3.62/1.70	WORST_CASE(NON_POLY, ?)
3.73/1.71	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.73/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.73/1.71	
3.73/1.71	
3.73/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.73/1.71	
3.73/1.71	(0) CpxTRS
3.73/1.71	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.73/1.71	(2) TRS for Loop Detection
3.73/1.71	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.73/1.71	(4) BEST
3.73/1.71	    (5) proven lower bound
3.73/1.71	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.73/1.71	        (7) BOUNDS(n^1, INF)
3.73/1.71	    (8) TRS for Loop Detection
3.73/1.71	        (9) DecreasingLoopProof [FINISHED, 12 ms]
3.73/1.71	        (10) BOUNDS(EXP, INF)
3.73/1.71	
3.73/1.71	
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(0)
3.73/1.71	Obligation:
3.73/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.73/1.71	
3.73/1.71	
3.73/1.71	The TRS R consists of the following rules:
3.73/1.71	
3.73/1.71	   a__from(X) -> cons(mark(X), from(s(X)))
3.73/1.71	   a__head(cons(X, XS)) -> mark(X)
3.73/1.71	   a__2nd(cons(X, XS)) -> a__head(mark(XS))
3.73/1.71	   a__take(0, XS) -> nil
3.73/1.71	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
3.73/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.73/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.73/1.71	   mark(from(X)) -> a__from(mark(X))
3.73/1.71	   mark(head(X)) -> a__head(mark(X))
3.73/1.71	   mark(2nd(X)) -> a__2nd(mark(X))
3.73/1.71	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.73/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.73/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.73/1.71	   mark(s(X)) -> s(mark(X))
3.73/1.71	   mark(0) -> 0
3.73/1.71	   mark(nil) -> nil
3.73/1.71	   a__from(X) -> from(X)
3.73/1.71	   a__head(X) -> head(X)
3.73/1.71	   a__2nd(X) -> 2nd(X)
3.73/1.71	   a__take(X1, X2) -> take(X1, X2)
3.73/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.73/1.71	
3.73/1.71	S is empty.
3.73/1.71	Rewrite Strategy: FULL
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.73/1.71	Transformed a relative TRS into a decreasing-loop problem.
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(2)
3.73/1.71	Obligation:
3.73/1.71	Analyzing the following TRS for decreasing loops:
3.73/1.71	
3.73/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.73/1.71	
3.73/1.71	
3.73/1.71	The TRS R consists of the following rules:
3.73/1.71	
3.73/1.71	   a__from(X) -> cons(mark(X), from(s(X)))
3.73/1.71	   a__head(cons(X, XS)) -> mark(X)
3.73/1.71	   a__2nd(cons(X, XS)) -> a__head(mark(XS))
3.73/1.71	   a__take(0, XS) -> nil
3.73/1.71	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
3.73/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.73/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.73/1.71	   mark(from(X)) -> a__from(mark(X))
3.73/1.71	   mark(head(X)) -> a__head(mark(X))
3.73/1.71	   mark(2nd(X)) -> a__2nd(mark(X))
3.73/1.71	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.73/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.73/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.73/1.71	   mark(s(X)) -> s(mark(X))
3.73/1.71	   mark(0) -> 0
3.73/1.71	   mark(nil) -> nil
3.73/1.71	   a__from(X) -> from(X)
3.73/1.71	   a__head(X) -> head(X)
3.73/1.71	   a__2nd(X) -> 2nd(X)
3.73/1.71	   a__take(X1, X2) -> take(X1, X2)
3.73/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.73/1.71	
3.73/1.71	S is empty.
3.73/1.71	Rewrite Strategy: FULL
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.73/1.71	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.73/1.71	
3.73/1.71	The rewrite sequence
3.73/1.71	
3.73/1.71	mark(from(X)) ->^+ a__from(mark(X))
3.73/1.71	
3.73/1.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.73/1.71	
3.73/1.71	The pumping substitution is [X / from(X)].
3.73/1.71	
3.73/1.71	The result substitution is [ ].
