12.75/4.10	YES
12.75/4.15	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.75/4.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.75/4.15	
12.75/4.15	
12.75/4.15	Termination w.r.t. Q of the given QTRS could be proven:
12.75/4.15	
12.75/4.15	(0) QTRS
12.75/4.15	(1) RootLabelingProof [EQUIVALENT, 0 ms]
12.75/4.15	(2) QTRS
12.75/4.15	(3) QTRSRRRProof [EQUIVALENT, 8 ms]
12.75/4.15	(4) QTRS
12.75/4.15	(5) DependencyPairsProof [EQUIVALENT, 0 ms]
12.75/4.15	(6) QDP
12.75/4.15	(7) DependencyGraphProof [EQUIVALENT, 1 ms]
12.75/4.15	(8) AND
12.75/4.15	    (9) QDP
12.75/4.15	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
12.75/4.15	        (11) QDP
12.75/4.15	        (12) QDPOrderProof [EQUIVALENT, 6 ms]
12.75/4.15	        (13) QDP
12.75/4.15	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
12.75/4.15	        (15) YES
12.75/4.15	    (16) QDP
12.75/4.15	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.75/4.15	        (18) YES
12.75/4.15	
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(0)
12.75/4.15	Obligation:
12.75/4.15	Q restricted rewrite system:
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   a(a(b(a(x1)))) -> a(b(a(a(x1))))
12.75/4.15	   b(a(b(b(x1)))) -> b(b(a(b(x1))))
12.75/4.15	   a(b(a(b(x1)))) -> a(b(b(b(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(1) RootLabelingProof (EQUIVALENT)
12.75/4.15	We used plain root labeling [ROOTLAB] with the following heuristic:
12.75/4.15	LabelAll: All function symbols get labeled
12.75/4.15	
12.75/4.15	As Q is empty the root labeling was sound AND complete.
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(2)
12.75/4.15	Obligation:
12.75/4.15	Q restricted rewrite system:
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.75/4.15	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(3) QTRSRRRProof (EQUIVALENT)
12.75/4.15	Used ordering:
12.75/4.15	Polynomial interpretation [POLO]:
12.75/4.15	
12.75/4.15	   POL(a_{a_1}(x_1)) = x_1
12.75/4.15	   POL(a_{b_1}(x_1)) = x_1
12.75/4.15	   POL(b_{a_1}(x_1)) = 1 + x_1
12.75/4.15	   POL(b_{b_1}(x_1)) = x_1
12.75/4.15	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
12.75/4.15	
12.75/4.15	   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
12.75/4.15	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	
12.75/4.15	
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(4)
12.75/4.15	Obligation:
12.75/4.15	Q restricted rewrite system:
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(5) DependencyPairsProof (EQUIVALENT)
12.75/4.15	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(6)
12.75/4.15	Obligation:
12.75/4.15	Q DP problem:
12.75/4.15	The TRS P consists of the following rules:
12.75/4.15	
12.75/4.15	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
12.75/4.15	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
12.75/4.15	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
12.75/4.15	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
12.75/4.15	
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	We have to consider all minimal (P,Q,R)-chains.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(7) DependencyGraphProof (EQUIVALENT)
12.75/4.15	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(8)
12.75/4.15	Complex Obligation (AND)
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(9)
12.75/4.15	Obligation:
12.75/4.15	Q DP problem:
12.75/4.15	The TRS P consists of the following rules:
12.75/4.15	
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
12.75/4.15	
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	We have to consider all minimal (P,Q,R)-chains.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(10) UsableRulesProof (EQUIVALENT)
12.75/4.15	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.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(11)
12.75/4.15	Obligation:
12.75/4.15	Q DP problem:
12.75/4.15	The TRS P consists of the following rules:
12.75/4.15	
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
12.75/4.15	
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	We have to consider all minimal (P,Q,R)-chains.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(12) QDPOrderProof (EQUIVALENT)
12.75/4.15	We use the reduction pair processor [LPAR04,JAR06].
12.75/4.15	
12.75/4.15	
12.75/4.15	The following pairs can be oriented strictly and are deleted.
12.75/4.15	
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
12.75/4.15	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
12.75/4.15	The remaining pairs can at least be oriented weakly.
12.75/4.15	Used ordering:  Polynomial interpretation [POLO]:
12.75/4.15	
12.75/4.15	   POL(B_{A_1}(x_1)) = x_1
12.75/4.15	   POL(a_{b_1}(x_1)) = x_1
12.75/4.15	   POL(b_{a_1}(x_1)) = 1 + x_1
12.75/4.15	   POL(b_{b_1}(x_1)) = 1 + x_1
12.75/4.15	
12.75/4.15	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.75/4.15	
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(13)
12.75/4.15	Obligation:
12.75/4.15	Q DP problem:
12.75/4.15	P is empty.
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	We have to consider all minimal (P,Q,R)-chains.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(14) PisEmptyProof (EQUIVALENT)
12.75/4.15	The TRS P is empty. Hence, there is no (P,Q,R) chain.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(15)
12.75/4.15	YES
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(16)
12.75/4.15	Obligation:
12.75/4.15	Q DP problem:
12.75/4.15	The TRS P consists of the following rules:
12.75/4.15	
12.75/4.15	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
12.75/4.15	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
12.75/4.15	
12.75/4.15	The TRS R consists of the following rules:
12.75/4.15	
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
12.75/4.15	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
12.75/4.15	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
12.75/4.15	
12.75/4.15	Q is empty.
12.75/4.15	We have to consider all minimal (P,Q,R)-chains.
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(17) QDPSizeChangeProof (EQUIVALENT)
12.75/4.15	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. 
12.75/4.15	
12.75/4.15	From the DPs we obtained the following set of size-change graphs:
12.75/4.15	*A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
12.75/4.15	The graph contains the following edges 1 > 1
12.75/4.15	
12.75/4.15	
12.75/4.15	*A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
12.75/4.15	The graph contains the following edges 1 > 1
12.75/4.15	
12.75/4.15	
12.75/4.15	----------------------------------------
12.75/4.15	
12.75/4.15	(18)
12.75/4.15	YES
13.09/4.23	EOF