MAYBE Problem: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(U101(tt(),N,XS)) -> splitAt#(N,XS) active#(U101(tt(),N,XS)) -> fst#(splitAt(N,XS)) active#(U11(tt(),N,XS)) -> splitAt#(N,XS) active#(U11(tt(),N,XS)) -> snd#(splitAt(N,XS)) active#(U41(tt(),N)) -> s#(N) active#(U41(tt(),N)) -> natsFrom#(s(N)) active#(U41(tt(),N)) -> cons#(N,natsFrom(s(N))) active#(U51(tt(),N,XS)) -> afterNth#(N,XS) active#(U51(tt(),N,XS)) -> head#(afterNth(N,XS)) active#(U71(tt(),XS)) -> pair#(nil(),XS) active#(U81(tt(),N,X,XS)) -> splitAt#(N,XS) active#(U81(tt(),N,X,XS)) -> U82#(splitAt(N,XS),X) active#(U82(pair(YS,ZS),X)) -> cons#(X,YS) active#(U82(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS) active#(afterNth(N,XS)) -> isLNat#(XS) active#(afterNth(N,XS)) -> isNatural#(N) active#(afterNth(N,XS)) -> and#(isNatural(N),isLNat(XS)) active#(afterNth(N,XS)) -> U11#(and(isNatural(N),isLNat(XS)),N,XS) active#(fst(pair(X,Y))) -> isLNat#(Y) active#(fst(pair(X,Y))) -> isLNat#(X) active#(fst(pair(X,Y))) -> and#(isLNat(X),isLNat(Y)) active#(fst(pair(X,Y))) -> U21#(and(isLNat(X),isLNat(Y)),X) active#(head(cons(N,XS))) -> isLNat#(XS) active#(head(cons(N,XS))) -> isNatural#(N) active#(head(cons(N,XS))) -> and#(isNatural(N),isLNat(XS)) active#(head(cons(N,XS))) -> U31#(and(isNatural(N),isLNat(XS)),N) active#(isLNat(afterNth(V1,V2))) -> isLNat#(V2) active#(isLNat(afterNth(V1,V2))) -> isNatural#(V1) active#(isLNat(afterNth(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isLNat(cons(V1,V2))) -> isLNat#(V2) active#(isLNat(cons(V1,V2))) -> isNatural#(V1) active#(isLNat(cons(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isLNat(fst(V1))) -> isPLNat#(V1) active#(isLNat(natsFrom(V1))) -> isNatural#(V1) active#(isLNat(snd(V1))) -> isPLNat#(V1) active#(isLNat(tail(V1))) -> isLNat#(V1) active#(isLNat(take(V1,V2))) -> isLNat#(V2) active#(isLNat(take(V1,V2))) -> isNatural#(V1) active#(isLNat(take(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isNatural(head(V1))) -> isLNat#(V1) active#(isNatural(s(V1))) -> isNatural#(V1) active#(isNatural(sel(V1,V2))) -> isLNat#(V2) active#(isNatural(sel(V1,V2))) -> isNatural#(V1) active#(isNatural(sel(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isPLNat(pair(V1,V2))) -> isLNat#(V2) active#(isPLNat(pair(V1,V2))) -> isLNat#(V1) active#(isPLNat(pair(V1,V2))) -> and#(isLNat(V1),isLNat(V2)) active#(isPLNat(splitAt(V1,V2))) -> isLNat#(V2) active#(isPLNat(splitAt(V1,V2))) -> isNatural#(V1) active#(isPLNat(splitAt(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(natsFrom(N)) -> isNatural#(N) active#(natsFrom(N)) -> U41#(isNatural(N),N) active#(sel(N,XS)) -> isLNat#(XS) active#(sel(N,XS)) -> isNatural#(N) active#(sel(N,XS)) -> and#(isNatural(N),isLNat(XS)) active#(sel(N,XS)) -> U51#(and(isNatural(N),isLNat(XS)),N,XS) active#(snd(pair(X,Y))) -> isLNat#(Y) active#(snd(pair(X,Y))) -> isLNat#(X) active#(snd(pair(X,Y))) -> and#(isLNat(X),isLNat(Y)) active#(snd(pair(X,Y))) -> U61#(and(isLNat(X),isLNat(Y)),Y) active#(splitAt(0(),XS)) -> isLNat#(XS) active#(splitAt(0(),XS)) -> U71#(isLNat(XS),XS) active#(splitAt(s(N),cons(X,XS))) -> isLNat#(XS) active#(splitAt(s(N),cons(X,XS))) -> isNatural#(X) active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(X),isLNat(XS)) active#(splitAt(s(N),cons(X,XS))) -> isNatural#(N) active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(N),and(isNatural(X),isLNat(XS))) active#(splitAt(s(N),cons(X,XS))) -> U81#(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS) active#(tail(cons(N,XS))) -> isLNat#(XS) active#(tail(cons(N,XS))) -> isNatural#(N) active#(tail(cons(N,XS))) -> and#(isNatural(N),isLNat(XS)) active#(tail(cons(N,XS))) -> U91#(and(isNatural(N),isLNat(XS)),XS) active#(take(N,XS)) -> isLNat#(XS) active#(take(N,XS)) -> isNatural#(N) active#(take(N,XS)) -> and#(isNatural(N),isLNat(XS)) active#(take(N,XS)) -> U101#(and(isNatural(N),isLNat(XS)),N,XS) active#(U101(X1,X2,X3)) -> active#(X1) active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3) active#(fst(X)) -> active#(X) active#(fst(X)) -> fst#(active(X)) active#(splitAt(X1,X2)) -> active#(X1) active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2) active#(splitAt(X1,X2)) -> active#(X2) active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2)) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(snd(X)) -> active#(X) active#(snd(X)) -> snd#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(natsFrom(X)) -> active