3.73/1.71	
3.73/1.71	
3.73/1.71	
3.73/1.71	
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(4)
3.73/1.71	Complex Obligation (BEST)
3.73/1.71	
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(5)
3.73/1.71	Obligation:
3.73/1.71	Proved the lower bound n^1 for the following obligation:
3.73/1.71	
3.73/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.73/1.71	
3.73/1.71	
3.73/1.71	The TRS R consists of the following rules:
3.73/1.71	
3.73/1.71	   a__from(X) -> cons(mark(X), from(s(X)))
3.73/1.71	   a__head(cons(X, XS)) -> mark(X)
3.73/1.71	   a__2nd(cons(X, XS)) -> a__head(mark(XS))
3.73/1.71	   a__take(0, XS) -> nil
3.73/1.71	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
3.73/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.73/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.73/1.71	   mark(from(X)) -> a__from(mark(X))
3.73/1.71	   mark(head(X)) -> a__head(mark(X))
3.73/1.71	   mark(2nd(X)) -> a__2nd(mark(X))
3.73/1.71	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.73/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.73/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.73/1.71	   mark(s(X)) -> s(mark(X))
3.73/1.71	   mark(0) -> 0
3.73/1.71	   mark(nil) -> nil
3.73/1.71	   a__from(X) -> from(X)
3.73/1.71	   a__head(X) -> head(X)
3.73/1.71	   a__2nd(X) -> 2nd(X)
3.73/1.71	   a__take(X1, X2) -> take(X1, X2)
3.73/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.73/1.71	
3.73/1.71	S is empty.
3.73/1.71	Rewrite Strategy: FULL
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(6) LowerBoundPropagationProof (FINISHED)
3.73/1.71	Propagated lower bound.
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(7)
3.73/1.71	BOUNDS(n^1, INF)
3.73/1.71	
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(8)
3.73/1.71	Obligation:
3.73/1.71	Analyzing the following TRS for decreasing loops:
3.73/1.71	
3.73/1.71	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.73/1.71	
3.73/1.71	
3.73/1.71	The TRS R consists of the following rules:
3.73/1.71	
3.73/1.71	   a__from(X) -> cons(mark(X), from(s(X)))
3.73/1.71	   a__head(cons(X, XS)) -> mark(X)
3.73/1.71	   a__2nd(cons(X, XS)) -> a__head(mark(XS))
3.73/1.71	   a__take(0, XS) -> nil
3.73/1.71	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
3.73/1.71	   a__sel(0, cons(X, XS)) -> mark(X)
3.73/1.71	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
3.73/1.71	   mark(from(X)) -> a__from(mark(X))
3.73/1.71	   mark(head(X)) -> a__head(mark(X))
3.73/1.71	   mark(2nd(X)) -> a__2nd(mark(X))
3.73/1.71	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.73/1.71	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
3.73/1.71	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.73/1.71	   mark(s(X)) -> s(mark(X))
3.73/1.71	   mark(0) -> 0
3.73/1.71	   mark(nil) -> nil
3.73/1.71	   a__from(X) -> from(X)
3.73/1.71	   a__head(X) -> head(X)
3.73/1.71	   a__2nd(X) -> 2nd(X)
3.73/1.71	   a__take(X1, X2) -> take(X1, X2)
3.73/1.71	   a__sel(X1, X2) -> sel(X1, X2)
3.73/1.71	
3.73/1.71	S is empty.
3.73/1.71	Rewrite Strategy: FULL
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(9) DecreasingLoopProof (FINISHED)
3.73/1.71	The following loop(s) give(s) rise to the lower bound EXP:
3.73/1.71	
3.73/1.71	The rewrite sequence
3.73/1.71	
3.73/1.71	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.73/1.71	
3.73/1.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.73/1.71	
3.73/1.71	The pumping substitution is [X / from(X)].
3.73/1.71	
3.73/1.71	The result substitution is [ ].
3.73/1.71	
3.73/1.71	
3.73/1.71	
3.73/1.71	The rewrite sequence
3.73/1.71	
3.73/1.71	mark(from(X)) ->^+ cons(mark(mark(X)), from(s(mark(X))))
3.73/1.71	
3.73/1.71	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.73/1.71	
3.73/1.71	The pumping substitution is [X / from(X)].
3.73/1.71	
3.73/1.71	The result substitution is [ ].
3.73/1.71	
3.73/1.71	
3.73/1.71	
3.73/1.71	
3.73/1.71	----------------------------------------
3.73/1.71	
3.73/1.71	(10)
3.73/1.71	BOUNDS(EXP, INF)
3.78/1.74	EOF