3.00/1.53	WORST_CASE(NON_POLY, ?)
3.00/1.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.00/1.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.00/1.54	
3.00/1.54	
3.00/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.00/1.54	
3.00/1.54	(0) CpxTRS
3.00/1.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.00/1.54	(2) TRS for Loop Detection
3.00/1.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.00/1.54	(4) BEST
3.00/1.54	    (5) proven lower bound
3.00/1.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.00/1.54	        (7) BOUNDS(n^1, INF)
3.00/1.54	    (8) TRS for Loop Detection
3.00/1.54	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.00/1.54	        (10) BOUNDS(INF, INF)
3.00/1.54	
3.00/1.54	
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(0)
3.00/1.54	Obligation:
3.00/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.00/1.54	
3.00/1.54	
3.00/1.54	The TRS R consists of the following rules:
3.00/1.54	
3.00/1.54	   eq(0, 0) -> true
3.00/1.54	   eq(s(X), s(Y)) -> eq(X, Y)
3.00/1.54	   eq(X, Y) -> false
3.00/1.54	   inf(X) -> cons(X, inf(s(X)))
3.00/1.54	   take(0, X) -> nil
3.00/1.54	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.00/1.54	   length(nil) -> 0
3.00/1.54	   length(cons(X, L)) -> s(length(L))
3.00/1.54	
3.00/1.54	S is empty.
3.00/1.54	Rewrite Strategy: INNERMOST
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.00/1.54	Transformed a relative TRS into a decreasing-loop problem.
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(2)
3.00/1.54	Obligation:
3.00/1.54	Analyzing the following TRS for decreasing loops:
3.00/1.54	
3.00/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.00/1.54	
3.00/1.54	
3.00/1.54	The TRS R consists of the following rules:
3.00/1.54	
3.00/1.54	   eq(0, 0) -> true
3.00/1.54	   eq(s(X), s(Y)) -> eq(X, Y)
3.00/1.54	   eq(X, Y) -> false
3.00/1.54	   inf(X) -> cons(X, inf(s(X)))
3.00/1.54	   take(0, X) -> nil
3.00/1.54	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.00/1.54	   length(nil) -> 0
3.00/1.54	   length(cons(X, L)) -> s(length(L))
3.00/1.54	
3.00/1.54	S is empty.
3.00/1.54	Rewrite Strategy: INNERMOST
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.00/1.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.00/1.54	
3.00/1.54	The rewrite sequence
3.00/1.54	
3.00/1.54	take(s(X), cons(Y, L)) ->^+ cons(Y, take(X, L))
3.00/1.54	
3.00/1.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.00/1.54	
3.00/1.54	The pumping substitution is [X / s(X), L / cons(Y, L)].
3.00/1.54	
3.00/1.54	The result substitution is [ ].
3.00/1.54	
3.00/1.54	
3.00/1.54	
3.00/1.54	
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(4)
3.00/1.54	Complex Obligation (BEST)
3.00/1.54	
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(5)
3.00/1.54	Obligation:
3.00/1.54	Proved the lower bound n^1 for the following obligation:
3.00/1.54	
3.00/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.00/1.54	
3.00/1.54	
3.00/1.54	The TRS R consists of the following rules:
3.00/1.54	
3.00/1.54	   eq(0, 0) -> true
3.00/1.54	   eq(s(X), s(Y)) -> eq(X, Y)
3.00/1.54	   eq(X, Y) -> false
3.00/1.54	   inf(X) -> cons(X, inf(s(X)))
3.00/1.54	   take(0, X) -> nil
3.00/1.54	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.00/1.54	   length(nil) -> 0
3.00/1.54	   length(cons(X, L)) -> s(length(L))
3.00/1.54	
3.00/1.54	S is empty.
3.00/1.54	Rewrite Strategy: INNERMOST
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(6) LowerBoundPropagationProof (FINISHED)
3.00/1.54	Propagated lower bound.
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(7)
3.00/1.54	BOUNDS(n^1, INF)
3.00/1.54	
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(8)
3.00/1.54	Obligation:
3.00/1.54	Analyzing the following TRS for decreasing loops:
3.00/1.54	
3.00/1.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.00/1.54	
3.00/1.54	
3.00/1.54	The TRS R consists of the following rules:
3.00/1.54	
3.00/1.54	   eq(0, 0) -> true
3.00/1.54	   eq(s(X), s(Y)) -> eq(X, Y)
3.00/1.54	   eq(X, Y) -> false
3.00/1.54	   inf(X) -> cons(X, inf(s(X)))
3.00/1.54	   take(0, X) -> nil
3.00/1.54	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.00/1.54	   length(nil) -> 0
3.00/1.54	   length(cons(X, L)) -> s(length(L))
3.00/1.54	
3.00/1.54	S is empty.
3.00/1.54	Rewrite Strategy: INNERMOST
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(9) InfiniteLowerBoundProof (FINISHED)
3.00/1.54	The following loop proves infinite runtime complexity:
3.00/1.54	
3.00/1.54	The rewrite sequence
3.00/1.54	
3.00/1.54	inf(X) ->^+ cons(X, inf(s(X)))
3.00/1.54	
3.00/1.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
3.00/1.54	
3.00/1.54	The pumping substitution is [ ].
3.00/1.54	
3.00/1.54	The result substitution is [X / s(X)].
3.00/1.54	
3.00/1.54	
3.00/1.54	
3.00/1.54	
3.00/1.54	----------------------------------------
3.00/1.54	
3.00/1.54	(10)
3.00/1.54	BOUNDS(INF, INF)
3.00/1.56	EOF