36.10/10.12	YES
36.10/10.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
36.10/10.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.10/10.14	
36.10/10.14	
36.10/10.14	Termination w.r.t. Q of the given QTRS could be proven:
36.10/10.14	
36.10/10.14	(0) QTRS
36.10/10.14	(1) QTRS Reverse [EQUIVALENT, 0 ms]
36.10/10.14	(2) QTRS
36.10/10.14	(3) FlatCCProof [EQUIVALENT, 0 ms]
36.10/10.14	(4) QTRS
36.10/10.14	(5) RootLabelingProof [EQUIVALENT, 0 ms]
36.10/10.14	(6) QTRS
36.10/10.14	(7) DependencyPairsProof [EQUIVALENT, 41 ms]
36.10/10.14	(8) QDP
36.10/10.14	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
36.10/10.14	(10) AND
36.10/10.14	    (11) QDP
36.10/10.14	        (12) UsableRulesProof [EQUIVALENT, 0 ms]
36.10/10.14	        (13) QDP
36.10/10.14	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
36.10/10.14	        (15) YES
36.10/10.14	    (16) QDP
36.10/10.14	        (17) QDPOrderProof [EQUIVALENT, 237 ms]
36.10/10.14	        (18) QDP
36.10/10.14	        (19) DependencyGraphProof [EQUIVALENT, 0 ms]
36.10/10.14	        (20) QDP
36.10/10.14	        (21) QDPOrderProof [EQUIVALENT, 134 ms]
36.10/10.14	        (22) QDP
36.10/10.14	        (23) PisEmptyProof [EQUIVALENT, 0 ms]
36.10/10.14	        (24) YES
36.10/10.14	
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(0)
36.10/10.14	Obligation:
36.10/10.14	Q restricted rewrite system:
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   b(b(a(a(x1)))) -> a(b(a(a(x1))))
36.10/10.14	   a(a(b(a(x1)))) -> b(b(a(a(x1))))
36.10/10.14	   a(a(b(a(x1)))) -> b(b(b(b(x1))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(1) QTRS Reverse (EQUIVALENT)
36.10/10.14	We applied the QTRS Reverse Processor [REVERSE].
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(2)
36.10/10.14	Obligation:
36.10/10.14	Q restricted rewrite system:
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
36.10/10.14	   a(b(a(a(x1)))) -> a(a(b(b(x1))))
36.10/10.14	   a(b(a(a(x1)))) -> b(b(b(b(x1))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(3) FlatCCProof (EQUIVALENT)
36.10/10.14	We used flat context closure [ROOTLAB]
36.10/10.14	As Q is empty the flat context closure was sound AND complete.
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(4)
36.10/10.14	Obligation:
36.10/10.14	Q restricted rewrite system:
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
36.10/10.14	   a(b(a(a(x1)))) -> a(a(b(b(x1))))
36.10/10.14	   a(a(b(a(a(x1))))) -> a(b(b(b(b(x1)))))
36.10/10.14	   b(a(b(a(a(x1))))) -> b(b(b(b(b(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(5) RootLabelingProof (EQUIVALENT)
36.10/10.14	We used plain root labeling [ROOTLAB] with the following heuristic:
36.10/10.14	LabelAll: All function symbols get labeled
36.10/10.14	
36.10/10.14	As Q is empty the root labeling was sound AND complete.
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(6)
36.10/10.14	Obligation:
36.10/10.14	Q restricted rewrite system:
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(7) DependencyPairsProof (EQUIVALENT)
36.10/10.14	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(8)
36.10/10.14	Obligation:
36.10/10.14	Q DP problem:
36.10/10.14	The TRS P consists of the following rules:
36.10/10.14	
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
36.10/10.14	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
36.10/10.14	
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	We have to consider all minimal (P,Q,R)-chains.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(9) DependencyGraphProof (EQUIVALENT)
36.10/10.14	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(10)
36.10/10.14	Complex Obligation (AND)
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(11)
36.10/10.14	Obligation:
36.10/10.14	Q DP problem:
36.10/10.14	The TRS P consists of the following rules:
36.10/10.14	
36.10/10.14	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
36.10/10.14	
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	We have to consider all minimal (P,Q,R)-chains.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(12) UsableRulesProof (EQUIVALENT)
36.10/10.14	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.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(13)
36.10/10.14	Obligation:
36.10/10.14	Q DP problem:
36.10/10.14	The TRS P consists of the following rules:
36.10/10.14	
36.10/10.14	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
36.10/10.14	
36.10/10.14	R is empty.
