12	
16.02/6.12	Termination w.r.t. Q of the given QTRS could be proven:
16.02/6.12	
16.02/6.12	(0) QTRS
16.02/6.12	(1) FlatCCProof [EQUIVALENT, 0 ms]
16.02/6.12	(2) QTRS
16.02/6.12	(3) RootLabelingProof [EQUIVALENT, 0 ms]
16.02/6.12	(4) QTRS
16.02/6.12	(5) QTRSRRRProof [EQUIVALENT, 25 ms]
16.02/6.12	(6) QTRS
16.02/6.12	(7) QTRSRRRProof [EQUIVALENT, 3 ms]
16.02/6.12	(8) QTRS
16.02/6.12	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
16.02/6.12	(10) QDP
16.02/6.12	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
16.02/6.12	(12) AND
16.02/6.12	    (13) QDP
16.02/6.12	        (14) UsableRulesProof [EQUIVALENT, 0 ms]
16.02/6.12	        (15) QDP
16.02/6.12	        (16) QDPSizeChangeProof [EQUIVALENT, 1 ms]
16.02/6.12	        (17) YES
16.02/6.12	    (18) QDP
16.02/6.12	        (19) QDPOrderProof [EQUIVALENT, 8 ms]
16.02/6.12	        (20) QDP
16.02/6.12	        (21) PisEmptyProof [EQUIVALENT, 0 ms]
16.02/6.12	        (22) YES
16.02/6.12	
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(0)
16.02/6.12	Obligation:
16.02/6.12	Q restricted rewrite system:
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
16.02/6.12	   b(a(b(b(x1)))) -> b(a(a(a(x1))))
16.02/6.12	   a(a(b(b(x1)))) -> a(b(b(a(x1))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(1) FlatCCProof (EQUIVALENT)
16.02/6.12	We used flat context closure [ROOTLAB]
16.02/6.12	As Q is empty the flat context closure was sound AND complete.
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(2)
16.02/6.12	Obligation:
16.02/6.12	Q restricted rewrite system:
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   b(a(b(b(x1)))) -> b(a(a(a(x1))))
16.02/6.12	   a(a(b(b(x1)))) -> a(b(b(a(x1))))
16.02/6.12	   b(b(a(a(a(x1))))) -> b(a(a(b(a(x1)))))
16.02/6.12	   a(b(a(a(a(x1))))) -> a(a(a(b(a(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(3) RootLabelingProof (EQUIVALENT)
16.02/6.12	We used plain root labeling [ROOTLAB] with the following heuristic:
16.02/6.12	LabelAll: All function symbols get labeled
16.02/6.12	
16.02/6.12	As Q is empty the root labeling was sound AND complete.
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(4)
16.02/6.12	Obligation:
16.02/6.12	Q restricted rewrite system:
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
16.02/6.12	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   b_{b_1}(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}(a_{a_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(5) QTRSRRRProof (EQUIVALENT)
16.02/6.12	Used ordering:
16.02/6.12	Polynomial interpretation [POLO]:
16.02/6.12	
16.02/6.12	   POL(a_{a_1}(x_1)) = x_1
16.02/6.12	   POL(a_{b_1}(x_1)) = 1 + x_1
16.02/6.12	   POL(b_{a_1}(x_1)) = x_1
16.02/6.12	   POL(b_{b_1}(x_1)) = 1 + x_1
16.02/6.12	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
16.02/6.12	
16.02/6.12	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
16.02/6.12	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
16.02/6.12	
16.02/6.12	
16.02/6.12	
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(6)
16.02/6.12	Obligation:
16.02/6.12	Q restricted rewrite system:
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   b_{b_1}(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}(a_{a_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(7) QTRSRRRProof (EQUIVALENT)
16.02/6.12	Used ordering:
16.02/6.12	Polynomial interpretation [POLO]:
16.02/6.12	
16.02/6.12	   POL(a_{a_1}(x_1)) = x_1
16.02/6.12	   POL(a_{b_1}(x_1)) = x_1
16.02/6.12	   POL(b_{a_1}(x_1)) = x_1
16.02/6.12	   POL(b_{b_1}(x_1)) = 1 + x_1
16.02/6.12	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
16.02/6.12	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   b_{b_1}(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}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	
16.02/6.12	
