47.53/12.92	YES
47.81/12.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
47.81/12.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
47.81/12.95	
47.81/12.95	
47.81/12.95	Termination w.r.t. Q of the given QTRS could be proven:
47.81/12.95	
47.81/12.95	(0) QTRS
47.81/12.95	(1) QTRS Reverse [EQUIVALENT, 0 ms]
47.81/12.95	(2) QTRS
47.81/12.95	(3) DependencyPairsProof [EQUIVALENT, 306 ms]
47.81/12.95	(4) QDP
47.81/12.95	(5) DependencyGraphProof [EQUIVALENT, 6 ms]
47.81/12.95	(6) AND
47.81/12.95	    (7) QDP
47.81/12.95	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
47.81/12.95	        (9) QDP
47.81/12.95	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
47.81/12.95	        (11) QDP
47.81/12.95	        (12) QDPOrderProof [EQUIVALENT, 438 ms]
47.81/12.95	        (13) QDP
47.81/12.95	        (14) QDPOrderProof [EQUIVALENT, 0 ms]
47.81/12.95	        (15) QDP
47.81/12.95	        (16) PisEmptyProof [EQUIVALENT, 0 ms]
47.81/12.95	        (17) YES
47.81/12.95	    (18) QDP
47.81/12.95	        (19) UsableRulesProof [EQUIVALENT, 0 ms]
47.81/12.95	        (20) QDP
47.81/12.95	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
47.81/12.95	        (22) YES
47.81/12.95	
47.81/12.95	
47.81/12.95	----------------------------------------
47.81/12.95	
47.81/12.95	(0)
47.81/12.95	Obligation:
47.81/12.95	Q restricted rewrite system:
47.81/12.95	The TRS R consists of the following rules:
47.81/12.95	
47.81/12.95	   0(1(1(x1))) -> 0(1(2(3(4(1(x1))))))
47.81/12.95	   0(1(1(x1))) -> 1(3(1(3(4(0(x1))))))
47.81/12.95	   0(1(1(x1))) -> 5(1(3(0(3(1(x1))))))
47.81/12.95	   0(1(4(x1))) -> 0(1(3(4(x1))))
47.81/12.95	   0(1(4(x1))) -> 1(3(4(2(0(x1)))))
47.81/12.95	   0(1(4(x1))) -> 1(5(3(4(0(x1)))))
47.81/12.95	   0(1(4(x1))) -> 3(1(3(4(0(x1)))))
47.81/12.95	   0(1(4(x1))) -> 1(1(5(3(4(0(x1))))))
47.81/12.95	   0(1(4(x1))) -> 1(2(3(4(3(0(x1))))))
47.81/12.95	   0(1(4(x1))) -> 1(3(3(4(4(0(x1))))))
47.81/12.95	   0(1(4(x1))) -> 1(3(4(5(0(5(x1))))))
47.81/12.95	   0(1(4(x1))) -> 3(4(5(5(0(1(x1))))))
47.81/12.95	   1(2(4(x1))) -> 1(2(3(4(1(x1)))))
47.81/12.95	   1(2(4(x1))) -> 5(3(4(1(2(x1)))))
47.81/12.95	   1(2(4(x1))) -> 1(5(2(3(4(3(x1))))))
47.81/12.95	   1(2(4(x1))) -> 2(3(3(4(5(1(x1))))))
47.81/12.95	   5(2(1(x1))) -> 1(2(2(3(5(4(x1))))))
47.81/12.95	   5(2(1(x1))) -> 1(3(2(5(3(4(x1))))))
47.81/12.95	   5(2(4(x1))) -> 0(5(2(3(4(x1)))))
47.81/12.95	   5(2(4(x1))) -> 5(5(3(4(2(x1)))))
47.81/12.95	   5(2(4(x1))) -> 0(3(4(4(5(2(x1))))))
47.81/12.95	   5(2(4(x1))) -> 2(2(5(3(4(4(x1))))))
47.81/12.95	   5(2(4(x1))) -> 2(3(3(4(5(5(x1))))))
47.81/12.95	   5(2(4(x1))) -> 2(3(4(3(5(3(x1))))))
47.81/12.95	   5(2(4(x1))) -> 2(5(3(4(0(5(x1))))))
47.81/12.95	   5(2(4(x1))) -> 3(1(2(5(3(4(x1))))))
47.81/12.95	   5(2(4(x1))) -> 5(2(3(4(0(3(x1))))))
47.81/12.95	   0(0(2(4(x1)))) -> 0(4(3(0(2(x1)))))
47.81/12.95	   0(1(1(5(x1)))) -> 0(1(3(5(1(2(x1))))))
47.81/12.95	   0(1(2(4(x1)))) -> 0(1(2(3(3(4(x1))))))
47.81/12.95	   0(1(4(5(x1)))) -> 3(4(0(5(1(x1)))))
47.81/12.95	   0(1(4(5(x1)))) -> 3(4(5(3(0(1(x1))))))
47.81/12.95	   0(4(2(1(x1)))) -> 0(4(1(2(3(4(x1))))))
47.81/12.95	   0(5(1(4(x1)))) -> 0(0(3(5(4(1(x1))))))
47.81/12.95	   1(0(1(4(x1)))) -> 3(4(0(1(2(1(x1))))))
47.81/12.95	   1(1(2(4(x1)))) -> 1(3(4(1(2(x1)))))
47.81/12.95	   1(2(2(4(x1)))) -> 2(2(2(3(4(1(x1))))))
47.81/12.95	   1(2(4(2(x1)))) -> 1(2(3(4(2(3(x1))))))
47.81/12.95	   1(5(2(1(x1)))) -> 1(0(3(5(1(2(x1))))))
47.81/12.95	   1(5(2(4(x1)))) -> 3(2(3(5(1(4(x1))))))
47.81/12.95	   1(5(2(4(x1)))) -> 5(2(3(3(1(4(x1))))))
47.81/12.95	   5(0(1(4(x1)))) -> 3(4(0(5(3(1(x1))))))
47.81/12.95	   5(2(5(1(x1)))) -> 3(5(1(5(2(x1)))))