#(X) active#(natsFrom(X)) -> natsFrom#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(afterNth(X1,X2)) -> active#(X1) active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2) active#(afterNth(X1,X2)) -> active#(X2) active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2)) active#(U61(X1,X2)) -> active#(X1) active#(U61(X1,X2)) -> U61#(active(X1),X2) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(pair(X1,X2)) -> active#(X1) active#(pair(X1,X2)) -> pair#(active(X1),X2) active#(pair(X1,X2)) -> active#(X2) active#(pair(X1,X2)) -> pair#(X1,active(X2)) active#(U81(X1,X2,X3,X4)) -> active#(X1) active#(U81(X1,X2,X3,X4)) -> U81#(active(X1),X2,X3,X4) active#(U82(X1,X2)) -> active#(X1) active#(U82(X1,X2)) -> U82#(active(X1),X2) active#(U91(X1,X2)) -> active#(X1) active#(U91(X1,X2)) -> U91#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3) fst#(mark(X)) -> fst#(X) splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) snd#(mark(X)) -> snd#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U41#(mark(X1),X2) -> U41#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) natsFrom#(mark(X)) -> natsFrom#(X) s#(mark(X)) -> s#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) head#(mark(X)) -> head#(X) afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) U61#(mark(X1),X2) -> U61#(X1,X2) U71#(mark(X1),X2) -> U71#(X1,X2) pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) U81#(mark(X1),X2,X3,X4) -> U81#(X1,X2,X3,X4) U82#(mark(X1),X2) -> U82#(X1,X2) U91#(mark(X1),X2) -> U91#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) tail#(mark(X)) -> tail#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) proper#(U101(X1,X2,X3)) -> proper#(X3) proper#(U101(X1,X2,X3)) -> proper#(X2) proper#(U101(X1,X2,X3)) -> proper#(X1) proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3)) proper#(fst(X)) -> proper#(X) proper#(fst(X)) -> fst#(proper(X)) proper#(splitAt(X1,X2)) -> proper#(X2) proper#(splitAt(X1,X2)) -> proper#(X1) proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(snd(X)) -> proper#(X) proper#(snd(X)) -> snd#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(natsFrom(X)) -> proper#(X) proper#(natsFrom(X)) -> natsFrom#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(afterNth(X1,X2)) -> proper#(X2) proper#(afterNth(X1,X2)) -> proper#(X1) proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2)) proper#(U61(X1,X2)) -> proper#(X2) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(pair(X1,X2)) -> proper#(X2) proper#(pair(X1,X2)) -> proper#(X1) proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2)) proper#(U81(X1,X2,X3,X4)) -> proper#(X4) proper#(U81(X1,X2,X3,X4)) -> proper#(X3) proper#(U81(X1,X2,X3,X4)) -> proper#(X2) proper#(U81(X1,X2,X3,X4)) -> proper#(X1) proper#(U81(X1,X2,X3,X4)) -> U81#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(U82(X1,X2)) -> proper#(X2) proper#(U82(X1,X2)) -> proper#(X1) proper#(U82(X1,X2)) -> U82#(proper(X1),proper(X2)) proper#(U91(X1,X2)) -> proper#(X2) proper#(U91(X1,X2)) -> proper#(X1) proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatural(X)) -> proper#(X) proper#(isNatural(X)) -> isNatural#(proper(X)) proper#(isLNat(X)) -> proper#(X) proper#(isLNat(X)) -> isLNat#(proper(X)) proper#(isPLNat(X)) -> proper#(X) proper#(isPLNat(X)) -> isPLNat#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3) fst#(ok(X)) -> fst#(X) splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) snd#(ok(X)) -> snd#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) natsFrom#(ok(X)) -> natsFrom#(X) s#(ok(X)) -> s#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) head#(ok(X)) -> head#(X) afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2) U61#(ok(X1),ok(X2)) -> U61#(X1,X2) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) pair#(ok(X1),ok(X2)) -> pair#(X1,X2) U81#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U81#(X1,X2,X3,X4) U82#(ok(X1),ok(X2)) -> U82#(X1,X2) U91#(ok(X1),ok(X2)) -> U91#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatural#(ok(X)) -> isNatural#(X) isLNat#(ok(X)) -> isLNat#(X) isPLNat#(ok(X)) -> isPLNat#(X) tail#(ok(X)) -> tail#(X) take#(ok(X1),ok(X2)) -> take#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(U101(tt(),N,XS)) -> splitAt#(N,XS) active#(U101(tt(),N,XS)) -> fst#(splitAt(N,XS)) active#(U11(tt(),N,XS)) -> splitAt#(N,XS) active#(U11(tt(),N,XS)) -> snd#(splitAt(N,XS)) active#(U41(tt(),N)) -> s#(N) active#(U41(tt(),N)) -> natsFrom#(s(N)) active#(U41(tt(),N)) -> cons#(N,natsFrom(s(N))) active#(U51(tt(),N,XS)) -> afterNth#(N,XS) active#(U51(tt(),N,XS)) -> head#(afterNth(N,XS)) active#(U71(tt(),XS)) -> pair#(nil(),XS) active#(U81(tt(),N,X,XS)) -> splitAt#(N,XS) active#(U81(tt(),N,X,XS)) -> U82#(splitAt(N,XS),X) active#(U82(pair(YS,ZS),X)) -> cons#(X,YS) active#(U82(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS) active#(afterNth(N,XS)) -> isLNat#(XS) active#(afterNth(N,XS)) -> isNatural#(N) active#(afterNth(N,XS)) -> and#(isNatural(N),isLNat(XS)) active#(afterNth(N,XS)) -> U11#(and(isNatural(N),isLNat(XS)),N,XS) active#(fst(pair(X,Y))) -> isLNat#(Y) active#(fst(pair(X,Y))) -> isLNat#(X) active#(fst(pair(X,Y))) -> and#(isLNat(X),isLNat(Y)) active#(fst(pair(X,Y))) -> U21#(and(isLNat(X),isLNat(Y)),X) active#(head(cons(N,XS))) -> isLNat#(XS) active#(head(cons(N,XS))) -> isNatural#(N) active#(head(cons(N,XS))) -> and#(isNatural(N),isLNat(XS)) active#(head(cons(N,XS))) -> U31#(and(isNatural(N),isLNat(XS)),N) active#(isLNat(afterNth(V1,V2))) -> isLNat#(V2) active#(isLNat(afterNth(V1,V2))) -> isNatural#(V1) active#(isLNat(afterNth(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isLNat(cons(V1,V2))) -> isLNat#(V2) active#(isLNat(cons(V1,V2))) -> isNatural#(V1) active#(isLNat(cons(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isLNat(fst(V1))) -> isPLNat#(V1) active#(isLNat(natsFrom(V1))) -> isNatural#(V1) active#(isLNat(snd(V1))) -> isPLNat#(V1) active#(isLNat(tail(V1))) -> isLNat#(V1) active#(isLNat(take(V1,V2))) -> isLNat#(V2) active#(isLNat(take(V1,V2))) -> isNatural#(V1) active#(isLNat(take(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isNatural(head(V1))) -> isLNat#(V1) active#(isNatural(s(V1))) -> isNatural#(V1) active#(isNatural(sel(V1,V2))) -> isLNat#(V2) active#(isNatural(sel(V1,V2))) -> isNatural#(V1) active#(isNatural(sel(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(isPLNat(pair(V1,V2))) -> isLNat#(V2) active#(isPLNat(pair(V1,V2))) -> isLNat#(V1) active#(isPLNat(pair(V1,V2))) -> and#(isLNat(V1),isLNat(V2)) active#(isPLNat(splitAt(V1,V2))) -> isLNat#(V2) active#(isPLNat(splitAt(V1,V2))) -> isNatural#(V1) active#(isPLNat(splitAt(V1,V2))) -> and#(isNatural(V1),isLNat(V2)) active#(natsFrom(N)) -> isNatural#(N) active#(natsFrom(N)) -> U41#(isNatural(N),N) active#(sel(N,XS)) -> isLNat#(XS) active#(sel(N,XS)) -> isNatural#(N) active#(sel(N,XS)) -> and#(isNatural(N),isLNat(XS)) active#(sel(N,XS)) -> U51#(and(isNatural(N),isLNat(XS)),N,XS) active#(snd(pair(X,Y))) -> isLNat#(Y) active#(snd(pair(X,Y))) -> isLNat#(X) active#(snd(pair(X,Y))) -> and#(isLNat(X),isLNat(Y)) active#(snd(pair(X,Y))) -> U61#(and(isLNat(X),isLNat(Y)),Y) active#(splitAt(0(),XS)) -> isLNat#(XS) active#(splitAt(0(),XS)) -> U71#(isLNat(XS),XS) active#(splitAt(s(N),cons(X,XS))) -> isLNat#(XS) active#(splitAt(s(N),cons(X,XS))) -> isNatural#(X) active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(X),isLNat(XS)) active#(splitAt(s(N),cons(X,XS))) -> isNatural#(N) active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(N),and(isNatural(X),isLNat(XS))) active#(splitAt(s(N),cons(X,XS))) -> U81#(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS) active#(tail(cons(N,XS))) -> isLNat#(XS) active#(tail(cons(N,XS))) -> isNatural#(N) active#(tail(cons(N,XS))) -> and#(isNatural(N),isLNat(XS)) active#(tail(cons(N,XS))) -> U91#(and(isNatural(N),isLNat(XS)),XS) active#(take(N,XS)) -> isLNat#(XS) active#(take(N,XS)) -> isNatural#(N) active#(take(N,XS)) -> and#(isNatural(N),isLNat(XS)) active#(take(N,XS)) -> U101#(and(isNatural(N),isLNat(XS)),N,XS) active#(U101(X1,X2,X3)) -> active#(X1) active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3) active#(fst(X)) -> active#(X) active#(fst(X)) -> fst#(active(X)) active#(splitAt(X1,X2)) -> active#(X1) active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2) active#(splitAt(X1,X2)) -> active#(X2) active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2)) active#(U11(X1,X2,X3)) -> active#(X1) active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3) active#(snd(X)) -> active#(X) active#(snd(X)) -> snd#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(natsFrom(X)) -> active#(X) active#(natsFrom(X)) -> natsFrom#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(afterNth(X1,X2)) -> active#(X1) active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2) active#(afterNth(X1,X2)) -> active#(X2) active