* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__U101(X1,X2,X3) -> U101(X1,X2,X3)
a__U101(tt(),V1,V2) -> a__U102(a__isNatural(V1),V2)
a__U102(X1,X2) -> U102(X1,X2)
a__U102(tt(),V2) -> a__U103(a__isLNat(V2))
a__U103(X) -> U103(X)
a__U103(tt()) -> tt()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U11(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
a__U111(X1,X2) -> U111(X1,X2)
a__U111(tt(),V1) -> a__U112(a__isLNat(V1))
a__U112(X) -> U112(X)
a__U112(tt()) -> tt()
a__U121(X1,X2) -> U121(X1,X2)
a__U121(tt(),V1) -> a__U122(a__isNatural(V1))
a__U122(X) -> U122(X)
a__U122(tt()) -> tt()
a__U131(X1,X2,X3) -> U131(X1,X2,X3)
a__U131(tt(),V1,V2) -> a__U132(a__isNatural(V1),V2)
a__U132(X1,X2) -> U132(X1,X2)
a__U132(tt(),V2) -> a__U133(a__isLNat(V2))
a__U133(X) -> U133(X)
a__U133(tt()) -> tt()
a__U141(X1,X2,X3) -> U141(X1,X2,X3)
a__U141(tt(),V1,V2) -> a__U142(a__isLNat(V1),V2)
a__U142(X1,X2) -> U142(X1,X2)
a__U142(tt(),V2) -> a__U143(a__isLNat(V2))
a__U143(X) -> U143(X)
a__U143(tt()) -> tt()
a__U151(X1,X2,X3) -> U151(X1,X2,X3)
a__U151(tt(),V1,V2) -> a__U152(a__isNatural(V1),V2)
a__U152(X1,X2) -> U152(X1,X2)
a__U152(tt(),V2) -> a__U153(a__isLNat(V2))
a__U153(X) -> U153(X)
a__U153(tt()) -> tt()
a__U161(X1,X2) -> U161(X1,X2)
a__U161(tt(),N) -> cons(mark(N),natsFrom(s(N)))
a__U171(X1,X2,X3) -> U171(X1,X2,X3)
a__U171(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
a__U181(X1,X2) -> U181(X1,X2)
a__U181(tt(),Y) -> mark(Y)
a__U191(X1,X2) -> U191(X1,X2)
a__U191(tt(),XS) -> pair(nil(),mark(XS))
a__U201(X1,X2,X3,X4) -> U201(X1,X2,X3,X4)
a__U201(tt(),N,X,XS) -> a__U202(a__splitAt(mark(N),mark(XS)),X)
a__U202(X1,X2) -> U202(X1,X2)
a__U202(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),X) -> mark(X)
a__U211(X1,X2) -> U211(X1,X2)
a__U211(tt(),XS) -> mark(XS)
a__U221(X1,X2,X3) -> U221(X1,X2,X3)
a__U221(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
a__U31(X1,X2) -> U31(X1,X2)
a__U31(tt(),N) -> mark(N)
a__U41(X1,X2,X3) -> U41(X1,X2,X3)
a__U41(tt(),V1,V2) -> a__U42(a__isNatural(V1),V2)
a__U42(X1,X2) -> U42(X1,X2)
a__U42(tt(),V2) -> a__U43(a__isLNat(V2))
a__U43(X) -> U43(X)
a__U43(tt()) -> tt()
a__U51(X1,X2,X3) -> U51(X1,X2,X3)
a__U51(tt(),V1,V2) -> a__U52(a__isNatural(V1),V2)
a__U52(X1,X2) -> U52(X1,X2)
a__U52(tt(),V2) -> a__U53(a__isLNat(V2))
a__U53(X) -> U53(X)
a__U53(tt()) -> tt()
a__U61(X1,X2) -> U61(X1,X2)
a__U61(tt(),V1) -> a__U62(a__isPLNat(V1))
a__U62(X) -> U62(X)
a__U62(tt()) -> tt()
a__U71(X1,X2) -> U71(X1,X2)
a__U71(tt(),V1) -> a__U72(a__isNatural(V1))
a__U72(X) -> U72(X)
a__U72(tt()) -> tt()
a__U81(X1,X2) -> U81(X1,X2)
a__U81(tt(),V1) -> a__U82(a__isPLNat(V1))
a__U82(X) -> U82(X)
a__U82(tt()) -> tt()
a__U91(X1,X2) -> U91(X1,X2)
a__U91(tt(),V1) -> a__U92(a__isLNat(V1))
a__U92(X) -> U92(X)
a__U92(tt()) -> tt()
a__afterNth(N,XS) -> a__U11(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(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__and(a__isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X)
a__head(X) -> head(X)
a__head(cons(N,XS)) -> a__U31(a__and(a__and(a__isNatural(N),isNaturalKind(N))
,and(isLNat(XS),isLNatKind(XS)))
,N)
a__isLNat(X) -> isLNat(X)