47.81/12.95	   0(1(1(5(4(x1))))) -> 0(0(5(4(1(1(x1))))))
47.81/12.95	   1(0(0(2(4(x1))))) -> 0(2(0(4(4(1(x1))))))
47.81/12.95	   1(0(4(2(1(x1))))) -> 1(2(3(4(1(0(x1))))))
47.81/12.95	   1(2(5(0(1(x1))))) -> 2(3(5(1(0(1(x1))))))
47.81/12.95	   1(5(3(0(4(x1))))) -> 5(1(3(4(0(4(x1))))))
47.81/12.95	   5(0(5(2(4(x1))))) -> 4(0(0(5(5(2(x1))))))
47.81/12.95	   5(3(2(4(2(x1))))) -> 2(1(5(3(4(2(x1))))))
47.81/12.95	
47.81/12.95	Q is empty.
47.81/12.95	
47.81/12.95	----------------------------------------
47.81/12.95	
47.81/12.95	(1) QTRS Reverse (EQUIVALENT)
47.81/12.95	We applied the QTRS Reverse Processor [REVERSE].
47.81/12.95	----------------------------------------
47.81/12.95	
47.81/12.95	(2)
47.81/12.95	Obligation:
47.81/12.95	Q restricted rewrite system:
47.81/12.95	The TRS R consists of the following rules:
47.81/12.95	
47.81/12.95	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.81/12.95	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.81/12.95	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.81/12.95	   4(1(0(x1))) -> 4(3(1(0(x1))))
47.81/12.95	   4(1(0(x1))) -> 0(2(4(3(1(x1)))))
47.81/12.95	   4(1(0(x1))) -> 0(4(3(5(1(x1)))))
47.81/12.95	   4(1(0(x1))) -> 0(4(3(1(3(x1)))))
47.81/12.95	   4(1(0(x1))) -> 0(4(3(5(1(1(x1))))))
47.81/12.95	   4(1(0(x1))) -> 0(3(4(3(2(1(x1))))))
47.81/12.95	   4(1(0(x1))) -> 0(4(4(3(3(1(x1))))))
47.81/12.95	   4(1(0(x1))) -> 5(0(5(4(3(1(x1))))))
47.81/12.95	   4(1(0(x1))) -> 1(0(5(5(4(3(x1))))))
47.81/12.95	   4(2(1(x1))) -> 1(4(3(2(1(x1)))))
47.81/12.95	   4(2(1(x1))) -> 2(1(4(3(5(x1)))))
47.81/12.95	   4(2(1(x1))) -> 3(4(3(2(5(1(x1))))))
47.81/12.95	   4(2(1(x1))) -> 1(5(4(3(3(2(x1))))))
47.81/12.95	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.81/12.95	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.81/12.95	   4(2(5(x1))) -> 4(3(2(5(0(x1)))))
47.81/12.95	   4(2(5(x1))) -> 2(4(3(5(5(x1)))))
47.81/12.95	   4(2(5(x1))) -> 2(5(4(4(3(0(x1))))))
47.81/12.95	   4(2(5(x1))) -> 4(4(3(5(2(2(x1))))))
47.81/12.95	   4(2(5(x1))) -> 5(5(4(3(3(2(x1))))))
47.81/12.95	   4(2(5(x1))) -> 3(5(3(4(3(2(x1))))))
47.81/12.95	   4(2(5(x1))) -> 5(0(4(3(5(2(x1))))))
47.81/12.95	   4(2(5(x1))) -> 4(3(5(2(1(3(x1))))))
47.81/12.95	   4(2(5(x1))) -> 3(0(4(3(2(5(x1))))))
47.81/12.95	   4(2(0(0(x1)))) -> 2(0(3(4(0(x1)))))
47.81/12.95	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.81/12.95	   4(2(1(0(x1)))) -> 4(3(3(2(1(0(x1))))))
47.81/12.95	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.81/12.95	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.81/12.95	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.81/12.95	   4(1(5(0(x1)))) -> 1(4(5(3(0(0(x1))))))
47.81/12.95	   4(1(0(1(x1)))) -> 1(2(1(0(4(3(x1))))))
47.81/12.95	   4(2(1(1(x1)))) -> 2(1(4(3(1(x1)))))
47.81/12.95	   4(2(2(1(x1)))) -> 1(4(3(2(2(2(x1))))))
47.81/12.95	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.81/12.95	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.81/12.95	   4(2(5(1(x1)))) -> 4(1(5(3(2(3(x1))))))
47.81/12.95	   4(2(5(1(x1)))) -> 4(1(3(3(2(5(x1))))))
47.81/12.95	   4(1(0(5(x1)))) -> 1(3(5(0(4(3(x1))))))
47.81/12.95	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.81/12.95	   4(5(1(1(0(x1))))) -> 1(1(4(5(0(0(x1))))))
47.81/12.95	   4(2(0(0(1(x1))))) -> 1(4(4(0(2(0(x1))))))
47.81/12.95	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.81/12.95	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.81/12.95	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.81/12.95	   4(2(5(0(5(x1))))) -> 2(5(5(0(0(4(x1))))))
47.81/12.95	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.81/12.95	
47.81/12.95	Q is empty.
