8.83/3.17	YES
9.23/3.23	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
9.23/3.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.23/3.23	
9.23/3.23	
9.23/3.23	Termination w.r.t. Q of the given QTRS could be proven:
9.23/3.23	
9.23/3.23	(0) QTRS
9.23/3.23	(1) RootLabelingProof [EQUIVALENT, 0 ms]
9.23/3.23	(2) QTRS
9.23/3.23	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
9.23/3.23	(4) QDP
9.23/3.23	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
9.23/3.23	(6) QDP
9.23/3.23	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.23/3.23	(8) YES
9.23/3.23	
9.23/3.23	
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(0)
9.23/3.23	Obligation:
9.23/3.23	Q restricted rewrite system:
9.23/3.23	The TRS R consists of the following rules:
9.23/3.23	
9.23/3.23	   a(b(a(x1))) -> a(a(b(b(a(a(x1))))))
9.23/3.23	   b(a(a(b(x1)))) -> b(a(b(x1)))
9.23/3.23	
9.23/3.23	Q is empty.
9.23/3.23	
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(1) RootLabelingProof (EQUIVALENT)
9.23/3.23	We used plain root labeling [ROOTLAB] with the following heuristic:
9.23/3.23	LabelAll: All function symbols get labeled
9.23/3.23	
9.23/3.23	As Q is empty the root labeling was sound AND complete.
9.23/3.23	
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(2)
9.23/3.23	Obligation:
9.23/3.23	Q restricted rewrite system:
9.23/3.23	The TRS R consists of the following rules:
9.23/3.23	
9.23/3.23	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
9.23/3.23	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
9.23/3.23	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
9.23/3.23	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
9.23/3.23	
9.23/3.23	Q is empty.
9.23/3.23	
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(3) DependencyPairsProof (EQUIVALENT)
9.23/3.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(4)
9.23/3.23	Obligation:
9.23/3.23	Q DP problem:
9.23/3.23	The TRS P consists of the following rules:
9.23/3.23	
9.23/3.23	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
9.23/3.23	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
9.23/3.23	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
9.23/3.23	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
9.23/3.23	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
9.23/3.23	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
9.23/3.23	
9.23/3.23	The TRS R consists of the following rules:
9.23/3.23	
9.23/3.23	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
9.23/3.23	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
9.23/3.23	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
9.23/3.23	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
9.23/3.23	
9.23/3.23	Q is empty.
9.23/3.23	We have to consider all minimal (P,Q,R)-chains.
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(5) DependencyGraphProof (EQUIVALENT)
9.23/3.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(6)
9.23/3.23	Obligation:
9.23/3.23	Q DP problem:
9.23/3.23	The TRS P consists of the following rules:
9.23/3.23	
9.23/3.23	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
9.23/3.23	
9.23/3.23	The TRS R consists of the following rules:
9.23/3.23	
9.23/3.23	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
9.23/3.23	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
9.23/3.23	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
9.23/3.23	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
9.23/3.23	
9.23/3.23	Q is empty.
9.23/3.23	We have to consider all minimal (P,Q,R)-chains.
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(7) QDPSizeChangeProof (EQUIVALENT)
9.23/3.23	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. 
9.23/3.23	
9.23/3.23	From the DPs we obtained the following set of size-change graphs:
9.23/3.23	*B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
9.23/3.23	The graph contains the following edges 1 > 1
9.23/3.23	
9.23/3.23	
9.23/3.23	----------------------------------------
9.23/3.23	
9.23/3.23	(8)
9.23/3.23	YES
9.23/3.30	EOF