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2)) active#(U61(X1,X2)) -> active#(X1) active#(U61(X1,X2)) -> U61#(active(X1),X2) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(pair(X1,X2)) -> active#(X1) active#(pair(X1,X2)) -> pair#(active(X1),X2) active#(pair(X1,X2)) -> active#(X2) active#(pair(X1,X2)) -> pair#(X1,active(X2)) active#(U81(X1,X2,X3,X4)) -> active#(X1) active#(U81(X1,X2,X3,X4)) -> U81#(active(X1),X2,X3,X4) active#(U82(X1,X2)) -> active#(X1) active#(U82(X1,X2)) -> U82#(active(X1),X2) active#(U91(X1,X2)) -> active#(X1) active#(U91(X1,X2)) -> U91#(active(X1),X2) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(tail(X)) -> active#(X) active#(tail(X)) -> tail#(active(X)) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> sel#(active(X1),X2) active#(sel(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> sel#(X1,active(X2)) U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3) fst#(mark(X)) -> fst#(X) splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) snd#(mark(X)) -> snd#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U41#(mark(X1),X2) -> U41#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) natsFrom#(mark(X)) -> natsFrom#(X) s#(mark(X)) -> s#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) head#(mark(X)) -> head#(X) afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) U61#(mark(X1),X2) -> U61#(X1,X2) U71#(mark(X1),X2) -> U71#(X1,X2) pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) U81#(mark(X1),X2,X3,X4) -> U81#(X1,X2,X3,X4) U82#(mark(X1),X2) -> U82#(X1,X2) U91#(mark(X1),X2) -> U91#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) tail#(mark(X)) -> tail#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) proper#(U101(X1,X2,X3)) -> proper#(X3) proper#(U101(X1,X2,X3)) -> proper#(X2) proper#(U101(X1,X2,X3)) -> proper#(X1) proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3)) proper#(fst(X)) -> proper#(X) proper#(fst(X)) -> fst#(proper(X)) proper#(splitAt(X1,X2)) -> proper#(X2) proper#(splitAt(X1,X2)) -> proper#(X1) proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2)) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3)) proper#(snd(X)) -> proper#(X) proper#(snd(X)) -> snd#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(natsFrom(X)) -> proper#(X) proper#(natsFrom(X)) -> natsFrom#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(afterNth(X1,X2)) -> proper#(X2) proper#(afterNth(X1,X2)) -> proper#(X1) proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2)) proper#(U61(X1,X2)) -> proper#(X2) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(pair(X1,X2)) -> proper#(X2) proper#(pair(X1,X2)) -> proper#(X1) proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2)) proper#(U81(X1,X2,X3,X4)) -> proper#(X4) proper#(U81(X1,X2,X3,X4)) -> proper#(X3) proper#(U81(X1,X2,X3,X4)) -> proper#(X2) proper#(U81(X1,X2,X3,X4)) -> proper#(X1) proper#(U81(X1,X2,X3,X4)) -> U81#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(U82(X1,X2)) -> proper#(X2) proper#(U82(X1,X2)) -> proper#(X1) proper#(U82(X1,X2)) -> U82#(proper(X1),proper(X2)) proper#(U91(X1,X2)) -> proper#(X2) proper#(U91(X1,X2)) -> proper#(X1) proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatural(X)) -> proper#(X) proper#(isNatural(X)) -> isNatural#(proper(X)) proper#(isLNat(X)) -> proper#(X) proper#(isLNat(X)) -> isLNat#(proper(X)) proper#(isPLNat(X)) -> proper#(X) proper#(isPLNat(X)) -> isPLNat#(proper(X)) proper#(tail(X)) -> proper#(X) proper#(tail(X)) -> tail#(proper(X)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3) fst#(ok(X)) -> fst#(X) splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) snd#(ok(X)) -> snd#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) natsFrom#(ok(X)) -> natsFrom#(X) s#(ok(X)) -> s#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) head#(ok(X)) -> head#(X) afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2) U61#(ok(X1),ok(X2)) -> U61#(X1,X2) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) pair#(ok(X1),ok(X2)) -> pair#(X1,X2) U81#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U81#(X1,X2,X3,X4) U82#(ok(X1),ok(X2)) -> U82#(X1,X2) U91#(ok(X1),ok(X2)) -> U91#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatural#(ok(X)) -> isNatural#(X) isLNat#(ok(X)) -> isLNat#(X) isPLNat#(ok(X)) -> isPLNat#(X) tail#(ok(X)) -> tail#(X) take#(ok(X1),ok(X2)) -> take#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 30 #rules: 137 #arcs: 