49.75/13.42	YES
49.75/13.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
49.75/13.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
49.75/13.47	
49.75/13.47	
49.75/13.47	Termination w.r.t. Q of the given QTRS could be proven:
49.75/13.47	
49.75/13.47	(0) QTRS
49.75/13.47	(1) QTRS Reverse [EQUIVALENT, 0 ms]
49.75/13.47	(2) QTRS
49.75/13.47	(3) FlatCCProof [EQUIVALENT, 0 ms]
49.75/13.47	(4) QTRS
49.75/13.47	(5) RootLabelingProof [EQUIVALENT, 0 ms]
49.75/13.47	(6) QTRS
49.75/13.47	(7) QTRSRRRProof [EQUIVALENT, 31 ms]
49.75/13.47	(8) QTRS
49.75/13.47	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
49.75/13.47	(10) QDP
49.75/13.47	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
49.75/13.47	(12) AND
49.75/13.47	    (13) QDP
49.75/13.47	        (14) UsableRulesProof [EQUIVALENT, 2 ms]
49.75/13.47	        (15) QDP
49.75/13.47	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
49.75/13.47	        (17) YES
49.75/13.47	    (18) QDP
49.75/13.47	        (19) QDPOrderProof [EQUIVALENT, 798 ms]
49.75/13.47	        (20) QDP
49.75/13.47	        (21) QDPOrderProof [EQUIVALENT, 14 ms]
49.75/13.47	        (22) QDP
49.75/13.47	        (23) QDPOrderProof [EQUIVALENT, 41 ms]
49.75/13.47	        (24) QDP
49.75/13.47	        (25) PisEmptyProof [EQUIVALENT, 0 ms]
49.75/13.47	        (26) YES
49.75/13.47	
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(0)
49.75/13.47	Obligation:
49.75/13.47	Q restricted rewrite system:
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   a(c(x1)) -> a(x1)
49.75/13.47	   a(c(b(c(x1)))) -> c(b(c(c(x1))))
49.75/13.47	   c(x1) -> b(a(a(x1)))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(1) QTRS Reverse (EQUIVALENT)
49.75/13.47	We applied the QTRS Reverse Processor [REVERSE].
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(2)
49.75/13.47	Obligation:
49.75/13.47	Q restricted rewrite system:
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c(a(x1)) -> a(x1)
49.75/13.47	   c(b(c(a(x1)))) -> c(c(b(c(x1))))
49.75/13.47	   c(x1) -> a(a(b(x1)))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(3) FlatCCProof (EQUIVALENT)
49.75/13.47	We used flat context closure [ROOTLAB]
49.75/13.47	As Q is empty the flat context closure was sound AND complete.
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(4)
49.75/13.47	Obligation:
49.75/13.47	Q restricted rewrite system:
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c(b(c(a(x1)))) -> c(c(b(c(x1))))
49.75/13.47	   c(c(a(x1))) -> c(a(x1))
49.75/13.47	   a(c(a(x1))) -> a(a(x1))
49.75/13.47	   b(c(a(x1))) -> b(a(x1))
49.75/13.47	   c(c(x1)) -> c(a(a(b(x1))))
49.75/13.47	   a(c(x1)) -> a(a(a(b(x1))))
49.75/13.47	   b(c(x1)) -> b(a(a(b(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(5) RootLabelingProof (EQUIVALENT)
49.75/13.47	We used plain root labeling [ROOTLAB] with the following heuristic:
49.75/13.47	LabelAll: All function symbols get labeled
49.75/13.47	
49.75/13.47	As Q is empty the root labeling was sound AND complete.