47.81/12.95	
47.81/12.95	----------------------------------------
47.81/12.95	
47.81/12.95	(3) DependencyPairsProof (EQUIVALENT)
47.81/12.95	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
47.81/12.95	----------------------------------------
47.81/12.95	
47.81/12.95	(4)
47.81/12.95	Obligation:
47.81/12.95	Q DP problem:
47.81/12.95	The TRS P consists of the following rules:
47.81/12.95	
47.81/12.95	   1^1(1(0(x1))) -> 1^1(4(3(2(1(0(x1))))))
47.81/12.95	   1^1(1(0(x1))) -> 4^1(3(2(1(0(x1)))))
47.81/12.95	   1^1(1(0(x1))) -> 2^1(1(0(x1)))
47.81/12.95	   1^1(1(0(x1))) -> 4^1(3(1(3(1(x1)))))
47.81/12.95	   1^1(1(0(x1))) -> 1^1(3(1(x1)))
47.81/12.95	   1^1(1(0(x1))) -> 1^1(x1)
47.81/12.95	   1^1(1(0(x1))) -> 1^1(3(0(3(1(5(x1))))))
47.81/12.95	   1^1(1(0(x1))) -> 1^1(5(x1))
47.81/12.95	   1^1(1(0(x1))) -> 5^1(x1)
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(1(0(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 2^1(4(3(1(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(1(x1)))
47.81/12.95	   4^1(1(0(x1))) -> 1^1(x1)
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(5(1(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 5^1(1(x1))
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(1(3(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 1^1(3(x1))
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(5(1(1(x1)))))
47.81/12.95	   4^1(1(0(x1))) -> 5^1(1(1(x1)))
47.81/12.95	   4^1(1(0(x1))) -> 1^1(1(x1))
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(2(1(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 2^1(1(x1))
47.81/12.95	   4^1(1(0(x1))) -> 4^1(4(3(3(1(x1)))))
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(3(1(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 5^1(0(5(4(3(1(x1))))))
47.81/12.95	   4^1(1(0(x1))) -> 5^1(4(3(1(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 1^1(0(5(5(4(3(x1))))))
47.81/12.95	   4^1(1(0(x1))) -> 5^1(5(4(3(x1))))
47.81/12.95	   4^1(1(0(x1))) -> 5^1(4(3(x1)))
47.81/12.95	   4^1(1(0(x1))) -> 4^1(3(x1))
47.81/12.95	   4^1(2(1(x1))) -> 1^1(4(3(2(1(x1)))))
47.81/12.95	   4^1(2(1(x1))) -> 4^1(3(2(1(x1))))
47.81/12.95	   4^1(2(1(x1))) -> 2^1(1(4(3(5(x1)))))
47.81/12.95	   4^1(2(1(x1))) -> 1^1(4(3(5(x1))))
47.81/12.95	   4^1(2(1(x1))) -> 4^1(3(5(x1)))
47.81/12.95	   4^1(2(1(x1))) -> 5^1(x1)
47.81/12.95	   4^1(2(1(x1))) -> 4^1(3(2(5(1(x1)))))
47.81/12.95	   4^1(2(1(x1))) -> 2^1(5(1(x1)))
47.81/12.95	   4^1(2(1(x1))) -> 5^1(1(x1))
47.81/12.95	   4^1(2(1(x1))) -> 1^1(5(4(3(3(2(x1))))))
47.81/12.95	   4^1(2(1(x1))) -> 5^1(4(3(3(2(x1)))))
47.81/12.95	   4^1(2(1(x1))) -> 4^1(3(3(2(x1))))
47.81/12.95	   4^1(2(1(x1))) -> 2^1(x1)
47.81/12.95	   1^1(2(5(x1))) -> 4^1(5(3(2(2(1(x1))))))
47.81/12.95	   1^1(2(5(x1))) -> 5^1(3(2(2(1(x1)))))
47.81/12.95	   1^1(2(5(x1))) -> 2^1(2(1(x1)))
47.81/12.95	   1^1(2(5(x1))) -> 2^1(1(x1))
47.81/12.95	   1^1(2(5(x1))) -> 1^1(x1)
47.81/12.95	   1^1(2(5(x1))) -> 4^1(3(5(2(3(1(x1))))))
47.81/12.95	   1^1(2(5(x1))) -> 5^1(2(3(1(x1))))
47.81/12.95	   1^1(2(5(x1))) -> 2^1(3(1(x1)))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(3(2(5(0(x1)))))
47.81/12.95	   4^1(2(5(x1))) -> 2^1(5(0(x1)))
47.81/12.95	   4^1(2(5(x1))) -> 5^1(0(x1))
47.81/12.95	   4^1(2(5(x1))) -> 2^1(4(3(5(5(x1)))))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(3(5(5(x1))))
47.81/12.95	   4^1(2(5(x1))) -> 5^1(5(x1))
47.81/12.95	   4^1(2(5(x1))) -> 2^1(5(4(4(3(0(x1))))))
47.81/12.95	   4^1(2(5(x1))) -> 5^1(4(4(3(0(x1)))))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(4(3(0(x1))))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(3(0(x1)))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(4(3(5(2(2(x1))))))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(3(5(2(2(x1)))))