8324/73441 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(U101(X1,X2,X3)) -> active#(X1) active#(fst(X)) -> active#(X) active#(splitAt(X1,X2)) -> active#(X1) active#(splitAt(X1,X2)) -> active#(X2) active#(U11(X1,X2,X3)) -> active#(X1) active#(snd(X)) -> active#(X) active#(U21(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) active#(natsFrom(X)) -> active#(X) active#(s(X)) -> active#(X) active#(U51(X1,X2,X3)) -> active#(X1) active#(head(X)) -> active#(X) active#(afterNth(X1,X2)) -> active#(X1) active#(afterNth(X1,X2)) -> active#(X2) active#(U61(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> active#(X1) active#(pair(X1,X2)) -> active#(X1) active#(pair(X1,X2)) -> active#(X2) active#(U81(X1,X2,X3,X4)) -> active#(X1) active#(U82(X1,X2)) -> active#(X1) active#(U91(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> active#(X1) active#(tail(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [sel](x0, x1) = x0 + x1 + 1, [0] = 0, [take](x0, x1) = x0 + x1 + 1, [tail](x0) = x0 + 1, [isPLNat](x0) = 1, [and](x0, x1) = x0 + 1, [isLNat](x0) = 1, [isNatural](x0) = 1, [U91](x0, x1) = x0 + 1, [U82](x0, x1) = x0 + 1, [U81](x0, x1, x2, x3) = x0 + 1, [pair](x0, x1) = x0 + x1 + 1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x0 + x1 + 1, [head](x0) = x0 + 1, [afterNth](x0, x1) = x0 + x1 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x0, [natsFrom](x0) = x0 + 1, [s](x0) = x0 + 1, [U41](x0, x1) = x0 + 1, [U31](x0, x1) = x0 + 1, [U21](x0, x1) = x0 + 1, [snd](x0) = x0 + 1, [U11](x0, x1, x2) = x0 + 1, [mark](x0) = 1, [fst](x0) = x0 + 1, [splitAt](x0, x1) = x0 + x1 + 1, [active](x0) = x0, [U101](x0, x1, x2) = x0 + 1, [tt] = 0 orientation: active#(U101(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(fst(X)) = X + 2 >= X + 1 = active#(X) active#(splitAt(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(splitAt(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(U11(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(snd(X)) = X + 2 >= X + 1 = active#(X) active#(U21(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U31(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U41(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active#(natsFrom(X)) = X + 2 >= X + 1 = active#(X) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(U51(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(head(X)) = X + 2 >= X + 1 = active#(X) active#(afterNth(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(afterNth(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(U61(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(U71(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(pair(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(pair(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(U81(X1,X2,X3,X4)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U82(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U91(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(and(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(tail(X)) = X + 2 >= X + 1 = active#(X) active#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active#(sel(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = active#(X1) active#(sel(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = active#(X2) active(U101(tt(),N,XS)) = 1 >= 1 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 1 >= 1 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = 1 >= 1 = mark(X) active(U31(tt(),N)) = 1 >= 1 = mark(N) active(U41(tt(),N)) = 1 >= 1 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 1 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y + 1 >= 1 = mark(Y) active(U71(tt(),XS)) = 1 >= 1 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = 1 >= 1 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = YS + ZS + 2 >= 1 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 1 >= 1 = mark(XS) active(afterNth(N,XS)) = N + XS + 1 >= 1 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 1 >= 1 = mark(X) active(fst(pair(X,Y))) = X + Y + 2 >= 1 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = N + 1 >= 1 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 1 >= 1 = mark(tt()) active(isLNat(afterNth(V1,V2))) = 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = 1 >= 1 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = 1 >= 1 = mark(isNatural(V1)) active(isLNat(snd(V1))) = 1 >= 1 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = 1 >= 1 