* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__U101(X1,X2,X3) -> U101(X1,X2,X3)
a__U101(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U11(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),X) -> mark(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U31(tt(),N) -> mark(N)
a__U41(X1,X2) -> U41(X1,X2)
a__U41(tt(),N) -> cons(mark(N),natsFrom(s(N)))
a__U51(X1,X2,X3) -> U51(X1,X2,X3)
a__U51(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
a__U61(X1,X2) -> U61(X1,X2)
a__U61(tt(),Y) -> mark(Y)
a__U71(X1,X2) -> U71(X1,X2)
a__U71(tt(),XS) -> pair(nil(),mark(XS))
a__U81(X1,X2,X3,X4) -> U81(X1,X2,X3,X4)
a__U81(tt(),N,X,XS) -> a__U82(a__splitAt(mark(N),mark(XS)),X)
a__U82(X1,X2) -> U82(X1,X2)
a__U82(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
a__U91(X1,X2) -> U91(X1,X2)
a__U91(tt(),XS) -> mark(XS)
a__afterNth(N,XS) -> a__U11(a__and(a__isNatural(N),isLNat(XS)),N,XS)
a__afterNth(X1,X2) -> afterNth(X1,X2)
a__and(X1,X2) -> and(X1,X2)
a__and(tt(),X) -> mark(X)
a__fst(X) -> fst(X)
a__fst(pair(X,Y)) -> a__U21(a__and(a__isLNat(X),isLNat(Y)),X)
a__head(X) -> head(X)
a__head(cons(N,XS)) -> a__U31(a__and(a__isNatural(N),isLNat(XS)),N)
a__isLNat(X) -> isLNat(X)
a__isLNat(afterNth(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isLNat(cons(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isLNat(fst(V1)) -> a__isPLNat(V1)
a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
a__isLNat(nil()) -> tt()
a__isLNat(snd(V1)) -> a__isPLNat(V1)
a__isLNat(tail(V1)) -> a__isLNat(V1)
a__isLNat(take(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isNatural(X) -> isNatural(X)
a__isNatural(0()) -> tt()
a__isNatural(head(V1)) -> a__isLNat(V1)
a__isNatural(s(V1)) -> a__isNatural(V1)
a__isNatural(sel(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isPLNat(X) -> isPLNat(X)
a__isPLNat(pair(V1,V2)) -> a__and(a__isLNat(V1),isLNat(V2))
a__isPLNat(splitAt(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__natsFrom(N) -> a__U41(a__isNatural(N),N)
a__natsFrom(X) -> natsFrom(X)
a__sel(N,XS) -> a__U51(a__and(a__isNatural(N),isLNat(XS)),N,XS)
a__sel(X1,X2) -> sel(X1,X2)
a__snd(X) -> snd(X)
a__snd(pair(X,Y)) -> a__U61(a__and(a__isLNat(X),isLNat(Y)),Y)
a__splitAt(X1,X2) -> splitAt(X1,X2)
a__splitAt(0(),XS) -> a__U71(a__isLNat(XS),XS)
a__splitAt(s(N),cons(X,XS)) -> a__U81(a__and(a__isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)
a__tail(X) -> tail(X)
a__tail(cons(N,XS)) -> a__U91(a__and(a__isNatural(N),isLNat(XS)),XS)
a__take(N,XS) -> a__U101(a__and(a__isNatural(N),isLNat(XS)),N,XS)
a__take(X1,X2) -> take(X1,X2)
mark(0()) -> 0()
mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
mark(U81(X1,X2,X3,X4)) -> a__U81(mark(X1),X2,X3,X4)
mark(U82(X1,X2)) -> a__U82(mark(X1),X2)
mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
mark(afterNth(X1,X2)) -> a__afterNth(mark(X1),mark(X2))
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(fst(X)) -> a__fst(mark(X))
