9.91/3.38	WORST_CASE(NON_POLY, ?)
9.91/3.39	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
9.91/3.39	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.91/3.39	
9.91/3.39	
9.91/3.39	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
9.91/3.39	
9.91/3.39	(0) CpxTRS
9.91/3.39	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
9.91/3.39	(2) TRS for Loop Detection
9.91/3.39	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
9.91/3.39	(4) BEST
9.91/3.39	    (5) proven lower bound
9.91/3.39	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
9.91/3.39	        (7) BOUNDS(n^1, INF)
9.91/3.39	    (8) TRS for Loop Detection
9.91/3.39	        (9) InfiniteLowerBoundProof [FINISHED, 1084 ms]
9.91/3.39	        (10) BOUNDS(INF, INF)
9.91/3.39	
9.91/3.39	
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(0)
9.91/3.39	Obligation:
9.91/3.39	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
9.91/3.39	
9.91/3.39	
9.91/3.39	The TRS R consists of the following rules:
9.91/3.39	
9.91/3.39	   a__zeros -> cons(0, zeros)
9.91/3.39	   a__U11(tt, L) -> a__U12(tt, L)
9.91/3.39	   a__U12(tt, L) -> s(a__length(mark(L)))
9.91/3.39	   a__length(nil) -> 0
9.91/3.39	   a__length(cons(N, L)) -> a__U11(tt, L)
9.91/3.39	   mark(zeros) -> a__zeros
9.91/3.39	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
9.91/3.39	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
9.91/3.39	   mark(length(X)) -> a__length(mark(X))
9.91/3.39	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
9.91/3.39	   mark(0) -> 0
9.91/3.39	   mark(tt) -> tt
9.91/3.39	   mark(s(X)) -> s(mark(X))
9.91/3.39	   mark(nil) -> nil
9.91/3.39	   a__zeros -> zeros
9.91/3.39	   a__U11(X1, X2) -> U11(X1, X2)
9.91/3.39	   a__U12(X1, X2) -> U12(X1, X2)
9.91/3.39	   a__length(X) -> length(X)
9.91/3.39	
9.91/3.39	S is empty.
9.91/3.39	Rewrite Strategy: INNERMOST
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
9.91/3.39	Transformed a relative TRS into a decreasing-loop problem.
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(2)
9.91/3.39	Obligation:
9.91/3.39	Analyzing the following TRS for decreasing loops:
9.91/3.39	
9.91/3.39	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
9.91/3.39	
9.91/3.39	
9.91/3.39	The TRS R consists of the following rules:
9.91/3.39	
9.91/3.39	   a__zeros -> cons(0, zeros)
9.91/3.39	   a__U11(tt, L) -> a__U12(tt, L)
9.91/3.39	   a__U12(tt, L) -> s(a__length(mark(L)))
9.91/3.39	   a__length(nil) -> 0
9.91/3.39	   a__length(cons(N, L)) -> a__U11(tt, L)
9.91/3.39	   mark(zeros) -> a__zeros
9.91/3.39	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
9.91/3.39	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
9.91/3.39	   mark(length(X)) -> a__length(mark(X))
9.91/3.39	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
9.91/3.39	   mark(0) -> 0
9.91/3.39	   mark(tt) -> tt
9.91/3.39	   mark(s(X)) -> s(mark(X))
9.91/3.39	   mark(nil) -> nil
9.91/3.39	   a__zeros -> zeros
9.91/3.39	   a__U11(X1, X2) -> U11(X1, X2)
9.91/3.39	   a__U12(X1, X2) -> U12(X1, X2)
9.91/3.39	   a__length(X) -> length(X)
9.91/3.39	
9.91/3.39	S is empty.
9.91/3.39	Rewrite Strategy: INNERMOST
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(3) DecreasingLoopProof (LOWER BOUND(ID))
9.91/3.39	The following loop(s) give(s) rise to the lower bound Omega(n^1):
9.91/3.39	
9.91/3.39	The rewrite sequence
9.91/3.39	
9.91/3.39	mark(length(X)) ->^+ a__length(mark(X))
9.91/3.39	
9.91/3.39	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
9.91/3.39	
9.91/3.39	The pumping substitution is [X / length(X)].