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(6)
49.75/13.47	Obligation:
49.75/13.47	Q restricted rewrite system:
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
49.75/13.47	   a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
49.75/13.47	   a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(7) QTRSRRRProof (EQUIVALENT)
49.75/13.47	Used ordering:
49.75/13.47	Polynomial interpretation [POLO]:
49.75/13.47	
49.75/13.47	   POL(a_{a_1}(x_1)) = x_1
49.75/13.47	   POL(a_{b_1}(x_1)) = x_1
49.75/13.47	   POL(a_{c_1}(x_1)) = 1 + x_1
49.75/13.47	   POL(b_{a_1}(x_1)) = x_1
49.75/13.47	   POL(b_{b_1}(x_1)) = x_1
49.75/13.47	   POL(b_{c_1}(x_1)) = x_1
49.75/13.47	   POL(c_{a_1}(x_1)) = x_1
49.75/13.47	   POL(c_{b_1}(x_1)) = x_1
49.75/13.47	   POL(c_{c_1}(x_1)) = x_1
49.75/13.47	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1))))
49.75/13.47	   a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1))
49.75/13.47	   a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1))
49.75/13.47	   a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1))
49.75/13.47	   a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	
49.75/13.47	
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(8)
49.75/13.47	Obligation:
49.75/13.47	Q restricted rewrite system:
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(9) DependencyPairsProof (EQUIVALENT)
49.75/13.47	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(10)
49.75/13.47	Obligation:
49.75/13.47	Q DP problem:
49.75/13.47	The TRS P consists of the following rules:
49.75/13.47	
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1)))
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{b_1}(x1))
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1)
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1)))
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{a_1}(x1))
49.75/13.47	   C_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
49.75/13.47	   B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
49.75/13.47	
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	We have to consider all minimal (P,Q,R)-chains.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(11) DependencyGraphProof (EQUIVALENT)
49.75/13.47	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(12)
49.75/13.47	Complex Obligation (AND)
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(13)
49.75/13.47	Obligation:
49.75/13.47	Q DP problem:
49.75/13.47	The TRS P consists of the following rules:
49.75/13.47	
49.75/13.47	   B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
49.75/13.47	
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	We have to consider all minimal (P,Q,R)-chains.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(14) UsableRulesProof (EQUIVALENT)
49.75/13.47	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.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(15)
49.75/13.47	Obligation:
49.75/13.47	Q DP problem:
49.75/13.47	The TRS P consists of the following rules:
49.75/13.47	
49.75/13.47	   B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
49.75/13.47	
49.75/13.47	R is empty.
49.75/13.47	Q is empty.
49.75/13.47	We have to consider all minimal (P,Q,R)-chains.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(16) QDPSizeChangeProof (EQUIVALENT)
49.75/13.47	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. 
49.75/13.47	
49.75/13.47	From the DPs we obtained the following set of size-change graphs:
49.75/13.47	*B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
49.75/13.47	The graph contains the following edges 1 > 1
49.75/13.47	
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(17)
49.75/13.47	YES
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(18)
49.75/13.47	Obligation:
49.75/13.47	Q DP problem:
49.75/13.47	The TRS P consists of the following rules:
49.75/13.47	
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1)
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1)))
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1)))
49.75/13.47	
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	We have to consider all minimal (P,Q,R)-chains.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(19) QDPOrderProof (EQUIVALENT)
49.75/13.47	We use the reduction pair processor [LPAR04,JAR06].
49.75/13.47	
49.75/13.47	
49.75/13.47	The following pairs can be oriented strictly and are deleted.
49.75/13.47	
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1)))
49.75/13.47	The remaining pairs can at least be oriented weakly.
49.75/13.47	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(C_{B_1}(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(b_{c_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(c_{a_1}(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, -I], [-I, 0A, 0A]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, -I, 0A], [-I, -I, -I]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(c_{b_1}(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(a_{a_1}(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, 1A], [0A, 1A, 0A]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(c_{c_1}(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 1A, 1A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(a_{c_1}(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	   <<<
49.75/13.47	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [-I, -I, -I]] 	* 	x_1
49.75/13.47	>>>
49.75/13.47	
49.75/13.47	
49.75/13.47	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(20)
49.75/13.47	Obligation:
49.75/13.47	Q DP problem:
49.75/13.47	The TRS P consists of the following rules:
49.75/13.47	
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1)
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1)))
49.75/13.47	
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	We have to consider all minimal (P,Q,R)-chains.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(21) QDPOrderProof (EQUIVALENT)
49.75/13.47	We use the reduction pair processor [LPAR04,JAR06].