47.81/12.95	   4^1(2(5(x1))) -> 5^1(2(2(x1)))
47.81/12.95	   4^1(2(5(x1))) -> 2^1(2(x1))
47.81/12.95	   4^1(2(5(x1))) -> 2^1(x1)
47.81/12.95	   4^1(2(5(x1))) -> 5^1(5(4(3(3(2(x1))))))
47.81/12.95	   4^1(2(5(x1))) -> 5^1(4(3(3(2(x1)))))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(3(3(2(x1))))
47.81/12.95	   4^1(2(5(x1))) -> 5^1(3(4(3(2(x1)))))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(3(2(x1)))
47.81/12.95	   4^1(2(5(x1))) -> 5^1(0(4(3(5(2(x1))))))
47.81/12.95	   4^1(2(5(x1))) -> 4^1(3(5(2(x1))))
47.98/12.99	   4^1(2(5(x1))) -> 5^1(2(x1))
47.98/12.99	   4^1(2(5(x1))) -> 4^1(3(5(2(1(3(x1))))))
47.98/12.99	   4^1(2(5(x1))) -> 5^1(2(1(3(x1))))
47.98/12.99	   4^1(2(5(x1))) -> 2^1(1(3(x1)))
47.98/12.99	   4^1(2(5(x1))) -> 1^1(3(x1))
47.98/12.99	   4^1(2(5(x1))) -> 4^1(3(2(5(x1))))
47.98/12.99	   4^1(2(0(0(x1)))) -> 2^1(0(3(4(0(x1)))))
47.98/12.99	   4^1(2(0(0(x1)))) -> 4^1(0(x1))
47.98/12.99	   5^1(1(1(0(x1)))) -> 2^1(1(5(3(1(0(x1))))))
47.98/12.99	   5^1(1(1(0(x1)))) -> 1^1(5(3(1(0(x1)))))
47.98/12.99	   5^1(1(1(0(x1)))) -> 5^1(3(1(0(x1))))
47.98/12.99	   4^1(2(1(0(x1)))) -> 4^1(3(3(2(1(0(x1))))))
47.98/12.99	   5^1(4(1(0(x1)))) -> 1^1(5(0(4(3(x1)))))
47.98/12.99	   5^1(4(1(0(x1)))) -> 5^1(0(4(3(x1))))
47.98/12.99	   5^1(4(1(0(x1)))) -> 4^1(3(x1))
47.98/12.99	   5^1(4(1(0(x1)))) -> 1^1(0(3(5(4(3(x1))))))
47.98/12.99	   5^1(4(1(0(x1)))) -> 5^1(4(3(x1)))
47.98/12.99	   1^1(2(4(0(x1)))) -> 4^1(3(2(1(4(0(x1))))))
47.98/12.99	   1^1(2(4(0(x1)))) -> 2^1(1(4(0(x1))))
47.98/12.99	   1^1(2(4(0(x1)))) -> 1^1(4(0(x1)))
47.98/12.99	   4^1(1(5(0(x1)))) -> 1^1(4(5(3(0(0(x1))))))
47.98/12.99	   4^1(1(5(0(x1)))) -> 4^1(5(3(0(0(x1)))))
47.98/12.99	   4^1(1(5(0(x1)))) -> 5^1(3(0(0(x1))))
47.98/12.99	   4^1(1(0(1(x1)))) -> 1^1(2(1(0(4(3(x1))))))
47.98/12.99	   4^1(1(0(1(x1)))) -> 2^1(1(0(4(3(x1)))))
47.98/12.99	   4^1(1(0(1(x1)))) -> 1^1(0(4(3(x1))))
47.98/12.99	   4^1(1(0(1(x1)))) -> 4^1(3(x1))
47.98/12.99	   4^1(2(1(1(x1)))) -> 2^1(1(4(3(1(x1)))))
47.98/12.99	   4^1(2(1(1(x1)))) -> 1^1(4(3(1(x1))))
47.98/12.99	   4^1(2(1(1(x1)))) -> 4^1(3(1(x1)))
47.98/12.99	   4^1(2(2(1(x1)))) -> 1^1(4(3(2(2(2(x1))))))
47.98/12.99	   4^1(2(2(1(x1)))) -> 4^1(3(2(2(2(x1)))))
47.98/12.99	   4^1(2(2(1(x1)))) -> 2^1(2(2(x1)))
47.98/12.99	   4^1(2(2(1(x1)))) -> 2^1(2(x1))
47.98/12.99	   4^1(2(2(1(x1)))) -> 2^1(x1)
47.98/12.99	   2^1(4(2(1(x1)))) -> 2^1(4(3(2(1(x1)))))
47.98/12.99	   2^1(4(2(1(x1)))) -> 4^1(3(2(1(x1))))
47.98/12.99	   1^1(2(5(1(x1)))) -> 2^1(1(5(3(0(1(x1))))))
47.98/12.99	   1^1(2(5(1(x1)))) -> 1^1(5(3(0(1(x1)))))
47.98/12.99	   1^1(2(5(1(x1)))) -> 5^1(3(0(1(x1))))
47.98/12.99	   4^1(2(5(1(x1)))) -> 4^1(1(5(3(2(3(x1))))))
47.98/12.99	   4^1(2(5(1(x1)))) -> 1^1(5(3(2(3(x1)))))
47.98/12.99	   4^1(2(5(1(x1)))) -> 5^1(3(2(3(x1))))
47.98/12.99	   4^1(2(5(1(x1)))) -> 2^1(3(x1))
47.98/12.99	   4^1(2(5(1(x1)))) -> 4^1(1(3(3(2(5(x1))))))
47.98/12.99	   4^1(2(5(1(x1)))) -> 1^1(3(3(2(5(x1)))))
47.98/12.99	   4^1(2(5(1(x1)))) -> 2^1(5(x1))
47.98/12.99	   4^1(2(5(1(x1)))) -> 5^1(x1)
47.98/12.99	   4^1(1(0(5(x1)))) -> 1^1(3(5(0(4(3(x1))))))
47.98/12.99	   4^1(1(0(5(x1)))) -> 5^1(0(4(3(x1))))
47.98/12.99	   4^1(1(0(5(x1)))) -> 4^1(3(x1))
47.98/12.99	   1^1(5(2(5(x1)))) -> 2^1(5(1(5(3(x1)))))
47.98/12.99	   1^1(5(2(5(x1)))) -> 5^1(1(5(3(x1))))
47.98/12.99	   1^1(5(2(5(x1)))) -> 1^1(5(3(x1)))
47.98/12.99	   1^1(5(2(5(x1)))) -> 5^1(3(x1))
47.98/12.99	   4^1(5(1(1(0(x1))))) -> 1^1(1(4(5(0(0(x1))))))
47.98/12.99	   4^1(5(1(1(0(x1))))) -> 1^1(4(5(0(0(x1)))))
47.98/12.99	   4^1(5(1(1(0(x1))))) -> 4^1(5(0(0(x1))))
47.98/12.99	   4^1(5(1(1(0(x1))))) -> 5^1(0(0(x1)))
