5.03/2.13	WORST_CASE(NON_POLY, ?)
5.03/2.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.03/2.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.03/2.13	
5.03/2.13	
5.03/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.03/2.13	
5.03/2.13	(0) CpxTRS
5.03/2.13	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
5.03/2.13	(2) TRS for Loop Detection
5.03/2.13	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
5.03/2.13	(4) BEST
5.03/2.13	    (5) proven lower bound
5.03/2.13	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
5.03/2.13	        (7) BOUNDS(n^1, INF)
5.03/2.13	    (8) TRS for Loop Detection
5.03/2.13	        (9) InfiniteLowerBoundProof [FINISHED, 145 ms]
5.03/2.13	        (10) BOUNDS(INF, INF)
5.03/2.13	
5.03/2.13	
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(0)
5.03/2.13	Obligation:
5.03/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.03/2.13	
5.03/2.13	
5.03/2.13	The TRS R consists of the following rules:
5.03/2.13	
5.03/2.13	   a__zeros -> cons(0, zeros)
5.03/2.13	   a__and(tt, X) -> mark(X)
5.03/2.13	   a__length(nil) -> 0
5.03/2.13	   a__length(cons(N, L)) -> s(a__length(mark(L)))
5.03/2.13	   a__take(0, IL) -> nil
5.03/2.13	   a__take(s(M), cons(N, IL)) -> cons(mark(N), take(M, IL))
5.03/2.13	   mark(zeros) -> a__zeros
5.03/2.13	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.03/2.13	   mark(length(X)) -> a__length(mark(X))
5.03/2.13	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
5.03/2.13	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
5.03/2.13	   mark(0) -> 0
5.03/2.13	   mark(tt) -> tt
5.03/2.13	   mark(nil) -> nil
5.03/2.13	   mark(s(X)) -> s(mark(X))
5.03/2.13	   a__zeros -> zeros
5.03/2.13	   a__and(X1, X2) -> and(X1, X2)
5.03/2.13	   a__length(X) -> length(X)
5.03/2.13	   a__take(X1, X2) -> take(X1, X2)
5.03/2.13	
5.03/2.13	S is empty.
5.03/2.13	Rewrite Strategy: FULL
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
5.03/2.13	Transformed a relative TRS into a decreasing-loop problem.
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(2)
5.03/2.13	Obligation:
5.03/2.13	Analyzing the following TRS for decreasing loops:
5.03/2.13	
5.03/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.03/2.13	
5.03/2.13	
5.03/2.13	The TRS R consists of the following rules:
5.03/2.13	
5.03/2.13	   a__zeros -> cons(0, zeros)
5.03/2.13	   a__and(tt, X) -> mark(X)
5.03/2.13	   a__length(nil) -> 0
5.03/2.13	   a__length(cons(N, L)) -> s(a__length(mark(L)))
5.03/2.13	   a__take(0, IL) -> nil
5.03/2.13	   a__take(s(M), cons(N, IL)) -> cons(mark(N), take(M, IL))
5.03/2.13	   mark(zeros) -> a__zeros
5.03/2.13	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.03/2.13	   mark(length(X)) -> a__length(mark(X))
5.03/2.13	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
5.03/2.13	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
5.03/2.13	   mark(0) -> 0
5.03/2.13	   mark(tt) -> tt
5.03/2.13	   mark(nil) -> nil
5.03/2.13	   mark(s(X)) -> s(mark(X))
5.03/2.13	   a__zeros -> zeros
5.03/2.13	   a__and(X1, X2) -> and(X1, X2)
5.03/2.13	   a__length(X) -> length(X)
5.03/2.13	   a__take(X1, X2) -> take(X1, X2)
5.03/2.13	
5.03/2.13	S is empty.
5.03/2.13	Rewrite Strategy: FULL
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(3) DecreasingLoopProof (LOWER BOUND(ID))
5.03/2.13	The following loop(s) give(s) rise to the lower bound Omega(n^1):
5.03/2.13	
5.03/2.13	The rewrite sequence
5.03/2.13	
5.03/2.13	mark(length(X)) ->^+ a__length(mark(X))
5.03/2.13	
5.03/2.13	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
5.03/2.13	
5.03/2.13	The pumping substitution is [X / length(X)].
