3.98/1.76	WORST_CASE(NON_POLY, ?)
3.98/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.98/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.98/1.77	
3.98/1.77	
3.98/1.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.77	
3.98/1.77	(0) CpxTRS
3.98/1.77	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.98/1.77	(2) TRS for Loop Detection
3.98/1.77	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.98/1.77	(4) BEST
3.98/1.77	    (5) proven lower bound
3.98/1.77	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.98/1.77	        (7) BOUNDS(n^1, INF)
3.98/1.77	    (8) TRS for Loop Detection
3.98/1.77	        (9) InfiniteLowerBoundProof [FINISHED, 66 ms]
3.98/1.77	        (10) BOUNDS(INF, INF)
3.98/1.77	
3.98/1.77	
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(0)
3.98/1.77	Obligation:
3.98/1.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.77	
3.98/1.77	
3.98/1.77	The TRS R consists of the following rules:
3.98/1.77	
3.98/1.77	   a__zeros -> cons(0, zeros)
3.98/1.77	   a__and(tt, X) -> mark(X)
3.98/1.77	   a__length(nil) -> 0
3.98/1.77	   a__length(cons(N, L)) -> s(a__length(mark(L)))
3.98/1.77	   mark(zeros) -> a__zeros
3.98/1.77	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
3.98/1.77	   mark(length(X)) -> a__length(mark(X))
3.98/1.77	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.98/1.77	   mark(0) -> 0
3.98/1.77	   mark(tt) -> tt
3.98/1.77	   mark(nil) -> nil
3.98/1.77	   mark(s(X)) -> s(mark(X))
3.98/1.77	   a__zeros -> zeros
3.98/1.77	   a__and(X1, X2) -> and(X1, X2)
3.98/1.77	   a__length(X) -> length(X)
3.98/1.77	
3.98/1.77	S is empty.
3.98/1.77	Rewrite Strategy: INNERMOST
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.98/1.77	Transformed a relative TRS into a decreasing-loop problem.
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(2)
3.98/1.77	Obligation:
3.98/1.77	Analyzing the following TRS for decreasing loops:
3.98/1.77	
3.98/1.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.77	
3.98/1.77	
3.98/1.77	The TRS R consists of the following rules:
3.98/1.77	
3.98/1.77	   a__zeros -> cons(0, zeros)
3.98/1.77	   a__and(tt, X) -> mark(X)
3.98/1.77	   a__length(nil) -> 0
3.98/1.77	   a__length(cons(N, L)) -> s(a__length(mark(L)))
3.98/1.77	   mark(zeros) -> a__zeros
3.98/1.77	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
3.98/1.77	   mark(length(X)) -> a__length(mark(X))
3.98/1.77	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.98/1.77	   mark(0) -> 0
3.98/1.77	   mark(tt) -> tt
3.98/1.77	   mark(nil) -> nil
3.98/1.77	   mark(s(X)) -> s(mark(X))
3.98/1.77	   a__zeros -> zeros
3.98/1.77	   a__and(X1, X2) -> and(X1, X2)
3.98/1.77	   a__length(X) -> length(X)
3.98/1.77	
3.98/1.77	S is empty.
3.98/1.77	Rewrite Strategy: INNERMOST
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.98/1.77	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.98/1.77	
3.98/1.77	The rewrite sequence
3.98/1.77	
3.98/1.77	mark(length(X)) ->^+ a__length(mark(X))
3.98/1.77	
3.98/1.77	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.98/1.77	
3.98/1.77	The pumping substitution is [X / length(X)].
3.98/1.77	
3.98/1.77	The result substitution is [ ].
3.98/1.77	
3.98/1.77	
3.98/1.77	
3.98/1.77	
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(4)
3.98/1.77	Complex Obligation (BEST)
3.98/1.77	
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(5)
3.98/1.77	Obligation:
3.98/1.77	Proved the lower bound n^1 for the following obligation:
3.98/1.77	
3.98/1.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.77	
3.98/1.77	
3.98/1.77	The TRS R consists of the following rules:
3.98/1.77	
3.98/1.77	   a__zeros -> cons(0, zeros)
3.98/1.77	   a__and(tt, X) -> mark(X)
3.98/1.77	   a__length(nil) -> 0
3.98/1.77	   a__length(cons(N, L)) -> s(a__length(mark(L)))
3.98/1.77	   mark(zeros) -> a__zeros
3.98/1.77	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
3.98/1.77	   mark(length(X)) -> a__length(mark(X))
3.98/1.77	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.98/1.77	   mark(0) -> 0
3.98/1.77	   mark(tt) -> tt
3.98/1.77	   mark(nil) -> nil
3.98/1.77	   mark(s(X)) -> s(mark(X))
3.98/1.77	   a__zeros -> zeros
3.98/1.77	   a__and(X1, X2) -> and(X1, X2)
3.98/1.77	   a__length(X) -> length(X)
3.98/1.77	
3.98/1.77	S is empty.
3.98/1.77	Rewrite Strategy: INNERMOST
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(6) LowerBoundPropagationProof (FINISHED)
3.98/1.77	Propagated lower bound.
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(7)
3.98/1.77	BOUNDS(n^1, INF)
3.98/1.77	
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(8)
3.98/1.77	Obligation:
3.98/1.77	Analyzing the following TRS for decreasing loops:
3.98/1.77	
3.98/1.77	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.98/1.77	
3.98/1.77	
3.98/1.77	The TRS R consists of the following rules:
3.98/1.77	
3.98/1.77	   a__zeros -> cons(0, zeros)
3.98/1.77	   a__and(tt, X) -> mark(X)
3.98/1.77	   a__length(nil) -> 0
3.98/1.77	   a__length(cons(N, L)) -> s(a__length(mark(L)))
3.98/1.77	   mark(zeros) -> a__zeros
3.98/1.77	   mark(and(X1, X2)) -> a__and(mark(X1), X2)
3.98/1.77	   mark(length(X)) -> a__length(mark(X))
3.98/1.77	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.98/1.77	   mark(0) -> 0
3.98/1.77	   mark(tt) -> tt
3.98/1.77	   mark(nil) -> nil
3.98/1.77	   mark(s(X)) -> s(mark(X))
3.98/1.77	   a__zeros -> zeros
3.98/1.77	   a__and(X1, X2) -> and(X1, X2)
3.98/1.77	   a__length(X) -> length(X)
3.98/1.77	
3.98/1.77	S is empty.
3.98/1.77	Rewrite Strategy: INNERMOST
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(9) InfiniteLowerBoundProof (FINISHED)
3.98/1.77	The following loop proves infinite runtime complexity:
3.98/1.77	
3.98/1.77	The rewrite sequence
3.98/1.77	
3.98/1.77	a__length(cons(N, zeros)) ->^+ s(a__length(cons(0, zeros)))
3.98/1.77	
3.98/1.77	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.98/1.77	
3.98/1.77	The pumping substitution is [ ].
3.98/1.77	
3.98/1.77	The result substitution is [N / 0].
3.98/1.77	
3.98/1.77	
3.98/1.77	
3.98/1.77	
3.98/1.77	----------------------------------------
3.98/1.77	
3.98/1.77	(10)
3.98/1.77	BOUNDS(INF, INF)
4.18/1.80	EOF