a__isLNat(afterNth(V1,V2)) -> a__U41(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNat(cons(V1,V2)) -> a__U51(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1),V1)
a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1),V1)
a__isLNat(nil()) -> tt()
a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1),V1)
a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1),V1)
a__isLNat(take(V1,V2)) -> a__U101(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNatKind(X) -> isLNatKind(X)
a__isLNatKind(afterNth(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isLNatKind(cons(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1)
a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1)
a__isLNatKind(nil()) -> tt()
a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1)
a__isLNatKind(tail(V1)) -> a__isLNatKind(V1)
a__isLNatKind(take(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isNatural(X) -> isNatural(X)
a__isNatural(0()) -> tt()
a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1),V1)
a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1),V1)
a__isNatural(sel(V1,V2)) -> a__U131(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isNaturalKind(X) -> isNaturalKind(X)
a__isNaturalKind(0()) -> tt()
a__isNaturalKind(head(V1)) -> a__isLNatKind(V1)
a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1)
a__isNaturalKind(sel(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isPLNat(X) -> isPLNat(X)
a__isPLNat(pair(V1,V2)) -> a__U141(a__and(a__isLNatKind(V1),isLNatKind(V2)),V1,V2)
a__isPLNat(splitAt(V1,V2)) -> a__U151(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isPLNatKind(X) -> isPLNatKind(X)
a__isPLNatKind(pair(V1,V2)) -> a__and(a__isLNatKind(V1),isLNatKind(V2))
a__isPLNatKind(splitAt(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__natsFrom(N) -> a__U161(a__and(a__isNatural(N),isNaturalKind(N)),N)
a__natsFrom(X) -> natsFrom(X)
a__sel(N,XS) -> a__U171(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
,N
,XS)
a__sel(X1,X2) -> sel(X1,X2)
a__snd(X) -> snd(X)
a__snd(pair(X,Y)) -> a__U181(a__and(a__and(a__isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y)
a__splitAt(X1,X2) -> splitAt(X1,X2)
a__splitAt(0(),XS) -> a__U191(a__and(a__isLNat(XS),isLNatKind(XS)),XS)
a__splitAt(s(N),cons(X,XS)) -> a__U201(a__and(a__and(a__isNatural(N),isNaturalKind(N))
,and(and(isNatural(X),isNaturalKind(X))
,and(isLNat(XS),isLNatKind(XS))))
,N
,X
,XS)
a__tail(X) -> tail(X)
a__tail(cons(N,XS)) -> a__U211(a__and(a__and(a__isNatural(N),isNaturalKind(N))
,and(isLNat(XS),isLNatKind(XS)))
,XS)
a__take(N,XS) -> a__U221(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(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(U102(X1,X2)) -> a__U102(mark(X1),X2)
mark(U103(X)) -> a__U103(mark(X))
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U111(X1,X2)) -> a__U111(mark(X1),X2)
mark(U112(X)) -> a__U112(mark(X))
mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
mark(U122(X)) -> a__U122(mark(X))
mark(U131(X1,X2,X3)) -> a__U131(mark(X1),X2,X3)
mark(U132(X1,X2)) -> a__U132(mark(X1),X2)
mark(U133(X)) -> a__U133(mark(X))
mark(U141(X1,X2,X3)) -> a__U141(mark(X1),X2,X3)
mark(U142(X1,X2)) -> a__U142(mark(X1),X2)
mark(U143(X)) -> a__U143(mark(X))