47.98/12.99	   4^1(2(0(0(1(x1))))) -> 1^1(4(4(0(2(0(x1))))))
47.98/12.99	   4^1(2(0(0(1(x1))))) -> 4^1(4(0(2(0(x1)))))
47.98/12.99	   4^1(2(0(0(1(x1))))) -> 4^1(0(2(0(x1))))
47.98/12.99	   4^1(2(0(0(1(x1))))) -> 2^1(0(x1))
47.98/12.99	   1^1(2(4(0(1(x1))))) -> 1^1(4(3(2(1(x1)))))
47.98/12.99	   1^1(2(4(0(1(x1))))) -> 4^1(3(2(1(x1))))
47.98/12.99	   1^1(2(4(0(1(x1))))) -> 2^1(1(x1))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 1^1(0(1(5(3(2(x1))))))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 1^1(5(3(2(x1))))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 5^1(3(2(x1)))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 2^1(x1)
47.98/12.99	   4^1(0(3(5(1(x1))))) -> 4^1(0(4(3(1(5(x1))))))
47.98/12.99	   4^1(0(3(5(1(x1))))) -> 4^1(3(1(5(x1))))
47.98/12.99	   4^1(0(3(5(1(x1))))) -> 1^1(5(x1))
47.98/12.99	   4^1(0(3(5(1(x1))))) -> 5^1(x1)
47.98/12.99	   4^1(2(5(0(5(x1))))) -> 2^1(5(5(0(0(4(x1))))))
47.98/12.99	   4^1(2(5(0(5(x1))))) -> 5^1(5(0(0(4(x1)))))
47.98/12.99	   4^1(2(5(0(5(x1))))) -> 5^1(0(0(4(x1))))
47.98/12.99	   4^1(2(5(0(5(x1))))) -> 4^1(x1)
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 2^1(4(3(5(1(2(x1))))))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 4^1(3(5(1(2(x1)))))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 5^1(1(2(x1)))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 1^1(2(x1))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 2^1(x1)
47.98/12.99	
47.98/12.99	The TRS R consists of the following rules:
47.98/12.99	
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   4(1(0(x1))) -> 4(3(1(0(x1))))
47.98/12.99	   4(1(0(x1))) -> 0(2(4(3(1(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(5(1(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(1(3(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(5(1(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 0(3(4(3(2(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 0(4(4(3(3(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 5(0(5(4(3(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 1(0(5(5(4(3(x1))))))
47.98/12.99	   4(2(1(x1))) -> 1(4(3(2(1(x1)))))
47.98/12.99	   4(2(1(x1))) -> 2(1(4(3(5(x1)))))
47.98/12.99	   4(2(1(x1))) -> 3(4(3(2(5(1(x1))))))
47.98/12.99	   4(2(1(x1))) -> 1(5(4(3(3(2(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(3(2(5(0(x1)))))
47.98/12.99	   4(2(5(x1))) -> 2(4(3(5(5(x1)))))
47.98/12.99	   4(2(5(x1))) -> 2(5(4(4(3(0(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(4(3(5(2(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 5(5(4(3(3(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 3(5(3(4(3(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 5(0(4(3(5(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(3(5(2(1(3(x1))))))
47.98/12.99	   4(2(5(x1))) -> 3(0(4(3(2(5(x1))))))
47.98/12.99	   4(2(0(0(x1)))) -> 2(0(3(4(0(x1)))))
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   4(2(1(0(x1)))) -> 4(3(3(2(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   4(1(5(0(x1)))) -> 1(4(5(3(0(0(x1))))))
47.98/12.99	   4(1(0(1(x1)))) -> 1(2(1(0(4(3(x1))))))
47.98/12.99	   4(2(1(1(x1)))) -> 2(1(4(3(1(x1)))))
47.98/12.99	   4(2(2(1(x1)))) -> 1(4(3(2(2(2(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   4(2(5(1(x1)))) -> 4(1(5(3(2(3(x1))))))
47.98/12.99	   4(2(5(1(x1)))) -> 4(1(3(3(2(5(x1))))))
47.98/12.99	   4(1(0(5(x1)))) -> 1(3(5(0(4(3(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	   4(5(1(1(0(x1))))) -> 1(1(4(5(0(0(x1))))))
47.98/12.99	   4(2(0(0(1(x1))))) -> 1(4(4(0(2(0(x1))))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	   4(2(5(0(5(x1))))) -> 2(5(5(0(0(4(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(5) DependencyGraphProof (EQUIVALENT)
