3.98/1.89	WORST_CASE(NON_POLY, ?)
3.98/1.90	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.98/1.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.98/1.90	
3.98/1.90	
3.98/1.90	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.98/1.90	
3.98/1.90	(0) CpxTRS
3.98/1.90	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.98/1.90	(2) TRS for Loop Detection
3.98/1.90	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.98/1.90	(4) BEST
3.98/1.90	    (5) proven lower bound
3.98/1.90	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.98/1.90	        (7) BOUNDS(n^1, INF)
3.98/1.90	    (8) TRS for Loop Detection
3.98/1.90	        (9) DecreasingLoopProof [FINISHED, 101 ms]
3.98/1.90	        (10) BOUNDS(EXP, INF)
3.98/1.90	
3.98/1.90	
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(0)
3.98/1.90	Obligation:
3.98/1.90	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.98/1.90	
3.98/1.90	
3.98/1.90	The TRS R consists of the following rules:
3.98/1.90	
3.98/1.90	   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
3.98/1.90	   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
3.98/1.90	   U21(tt, X) -> U22(tt, activate(X))
3.98/1.90	   U22(tt, X) -> activate(X)
3.98/1.90	   U31(tt, N) -> U32(tt, activate(N))
3.98/1.90	   U32(tt, N) -> activate(N)
3.98/1.90	   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
3.98/1.90	   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
3.98/1.90	   U51(tt, Y) -> U52(tt, activate(Y))
3.98/1.90	   U52(tt, Y) -> activate(Y)
3.98/1.90	   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
3.98/1.90	   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.98/1.90	   U71(tt, XS) -> U72(tt, activate(XS))
3.98/1.90	   U72(tt, XS) -> activate(XS)
3.98/1.90	   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
3.98/1.90	   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
3.98/1.90	   afterNth(N, XS) -> U11(tt, N, XS)
3.98/1.90	   fst(pair(X, Y)) -> U21(tt, X)
3.98/1.90	   head(cons(N, XS)) -> U31(tt, N)
3.98/1.90	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.98/1.90	   sel(N, XS) -> U41(tt, N, XS)
3.98/1.90	   snd(pair(X, Y)) -> U51(tt, Y)
3.98/1.90	   splitAt(0, XS) -> pair(nil, XS)
3.98/1.90	   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
3.98/1.90	   tail(cons(N, XS)) -> U71(tt, activate(XS))
3.98/1.90	   take(N, XS) -> U81(tt, N, XS)
3.98/1.90	   natsFrom(X) -> n__natsFrom(X)
3.98/1.90	   s(X) -> n__s(X)
3.98/1.90	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.98/1.90	   activate(n__s(X)) -> s(activate(X))
3.98/1.90	   activate(X) -> X
3.98/1.90	
3.98/1.90	S is empty.
3.98/1.90	Rewrite Strategy: FULL
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.98/1.90	Transformed a relative TRS into a decreasing-loop problem.
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(2)
3.98/1.90	Obligation:
3.98/1.90	Analyzing the following TRS for decreasing loops:
3.98/1.90	
3.98/1.90	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.98/1.90	
3.98/1.90	
3.98/1.90	The TRS R consists of the following rules:
3.98/1.90	
3.98/1.90	   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
3.98/1.90	   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
3.98/1.90	   U21(tt, X) -> U22(tt, activate(X))
3.98/1.90	   U22(tt, X) -> activate(X)
3.98/1.90	   U31(tt, N) -> U32(tt, activate(N))
3.98/1.90	   U32(tt, N) -> activate(N)
3.98/1.90	   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
3.98/1.90	   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
3.98/1.90	   U51(tt, Y) -> U52(tt, activate(Y))
3.98/1.90	   U52(tt, Y) -> activate(Y)
3.98/1.90	   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
3.98/1.90	   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.98/1.90	   U71(tt, XS) -> U72(tt, activate(XS))
3.98/1.90	   U72(tt, XS) -> activate(XS)
3.98/1.90	   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
3.98/1.90	   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
3.98/1.90	   afterNth(N, XS) -> U11(tt, N, XS)
3.98/1.90	   fst(pair(X, Y)) -> U21(tt, X)
3.98/1.90	   head(cons(N, XS)) -> U31(tt, N)
3.98/1.90	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.98/1.90	   sel(N, XS) -> U41(tt, N, XS)
3.98/1.90	   snd(pair(X, Y)) -> U51(tt, Y)
3.98/1.90	   splitAt(0, XS) -> pair(nil, XS)
3.98/1.90	   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
3.98/1.90	   tail(cons(N, XS)) -> U71(tt, activate(XS))
3.98/1.90	   take(N, XS) -> U81(tt, N, XS)
3.98/1.90	   natsFrom(X) -> n__natsFrom(X)
3.98/1.90	   s(X) -> n__s(X)
3.98/1.90	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.98/1.90	   activate(n__s(X)) -> s(activate(X))
3.98/1.90	   activate(X) -> X
3.98/1.90	
3.98/1.90	S is empty.
3.98/1.90	Rewrite Strategy: FULL
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.98/1.90	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.98/1.90	
3.98/1.90	The rewrite sequence
3.98/1.90	
3.98/1.90	activate(n__s(X)) ->^+ s(activate(X))
3.98/1.90	
3.98/1.90	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.98/1.90	
3.98/1.90	The pumping substitution is [X / n__s(X)].
3.98/1.90	
3.98/1.90	The result substitution is [ ].