49.75/13.47	
49.75/13.47	
49.75/13.47	The following pairs can be oriented strictly and are deleted.
49.75/13.47	
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1)
49.75/13.47	The remaining pairs can at least be oriented weakly.
49.75/13.47	Used ordering:  Polynomial interpretation [POLO]:
49.75/13.47	
49.75/13.47	   POL(C_{B_1}(x_1)) = x_1
49.75/13.47	   POL(a_{a_1}(x_1)) = 0
49.75/13.47	   POL(a_{b_1}(x_1)) = x_1
49.75/13.47	   POL(a_{c_1}(x_1)) = x_1
49.75/13.47	   POL(b_{a_1}(x_1)) = 0
49.75/13.47	   POL(b_{b_1}(x_1)) = x_1
49.75/13.47	   POL(b_{c_1}(x_1)) = 1 + x_1
49.75/13.47	   POL(c_{a_1}(x_1)) = x_1
49.75/13.47	   POL(c_{b_1}(x_1)) = 0
49.75/13.47	   POL(c_{c_1}(x_1)) = x_1
49.75/13.47	
49.75/13.47	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(22)
49.75/13.47	Obligation:
49.75/13.47	Q DP problem:
49.75/13.47	The TRS P consists of the following rules:
49.75/13.47	
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1)))
49.75/13.47	
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	We have to consider all minimal (P,Q,R)-chains.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(23) QDPOrderProof (EQUIVALENT)
49.75/13.47	We use the reduction pair processor [LPAR04,JAR06].
49.75/13.47	
49.75/13.47	
49.75/13.47	The following pairs can be oriented strictly and are deleted.
49.75/13.47	
49.75/13.47	   C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1)))
49.75/13.47	The remaining pairs can at least be oriented weakly.
49.75/13.47	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
49.75/13.47	
49.75/13.47	POL( C_{B_1}_1(x_1) ) = max{0, x_1 - 2}
49.75/13.47	POL( b_{c_1}_1(x_1) ) = 2x_1 + 2
49.75/13.47	POL( c_{b_1}_1(x_1) ) = max{0, -2}
49.75/13.47	POL( c_{a_1}_1(x_1) ) = x_1
49.75/13.47	POL( a_{b_1}_1(x_1) ) = x_1 + 2
49.75/13.47	POL( c_{c_1}_1(x_1) ) = x_1
49.75/13.47	POL( a_{a_1}_1(x_1) ) = max{0, -2}
49.75/13.47	POL( a_{c_1}_1(x_1) ) = 2
49.75/13.47	POL( b_{a_1}_1(x_1) ) = 1
49.75/13.47	POL( b_{b_1}_1(x_1) ) = 0
49.75/13.47	
49.75/13.47	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(24)
49.75/13.47	Obligation:
49.75/13.47	Q DP problem:
49.75/13.47	P is empty.
49.75/13.47	The TRS R consists of the following rules:
49.75/13.47	
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1))))
49.75/13.47	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1))
49.75/13.47	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1))
49.75/13.47	   b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1))
49.75/13.47	   c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
49.75/13.47	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
49.75/13.47	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
49.75/13.47	
49.75/13.47	Q is empty.
49.75/13.47	We have to consider all minimal (P,Q,R)-chains.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(25) PisEmptyProof (EQUIVALENT)
49.75/13.47	The TRS P is empty. Hence, there is no (P,Q,R) chain.
49.75/13.47	----------------------------------------
49.75/13.47	
49.75/13.47	(26)
49.75/13.47	YES
50.19/13.59	EOF