33.80/9.57	YES
33.80/9.58	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
33.80/9.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.80/9.58	
33.80/9.58	
33.80/9.58	Termination w.r.t. Q of the given QTRS could be proven:
33.80/9.58	
33.80/9.58	(0) QTRS
33.80/9.58	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.80/9.58	(2) QTRS
33.80/9.58	(3) FlatCCProof [EQUIVALENT, 0 ms]
33.80/9.58	(4) QTRS
33.80/9.58	(5) RootLabelingProof [EQUIVALENT, 1 ms]
33.80/9.58	(6) QTRS
33.80/9.58	(7) QTRSRRRProof [EQUIVALENT, 60 ms]
33.80/9.58	(8) QTRS
33.80/9.58	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
33.80/9.58	(10) QDP
33.80/9.58	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
33.80/9.58	(12) QDP
33.80/9.58	(13) QDPOrderProof [EQUIVALENT, 223 ms]
33.80/9.58	(14) QDP
33.80/9.58	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
33.80/9.58	(16) AND
33.80/9.58	    (17) QDP
33.80/9.58	        (18) UsableRulesProof [EQUIVALENT, 0 ms]
33.80/9.58	        (19) QDP
33.80/9.58	        (20) QDPSizeChangeProof [EQUIVALENT, 0 ms]
33.80/9.58	        (21) YES
33.80/9.58	    (22) QDP
33.80/9.58	        (23) UsableRulesProof [EQUIVALENT, 0 ms]
33.80/9.58	        (24) QDP
33.80/9.58	        (25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
33.80/9.58	        (26) YES
33.80/9.58	
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(0)
33.80/9.58	Obligation:
33.80/9.58	Q restricted rewrite system:
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a(x1) -> x1
33.80/9.58	   a(x1) -> b(c(x1))
33.80/9.58	   b(b(x1)) -> a(a(x1))
33.80/9.58	   c(c(c(x1))) -> b(x1)
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(1) QTRS Reverse (EQUIVALENT)
33.80/9.58	We applied the QTRS Reverse Processor [REVERSE].
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(2)
33.80/9.58	Obligation:
33.80/9.58	Q restricted rewrite system:
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a(x1) -> x1
33.80/9.58	   a(x1) -> c(b(x1))
33.80/9.58	   b(b(x1)) -> a(a(x1))
33.80/9.58	   c(c(c(x1))) -> b(x1)
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(3) FlatCCProof (EQUIVALENT)
33.80/9.58	We used flat context closure [ROOTLAB]
33.80/9.58	As Q is empty the flat context closure was sound AND complete.
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(4)
33.80/9.58	Obligation:
33.80/9.58	Q restricted rewrite system:
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a(a(x1)) -> a(x1)
33.80/9.58	   c(a(x1)) -> c(x1)
33.80/9.58	   b(a(x1)) -> b(x1)
33.80/9.58	   a(a(x1)) -> a(c(b(x1)))
33.80/9.58	   c(a(x1)) -> c(c(b(x1)))
33.80/9.58	   b(a(x1)) -> b(c(b(x1)))
33.80/9.58	   a(b(b(x1))) -> a(a(a(x1)))
33.80/9.58	   c(b(b(x1))) -> c(a(a(x1)))
33.80/9.58	   b(b(b(x1))) -> b(a(a(x1)))
33.80/9.58	   a(c(c(c(x1)))) -> a(b(x1))
33.80/9.58	   c(c(c(c(x1)))) -> c(b(x1))
33.80/9.58	   b(c(c(c(x1)))) -> b(b(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(5) RootLabelingProof (EQUIVALENT)
33.80/9.58	We used plain root labeling [ROOTLAB] with the following heuristic:
33.80/9.58	LabelAll: All function symbols get labeled
33.80/9.58	
33.80/9.58	As Q is empty the root labeling was sound AND complete.
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(6)
33.80/9.58	Obligation:
33.80/9.58	Q restricted rewrite system:
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(x1))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(x1))
33.80/9.58	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(x1))
33.80/9.58	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(7) QTRSRRRProof (EQUIVALENT)
33.80/9.58	Used ordering:
33.80/9.58	Polynomial interpretation [POLO]:
33.80/9.58	
33.80/9.58	   POL(a_{a_1}(x_1)) = x_1
33.80/9.58	   POL(a_{b_1}(x_1)) = 1 + x_1
33.80/9.58	   POL(a_{c_1}(x_1)) = x_1
33.80/9.58	   POL(b_{a_1}(x_1)) = x_1
33.80/9.58	   POL(b_{b_1}(x_1)) = 1 + x_1
33.80/9.58	   POL(b_{c_1}(x_1)) = x_1
33.80/9.58	   POL(c_{a_1}(x_1)) = 1 + x_1
33.80/9.58	   POL(c_{b_1}(x_1)) = x_1
33.80/9.58	   POL(c_{c_1}(x_1)) = 1 + x_1
33.80/9.58	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
33.80/9.58	
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
33.80/9.58	   a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(x1))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(x1))
33.80/9.58	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(x1))
33.80/9.58	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(x1))
33.80/9.58	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(x1))
33.80/9.58	
33.80/9.58	
33.80/9.58	
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(8)
33.80/9.58	Obligation:
33.80/9.58	Q restricted rewrite system:
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(9) DependencyPairsProof (EQUIVALENT)
33.80/9.58	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(10)
33.80/9.58	Obligation:
33.80/9.58	Q DP problem:
33.80/9.58	The TRS P consists of the following rules:
33.80/9.58	
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
33.80/9.58	   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
33.80/9.58	   A_{A_1}(a_{b_1}(x1)) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   A_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
33.80/9.58	   C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   B_{A_1}(a_{b_1}(x1)) -> B_{C_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   B_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
33.80/9.58	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1)
33.80/9.58	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
33.80/9.58	
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	We have to consider all minimal (P,Q,R)-chains.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(11) DependencyGraphProof (EQUIVALENT)
33.80/9.58	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(12)
33.80/9.58	Obligation:
33.80/9.58	Q DP problem:
33.80/9.58	The TRS P consists of the following rules:
33.80/9.58	
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1))
33.80/9.58	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
33.80/9.58	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1))
33.80/9.58	   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
33.80/9.58	   A_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	   B_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	We have to consider all minimal (P,Q,R)-chains.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(13) QDPOrderProof (EQUIVALENT)
33.80/9.58	We use the reduction pair processor [LPAR04,JAR06].
