12.13/4.72	YES
12.66/4.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.66/4.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.66/4.83	
12.66/4.83	
12.66/4.83	Termination w.r.t. Q of the given QTRS could be proven:
12.66/4.83	
12.66/4.83	(0) QTRS
12.66/4.83	(1) QTRS Reverse [EQUIVALENT, 0 ms]
12.66/4.83	(2) QTRS
12.66/4.83	(3) FlatCCProof [EQUIVALENT, 0 ms]
12.66/4.83	(4) QTRS
12.66/4.83	(5) RootLabelingProof [EQUIVALENT, 0 ms]
12.66/4.83	(6) QTRS
12.66/4.83	(7) QTRSRRRProof [EQUIVALENT, 3 ms]
12.66/4.83	(8) QTRS
12.66/4.83	(9) DependencyPairsProof [EQUIVALENT, 3 ms]
12.66/4.83	(10) QDP
12.66/4.83	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
12.66/4.83	(12) AND
12.66/4.83	    (13) QDP
12.66/4.83	        (14) UsableRulesProof [EQUIVALENT, 0 ms]
12.66/4.83	        (15) QDP
12.66/4.83	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.66/4.83	        (17) YES
12.66/4.83	    (18) QDP
12.66/4.83	        (19) UsableRulesProof [EQUIVALENT, 0 ms]
12.66/4.83	        (20) QDP
12.66/4.83	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.66/4.83	        (22) YES
12.66/4.83	
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(0)
12.66/4.83	Obligation:
12.66/4.83	Q restricted rewrite system:
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   a(b(x1)) -> b(b(b(b(x1))))
12.66/4.83	   b(a(x1)) -> a(a(a(a(x1))))
12.66/4.83	   a(x1) -> x1
12.66/4.83	   b(x1) -> x1
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(1) QTRS Reverse (EQUIVALENT)
12.66/4.83	We applied the QTRS Reverse Processor [REVERSE].
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(2)
12.66/4.83	Obligation:
12.66/4.83	Q restricted rewrite system:
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   b(a(x1)) -> b(b(b(b(x1))))
12.66/4.83	   a(b(x1)) -> a(a(a(a(x1))))
12.66/4.83	   a(x1) -> x1
12.66/4.83	   b(x1) -> x1
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(3) FlatCCProof (EQUIVALENT)
12.66/4.83	We used flat context closure [ROOTLAB]
12.66/4.83	As Q is empty the flat context closure was sound AND complete.
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(4)
12.66/4.83	Obligation:
12.66/4.83	Q restricted rewrite system:
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   b(a(x1)) -> b(b(b(b(x1))))
12.66/4.83	   a(b(x1)) -> a(a(a(a(x1))))
12.66/4.83	   b(a(x1)) -> b(x1)
12.66/4.83	   a(a(x1)) -> a(x1)
12.66/4.83	   b(b(x1)) -> b(x1)
12.66/4.83	   a(b(x1)) -> a(x1)
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(5) RootLabelingProof (EQUIVALENT)
12.66/4.83	We used plain root labeling [ROOTLAB] with the following heuristic:
12.66/4.83	LabelAll: All function symbols get labeled
12.66/4.83	
12.66/4.83	As Q is empty the root labeling was sound AND complete.
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(6)
12.66/4.83	Obligation:
12.66/4.83	Q restricted rewrite system:
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.66/4.83	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
12.66/4.83	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
12.66/4.83	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
12.66/4.83	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(7) QTRSRRRProof (EQUIVALENT)
12.66/4.83	Used ordering:
12.66/4.83	Polynomial interpretation [POLO]:
12.66/4.83	
12.66/4.83	   POL(a_{a_1}(x_1)) = x_1
12.66/4.83	   POL(a_{b_1}(x_1)) = 1 + x_1
12.66/4.83	   POL(b_{a_1}(x_1)) = x_1
12.66/4.83	   POL(b_{b_1}(x_1)) = x_1
12.66/4.83	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
12.66/4.83	
12.66/4.83	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
12.66/4.83	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
12.66/4.83	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
12.66/4.83	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
12.66/4.83	
12.66/4.83	
12.66/4.83	
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(8)
12.66/4.83	Obligation:
12.66/4.83	Q restricted rewrite system:
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
12.66/4.83	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
12.66/4.83	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(9) DependencyPairsProof (EQUIVALENT)
12.66/4.83	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(10)
12.66/4.83	Obligation:
12.66/4.83	Q DP problem:
12.66/4.83	The TRS P consists of the following rules:
12.66/4.83	
12.66/4.83	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.66/4.83	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
12.66/4.83	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
12.66/4.83	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
12.66/4.83	   A_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.66/4.83	   A_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
12.66/4.83	   A_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{b_1}(x1))
12.66/4.83	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
12.66/4.83	
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
12.66/4.83	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
12.66/4.83	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	We have to consider all minimal (P,Q,R)-chains.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(11) DependencyGraphProof (EQUIVALENT)
12.66/4.83	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(12)
12.66/4.83	Complex Obligation (AND)
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(13)
12.66/4.83	Obligation:
12.66/4.83	Q DP problem:
12.66/4.83	The TRS P consists of the following rules:
12.66/4.83	
12.66/4.83	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
12.66/4.83	
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
12.66/4.83	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
12.66/4.83	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	We have to consider all minimal (P,Q,R)-chains.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(14) UsableRulesProof (EQUIVALENT)
12.66/4.83	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.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(15)
12.66/4.83	Obligation:
12.66/4.83	Q DP problem:
12.66/4.83	The TRS P consists of the following rules:
12.66/4.83	
12.66/4.83	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
12.66/4.83	
12.66/4.83	R is empty.
12.66/4.83	Q is empty.
12.66/4.83	We have to consider all minimal (P,Q,R)-chains.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(16) QDPSizeChangeProof (EQUIVALENT)
12.66/4.83	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. 
12.66/4.83	
12.66/4.83	From the DPs we obtained the following set of size-change graphs:
12.66/4.83	*A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
12.66/4.83	The graph contains the following edges 1 > 1
12.66/4.83	
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(17)
12.66/4.83	YES
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(18)
12.66/4.83	Obligation:
12.66/4.83	Q DP problem:
12.66/4.83	The TRS P consists of the following rules:
12.66/4.83	
12.66/4.83	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
12.66/4.83	
12.66/4.83	The TRS R consists of the following rules:
12.66/4.83	
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.66/4.83	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
12.66/4.83	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
12.66/4.83	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
12.66/4.83	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
12.66/4.83	
12.66/4.83	Q is empty.
12.66/4.83	We have to consider all minimal (P,Q,R)-chains.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(19) UsableRulesProof (EQUIVALENT)
12.66/4.83	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.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(20)
12.66/4.83	Obligation:
12.66/4.83	Q DP problem:
12.66/4.83	The TRS P consists of the following rules:
12.66/4.83	
12.66/4.83	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
12.66/4.83	
12.66/4.83	R is empty.
12.66/4.83	Q is empty.
12.66/4.83	We have to consider all minimal (P,Q,R)-chains.
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(21) QDPSizeChangeProof (EQUIVALENT)
12.66/4.83	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. 
12.66/4.83	
12.66/4.83	From the DPs we obtained the following set of size-change graphs:
12.66/4.83	*B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
12.66/4.83	The graph contains the following edges 1 > 1
12.66/4.83	
12.66/4.83	
12.66/4.83	----------------------------------------
12.66/4.83	
12.66/4.83	(22)
12.66/4.83	YES
12.83/4.90	EOF