47.98/12.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 143 less nodes.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(6)
47.98/12.99	Complex Obligation (AND)
47.98/12.99	
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(7)
47.98/12.99	Obligation:
47.98/12.99	Q DP problem:
47.98/12.99	The TRS P consists of the following rules:
47.98/12.99	
47.98/12.99	   1^1(1(0(x1))) -> 2^1(1(0(x1)))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 1^1(2(x1))
47.98/12.99	   1^1(1(0(x1))) -> 1^1(x1)
47.98/12.99	   1^1(1(0(x1))) -> 1^1(5(x1))
47.98/12.99	   1^1(2(5(x1))) -> 2^1(2(1(x1)))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 2^1(x1)
47.98/12.99	   1^1(2(5(x1))) -> 2^1(1(x1))
47.98/12.99	   1^1(2(5(x1))) -> 1^1(x1)
47.98/12.99	   1^1(2(4(0(x1)))) -> 2^1(1(4(0(x1))))
47.98/12.99	   1^1(2(4(0(x1)))) -> 1^1(4(0(x1)))
47.98/12.99	   1^1(2(4(0(1(x1))))) -> 2^1(1(x1))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 2^1(x1)
47.98/12.99	
47.98/12.99	The TRS R consists of the following rules:
47.98/12.99	
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   4(1(0(x1))) -> 4(3(1(0(x1))))
47.98/12.99	   4(1(0(x1))) -> 0(2(4(3(1(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(5(1(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(1(3(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(5(1(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 0(3(4(3(2(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 0(4(4(3(3(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 5(0(5(4(3(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 1(0(5(5(4(3(x1))))))
47.98/12.99	   4(2(1(x1))) -> 1(4(3(2(1(x1)))))
47.98/12.99	   4(2(1(x1))) -> 2(1(4(3(5(x1)))))
47.98/12.99	   4(2(1(x1))) -> 3(4(3(2(5(1(x1))))))
47.98/12.99	   4(2(1(x1))) -> 1(5(4(3(3(2(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(3(2(5(0(x1)))))
47.98/12.99	   4(2(5(x1))) -> 2(4(3(5(5(x1)))))
47.98/12.99	   4(2(5(x1))) -> 2(5(4(4(3(0(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(4(3(5(2(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 5(5(4(3(3(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 3(5(3(4(3(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 5(0(4(3(5(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(3(5(2(1(3(x1))))))
47.98/12.99	   4(2(5(x1))) -> 3(0(4(3(2(5(x1))))))
47.98/12.99	   4(2(0(0(x1)))) -> 2(0(3(4(0(x1)))))
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   4(2(1(0(x1)))) -> 4(3(3(2(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   4(1(5(0(x1)))) -> 1(4(5(3(0(0(x1))))))
47.98/12.99	   4(1(0(1(x1)))) -> 1(2(1(0(4(3(x1))))))
47.98/12.99	   4(2(1(1(x1)))) -> 2(1(4(3(1(x1)))))
47.98/12.99	   4(2(2(1(x1)))) -> 1(4(3(2(2(2(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   4(2(5(1(x1)))) -> 4(1(5(3(2(3(x1))))))
47.98/12.99	   4(2(5(1(x1)))) -> 4(1(3(3(2(5(x1))))))
47.98/12.99	   4(1(0(5(x1)))) -> 1(3(5(0(4(3(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	   4(5(1(1(0(x1))))) -> 1(1(4(5(0(0(x1))))))
47.98/12.99	   4(2(0(0(1(x1))))) -> 1(4(4(0(2(0(x1))))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	   4(2(5(0(5(x1))))) -> 2(5(5(0(0(4(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(8) UsableRulesProof (EQUIVALENT)
47.98/12.99	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.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(9)
47.98/12.99	Obligation:
47.98/12.99	Q DP problem:
47.98/12.99	The TRS P consists of the following rules:
47.98/12.99	
47.98/12.99	   1^1(1(0(x1))) -> 2^1(1(0(x1)))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 1^1(2(x1))
47.98/12.99	   1^1(1(0(x1))) -> 1^1(x1)