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(8)
16.02/6.12	Obligation:
16.02/6.12	Q restricted rewrite system:
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(9) DependencyPairsProof (EQUIVALENT)
16.02/6.12	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(10)
16.02/6.12	Obligation:
16.02/6.12	Q DP problem:
16.02/6.12	The TRS P consists of the following rules:
16.02/6.12	
16.02/6.12	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
16.02/6.12	
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	We have to consider all minimal (P,Q,R)-chains.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(11) DependencyGraphProof (EQUIVALENT)
16.02/6.12	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(12)
16.02/6.12	Complex Obligation (AND)
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(13)
16.02/6.12	Obligation:
16.02/6.12	Q DP problem:
16.02/6.12	The TRS P consists of the following rules:
16.02/6.12	
16.02/6.12	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
16.02/6.12	
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	We have to consider all minimal (P,Q,R)-chains.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(14) UsableRulesProof (EQUIVALENT)
16.02/6.12	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.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(15)
16.02/6.12	Obligation:
16.02/6.12	Q DP problem:
16.02/6.12	The TRS P consists of the following rules:
16.02/6.12	
16.02/6.12	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
16.02/6.12	
16.02/6.12	R is empty.
16.02/6.12	Q is empty.
16.02/6.12	We have to consider all minimal (P,Q,R)-chains.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(16) QDPSizeChangeProof (EQUIVALENT)
16.02/6.12	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. 
16.02/6.12	
16.02/6.12	From the DPs we obtained the following set of size-change graphs:
16.02/6.12	*A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
16.02/6.12	The graph contains the following edges 1 > 1
16.02/6.12	
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(17)
16.02/6.12	YES
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(18)
16.02/6.12	Obligation:
16.02/6.12	Q DP problem:
16.02/6.12	The TRS P consists of the following rules:
16.02/6.12	
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
16.02/6.12	
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	We have to consider all minimal (P,Q,R)-chains.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(19) QDPOrderProof (EQUIVALENT)
16.02/6.12	We use the reduction pair processor [LPAR04,JAR06].
16.02/6.12	
16.02/6.12	
16.02/6.12	The following pairs can be oriented strictly and are deleted.
16.02/6.12	
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
16.02/6.12	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
16.02/6.12	The remaining pairs can at least be oriented weakly.
16.02/6.12	Used ordering:  Polynomial interpretation [POLO]:
16.02/6.12	
16.02/6.12	   POL(A_{B_1}(x_1)) = x_1
16.02/6.12	   POL(a_{a_1}(x_1)) = 1 + x_1
16.02/6.12	   POL(a_{b_1}(x_1)) = x_1
16.02/6.12	   POL(b_{a_1}(x_1)) = x_1
16.02/6.12	   POL(b_{b_1}(x_1)) = 1
16.02/6.12	
16.02/6.12	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(20)
16.02/6.12	Obligation:
16.02/6.12	Q DP problem:
16.02/6.12	P is empty.
16.02/6.12	The TRS R consists of the following rules:
16.02/6.12	
16.02/6.12	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
16.02/6.12	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
16.02/6.12	
16.02/6.12	Q is empty.
16.02/6.12	We have to consider all minimal (P,Q,R)-chains.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(21) PisEmptyProof (EQUIVALENT)
16.02/6.12	The TRS P is empty. Hence, there is no (P,Q,R) chain.
16.02/6.12	----------------------------------------
16.02/6.12	
16.02/6.12	(22)
16.02/6.12	YES
16.06/7.44	EOF