mark(U151(X1,X2,X3)) -> a__U151(mark(X1),X2,X3)
mark(U152(X1,X2)) -> a__U152(mark(X1),X2)
mark(U153(X)) -> a__U153(mark(X))
mark(U161(X1,X2)) -> a__U161(mark(X1),X2)
mark(U171(X1,X2,X3)) -> a__U171(mark(X1),X2,X3)
mark(U181(X1,X2)) -> a__U181(mark(X1),X2)
mark(U191(X1,X2)) -> a__U191(mark(X1),X2)
mark(U201(X1,X2,X3,X4)) -> a__U201(mark(X1),X2,X3,X4)
mark(U202(X1,X2)) -> a__U202(mark(X1),X2)
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U211(X1,X2)) -> a__U211(mark(X1),X2)
mark(U221(X1,X2,X3)) -> a__U221(mark(X1),X2,X3)
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
mark(U43(X)) -> a__U43(mark(X))
mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U53(X)) -> a__U53(mark(X))
mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
mark(U62(X)) -> a__U62(mark(X))
mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
mark(U72(X)) -> a__U72(mark(X))
mark(U81(X1,X2)) -> a__U81(mark(X1),X2)
mark(U82(X)) -> a__U82(mark(X))
mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
mark(U92(X)) -> a__U92(mark(X))
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(isLNatKind(X)) -> a__isLNatKind(X)
mark(isNatural(X)) -> a__isNatural(X)
mark(isNaturalKind(X)) -> a__isNaturalKind(X)
mark(isPLNat(X)) -> a__isPLNat(X)
mark(isPLNatKind(X)) -> a__isPLNatKind(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__U102/2,a__U103/1,a__U11/3,a__U111/2,a__U112/1,a__U121/2,a__U122/1,a__U131/3,a__U132/2
,a__U133/1,a__U141/3,a__U142/2,a__U143/1,a__U151/3,a__U152/2,a__U153/1,a__U161/2,a__U171/3,a__U181/2
,a__U191/2,a__U201/4,a__U202/2,a__U21/2,a__U211/2,a__U221/3,a__U31/2,a__U41/3,a__U42/2,a__U43/1,a__U51/3
,a__U52/2,a__U53/1,a__U61/2,a__U62/1,a__U71/2,a__U72/1,a__U81/2,a__U82/1,a__U91/2,a__U92/1,a__afterNth/2
,a__and/2,a__fst/1,a__head/1,a__isLNat/1,a__isLNatKind/1,a__isNatural/1,a__isNaturalKind/1,a__isPLNat/1
,a__isPLNatKind/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
,U102/2,U103/1,U11/3,U111/2,U112/1,U121/2,U122/1,U131/3,U132/2,U133/1,U141/3,U142/2,U143/1,U151/3,U152/2
,U153/1,U161/2,U171/3,U181/2,U191/2,U201/4,U202/2,U21/2,U211/2,U221/3,U31/2,U41/3,U42/2,U43/1,U51/3,U52/2
,U53/1,U61/2,U62/1,U71/2,U72/1,U81/2,U82/1,U91/2,U92/1,afterNth/2,and/2,cons/2,fst/1,head/1,isLNat/1
,isLNatKind/1,isNatural/1,isNaturalKind/1,isPLNat/1,isPLNatKind/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__U102,a__U103,a__U11,a__U111,a__U112,a__U121
,a__U122,a__U131,a__U132,a__U133,a__U141,a__U142,a__U143,a__U151,a__U152,a__U153,a__U161,a__U171,a__U181
,a__U191,a__U201,a__U202,a__U21,a__U211,a__U221,a__U31,a__U41,a__U42,a__U43,a__U51,a__U52,a__U53,a__U61
,a__U62,a__U71,a__U72,a__U81,a__U82,a__U91,a__U92,a__afterNth,a__and,a__fst,a__head,a__isLNat,a__isLNatKind
,a__isNatural,a__isNaturalKind,a__isPLNat,a__isPLNatKind,a__natsFrom,a__sel,a__snd,a__splitAt,a__tail
,a__take,mark} and constructors {0,U101,U102,U103,U11,U111,U112,U121,U122,U131,U132,U133,U141,U142,U143,U151
,U152,U153,U161,U171,U181,U191,U201,U202,U21,U211,U221,U31,U41,U42,U43,U51,U52,U53,U61,U62,U71,U72,U81,U82
,U91,U92,afterNth,and,cons,fst,head,isLNat,isLNatKind,isNatural,isNaturalKind,isPLNat,isPLNatKind,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(),V1,V2) -> a__U102(a__isNatural(V1),V2)