47.98/12.99	   1^1(1(0(x1))) -> 1^1(5(x1))
47.98/12.99	   1^1(2(5(x1))) -> 2^1(2(1(x1)))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 2^1(x1)
47.98/12.99	   1^1(2(5(x1))) -> 2^1(1(x1))
47.98/12.99	   1^1(2(5(x1))) -> 1^1(x1)
47.98/12.99	   1^1(2(4(0(x1)))) -> 2^1(1(4(0(x1))))
47.98/12.99	   1^1(2(4(0(x1)))) -> 1^1(4(0(x1)))
47.98/12.99	   1^1(2(4(0(1(x1))))) -> 2^1(1(x1))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 2^1(x1)
47.98/12.99	
47.98/12.99	The TRS R consists of the following rules:
47.98/12.99	
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(10) DependencyGraphProof (EQUIVALENT)
47.98/12.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(11)
47.98/12.99	Obligation:
47.98/12.99	Q DP problem:
47.98/12.99	The TRS P consists of the following rules:
47.98/12.99	
47.98/12.99	   1^1(1(0(x1))) -> 1^1(x1)
47.98/12.99	   1^1(1(0(x1))) -> 1^1(5(x1))
47.98/12.99	   1^1(2(5(x1))) -> 1^1(x1)
47.98/12.99	   1^1(2(4(0(x1)))) -> 1^1(4(0(x1)))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 2^1(x1)
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 1^1(2(x1))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 2^1(x1)
47.98/12.99	
47.98/12.99	The TRS R consists of the following rules:
47.98/12.99	
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(12) QDPOrderProof (EQUIVALENT)
47.98/12.99	We use the reduction pair processor [LPAR04,JAR06].
47.98/12.99	
47.98/12.99	
47.98/12.99	The following pairs can be oriented strictly and are deleted.
47.98/12.99	
47.98/12.99	   1^1(1(0(x1))) -> 1^1(x1)
47.98/12.99	   1^1(2(5(x1))) -> 1^1(x1)
47.98/12.99	   1^1(2(4(0(x1)))) -> 1^1(4(0(x1)))
47.98/12.99	   1^1(0(5(2(1(x1))))) -> 2^1(x1)
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 1^1(2(x1))
47.98/12.99	   2^1(4(2(3(5(x1))))) -> 2^1(x1)
47.98/12.99	The remaining pairs can at least be oriented weakly.
47.98/12.99	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
47.98/12.99	
47.98/12.99	POL( 1^1_1(x_1) ) = max{0, 2x_1 - 2}
47.98/12.99	POL( 5_1(x_1) ) = x_1 + 2
47.98/12.99	POL( 1_1(x_1) ) = x_1
47.98/12.99	POL( 0_1(x_1) ) = x_1 + 2
47.98/12.99	POL( 2_1(x_1) ) = x_1 + 2
47.98/12.99	POL( 3_1(x_1) ) = max{0, x_1 - 2}
47.98/12.99	POL( 4_1(x_1) ) = x_1
47.98/12.99	POL( 2^1_1(x_1) ) = 2x_1
47.98/12.99	
47.98/12.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
47.98/12.99	
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	
47.98/12.99	
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(13)
47.98/12.99	Obligation:
47.98/12.99	Q DP problem:
47.98/12.99	The TRS P consists of the following rules:
47.98/12.99	
47.98/12.99	   1^1(1(0(x1))) -> 1^1(5(x1))
47.98/12.99	
47.98/12.99	The TRS R consists of the following rules:
47.98/12.99	
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(14) QDPOrderProof (EQUIVALENT)
47.98/12.99	We use the reduction pair processor [LPAR04,JAR06].
47.98/12.99	
47.98/12.99	
47.98/12.99	The following pairs can be oriented strictly and are deleted.
47.98/12.99	
47.98/12.99	   1^1(1(0(x1))) -> 1^1(5(x1))
47.98/12.99	The remaining pairs can at least be oriented weakly.
47.98/12.99	Used ordering:  Polynomial interpretation [POLO]:
47.98/12.99	
47.98/12.99	   POL(0(x_1)) = 1 + x_1
47.98/12.99	   POL(1(x_1)) = x_1
47.98/12.99	   POL(1^1(x_1)) = x_1
47.98/12.99	   POL(2(x_1)) = 0
47.98/12.99	   POL(3(x_1)) = 0
47.98/12.99	   POL(4(x_1)) = x_1
47.98/12.99	   POL(5(x_1)) = x_1
47.98/12.99	
47.98/12.99	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
47.98/12.99	
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	
47.98/12.99	
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(15)
47.98/12.99	Obligation:
47.98/12.99	Q DP problem:
47.98/12.99	P is empty.