mark(head(X)) -> a__head(mark(X))
mark(isLNat(X)) -> a__isLNat(X)
mark(isNatural(X)) -> a__isNatural(X)
mark(isPLNat(X)) -> a__isPLNat(X)
mark(natsFrom(X)) -> a__natsFrom(mark(X))
mark(nil()) -> nil()
mark(pair(X1,X2)) -> pair(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
mark(snd(X)) -> a__snd(mark(X))
mark(splitAt(X1,X2)) -> a__splitAt(mark(X1),mark(X2))
mark(tail(X)) -> a__tail(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(tt()) -> tt()
- Signature:
{a__U101/3,a__U11/3,a__U21/2,a__U31/2,a__U41/2,a__U51/3,a__U61/2,a__U71/2,a__U81/4,a__U82/2,a__U91/2
,a__afterNth/2,a__and/2,a__fst/1,a__head/1,a__isLNat/1,a__isNatural/1,a__isPLNat/1,a__natsFrom/1,a__sel/2
,a__snd/1,a__splitAt/2,a__tail/1,a__take/2,mark/1} / {0/0,U101/3,U11/3,U21/2,U31/2,U41/2,U51/3,U61/2,U71/2
,U81/4,U82/2,U91/2,afterNth/2,and/2,cons/2,fst/1,head/1,isLNat/1,isNatural/1,isPLNat/1,natsFrom/1,nil/0
,pair/2,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U101,a__U11,a__U21,a__U31,a__U41,a__U51,a__U61,a__U71
,a__U81,a__U82,a__U91,a__afterNth,a__and,a__fst,a__head,a__isLNat,a__isNatural,a__isPLNat,a__natsFrom,a__sel
,a__snd,a__splitAt,a__tail,a__take,mark} and constructors {0,U101,U11,U21,U31,U41,U51,U61,U71,U81,U82,U91
,afterNth,and,cons,fst,head,isLNat,isNatural,isPLNat,natsFrom,nil,pair,s,sel,snd,splitAt,tail,take,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__U101(X1,X2,X3) -> U101(X1,X2,X3)
a__U101(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U11(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),X) -> mark(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U31(tt(),N) -> mark(N)
a__U41(X1,X2) -> U41(X1,X2)
a__U41(tt(),N) -> cons(mark(N),natsFrom(s(N)))
a__U51(X1,X2,X3) -> U51(X1,X2,X3)
a__U51(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
a__U61(X1,X2) -> U61(X1,X2)
a__U61(tt(),Y) -> mark(Y)
a__U71(X1,X2) -> U71(X1,X2)
a__U71(tt(),XS) -> pair(nil(),mark(XS))
a__U81(X1,X2,X3,X4) -> U81(X1,X2,X3,X4)
a__U81(tt(),N,X,XS) -> a__U82(a__splitAt(mark(N),mark(XS)),X)
a__U82(X1,X2) -> U82(X1,X2)
a__U82(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
a__U91(X1,X2) -> U91(X1,X2)
a__U91(tt(),XS) -> mark(XS)
a__afterNth(N,XS) -> a__U11(a__and(a__isNatural(N),isLNat(XS)),N,XS)
a__afterNth(X1,X2) -> afterNth(X1,X2)
a__and(X1,X2) -> and(X1,X2)
a__and(tt(),X) -> mark(X)
a__fst(X) -> fst(X)
a__fst(pair(X,Y)) -> a__U21(a__and(a__isLNat(X),isLNat(Y)),X)
a__head(X) -> head(X)
a__head(cons(N,XS)) -> a__U31(a__and(a__isNatural(N),isLNat(XS)),N)
a__isLNat(X) -> isLNat(X)
a__isLNat(afterNth(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isLNat(cons(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isLNat(fst(V1)) -> a__isPLNat(V1)
a__isLNat(natsFrom(V1)) -> a__isNatural(V1)
a__isLNat(nil()) -> tt()
a__isLNat(snd(V1)) -> a__isPLNat(V1)
a__isLNat(tail(V1)) -> a__isLNat(V1)