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 1 >= 1 = mark(tt()) active(isNatural(head(V1))) = 1 >= 1 = mark(isLNat(V1)) active(isNatural(s(V1))) = 1 >= 1 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = 1 >= 1 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N + 1 >= 1 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N + XS + 1 >= 1 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = X + Y + 2 >= 1 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS + 1 >= 1 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N + X + 2 >= 1 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = N + 1 >= 1 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = N + XS + 1 >= 1 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(active(X1),X2,X3) active(fst(X)) = X + 1 >= X + 1 = fst(active(X)) active(splitAt(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2,X3) active(snd(X)) = X + 1 >= X + 1 = snd(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(afterNth(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = pair(active(X1),X2) active(pair(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X1 + 1 >= X1 + 1 = U82(active(X1),X2) active(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(active(X1),X2) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 2 >= 1 = mark(U101(X1,X2,X3)) fst(mark(X)) = 2 >= 1 = mark(fst(X)) splitAt(mark(X1),X2) = X2 + 2 >= 1 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 + 2 >= 1 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 2 >= 1 = mark(U11(X1,X2,X3)) snd(mark(X)) = 2 >= 1 = mark(snd(X)) U21(mark(X1),X2) = 2 >= 1 = mark(U21(X1,X2)) U31(mark(X1),X2) = 2 >= 1 = mark(U31(X1,X2)) U41(mark(X1),X2) = 2 >= 1 = mark(U41(X1,X2)) cons(mark(X1),X2) = 1 >= 1 = mark(cons(X1,X2)) natsFrom(mark(X)) = 2 >= 1 = mark(natsFrom(X)) s(mark(X)) = 2 >= 1 = mark(s(X)) U51(mark(X1),X2,X3) = 2 >= 1 = mark(U51(X1,X2,X3)) head(mark(X)) = 2 >= 1 = mark(head(X)) afterNth(mark(X1),X2) = X2 + 2 >= 1 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 2 >= 1 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 + 2 >= 1 = mark(U61(X1,X2)) U71(mark(X1),X2) = 2 >= 1 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 + 2 >= 1 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 + 2 >= 1 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = 2 >= 1 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = 2 >= 1 = mark(U82(X1,X2)) U91(mark(X1),X2) = 2 >= 1 = mark(U91(X1,X2)) and(mark(X1),X2) = 2 >= 1 = mark(and(X1,X2)) tail(mark(X)) = 2 >= 1 = mark(tail(X)) take(mark(X1),X2) = X2 + 2 >= 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2 >= 1 = mark(take(X1,X2)) sel(mark(X1),X2) = X2 + 2 >= 1 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 + 2 >= 1 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X1 >= X1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X1 + 1 >= X1 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X1 + 1 >= X1 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = 1 >= 1 = isNatural(proper(X)) proper(isLNat(X)) = 1 >= 1 = isLNat(proper(X)) proper(isPLNat(X)) = 1 >= 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X1 >= X1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + 1 >= X1 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = 1 >= 1 = ok(isNatural(X)) isLNat(ok(X)) = 1 >= 1 = ok(isLNat(X)) isPLNat(ok(X)) = 1 >= 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(cons(X1,X2)) -> active#(X1) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [sel](x0, x1) = 0, [0] = 0, [take](x0, x1) = 0, [tail](x0) = 0, [isPLNat](x0) = 0, [and](x0, x1) = 0, [isLNat](x0) = x0, [isNatural](x0) = 0, [U91](x0, x1) = 0, [U82](x0, x1) = 0, [U81](x0, x1, x2, x3) = 0, [pair](x0, x1) = 0, [nil] = 0, [U71](x0, x1) = 0, [U61](x0, x1) = 0, [head](x0) = 1, [afterNth](x0, x1) = 0, [U51](x0, x1, x2) = 0, [cons](x0, x1) = x0 + 1, [natsFrom](x0) = 1, [s](x0) = 0, [U41](x0, x1) = 0, [U31](x0, x1) = x0 + 1, [U21](x0, x1) = 0, [snd](x0) = 0, [U11](x0, x1, x2) = 0, [mark](x0) = 0, [fst](x0) = 0, [splitAt](x0, x1) = 0, [active](x0) = x0, [U101](x0, x1, x2) = 0, [tt] = 1 orientation: active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = 0 >= 0 = mark(X) active(U31(tt(),N)) = 2 >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 0 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = 0 >= 0 = mark(Y) active(U71(tt(),XS)) = 0 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = 0 >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = 0 >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = 0 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = 0 >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = 1 >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = 0 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = 1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = 0 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = 0 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = 0 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = 0 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = 0 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = 1 >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = 0 >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = 0 >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = 0 >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = 0 >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = 0 >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = 0 >= 0 = U101(active(X1),X2,X3) active(fst(X)) = 0 >= 0 = fst(active(X)) active(splitAt(X1,X2)) = 0 >= 0 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = 0 >= 0 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = 0 >= 0 = U11(active(X1),X2,X3) active(snd(X)) = 0 >= 0 = snd(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(natsFrom(X)) = 1 >= 1 = natsFrom(active(X)) active(s(X)) = 0 >= 0 = s(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(head(X)) = 1 >= 1 = head(active(X)) active(afterNth(X1,X2)) = 0 >= 0 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = 0 >= 0 = afterNth(X1,active(X2)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(U71(X1,X2)) = 0 >= 0 = U71(active(X1),X2) active(pair(X1,X2)) = 0 >= 0 = pair(active(X1),X2) active(pair(X1,X2)) = 0 >= 0 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = 0 >= 0 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = 0 >= 0 = U82(active(X1),X2) active(U91(X1,X2)) = 0 >= 0 = U91(active(X1),X2) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) active(tail(X)) = 0 >= 0 = tail(active(X)) active(take(X1,X2)) = 0 >= 0 = take(active(X1),X2) active(take(X1,X2)) = 0 >= 0 = take(X1,active(X2)) active(sel(X1,X2)) = 0 >= 0 = sel(active(X1),X2) active(sel(X1,X2)) = 0 >= 0 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = 1 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 1 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 1 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 0 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = 0 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 0 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = 0 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = 0 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = 0 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = 0 >= 0 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = 0 >= 0 = fst(proper(X)) proper(splitAt(X1,X2)) = 0 >= 0 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = 0 >= 0 = snd(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = 1 >= 1 = natsFrom(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = 1 >= 1 = head(proper(X)) proper(afterNth(X1,X2)) = 0 >= 0 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = 0 >= 0 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = 0 >= 0 = pair(proper(X1),proper(X2)) proper(nil()) = 0 >= 0 = ok(nil()) proper(U81(X1,X2,X3,X4)) = 0 >= 0 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = 0 >= 0 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = 0 >= 0 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatural(X)) = 0 >= 0 = isNatural(proper(X)) proper(isLNat(X)) = X >= X = isLNat(proper(X)) proper(isPLNat(X)) = 0 >= 0 = isPLNat(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(sel(X1,X2)) = 0 >= 0 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U101(X1,X2,X3)) fst(ok(X)) = 0 >= 0 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = 0 >= 0 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) snd(ok(X)) = 0 >= 0 = ok(snd(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = 1 >= 1 = ok(natsFrom(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) head(ok(X)) = 1 >= 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = 0 >= 0 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = 0 >= 0 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = 0 >= 0 