3.98/1.90	
3.98/1.90	
3.98/1.90	
3.98/1.90	
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(4)
3.98/1.90	Complex Obligation (BEST)
3.98/1.90	
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(5)
3.98/1.90	Obligation:
3.98/1.90	Proved the lower bound n^1 for the following obligation:
3.98/1.90	
3.98/1.90	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.98/1.90	
3.98/1.90	
3.98/1.90	The TRS R consists of the following rules:
3.98/1.90	
3.98/1.90	   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
3.98/1.90	   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
3.98/1.90	   U21(tt, X) -> U22(tt, activate(X))
3.98/1.90	   U22(tt, X) -> activate(X)
3.98/1.90	   U31(tt, N) -> U32(tt, activate(N))
3.98/1.90	   U32(tt, N) -> activate(N)
3.98/1.90	   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
3.98/1.90	   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
3.98/1.90	   U51(tt, Y) -> U52(tt, activate(Y))
3.98/1.90	   U52(tt, Y) -> activate(Y)
3.98/1.90	   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
3.98/1.90	   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.98/1.90	   U71(tt, XS) -> U72(tt, activate(XS))
3.98/1.90	   U72(tt, XS) -> activate(XS)
3.98/1.90	   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
3.98/1.90	   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
3.98/1.90	   afterNth(N, XS) -> U11(tt, N, XS)
3.98/1.90	   fst(pair(X, Y)) -> U21(tt, X)
3.98/1.90	   head(cons(N, XS)) -> U31(tt, N)
3.98/1.90	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.98/1.90	   sel(N, XS) -> U41(tt, N, XS)
3.98/1.90	   snd(pair(X, Y)) -> U51(tt, Y)
3.98/1.90	   splitAt(0, XS) -> pair(nil, XS)
3.98/1.90	   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
3.98/1.90	   tail(cons(N, XS)) -> U71(tt, activate(XS))
3.98/1.90	   take(N, XS) -> U81(tt, N, XS)
3.98/1.90	   natsFrom(X) -> n__natsFrom(X)
3.98/1.90	   s(X) -> n__s(X)
3.98/1.90	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.98/1.90	   activate(n__s(X)) -> s(activate(X))
3.98/1.90	   activate(X) -> X
3.98/1.90	
3.98/1.90	S is empty.
3.98/1.90	Rewrite Strategy: FULL
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(6) LowerBoundPropagationProof (FINISHED)
3.98/1.90	Propagated lower bound.
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(7)
3.98/1.90	BOUNDS(n^1, INF)
3.98/1.90	
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(8)
3.98/1.90	Obligation:
3.98/1.90	Analyzing the following TRS for decreasing loops:
3.98/1.90	
3.98/1.90	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.98/1.90	
3.98/1.90	
3.98/1.90	The TRS R consists of the following rules:
3.98/1.90	
3.98/1.90	   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
3.98/1.90	   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
3.98/1.90	   U21(tt, X) -> U22(tt, activate(X))
3.98/1.90	   U22(tt, X) -> activate(X)
3.98/1.90	   U31(tt, N) -> U32(tt, activate(N))
3.98/1.90	   U32(tt, N) -> activate(N)
3.98/1.90	   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
3.98/1.90	   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
3.98/1.90	   U51(tt, Y) -> U52(tt, activate(Y))
3.98/1.90	   U52(tt, Y) -> activate(Y)
3.98/1.90	   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
3.98/1.90	   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
3.98/1.90	   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.98/1.90	   U71(tt, XS) -> U72(tt, activate(XS))
3.98/1.90	   U72(tt, XS) -> activate(XS)
3.98/1.90	   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
3.98/1.90	   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
3.98/1.90	   afterNth(N, XS) -> U11(tt, N, XS)
3.98/1.90	   fst(pair(X, Y)) -> U21(tt, X)
3.98/1.90	   head(cons(N, XS)) -> U31(tt, N)
3.98/1.90	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.98/1.90	   sel(N, XS) -> U41(tt, N, XS)
3.98/1.90	   snd(pair(X, Y)) -> U51(tt, Y)
3.98/1.90	   splitAt(0, XS) -> pair(nil, XS)
3.98/1.90	   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
3.98/1.90	   tail(cons(N, XS)) -> U71(tt, activate(XS))
3.98/1.90	   take(N, XS) -> U81(tt, N, XS)
3.98/1.90	   natsFrom(X) -> n__natsFrom(X)
3.98/1.90	   s(X) -> n__s(X)
3.98/1.90	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.98/1.90	   activate(n__s(X)) -> s(activate(X))
3.98/1.90	   activate(X) -> X
3.98/1.90	
3.98/1.90	S is empty.
3.98/1.90	Rewrite Strategy: FULL
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(9) DecreasingLoopProof (FINISHED)
3.98/1.90	The following loop(s) give(s) rise to the lower bound EXP:
3.98/1.90	
3.98/1.90	The rewrite sequence
3.98/1.90	
3.98/1.90	activate(n__natsFrom(X)) ->^+ cons(activate(X), n__natsFrom(n__s(activate(X))))
3.98/1.90	
3.98/1.90	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.98/1.90	
3.98/1.90	The pumping substitution is [X / n__natsFrom(X)].
3.98/1.90	
3.98/1.90	The result substitution is [ ].
3.98/1.90	
3.98/1.90	
3.98/1.90	
3.98/1.90	The rewrite sequence
3.98/1.90	
3.98/1.90	activate(n__natsFrom(X)) ->^+ cons(activate(X), n__natsFrom(n__s(activate(X))))
3.98/1.90	
3.98/1.90	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.98/1.90	
3.98/1.90	The pumping substitution is [X / n__natsFrom(X)].
3.98/1.90	
3.98/1.90	The result substitution is [ ].
3.98/1.90	
3.98/1.90	
3.98/1.90	
3.98/1.90	
3.98/1.90	----------------------------------------
3.98/1.90	
3.98/1.90	(10)
3.98/1.90	BOUNDS(EXP, INF)
4.23/1.94	EOF