a__U102(X1,X2) -> U102(X1,X2)
a__U102(tt(),V2) -> a__U103(a__isLNat(V2))
a__U103(X) -> U103(X)
a__U103(tt()) -> tt()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U11(tt(),N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
a__U111(X1,X2) -> U111(X1,X2)
a__U111(tt(),V1) -> a__U112(a__isLNat(V1))
a__U112(X) -> U112(X)
a__U112(tt()) -> tt()
a__U121(X1,X2) -> U121(X1,X2)
a__U121(tt(),V1) -> a__U122(a__isNatural(V1))
a__U122(X) -> U122(X)
a__U122(tt()) -> tt()
a__U131(X1,X2,X3) -> U131(X1,X2,X3)
a__U131(tt(),V1,V2) -> a__U132(a__isNatural(V1),V2)
a__U132(X1,X2) -> U132(X1,X2)
a__U132(tt(),V2) -> a__U133(a__isLNat(V2))
a__U133(X) -> U133(X)
a__U133(tt()) -> tt()
a__U141(X1,X2,X3) -> U141(X1,X2,X3)
a__U141(tt(),V1,V2) -> a__U142(a__isLNat(V1),V2)
a__U142(X1,X2) -> U142(X1,X2)
a__U142(tt(),V2) -> a__U143(a__isLNat(V2))
a__U143(X) -> U143(X)
a__U143(tt()) -> tt()
a__U151(X1,X2,X3) -> U151(X1,X2,X3)
a__U151(tt(),V1,V2) -> a__U152(a__isNatural(V1),V2)
a__U152(X1,X2) -> U152(X1,X2)
a__U152(tt(),V2) -> a__U153(a__isLNat(V2))
a__U153(X) -> U153(X)
a__U153(tt()) -> tt()
a__U161(X1,X2) -> U161(X1,X2)
a__U161(tt(),N) -> cons(mark(N),natsFrom(s(N)))
a__U171(X1,X2,X3) -> U171(X1,X2,X3)
a__U171(tt(),N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
a__U181(X1,X2) -> U181(X1,X2)
a__U181(tt(),Y) -> mark(Y)
a__U191(X1,X2) -> U191(X1,X2)
a__U191(tt(),XS) -> pair(nil(),mark(XS))
a__U201(X1,X2,X3,X4) -> U201(X1,X2,X3,X4)
a__U201(tt(),N,X,XS) -> a__U202(a__splitAt(mark(N),mark(XS)),X)
a__U202(X1,X2) -> U202(X1,X2)
a__U202(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),X) -> mark(X)
a__U211(X1,X2) -> U211(X1,X2)
a__U211(tt(),XS) -> mark(XS)
a__U221(X1,X2,X3) -> U221(X1,X2,X3)
a__U221(tt(),N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
a__U31(X1,X2) -> U31(X1,X2)
a__U31(tt(),N) -> mark(N)
a__U41(X1,X2,X3) -> U41(X1,X2,X3)
a__U41(tt(),V1,V2) -> a__U42(a__isNatural(V1),V2)
a__U42(X1,X2) -> U42(X1,X2)
a__U42(tt(),V2) -> a__U43(a__isLNat(V2))
a__U43(X) -> U43(X)
a__U43(tt()) -> tt()
a__U51(X1,X2,X3) -> U51(X1,X2,X3)
a__U51(tt(),V1,V2) -> a__U52(a__isNatural(V1),V2)
a__U52(X1,X2) -> U52(X1,X2)
a__U52(tt(),V2) -> a__U53(a__isLNat(V2))
a__U53(X) -> U53(X)
a__U53(tt()) -> tt()
a__U61(X1,X2) -> U61(X1,X2)
a__U61(tt(),V1) -> a__U62(a__isPLNat(V1))
a__U62(X) -> U62(X)
a__U62(tt()) -> tt()
a__U71(X1,X2) -> U71(X1,X2)
a__U71(tt(),V1) -> a__U72(a__isNatural(V1))
a__U72(X) -> U72(X)
a__U72(tt()) -> tt()
a__U81(X1,X2) -> U81(X1,X2)
a__U81(tt(),V1) -> a__U82(a__isPLNat(V1))
a__U82(X) -> U82(X)
a__U82(tt()) -> tt()
a__U91(X1,X2) -> U91(X1,X2)
a__U91(tt(),V1) -> a__U92(a__isLNat(V1))
a__U92(X) -> U92(X)
a__U92(tt()) -> tt()
a__afterNth(N,XS) -> a__U11(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(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__and(a__isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X)
a__head(X) -> head(X)
a__head(cons(N,XS)) -> a__U31(a__and(a__and(a__isNatural(N),isNaturalKind(N))
,and(isLNat(XS),isLNatKind(XS)))
,N)
a__isLNat(X) -> isLNat(X)
a__isLNat(afterNth(V1,V2)) -> a__U41(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNat(cons(V1,V2)) -> a__U51(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1),V1)
