15.39/4.83	WORST_CASE(NON_POLY, ?)
15.39/4.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
15.39/4.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.39/4.84	
15.39/4.84	
15.39/4.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
15.39/4.84	
15.39/4.84	(0) CpxTRS
15.39/4.84	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
15.39/4.84	(2) TRS for Loop Detection
15.39/4.84	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
15.39/4.84	(4) BEST
15.39/4.84	    (5) proven lower bound
15.39/4.84	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
15.39/4.84	        (7) BOUNDS(n^1, INF)
15.39/4.84	    (8) TRS for Loop Detection
15.39/4.84	        (9) InfiniteLowerBoundProof [FINISHED, 1931 ms]
15.39/4.84	        (10) BOUNDS(INF, INF)
15.39/4.84	
15.39/4.84	
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(0)
15.39/4.84	Obligation:
15.39/4.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
15.39/4.84	
15.39/4.84	
15.39/4.84	The TRS R consists of the following rules:
15.39/4.84	
15.39/4.84	   a__zeros -> cons(0, zeros)
15.39/4.84	   a__U11(tt, L) -> a__U12(tt, L)
15.39/4.84	   a__U12(tt, L) -> s(a__length(mark(L)))
15.39/4.84	   a__length(nil) -> 0
15.39/4.84	   a__length(cons(N, L)) -> a__U11(tt, L)
15.39/4.84	   mark(zeros) -> a__zeros
15.39/4.84	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
15.39/4.84	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
15.39/4.84	   mark(length(X)) -> a__length(mark(X))
15.39/4.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
15.39/4.84	   mark(0) -> 0
15.39/4.84	   mark(tt) -> tt
15.39/4.84	   mark(s(X)) -> s(mark(X))
15.39/4.84	   mark(nil) -> nil
15.39/4.84	   a__zeros -> zeros
15.39/4.84	   a__U11(X1, X2) -> U11(X1, X2)
15.39/4.84	   a__U12(X1, X2) -> U12(X1, X2)
15.39/4.84	   a__length(X) -> length(X)
15.39/4.84	
15.39/4.84	S is empty.
15.39/4.84	Rewrite Strategy: FULL
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
15.39/4.84	Transformed a relative TRS into a decreasing-loop problem.
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(2)
15.39/4.84	Obligation:
15.39/4.84	Analyzing the following TRS for decreasing loops:
15.39/4.84	
15.39/4.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
15.39/4.84	
15.39/4.84	
15.39/4.84	The TRS R consists of the following rules:
15.39/4.84	
15.39/4.84	   a__zeros -> cons(0, zeros)
15.39/4.84	   a__U11(tt, L) -> a__U12(tt, L)
15.39/4.84	   a__U12(tt, L) -> s(a__length(mark(L)))
15.39/4.84	   a__length(nil) -> 0
15.39/4.84	   a__length(cons(N, L)) -> a__U11(tt, L)
15.39/4.84	   mark(zeros) -> a__zeros
15.39/4.84	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
15.39/4.84	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
15.39/4.84	   mark(length(X)) -> a__length(mark(X))
15.39/4.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
15.39/4.84	   mark(0) -> 0
15.39/4.84	   mark(tt) -> tt
15.39/4.84	   mark(s(X)) -> s(mark(X))
15.39/4.84	   mark(nil) -> nil
15.39/4.84	   a__zeros -> zeros
15.39/4.84	   a__U11(X1, X2) -> U11(X1, X2)
15.39/4.84	   a__U12(X1, X2) -> U12(X1, X2)
15.39/4.84	   a__length(X) -> length(X)
15.39/4.84	
15.39/4.84	S is empty.
15.39/4.84	Rewrite Strategy: FULL
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(3) DecreasingLoopProof (LOWER BOUND(ID))
15.39/4.84	The following loop(s) give(s) rise to the lower bound Omega(n^1):
15.39/4.84	
15.39/4.84	The rewrite sequence
15.39/4.84	
15.39/4.84	mark(length(X)) ->^+ a__length(mark(X))
15.39/4.84	
15.39/4.84	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
15.39/4.84	
15.39/4.84	The pumping substitution is [X / length(X)].