a__isLNat(take(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isNatural(X) -> isNatural(X)
a__isNatural(0()) -> tt()
a__isNatural(head(V1)) -> a__isLNat(V1)
a__isNatural(s(V1)) -> a__isNatural(V1)
a__isNatural(sel(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__isPLNat(X) -> isPLNat(X)
a__isPLNat(pair(V1,V2)) -> a__and(a__isLNat(V1),isLNat(V2))
a__isPLNat(splitAt(V1,V2)) -> a__and(a__isNatural(V1),isLNat(V2))
a__natsFrom(N) -> a__U41(a__isNatural(N),N)
a__natsFrom(X) -> natsFrom(X)
a__sel(N,XS) -> a__U51(a__and(a__isNatural(N),isLNat(XS)),N,XS)
a__sel(X1,X2) -> sel(X1,X2)
a__snd(X) -> snd(X)
a__snd(pair(X,Y)) -> a__U61(a__and(a__isLNat(X),isLNat(Y)),Y)
a__splitAt(X1,X2) -> splitAt(X1,X2)
a__splitAt(0(),XS) -> a__U71(a__isLNat(XS),XS)
a__splitAt(s(N),cons(X,XS)) -> a__U81(a__and(a__isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)
a__tail(X) -> tail(X)
a__tail(cons(N,XS)) -> a__U91(a__and(a__isNatural(N),isLNat(XS)),XS)
a__take(N,XS) -> a__U101(a__and(a__isNatural(N),isLNat(XS)),N,XS)
a__take(X1,X2) -> take(X1,X2)
mark(0()) -> 0()
mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
mark(U81(X1,X2,X3,X4)) -> a__U81(mark(X1),X2,X3,X4)
mark(U82(X1,X2)) -> a__U82(mark(X1),X2)
mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
mark(afterNth(X1,X2)) -> a__afterNth(mark(X1),mark(X2))
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(fst(X)) -> a__fst(mark(X))
mark(head(X)) -> a__head(mark(X))
mark(isLNat(X)) -> a__isLNat(X)
mark(isNatural(X)) -> a__isNatural(X)
mark(isPLNat(X)) -> a__isPLNat(X)
mark(natsFrom(X)) -> a__natsFrom(mark(X))
mark(nil()) -> nil()
mark(pair(X1,X2)) -> pair(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
mark(snd(X)) -> a__snd(mark(X))
mark(splitAt(X1,X2)) -> a__splitAt(mark(X1),mark(X2))
mark(tail(X)) -> a__tail(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(tt()) -> tt()
- Signature:
{a__U101/3,a__U11/3,a__U21/2,a__U31/2,a__U41/2,a__U51/3,a__U61/2,a__U71/2,a__U81/4,a__U82/2,a__U91/2
,a__afterNth/2,a__and/2,a__fst/1,a__head/1,a__isLNat/1,a__isNatural/1,a__isPLNat/1,a__natsFrom/1,a__sel/2
,a__snd/1,a__splitAt/2,a__tail/1,a__take/2,mark/1} / {0/0,U101/3,U11/3,U21/2,U31/2,U41/2,U51/3,U61/2,U71/2
,U81/4,U82/2,U91/2,afterNth/2,and/2,cons/2,fst/1,head/1,isLNat/1,isNatural/1,isPLNat/1,natsFrom/1,nil/0
,pair/2,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U101,a__U11,a__U21,a__U31,a__U41,a__U51,a__U61,a__U71
,a__U81,a__U82,a__U91,a__afterNth,a__and,a__fst,a__head,a__isLNat,a__isNatural,a__isPLNat,a__natsFrom,a__sel
,a__snd,a__splitAt,a__tail,a__take,mark} and constructors {0,U101,U11,U21,U31,U41,U51,U61,U71,U81,U82,U91
,afterNth,and,cons,fst,head,isLNat,isNatural,isPLNat,natsFrom,nil,pair,s,sel,snd,splitAt,tail,take,tt}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
a__isLNat(x){x -> tail(x)} =
a__isLNat(tail(x)) ->^+ a__isLNat(x)
= C[a__isLNat(x) = a__isLNat(x){}]
WORST_CASE(Omega(n^1),?)