36.10/10.14	Q is empty.
36.10/10.14	We have to consider all minimal (P,Q,R)-chains.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(14) QDPSizeChangeProof (EQUIVALENT)
36.10/10.14	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. 
36.10/10.14	
36.10/10.14	From the DPs we obtained the following set of size-change graphs:
36.10/10.14	*B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
36.10/10.14	The graph contains the following edges 1 > 1
36.10/10.14	
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(15)
36.10/10.14	YES
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(16)
36.10/10.14	Obligation:
36.10/10.14	Q DP problem:
36.10/10.14	The TRS P consists of the following rules:
36.10/10.14	
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
36.10/10.14	
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	We have to consider all minimal (P,Q,R)-chains.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(17) QDPOrderProof (EQUIVALENT)
36.10/10.14	We use the reduction pair processor [LPAR04,JAR06].
36.10/10.14	
36.10/10.14	
36.10/10.14	The following pairs can be oriented strictly and are deleted.
36.10/10.14	
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
36.10/10.14	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
36.10/10.14	The remaining pairs can at least be oriented weakly.
36.10/10.14	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
36.10/10.14	
36.10/10.14	POL( A_{A_1}_1(x_1) ) = max{0, x_1 - 2}
36.10/10.14	POL( A_{B_1}_1(x_1) ) = x_1
36.10/10.14	POL( b_{a_1}_1(x_1) ) = x_1
36.10/10.14	POL( a_{b_1}_1(x_1) ) = x_1 + 2
36.10/10.14	POL( a_{a_1}_1(x_1) ) = x_1 + 1
36.10/10.14	POL( b_{b_1}_1(x_1) ) = x_1 + 1
36.10/10.14	
36.10/10.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.10/10.14	
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(18)
36.10/10.14	Obligation:
36.10/10.14	Q DP problem:
36.10/10.14	The TRS P consists of the following rules:
36.10/10.14	
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
36.10/10.14	
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	We have to consider all minimal (P,Q,R)-chains.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(19) DependencyGraphProof (EQUIVALENT)
36.10/10.14	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(20)
36.10/10.14	Obligation:
36.10/10.14	Q DP problem:
36.10/10.14	The TRS P consists of the following rules:
36.10/10.14	
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	We have to consider all minimal (P,Q,R)-chains.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(21) QDPOrderProof (EQUIVALENT)
36.10/10.14	We use the reduction pair processor [LPAR04,JAR06].
36.10/10.14	
36.10/10.14	
36.10/10.14	The following pairs can be oriented strictly and are deleted.
36.10/10.14	
36.10/10.14	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	The remaining pairs can at least be oriented weakly.
36.10/10.14	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.10/10.14	
36.10/10.14	   <<<
36.10/10.14	 POL(A_{A_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
36.10/10.14	>>>
36.10/10.14	
36.10/10.14	   <<<
36.10/10.14	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
36.10/10.14	>>>
36.10/10.14	
36.10/10.14	   <<<
36.10/10.14	 POL(b_{b_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
36.10/10.14	>>>
36.10/10.14	
36.10/10.14	   <<<
36.10/10.14	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[1A, 0A, 1A], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
36.10/10.14	>>>
36.10/10.14	
36.10/10.14	   <<<
36.10/10.14	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
36.10/10.14	>>>
36.10/10.14	
36.10/10.14	
36.10/10.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.10/10.14	
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(22)
36.10/10.14	Obligation:
36.10/10.14	Q DP problem:
36.10/10.14	P is empty.
36.10/10.14	The TRS R consists of the following rules:
36.10/10.14	
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
36.10/10.14	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
36.10/10.14	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
36.10/10.14	
36.10/10.14	Q is empty.
36.10/10.14	We have to consider all minimal (P,Q,R)-chains.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(23) PisEmptyProof (EQUIVALENT)
36.10/10.14	The TRS P is empty. Hence, there is no (P,Q,R) chain.
36.10/10.14	----------------------------------------
36.10/10.14	
36.10/10.14	(24)
36.10/10.14	YES
36.30/10.28	EOF