a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1),V1)
a__isLNat(nil()) -> tt()
a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1),V1)
a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1),V1)
a__isLNat(take(V1,V2)) -> a__U101(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNatKind(X) -> isLNatKind(X)
a__isLNatKind(afterNth(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isLNatKind(cons(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1)
a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1)
a__isLNatKind(nil()) -> tt()
a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1)
a__isLNatKind(tail(V1)) -> a__isLNatKind(V1)
a__isLNatKind(take(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isNatural(X) -> isNatural(X)
a__isNatural(0()) -> tt()
a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1),V1)
a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1),V1)
a__isNatural(sel(V1,V2)) -> a__U131(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isNaturalKind(X) -> isNaturalKind(X)
a__isNaturalKind(0()) -> tt()
a__isNaturalKind(head(V1)) -> a__isLNatKind(V1)
a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1)
a__isNaturalKind(sel(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isPLNat(X) -> isPLNat(X)
a__isPLNat(pair(V1,V2)) -> a__U141(a__and(a__isLNatKind(V1),isLNatKind(V2)),V1,V2)
a__isPLNat(splitAt(V1,V2)) -> a__U151(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isPLNatKind(X) -> isPLNatKind(X)
a__isPLNatKind(pair(V1,V2)) -> a__and(a__isLNatKind(V1),isLNatKind(V2))
a__isPLNatKind(splitAt(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__natsFrom(N) -> a__U161(a__and(a__isNatural(N),isNaturalKind(N)),N)
a__natsFrom(X) -> natsFrom(X)
a__sel(N,XS) -> a__U171(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
,N
,XS)
a__sel(X1,X2) -> sel(X1,X2)
a__snd(X) -> snd(X)
a__snd(pair(X,Y)) -> a__U181(a__and(a__and(a__isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y)
a__splitAt(X1,X2) -> splitAt(X1,X2)
a__splitAt(0(),XS) -> a__U191(a__and(a__isLNat(XS),isLNatKind(XS)),XS)
a__splitAt(s(N),cons(X,XS)) -> a__U201(a__and(a__and(a__isNatural(N),isNaturalKind(N))
,and(and(isNatural(X),isNaturalKind(X))
,and(isLNat(XS),isLNatKind(XS))))
,N
,X
,XS)
a__tail(X) -> tail(X)
a__tail(cons(N,XS)) -> a__U211(a__and(a__and(a__isNatural(N),isNaturalKind(N))
,and(isLNat(XS),isLNatKind(XS)))
,XS)
a__take(N,XS) -> a__U221(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(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(U102(X1,X2)) -> a__U102(mark(X1),X2)
mark(U103(X)) -> a__U103(mark(X))
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U111(X1,X2)) -> a__U111(mark(X1),X2)
mark(U112(X)) -> a__U112(mark(X))
mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
mark(U122(X)) -> a__U122(mark(X))
mark(U131(X1,X2,X3)) -> a__U131(mark(X1),X2,X3)
mark(U132(X1,X2)) -> a__U132(mark(X1),X2)
mark(U133(X)) -> a__U133(mark(X))
mark(U141(X1,X2,X3)) -> a__U141(mark(X1),X2,X3)
mark(U142(X1,X2)) -> a__U142(mark(X1),X2)
mark(U143(X)) -> a__U143(mark(X))
mark(U151(X1,X2,X3)) -> a__U151(mark(X1),X2,X3)
mark(U152(X1,X2)) -> a__U152(mark(X1),X2)
mark(U153(X)) -> a__U153(mark(X))