33.80/9.58	
33.80/9.58	
33.80/9.58	The following pairs can be oriented strictly and are deleted.
33.80/9.58	
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1)
33.80/9.58	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
33.80/9.58	The remaining pairs can at least be oriented weakly.
33.80/9.58	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
33.80/9.58	
33.80/9.58	POL( C_{B_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	POL( A_{A_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	POL( a_{c_1}_1(x_1) ) = x_1
33.80/9.58	POL( c_{b_1}_1(x_1) ) = x_1
33.80/9.58	POL( C_{A_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	POL( b_{a_1}_1(x_1) ) = x_1
33.80/9.58	POL( a_{a_1}_1(x_1) ) = x_1
33.80/9.58	POL( b_{c_1}_1(x_1) ) = x_1
33.80/9.58	POL( a_{b_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	POL( b_{b_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	POL( c_{a_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	POL( c_{c_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	POL( B_{A_1}_1(x_1) ) = 2x_1 + 2
33.80/9.58	
33.80/9.58	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.80/9.58	
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(14)
33.80/9.58	Obligation:
33.80/9.58	Q DP problem:
33.80/9.58	The TRS P consists of the following rules:
33.80/9.58	
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
33.80/9.58	   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
33.80/9.58	   C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	   A_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	   B_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	We have to consider all minimal (P,Q,R)-chains.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(15) DependencyGraphProof (EQUIVALENT)
33.80/9.58	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 11 less nodes.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(16)
33.80/9.58	Complex Obligation (AND)
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(17)
33.80/9.58	Obligation:
33.80/9.58	Q DP problem:
33.80/9.58	The TRS P consists of the following rules:
33.80/9.58	
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	We have to consider all minimal (P,Q,R)-chains.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(18) UsableRulesProof (EQUIVALENT)
33.80/9.58	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.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(19)
33.80/9.58	Obligation:
33.80/9.58	Q DP problem:
33.80/9.58	The TRS P consists of the following rules:
33.80/9.58	
33.80/9.58	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	
33.80/9.58	R is empty.
33.80/9.58	Q is empty.
33.80/9.58	We have to consider all minimal (P,Q,R)-chains.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(20) QDPSizeChangeProof (EQUIVALENT)
33.80/9.58	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. 
33.80/9.58	
33.80/9.58	From the DPs we obtained the following set of size-change graphs:
33.80/9.58	*B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
33.80/9.58	The graph contains the following edges 1 > 1
33.80/9.58	
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(21)
33.80/9.58	YES
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(22)
33.80/9.58	Obligation:
33.80/9.58	Q DP problem:
33.80/9.58	The TRS P consists of the following rules:
33.80/9.58	
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
33.80/9.58	
33.80/9.58	The TRS R consists of the following rules:
33.80/9.58	
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
33.80/9.58	   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
33.80/9.58	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
33.80/9.58	   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
33.80/9.58	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
33.80/9.58	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(x1))
33.80/9.58	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(x1))
33.80/9.58	
33.80/9.58	Q is empty.
33.80/9.58	We have to consider all minimal (P,Q,R)-chains.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(23) UsableRulesProof (EQUIVALENT)
33.80/9.58	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.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(24)
33.80/9.58	Obligation:
33.80/9.58	Q DP problem:
33.80/9.58	The TRS P consists of the following rules:
33.80/9.58	
33.80/9.58	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
33.80/9.58	
33.80/9.58	R is empty.
33.80/9.58	Q is empty.
33.80/9.58	We have to consider all minimal (P,Q,R)-chains.
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(25) QDPSizeChangeProof (EQUIVALENT)
33.80/9.58	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. 
33.80/9.58	
33.80/9.58	From the DPs we obtained the following set of size-change graphs:
33.80/9.58	*C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
33.80/9.58	The graph contains the following edges 1 > 1
33.80/9.58	
33.80/9.58	
33.80/9.58	----------------------------------------
33.80/9.58	
33.80/9.58	(26)
33.80/9.58	YES
33.89/9.62	EOF