14.33/5.15	YES
16.13/5.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
16.13/5.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.13/5.59	
16.13/5.59	
16.13/5.59	Termination w.r.t. Q of the given QTRS could be proven:
16.13/5.59	
16.13/5.59	(0) QTRS
16.13/5.59	(1) QTRS Reverse [EQUIVALENT, 0 ms]
16.13/5.59	(2) QTRS
16.13/5.59	(3) FlatCCProof [EQUIVALENT, 0 ms]
16.13/5.59	(4) QTRS
16.13/5.59	(5) RootLabelingProof [EQUIVALENT, 0 ms]
16.13/5.59	(6) QTRS
16.13/5.59	(7) QTRSRRRProof [EQUIVALENT, 3 ms]
16.13/5.59	(8) QTRS
16.13/5.59	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
16.13/5.59	(10) QDP
16.13/5.59	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
16.13/5.59	(12) AND
16.13/5.59	    (13) QDP
16.13/5.59	        (14) UsableRulesProof [EQUIVALENT, 0 ms]
16.13/5.59	        (15) QDP
16.13/5.59	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.13/5.59	        (17) YES
16.13/5.59	    (18) QDP
16.13/5.59	        (19) UsableRulesProof [EQUIVALENT, 0 ms]
16.13/5.59	        (20) QDP
16.13/5.59	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.13/5.59	        (22) YES
16.13/5.59	
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(0)
16.13/5.59	Obligation:
16.13/5.59	Q restricted rewrite system:
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   a(b(x1)) -> b(b(b(x1)))
16.13/5.59	   b(a(x1)) -> a(a(a(x1)))
16.13/5.59	   a(x1) -> x1
16.13/5.59	   b(x1) -> x1
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(1) QTRS Reverse (EQUIVALENT)
16.13/5.59	We applied the QTRS Reverse Processor [REVERSE].
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(2)
16.13/5.59	Obligation:
16.13/5.59	Q restricted rewrite system:
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   b(a(x1)) -> b(b(b(x1)))
16.13/5.59	   a(b(x1)) -> a(a(a(x1)))
16.13/5.59	   a(x1) -> x1
16.13/5.59	   b(x1) -> x1
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(3) FlatCCProof (EQUIVALENT)
16.13/5.59	We used flat context closure [ROOTLAB]
16.13/5.59	As Q is empty the flat context closure was sound AND complete.
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(4)
16.13/5.59	Obligation:
16.13/5.59	Q restricted rewrite system:
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   b(a(x1)) -> b(b(b(x1)))
16.13/5.59	   a(b(x1)) -> a(a(a(x1)))
16.13/5.59	   b(a(x1)) -> b(x1)
16.13/5.59	   a(a(x1)) -> a(x1)
16.13/5.59	   b(b(x1)) -> b(x1)
16.13/5.59	   a(b(x1)) -> a(x1)
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(5) RootLabelingProof (EQUIVALENT)
16.13/5.59	We used plain root labeling [ROOTLAB] with the following heuristic:
16.13/5.59	LabelAll: All function symbols get labeled
16.13/5.59	
16.13/5.59	As Q is empty the root labeling was sound AND complete.
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(6)
16.13/5.59	Obligation:
16.13/5.59	Q restricted rewrite system:
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
16.13/5.59	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
16.13/5.59	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
16.13/5.59	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
16.13/5.59	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(7) QTRSRRRProof (EQUIVALENT)
16.13/5.59	Used ordering:
16.13/5.59	Polynomial interpretation [POLO]:
16.13/5.59	
16.13/5.59	   POL(a_{a_1}(x_1)) = x_1
16.13/5.59	   POL(a_{b_1}(x_1)) = 1 + x_1
16.13/5.59	   POL(b_{a_1}(x_1)) = x_1
16.13/5.59	   POL(b_{b_1}(x_1)) = x_1
16.13/5.59	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
16.13/5.59	
16.13/5.59	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
16.13/5.59	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
16.13/5.59	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
16.13/5.59	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
16.13/5.59	
16.13/5.59	
16.13/5.59	
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(8)
16.13/5.59	Obligation:
16.13/5.59	Q restricted rewrite system:
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
16.13/5.59	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
16.13/5.59	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(9) DependencyPairsProof (EQUIVALENT)
16.13/5.59	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(10)
16.13/5.59	Obligation:
16.13/5.59	Q DP problem:
16.13/5.59	The TRS P consists of the following rules:
16.13/5.59	
16.13/5.59	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
16.13/5.59	   B_{A_1}(a_{a_1}(x1)) -> B_{B_1}(b_{a_1}(x1))
16.13/5.59	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
16.13/5.59	   A_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
16.13/5.59	   A_{B_1}(b_{b_1}(x1)) -> A_{A_1}(a_{b_1}(x1))
16.13/5.59	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
16.13/5.59	
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
16.13/5.59	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
16.13/5.59	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	We have to consider all minimal (P,Q,R)-chains.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(11) DependencyGraphProof (EQUIVALENT)
16.13/5.59	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(12)
16.13/5.59	Complex Obligation (AND)
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(13)
16.13/5.59	Obligation:
16.13/5.59	Q DP problem:
16.13/5.59	The TRS P consists of the following rules:
16.13/5.59	
16.13/5.59	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
16.13/5.59	
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
16.13/5.59	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
16.13/5.59	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	We have to consider all minimal (P,Q,R)-chains.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(14) UsableRulesProof (EQUIVALENT)
16.13/5.59	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.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(15)
16.13/5.59	Obligation:
16.13/5.59	Q DP problem:
16.13/5.59	The TRS P consists of the following rules:
16.13/5.59	
16.13/5.59	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
16.13/5.59	
16.13/5.59	R is empty.
16.13/5.59	Q is empty.
16.13/5.59	We have to consider all minimal (P,Q,R)-chains.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(16) QDPSizeChangeProof (EQUIVALENT)
16.13/5.59	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. 
16.13/5.59	
16.13/5.59	From the DPs we obtained the following set of size-change graphs:
16.13/5.59	*A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
16.13/5.59	The graph contains the following edges 1 > 1
16.13/5.59	
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(17)
16.13/5.59	YES
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(18)
16.13/5.59	Obligation:
16.13/5.59	Q DP problem:
16.13/5.59	The TRS P consists of the following rules:
16.13/5.59	
16.13/5.59	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
16.13/5.59	
16.13/5.59	The TRS R consists of the following rules:
16.13/5.59	
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
16.13/5.59	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
16.13/5.59	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
16.13/5.59	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
16.13/5.59	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
16.13/5.59	
16.13/5.59	Q is empty.
16.13/5.59	We have to consider all minimal (P,Q,R)-chains.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(19) UsableRulesProof (EQUIVALENT)
16.13/5.59	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.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(20)
16.13/5.59	Obligation:
16.13/5.59	Q DP problem:
16.13/5.59	The TRS P consists of the following rules:
16.13/5.59	
16.13/5.59	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
16.13/5.59	
16.13/5.59	R is empty.
16.13/5.59	Q is empty.
16.13/5.59	We have to consider all minimal (P,Q,R)-chains.
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(21) QDPSizeChangeProof (EQUIVALENT)
16.13/5.59	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. 
16.13/5.59	
16.13/5.59	From the DPs we obtained the following set of size-change graphs:
16.13/5.59	*B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
16.13/5.59	The graph contains the following edges 1 > 1
16.13/5.59	
16.13/5.59	
16.13/5.59	----------------------------------------
16.13/5.59	
16.13/5.59	(22)
16.13/5.59	YES
16.52/5.71	EOF