15.39/4.84	
15.39/4.84	The result substitution is [ ].
15.39/4.84	
15.39/4.84	
15.39/4.84	
15.39/4.84	
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(4)
15.39/4.84	Complex Obligation (BEST)
15.39/4.84	
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(5)
15.39/4.84	Obligation:
15.39/4.84	Proved the lower bound n^1 for the following obligation:
15.39/4.84	
15.39/4.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
15.39/4.84	
15.39/4.84	
15.39/4.84	The TRS R consists of the following rules:
15.39/4.84	
15.39/4.84	   a__zeros -> cons(0, zeros)
15.39/4.84	   a__U11(tt, L) -> a__U12(tt, L)
15.39/4.84	   a__U12(tt, L) -> s(a__length(mark(L)))
15.39/4.84	   a__length(nil) -> 0
15.39/4.84	   a__length(cons(N, L)) -> a__U11(tt, L)
15.39/4.84	   mark(zeros) -> a__zeros
15.39/4.84	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
15.39/4.84	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
15.39/4.84	   mark(length(X)) -> a__length(mark(X))
15.39/4.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
15.39/4.84	   mark(0) -> 0
15.39/4.84	   mark(tt) -> tt
15.39/4.84	   mark(s(X)) -> s(mark(X))
15.39/4.84	   mark(nil) -> nil
15.39/4.84	   a__zeros -> zeros
15.39/4.84	   a__U11(X1, X2) -> U11(X1, X2)
15.39/4.84	   a__U12(X1, X2) -> U12(X1, X2)
15.39/4.84	   a__length(X) -> length(X)
15.39/4.84	
15.39/4.84	S is empty.
15.39/4.84	Rewrite Strategy: FULL
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(6) LowerBoundPropagationProof (FINISHED)
15.39/4.84	Propagated lower bound.
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(7)
15.39/4.84	BOUNDS(n^1, INF)
15.39/4.84	
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(8)
15.39/4.84	Obligation:
15.39/4.84	Analyzing the following TRS for decreasing loops:
15.39/4.84	
15.39/4.84	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
15.39/4.84	
15.39/4.84	
15.39/4.84	The TRS R consists of the following rules:
15.39/4.84	
15.39/4.84	   a__zeros -> cons(0, zeros)
15.39/4.84	   a__U11(tt, L) -> a__U12(tt, L)
15.39/4.84	   a__U12(tt, L) -> s(a__length(mark(L)))
15.39/4.84	   a__length(nil) -> 0
15.39/4.84	   a__length(cons(N, L)) -> a__U11(tt, L)
15.39/4.84	   mark(zeros) -> a__zeros
15.39/4.84	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
15.39/4.84	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
15.39/4.84	   mark(length(X)) -> a__length(mark(X))
15.39/4.84	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
15.39/4.84	   mark(0) -> 0
15.39/4.84	   mark(tt) -> tt
15.39/4.84	   mark(s(X)) -> s(mark(X))
15.39/4.84	   mark(nil) -> nil
15.39/4.84	   a__zeros -> zeros
15.39/4.84	   a__U11(X1, X2) -> U11(X1, X2)
15.39/4.84	   a__U12(X1, X2) -> U12(X1, X2)
15.39/4.84	   a__length(X) -> length(X)
15.39/4.84	
15.39/4.84	S is empty.
15.39/4.84	Rewrite Strategy: FULL
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(9) InfiniteLowerBoundProof (FINISHED)
15.39/4.84	The following loop proves infinite runtime complexity:
15.39/4.84	
15.39/4.84	The rewrite sequence
15.39/4.84	
15.39/4.84	a__U11(tt, zeros) ->^+ s(a__U11(tt, zeros))
15.39/4.84	
15.39/4.84	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
15.39/4.84	
15.39/4.84	The pumping substitution is [ ].
15.39/4.84	
15.39/4.84	The result substitution is [ ].
15.39/4.84	
15.39/4.84	
15.39/4.84	
15.39/4.84	
15.39/4.84	----------------------------------------
15.39/4.84	
15.39/4.84	(10)
15.39/4.84	BOUNDS(INF, INF)
15.87/4.93	EOF