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