311.28/82.18	WORST_CASE(NON_POLY, ?)
311.51/82.20	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
311.51/82.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
311.51/82.20	
311.51/82.20	
311.51/82.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
311.51/82.20	
311.51/82.20	(0) CpxTRS
311.51/82.20	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
311.51/82.20	(2) TRS for Loop Detection
311.51/82.20	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
311.51/82.20	(4) BEST
311.51/82.20	    (5) proven lower bound
311.51/82.20	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
311.51/82.20	        (7) BOUNDS(n^1, INF)
311.51/82.20	    (8) TRS for Loop Detection
311.51/82.20	        (9) InfiniteLowerBoundProof [FINISHED, 47.1 s]
311.51/82.20	        (10) BOUNDS(INF, INF)
311.51/82.20	
311.51/82.20	
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(0)
311.51/82.20	Obligation:
311.51/82.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
311.51/82.20	
311.51/82.20	
311.51/82.20	The TRS R consists of the following rules:
311.51/82.20	
311.51/82.20	   a__zeros -> cons(0, zeros)
311.51/82.20	   a__U11(tt, L) -> a__U12(tt, L)
311.51/82.20	   a__U12(tt, L) -> s(a__length(mark(L)))
311.51/82.20	   a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N)
311.51/82.20	   a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N)
311.51/82.20	   a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL))
311.51/82.20	   a__length(nil) -> 0
311.51/82.20	   a__length(cons(N, L)) -> a__U11(tt, L)
311.51/82.20	   a__take(0, IL) -> nil
311.51/82.20	   a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N)
311.51/82.20	   mark(zeros) -> a__zeros
311.51/82.20	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
311.51/82.20	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
311.51/82.20	   mark(length(X)) -> a__length(mark(X))
311.51/82.20	   mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4)
311.51/82.20	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
311.51/82.20	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
311.51/82.20	   mark(0) -> 0
311.51/82.20	   mark(tt) -> tt
311.51/82.20	   mark(s(X)) -> s(mark(X))
311.51/82.20	   mark(nil) -> nil
311.51/82.20	   a__zeros -> zeros
311.51/82.20	   a__U11(X1, X2) -> U11(X1, X2)
311.51/82.20	   a__U12(X1, X2) -> U12(X1, X2)
311.51/82.20	   a__length(X) -> length(X)
311.51/82.20	   a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4)
311.51/82.20	   a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4)
311.51/82.20	   a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4)
311.51/82.20	   a__take(X1, X2) -> take(X1, X2)
311.51/82.20	
311.51/82.20	S is empty.
311.51/82.20	Rewrite Strategy: FULL
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
311.51/82.20	Transformed a relative TRS into a decreasing-loop problem.
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(2)
311.51/82.20	Obligation:
311.51/82.20	Analyzing the following TRS for decreasing loops:
311.51/82.20	
311.51/82.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
311.51/82.20	
311.51/82.20	
311.51/82.20	The TRS R consists of the following rules:
311.51/82.20	
311.51/82.20	   a__zeros -> cons(0, zeros)
311.51/82.20	   a__U11(tt, L) -> a__U12(tt, L)
311.51/82.20	   a__U12(tt, L) -> s(a__length(mark(L)))
311.51/82.20	   a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N)
311.51/82.20	   a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N)
311.51/82.20	   a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL))
311.51/82.20	   a__length(nil) -> 0
311.51/82.20	   a__length(cons(N, L)) -> a__U11(tt, L)
311.51/82.20	   a__take(0, IL) -> nil
311.51/82.20	   a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N)
311.51/82.20	   mark(zeros) -> a__zeros
311.51/82.20	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
311.51/82.20	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
311.51/82.20	   mark(length(X)) -> a__length(mark(X))
311.51/82.20	   mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4)
311.51/82.20	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
311.51/82.20	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
311.51/82.20	   mark(0) -> 0
311.51/82.20	   mark(tt) -> tt
311.51/82.20	   mark(s(X)) -> s(mark(X))
311.51/82.20	   mark(nil) -> nil
311.51/82.20	   a__zeros -> zeros
311.51/82.20	   a__U11(X1, X2) -> U11(X1, X2)
311.51/82.20	   a__U12(X1, X2) -> U12(X1, X2)
311.51/82.20	   a__length(X) -> length(X)
311.51/82.20	   a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4)
311.51/82.20	   a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4)
311.51/82.20	   a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4)
311.51/82.20	   a__take(X1, X2) -> take(X1, X2)
311.51/82.20	
311.51/82.20	S is empty.
311.51/82.20	Rewrite Strategy: FULL
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(3) DecreasingLoopProof (LOWER BOUND(ID))
311.51/82.20	The following loop(s) give(s) rise to the lower bound Omega(n^1):
311.51/82.20	
311.51/82.20	The rewrite sequence
311.51/82.20	
311.51/82.20	mark(U21(X1, X2, X3, X4)) ->^+ a__U21(mark(X1), X2, X3, X4)
311.51/82.20	
311.51/82.20	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
311.51/82.20	
311.51/82.20	The pumping substitution is [X1 / U21(X1, X2, X3, X4)].
311.51/82.20	
311.51/82.20	The result substitution is [ ].
