3.51/2.21	WORST_CASE(NON_POLY, ?)
3.51/2.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.51/2.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.51/2.22	
3.51/2.22	
3.51/2.22	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.51/2.22	
3.51/2.22	(0) CpxTRS
3.51/2.22	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.51/2.22	(2) TRS for Loop Detection
3.51/2.22	(3) DecreasingLoopProof [LOWER BOUND(ID), 31 ms]
3.51/2.22	(4) BEST
3.51/2.22	    (5) proven lower bound
3.51/2.22	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.51/2.22	        (7) BOUNDS(n^1, INF)
3.51/2.22	    (8) TRS for Loop Detection
3.51/2.22	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.51/2.22	        (10) BOUNDS(INF, INF)
3.51/2.22	
3.51/2.22	
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(0)
3.51/2.22	Obligation:
3.51/2.22	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.51/2.22	
3.51/2.22	
3.51/2.22	The TRS R consists of the following rules:
3.51/2.22	
3.51/2.22	   from(X) -> cons(X, n__from(s(X)))
3.51/2.22	   length(n__nil) -> 0
3.51/2.22	   length(n__cons(X, Y)) -> s(length1(activate(Y)))
3.51/2.22	   length1(X) -> length(activate(X))
3.51/2.22	   from(X) -> n__from(X)
3.51/2.22	   nil -> n__nil
3.51/2.22	   cons(X1, X2) -> n__cons(X1, X2)
3.51/2.22	   activate(n__from(X)) -> from(X)
3.51/2.22	   activate(n__nil) -> nil
3.51/2.22	   activate(n__cons(X1, X2)) -> cons(X1, X2)
3.51/2.22	   activate(X) -> X
3.51/2.22	
3.51/2.22	S is empty.
3.51/2.22	Rewrite Strategy: INNERMOST
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.51/2.22	Transformed a relative TRS into a decreasing-loop problem.
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(2)
3.51/2.22	Obligation:
3.51/2.22	Analyzing the following TRS for decreasing loops:
3.51/2.22	
3.51/2.22	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.51/2.22	
3.51/2.22	
3.51/2.22	The TRS R consists of the following rules:
3.51/2.22	
3.51/2.22	   from(X) -> cons(X, n__from(s(X)))
3.51/2.22	   length(n__nil) -> 0
3.51/2.22	   length(n__cons(X, Y)) -> s(length1(activate(Y)))
3.51/2.22	   length1(X) -> length(activate(X))
3.51/2.22	   from(X) -> n__from(X)
3.51/2.22	   nil -> n__nil
3.51/2.22	   cons(X1, X2) -> n__cons(X1, X2)
3.51/2.22	   activate(n__from(X)) -> from(X)
3.51/2.22	   activate(n__nil) -> nil
3.51/2.22	   activate(n__cons(X1, X2)) -> cons(X1, X2)
3.51/2.22	   activate(X) -> X
3.51/2.22	
3.51/2.22	S is empty.
3.51/2.22	Rewrite Strategy: INNERMOST
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.51/2.22	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.51/2.22	
3.51/2.22	The rewrite sequence
3.51/2.22	
3.51/2.22	length(n__cons(X, Y)) ->^+ s(length(Y))
3.51/2.22	
3.51/2.22	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.51/2.22	
3.51/2.22	The pumping substitution is [Y / n__cons(X, Y)].
3.51/2.22	
3.51/2.22	The result substitution is [ ].
3.51/2.22	
3.51/2.22	
3.51/2.22	
3.51/2.22	
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(4)
3.51/2.22	Complex Obligation (BEST)
3.51/2.22	
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(5)
3.51/2.22	Obligation:
3.51/2.22	Proved the lower bound n^1 for the following obligation:
3.51/2.22	
3.51/2.22	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.51/2.22	
3.51/2.22	
3.51/2.22	The TRS R consists of the following rules:
3.51/2.22	
3.51/2.22	   from(X) -> cons(X, n__from(s(X)))
3.51/2.22	   length(n__nil) -> 0
3.51/2.22	   length(n__cons(X, Y)) -> s(length1(activate(Y)))
3.51/2.22	   length1(X) -> length(activate(X))
3.51/2.22	   from(X) -> n__from(X)
3.51/2.22	   nil -> n__nil
3.51/2.22	   cons(X1, X2) -> n__cons(X1, X2)
3.51/2.22	   activate(n__from(X)) -> from(X)
3.51/2.22	   activate(n__nil) -> nil
3.51/2.22	   activate(n__cons(X1, X2)) -> cons(X1, X2)
3.51/2.22	   activate(X) -> X
3.51/2.22	
3.51/2.22	S is empty.
3.51/2.22	Rewrite Strategy: INNERMOST
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(6) LowerBoundPropagationProof (FINISHED)
3.51/2.22	Propagated lower bound.
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(7)
3.51/2.22	BOUNDS(n^1, INF)
3.51/2.22	
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(8)
3.51/2.22	Obligation:
3.51/2.22	Analyzing the following TRS for decreasing loops:
3.51/2.22	
3.51/2.22	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.51/2.22	
3.51/2.22	
3.51/2.22	The TRS R consists of the following rules:
3.51/2.22	
3.51/2.22	   from(X) -> cons(X, n__from(s(X)))
3.51/2.22	   length(n__nil) -> 0
3.51/2.22	   length(n__cons(X, Y)) -> s(length1(activate(Y)))
3.51/2.22	   length1(X) -> length(activate(X))
3.51/2.22	   from(X) -> n__from(X)
3.51/2.22	   nil -> n__nil
3.51/2.22	   cons(X1, X2) -> n__cons(X1, X2)
3.51/2.22	   activate(n__from(X)) -> from(X)
3.51/2.22	   activate(n__nil) -> nil
3.51/2.22	   activate(n__cons(X1, X2)) -> cons(X1, X2)
3.51/2.22	   activate(X) -> X
3.51/2.22	
3.51/2.22	S is empty.
3.51/2.22	Rewrite Strategy: INNERMOST
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(9) InfiniteLowerBoundProof (FINISHED)
3.51/2.22	The following loop proves infinite runtime complexity:
3.51/2.22	
3.51/2.22	The rewrite sequence
3.51/2.22	
3.51/2.22	length(n__cons(X, n__from(X1_0))) ->^+ s(length(n__cons(X1_0, n__from(s(X1_0)))))
3.51/2.22	
3.51/2.22	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.51/2.22	
3.51/2.22	The pumping substitution is [ ].
3.51/2.22	
3.51/2.22	The result substitution is [X / X1_0, X1_0 / s(X1_0)].
3.51/2.22	
3.51/2.22	
3.51/2.22	
3.51/2.22	
3.51/2.22	----------------------------------------
3.51/2.22	
3.51/2.22	(10)
3.51/2.22	BOUNDS(INF, INF)
3.76/2.25	EOF