9.91/3.39	
9.91/3.39	The result substitution is [ ].
9.91/3.39	
9.91/3.39	
9.91/3.39	
9.91/3.39	
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(4)
9.91/3.39	Complex Obligation (BEST)
9.91/3.39	
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(5)
9.91/3.39	Obligation:
9.91/3.39	Proved the lower bound n^1 for the following obligation:
9.91/3.39	
9.91/3.39	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
9.91/3.39	
9.91/3.39	
9.91/3.39	The TRS R consists of the following rules:
9.91/3.39	
9.91/3.39	   a__zeros -> cons(0, zeros)
9.91/3.39	   a__U11(tt, L) -> a__U12(tt, L)
9.91/3.39	   a__U12(tt, L) -> s(a__length(mark(L)))
9.91/3.39	   a__length(nil) -> 0
9.91/3.39	   a__length(cons(N, L)) -> a__U11(tt, L)
9.91/3.39	   mark(zeros) -> a__zeros
9.91/3.39	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
9.91/3.39	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
9.91/3.39	   mark(length(X)) -> a__length(mark(X))
9.91/3.39	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
9.91/3.39	   mark(0) -> 0
9.91/3.39	   mark(tt) -> tt
9.91/3.39	   mark(s(X)) -> s(mark(X))
9.91/3.39	   mark(nil) -> nil
9.91/3.39	   a__zeros -> zeros
9.91/3.39	   a__U11(X1, X2) -> U11(X1, X2)
9.91/3.39	   a__U12(X1, X2) -> U12(X1, X2)
9.91/3.39	   a__length(X) -> length(X)
9.91/3.39	
9.91/3.39	S is empty.
9.91/3.39	Rewrite Strategy: INNERMOST
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(6) LowerBoundPropagationProof (FINISHED)
9.91/3.39	Propagated lower bound.
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(7)
9.91/3.39	BOUNDS(n^1, INF)
9.91/3.39	
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(8)
9.91/3.39	Obligation:
9.91/3.39	Analyzing the following TRS for decreasing loops:
9.91/3.39	
9.91/3.39	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
9.91/3.39	
9.91/3.39	
9.91/3.39	The TRS R consists of the following rules:
9.91/3.39	
9.91/3.39	   a__zeros -> cons(0, zeros)
9.91/3.39	   a__U11(tt, L) -> a__U12(tt, L)
9.91/3.39	   a__U12(tt, L) -> s(a__length(mark(L)))
9.91/3.39	   a__length(nil) -> 0
9.91/3.39	   a__length(cons(N, L)) -> a__U11(tt, L)
9.91/3.39	   mark(zeros) -> a__zeros
9.91/3.39	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
9.91/3.39	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
9.91/3.39	   mark(length(X)) -> a__length(mark(X))
9.91/3.39	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
9.91/3.39	   mark(0) -> 0
9.91/3.39	   mark(tt) -> tt
9.91/3.39	   mark(s(X)) -> s(mark(X))
9.91/3.39	   mark(nil) -> nil
9.91/3.39	   a__zeros -> zeros
9.91/3.39	   a__U11(X1, X2) -> U11(X1, X2)
9.91/3.39	   a__U12(X1, X2) -> U12(X1, X2)
9.91/3.39	   a__length(X) -> length(X)
9.91/3.39	
9.91/3.39	S is empty.
9.91/3.39	Rewrite Strategy: INNERMOST
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(9) InfiniteLowerBoundProof (FINISHED)
9.91/3.39	The following loop proves infinite runtime complexity:
9.91/3.39	
9.91/3.39	The rewrite sequence
9.91/3.39	
9.91/3.39	a__U11(tt, zeros) ->^+ s(a__U11(tt, zeros))
9.91/3.39	
9.91/3.39	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
9.91/3.39	
9.91/3.39	The pumping substitution is [ ].
9.91/3.39	
9.91/3.39	The result substitution is [ ].
9.91/3.39	
9.91/3.39	
9.91/3.39	
9.91/3.39	
9.91/3.39	----------------------------------------
9.91/3.39	
9.91/3.39	(10)
9.91/3.39	BOUNDS(INF, INF)
10.07/3.45	EOF