311.51/82.20	
311.51/82.20	
311.51/82.20	
311.51/82.20	
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(4)
311.51/82.20	Complex Obligation (BEST)
311.51/82.20	
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(5)
311.51/82.20	Obligation:
311.51/82.20	Proved the lower bound n^1 for the following obligation:
311.51/82.20	
311.51/82.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
311.51/82.20	
311.51/82.20	
311.51/82.20	The TRS R consists of the following rules:
311.51/82.20	
311.51/82.20	   a__zeros -> cons(0, zeros)
311.51/82.20	   a__U11(tt, L) -> a__U12(tt, L)
311.51/82.20	   a__U12(tt, L) -> s(a__length(mark(L)))
311.51/82.20	   a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N)
311.51/82.20	   a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N)
311.51/82.20	   a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL))
311.51/82.20	   a__length(nil) -> 0
311.51/82.20	   a__length(cons(N, L)) -> a__U11(tt, L)
311.51/82.20	   a__take(0, IL) -> nil
311.51/82.20	   a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N)
311.51/82.20	   mark(zeros) -> a__zeros
311.51/82.20	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
311.51/82.20	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
311.51/82.20	   mark(length(X)) -> a__length(mark(X))
311.51/82.20	   mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4)
311.51/82.20	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
311.51/82.20	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
311.51/82.20	   mark(0) -> 0
311.51/82.20	   mark(tt) -> tt
311.51/82.20	   mark(s(X)) -> s(mark(X))
311.51/82.20	   mark(nil) -> nil
311.51/82.20	   a__zeros -> zeros
311.51/82.20	   a__U11(X1, X2) -> U11(X1, X2)
311.51/82.20	   a__U12(X1, X2) -> U12(X1, X2)
311.51/82.20	   a__length(X) -> length(X)
311.51/82.20	   a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4)
311.51/82.20	   a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4)
311.51/82.20	   a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4)
311.51/82.20	   a__take(X1, X2) -> take(X1, X2)
311.51/82.20	
311.51/82.20	S is empty.
311.51/82.20	Rewrite Strategy: FULL
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(6) LowerBoundPropagationProof (FINISHED)
311.51/82.20	Propagated lower bound.
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(7)
311.51/82.20	BOUNDS(n^1, INF)
311.51/82.20	
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(8)
311.51/82.20	Obligation:
311.51/82.20	Analyzing the following TRS for decreasing loops:
311.51/82.20	
311.51/82.20	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
311.51/82.20	
311.51/82.20	
311.51/82.20	The TRS R consists of the following rules:
311.51/82.20	
311.51/82.20	   a__zeros -> cons(0, zeros)
311.51/82.20	   a__U11(tt, L) -> a__U12(tt, L)
311.51/82.20	   a__U12(tt, L) -> s(a__length(mark(L)))
311.51/82.20	   a__U21(tt, IL, M, N) -> a__U22(tt, IL, M, N)
311.51/82.20	   a__U22(tt, IL, M, N) -> a__U23(tt, IL, M, N)
311.51/82.20	   a__U23(tt, IL, M, N) -> cons(mark(N), take(M, IL))
311.51/82.20	   a__length(nil) -> 0
311.51/82.20	   a__length(cons(N, L)) -> a__U11(tt, L)
311.51/82.20	   a__take(0, IL) -> nil
311.51/82.20	   a__take(s(M), cons(N, IL)) -> a__U21(tt, IL, M, N)
311.51/82.20	   mark(zeros) -> a__zeros
311.51/82.20	   mark(U11(X1, X2)) -> a__U11(mark(X1), X2)
311.51/82.20	   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
311.51/82.20	   mark(length(X)) -> a__length(mark(X))
311.51/82.20	   mark(U21(X1, X2, X3, X4)) -> a__U21(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U22(X1, X2, X3, X4)) -> a__U22(mark(X1), X2, X3, X4)
311.51/82.20	   mark(U23(X1, X2, X3, X4)) -> a__U23(mark(X1), X2, X3, X4)
311.51/82.20	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
311.51/82.20	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
311.51/82.20	   mark(0) -> 0
311.51/82.20	   mark(tt) -> tt
311.51/82.20	   mark(s(X)) -> s(mark(X))
311.51/82.20	   mark(nil) -> nil
311.51/82.20	   a__zeros -> zeros
311.51/82.20	   a__U11(X1, X2) -> U11(X1, X2)
311.51/82.20	   a__U12(X1, X2) -> U12(X1, X2)
311.51/82.20	   a__length(X) -> length(X)
311.51/82.20	   a__U21(X1, X2, X3, X4) -> U21(X1, X2, X3, X4)
311.51/82.20	   a__U22(X1, X2, X3, X4) -> U22(X1, X2, X3, X4)
311.51/82.20	   a__U23(X1, X2, X3, X4) -> U23(X1, X2, X3, X4)
311.51/82.20	   a__take(X1, X2) -> take(X1, X2)
311.51/82.20	
311.51/82.20	S is empty.
311.51/82.20	Rewrite Strategy: FULL
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(9) InfiniteLowerBoundProof (FINISHED)
311.51/82.20	The following loop proves infinite runtime complexity:
311.51/82.20	
311.51/82.20	The rewrite sequence
311.51/82.20	
311.51/82.20	a__U11(tt, zeros) ->^+ s(a__U11(tt, zeros))
311.51/82.20	
311.51/82.20	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
311.51/82.20	
311.51/82.20	The pumping substitution is [ ].
311.51/82.20	
311.51/82.20	The result substitution is [ ].
311.51/82.20	
311.51/82.20	
311.51/82.20	
311.51/82.20	
311.51/82.20	----------------------------------------
311.51/82.20	
311.51/82.20	(10)
311.51/82.20	BOUNDS(INF, INF)
311.62/82.26	EOF