5.03/2.13	
5.03/2.13	The result substitution is [ ].
5.03/2.13	
5.03/2.13	
5.03/2.13	
5.03/2.13	
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(4)
5.03/2.13	Complex Obligation (BEST)
5.03/2.13	
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(5)
5.03/2.13	Obligation:
5.03/2.13	Proved the lower bound n^1 for the following obligation:
5.03/2.13	
5.03/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.03/2.13	
5.03/2.13	
5.03/2.13	The TRS R consists of the following rules:
5.03/2.13	
5.03/2.13	   a__zeros -> cons(0, zeros)
5.03/2.13	   a__and(tt, X) -> mark(X)
5.03/2.13	   a__length(nil) -> 0
5.03/2.13	   a__length(cons(N, L)) -> s(a__length(mark(L)))
5.03/2.13	   a__take(0, IL) -> nil
5.03/2.13	   a__take(s(M), cons(N, IL)) -> cons(mark(N), take(M, IL))
5.03/2.13	   mark(zeros) -> a__zeros
5.03/2.13	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.03/2.13	   mark(length(X)) -> a__length(mark(X))
5.03/2.13	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
5.03/2.13	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
5.03/2.13	   mark(0) -> 0
5.03/2.13	   mark(tt) -> tt
5.03/2.13	   mark(nil) -> nil
5.03/2.13	   mark(s(X)) -> s(mark(X))
5.03/2.13	   a__zeros -> zeros
5.03/2.13	   a__and(X1, X2) -> and(X1, X2)
5.03/2.13	   a__length(X) -> length(X)
5.03/2.13	   a__take(X1, X2) -> take(X1, X2)
5.03/2.13	
5.03/2.13	S is empty.
5.03/2.13	Rewrite Strategy: FULL
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(6) LowerBoundPropagationProof (FINISHED)
5.03/2.13	Propagated lower bound.
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(7)
5.03/2.13	BOUNDS(n^1, INF)
5.03/2.13	
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(8)
5.03/2.13	Obligation:
5.03/2.13	Analyzing the following TRS for decreasing loops:
5.03/2.13	
5.03/2.13	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
5.03/2.13	
5.03/2.13	
5.03/2.13	The TRS R consists of the following rules:
5.03/2.13	
5.03/2.13	   a__zeros -> cons(0, zeros)
5.03/2.13	   a__and(tt, X) -> mark(X)
5.03/2.13	   a__length(nil) -> 0
5.03/2.13	   a__length(cons(N, L)) -> s(a__length(mark(L)))
5.03/2.13	   a__take(0, IL) -> nil
5.03/2.13	   a__take(s(M), cons(N, IL)) -> cons(mark(N), take(M, IL))
5.03/2.13	   mark(zeros) -> a__zeros
5.03/2.13	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
5.03/2.13	   mark(length(X)) -> a__length(mark(X))
5.03/2.13	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
5.03/2.13	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
5.03/2.13	   mark(0) -> 0
5.03/2.13	   mark(tt) -> tt
5.03/2.13	   mark(nil) -> nil
5.03/2.13	   mark(s(X)) -> s(mark(X))
5.03/2.13	   a__zeros -> zeros
5.03/2.13	   a__and(X1, X2) -> and(X1, X2)
5.03/2.13	   a__length(X) -> length(X)
5.03/2.13	   a__take(X1, X2) -> take(X1, X2)
5.03/2.13	
5.03/2.13	S is empty.
5.03/2.13	Rewrite Strategy: FULL
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(9) InfiniteLowerBoundProof (FINISHED)
5.03/2.13	The following loop proves infinite runtime complexity:
5.03/2.13	
5.03/2.13	The rewrite sequence
5.03/2.13	
5.03/2.13	a__length(cons(N, zeros)) ->^+ s(a__length(cons(0, zeros)))
5.03/2.13	
5.03/2.13	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
5.03/2.13	
5.03/2.13	The pumping substitution is [ ].
5.03/2.13	
5.03/2.13	The result substitution is [N / 0].
5.03/2.13	
5.03/2.13	
5.03/2.13	
5.03/2.13	
5.03/2.13	----------------------------------------
5.03/2.13	
5.03/2.13	(10)
5.03/2.13	BOUNDS(INF, INF)
5.03/2.16	EOF