47.98/12.99	The TRS R consists of the following rules:
47.98/12.99	
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(16) PisEmptyProof (EQUIVALENT)
47.98/12.99	The TRS P is empty. Hence, there is no (P,Q,R) chain.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(17)
47.98/12.99	YES
47.98/12.99	
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(18)
47.98/12.99	Obligation:
47.98/12.99	Q DP problem:
47.98/12.99	The TRS P consists of the following rules:
47.98/12.99	
47.98/12.99	   4^1(2(5(0(5(x1))))) -> 4^1(x1)
47.98/12.99	
47.98/12.99	The TRS R consists of the following rules:
47.98/12.99	
47.98/12.99	   1(1(0(x1))) -> 1(4(3(2(1(0(x1))))))
47.98/12.99	   1(1(0(x1))) -> 0(4(3(1(3(1(x1))))))
47.98/12.99	   1(1(0(x1))) -> 1(3(0(3(1(5(x1))))))
47.98/12.99	   4(1(0(x1))) -> 4(3(1(0(x1))))
47.98/12.99	   4(1(0(x1))) -> 0(2(4(3(1(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(5(1(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(1(3(x1)))))
47.98/12.99	   4(1(0(x1))) -> 0(4(3(5(1(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 0(3(4(3(2(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 0(4(4(3(3(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 5(0(5(4(3(1(x1))))))
47.98/12.99	   4(1(0(x1))) -> 1(0(5(5(4(3(x1))))))
47.98/12.99	   4(2(1(x1))) -> 1(4(3(2(1(x1)))))
47.98/12.99	   4(2(1(x1))) -> 2(1(4(3(5(x1)))))
47.98/12.99	   4(2(1(x1))) -> 3(4(3(2(5(1(x1))))))
47.98/12.99	   4(2(1(x1))) -> 1(5(4(3(3(2(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(5(3(2(2(1(x1))))))
47.98/12.99	   1(2(5(x1))) -> 4(3(5(2(3(1(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(3(2(5(0(x1)))))
47.98/12.99	   4(2(5(x1))) -> 2(4(3(5(5(x1)))))
47.98/12.99	   4(2(5(x1))) -> 2(5(4(4(3(0(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(4(3(5(2(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 5(5(4(3(3(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 3(5(3(4(3(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 5(0(4(3(5(2(x1))))))
47.98/12.99	   4(2(5(x1))) -> 4(3(5(2(1(3(x1))))))
47.98/12.99	   4(2(5(x1))) -> 3(0(4(3(2(5(x1))))))
47.98/12.99	   4(2(0(0(x1)))) -> 2(0(3(4(0(x1)))))
47.98/12.99	   5(1(1(0(x1)))) -> 2(1(5(3(1(0(x1))))))
47.98/12.99	   4(2(1(0(x1)))) -> 4(3(3(2(1(0(x1))))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(5(0(4(3(x1)))))
47.98/12.99	   5(4(1(0(x1)))) -> 1(0(3(5(4(3(x1))))))
47.98/12.99	   1(2(4(0(x1)))) -> 4(3(2(1(4(0(x1))))))
47.98/12.99	   4(1(5(0(x1)))) -> 1(4(5(3(0(0(x1))))))
47.98/12.99	   4(1(0(1(x1)))) -> 1(2(1(0(4(3(x1))))))
47.98/12.99	   4(2(1(1(x1)))) -> 2(1(4(3(1(x1)))))
47.98/12.99	   4(2(2(1(x1)))) -> 1(4(3(2(2(2(x1))))))
47.98/12.99	   2(4(2(1(x1)))) -> 3(2(4(3(2(1(x1))))))
47.98/12.99	   1(2(5(1(x1)))) -> 2(1(5(3(0(1(x1))))))
47.98/12.99	   4(2(5(1(x1)))) -> 4(1(5(3(2(3(x1))))))
47.98/12.99	   4(2(5(1(x1)))) -> 4(1(3(3(2(5(x1))))))
47.98/12.99	   4(1(0(5(x1)))) -> 1(3(5(0(4(3(x1))))))
47.98/12.99	   1(5(2(5(x1)))) -> 2(5(1(5(3(x1)))))
47.98/12.99	   4(5(1(1(0(x1))))) -> 1(1(4(5(0(0(x1))))))
47.98/12.99	   4(2(0(0(1(x1))))) -> 1(4(4(0(2(0(x1))))))
47.98/12.99	   1(2(4(0(1(x1))))) -> 0(1(4(3(2(1(x1))))))
47.98/12.99	   1(0(5(2(1(x1))))) -> 1(0(1(5(3(2(x1))))))
47.98/12.99	   4(0(3(5(1(x1))))) -> 4(0(4(3(1(5(x1))))))
47.98/12.99	   4(2(5(0(5(x1))))) -> 2(5(5(0(0(4(x1))))))
47.98/12.99	   2(4(2(3(5(x1))))) -> 2(4(3(5(1(2(x1))))))
47.98/12.99	
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(19) UsableRulesProof (EQUIVALENT)
47.98/12.99	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.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(20)
47.98/12.99	Obligation:
47.98/12.99	Q DP problem:
47.98/12.99	The TRS P consists of the following rules:
47.98/12.99	
47.98/12.99	   4^1(2(5(0(5(x1))))) -> 4^1(x1)
47.98/12.99	
47.98/12.99	R is empty.
47.98/12.99	Q is empty.
47.98/12.99	We have to consider all minimal (P,Q,R)-chains.
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(21) QDPSizeChangeProof (EQUIVALENT)
47.98/12.99	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. 
47.98/12.99	
47.98/12.99	From the DPs we obtained the following set of size-change graphs:
47.98/12.99	*4^1(2(5(0(5(x1))))) -> 4^1(x1)
47.98/12.99	The graph contains the following edges 1 > 1
47.98/12.99	
47.98/12.99	
47.98/12.99	----------------------------------------
47.98/12.99	
47.98/12.99	(22)
47.98/12.99	YES
48.37/13.16	EOF