mark(U161(X1,X2)) -> a__U161(mark(X1),X2)
mark(U171(X1,X2,X3)) -> a__U171(mark(X1),X2,X3)
mark(U181(X1,X2)) -> a__U181(mark(X1),X2)
mark(U191(X1,X2)) -> a__U191(mark(X1),X2)
mark(U201(X1,X2,X3,X4)) -> a__U201(mark(X1),X2,X3,X4)
mark(U202(X1,X2)) -> a__U202(mark(X1),X2)
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U211(X1,X2)) -> a__U211(mark(X1),X2)
mark(U221(X1,X2,X3)) -> a__U221(mark(X1),X2,X3)
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
mark(U43(X)) -> a__U43(mark(X))
mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U53(X)) -> a__U53(mark(X))
mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
mark(U62(X)) -> a__U62(mark(X))
mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
mark(U72(X)) -> a__U72(mark(X))
mark(U81(X1,X2)) -> a__U81(mark(X1),X2)
mark(U82(X)) -> a__U82(mark(X))
mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
mark(U92(X)) -> a__U92(mark(X))
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(isLNatKind(X)) -> a__isLNatKind(X)
mark(isNatural(X)) -> a__isNatural(X)
mark(isNaturalKind(X)) -> a__isNaturalKind(X)
mark(isPLNat(X)) -> a__isPLNat(X)
mark(isPLNatKind(X)) -> a__isPLNatKind(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__U102/2,a__U103/1,a__U11/3,a__U111/2,a__U112/1,a__U121/2,a__U122/1,a__U131/3,a__U132/2
,a__U133/1,a__U141/3,a__U142/2,a__U143/1,a__U151/3,a__U152/2,a__U153/1,a__U161/2,a__U171/3,a__U181/2
,a__U191/2,a__U201/4,a__U202/2,a__U21/2,a__U211/2,a__U221/3,a__U31/2,a__U41/3,a__U42/2,a__U43/1,a__U51/3
,a__U52/2,a__U53/1,a__U61/2,a__U62/1,a__U71/2,a__U72/1,a__U81/2,a__U82/1,a__U91/2,a__U92/1,a__afterNth/2
,a__and/2,a__fst/1,a__head/1,a__isLNat/1,a__isLNatKind/1,a__isNatural/1,a__isNaturalKind/1,a__isPLNat/1
,a__isPLNatKind/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
,U102/2,U103/1,U11/3,U111/2,U112/1,U121/2,U122/1,U131/3,U132/2,U133/1,U141/3,U142/2,U143/1,U151/3,U152/2
,U153/1,U161/2,U171/3,U181/2,U191/2,U201/4,U202/2,U21/2,U211/2,U221/3,U31/2,U41/3,U42/2,U43/1,U51/3,U52/2
,U53/1,U61/2,U62/1,U71/2,U72/1,U81/2,U82/1,U91/2,U92/1,afterNth/2,and/2,cons/2,fst/1,head/1,isLNat/1
,isLNatKind/1,isNatural/1,isNaturalKind/1,isPLNat/1,isPLNatKind/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__U102,a__U103,a__U11,a__U111,a__U112,a__U121
,a__U122,a__U131,a__U132,a__U133,a__U141,a__U142,a__U143,a__U151,a__U152,a__U153,a__U161,a__U171,a__U181
,a__U191,a__U201,a__U202,a__U21,a__U211,a__U221,a__U31,a__U41,a__U42,a__U43,a__U51,a__U52,a__U53,a__U61
,a__U62,a__U71,a__U72,a__U81,a__U82,a__U91,a__U92,a__afterNth,a__and,a__fst,a__head,a__isLNat,a__isLNatKind
,a__isNatural,a__isNaturalKind,a__isPLNat,a__isPLNatKind,a__natsFrom,a__sel,a__snd,a__splitAt,a__tail
,a__take,mark} and constructors {0,U101,U102,U103,U11,U111,U112,U121,U122,U131,U132,U133,U141,U142,U143,U151
,U152,U153,U161,U171,U181,U191,U201,U202,U21,U211,U221,U31,U41,U42,U43,U51,U52,U53,U61,U62,U71,U72,U81,U82
,U91,U92,afterNth,and,cons,fst,head,isLNat,isLNatKind,isNatural,isNaturalKind,isPLNat,isPLNatKind,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__isLNatKind(x){x -> tail(x)} =
a__isLNatKind(tail(x)) ->^+ a__isLNatKind(x)
= C[a__isLNatKind(x) = a__isLNatKind(x){}]
WORST_CASE(Omega(n^1),?)