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = 0 >= 0 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = 0 >= 0 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatural(ok(X)) = 0 >= 0 = ok(isNatural(X)) isLNat(ok(X)) = X >= X = ok(isLNat(X)) isPLNat(ok(X)) = 0 >= 0 = ok(isPLNat(X)) tail(ok(X)) = 0 >= 0 = ok(tail(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = 0 >= 0 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(U101(X1,X2,X3)) -> proper#(X3) proper#(U101(X1,X2,X3)) -> proper#(X2) proper#(U101(X1,X2,X3)) -> proper#(X1) proper#(fst(X)) -> proper#(X) proper#(splitAt(X1,X2)) -> proper#(X2) proper#(splitAt(X1,X2)) -> proper#(X1) proper#(U11(X1,X2,X3)) -> proper#(X3) proper#(U11(X1,X2,X3)) -> proper#(X2) proper#(U11(X1,X2,X3)) -> proper#(X1) proper#(snd(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(natsFrom(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(head(X)) -> proper#(X) proper#(afterNth(X1,X2)) -> proper#(X2) proper#(afterNth(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U61(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(pair(X1,X2)) -> proper#(X2) proper#(pair(X1,X2)) -> proper#(X1) proper#(U81(X1,X2,X3,X4)) -> proper#(X4) proper#(U81(X1,X2,X3,X4)) -> proper#(X3) proper#(U81(X1,X2,X3,X4)) -> proper#(X2) proper#(U81(X1,X2,X3,X4)) -> proper#(X1) proper#(U82(X1,X2)) -> proper#(X2) proper#(U82(X1,X2)) -> proper#(X1) proper#(U91(X1,X2)) -> proper#(X2) proper#(U91(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(isNatural(X)) -> proper#(X) proper#(isLNat(X)) -> proper#(X) proper#(isPLNat(X)) -> proper#(X) proper#(tail(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [sel](x0, x1) = x0 + x1 + 1, [0] = 1, [take](x0, x1) = x0 + x1 + 1, [tail](x0) = x0 + 1, [isPLNat](x0) = x0 + 1, [and](x0, x1) = x0 + x1 + 1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0 + x1 + 1, [U82](x0, x1) = x0 + x1 + 1, [U81](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [pair](x0, x1) = x0 + x1 + 1, [nil] = 1, [U71](x0, x1) = x0 + x1 + 1, [U61](x0, x1) = x0 + x1 + 1, [head](x0) = x0 + 1, [afterNth](x0, x1) = x0 + x1 + 1, [U51](x0, x1, x2) = x0 + x1 + x2 + 1, [cons](x0, x1) = x0 + x1 + 1, [natsFrom](x0) = x0 + 1, [s](x0) = x0 + 1, [U41](x0, x1) = x0 + x1 + 1, [U31](x0, x1) = x0 + x1 + 1, [U21](x0, x1) = x0 + x1 + 1, [snd](x0) = x0 + 1, [U11](x0, x1, x2) = x0 + x1 + x2 + 1, [mark](x0) = 1, [fst](x0) = x0 + 1, [splitAt](x0, x1) = x0 + x1, [active](x0) = x0, [U101](x0, x1, x2) = x0 + x1 + x2 + 1, [tt] = 0 orientation: proper#(U101(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(U101(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U101(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(fst(X)) = X + 2 >= X + 1 = proper#(X) proper#(splitAt(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(splitAt(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(snd(X)) = X + 2 >= X + 1 = proper#(X) proper#(U21(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U21(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U31(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U41(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(natsFrom(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(head(X)) = X + 2 >= X + 1 = proper#(X) proper#(afterNth(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(afterNth(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U61(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U61(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U71(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U71(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(pair(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(pair(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U81(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X4 + 1 = proper#(X4) proper#(U81(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X3 + 1 = proper#(X3) proper#(U81(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X2 + 1 = proper#(X2) proper#(U81(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 2 >= X1 + 1 = proper#(X1) proper#(U82(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U82(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U91(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U91(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(and(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(isNatural(X)) = X + 1 >= X + 1 = proper#(X) proper#(isLNat(X)) = X + 1 >= X + 1 = proper#(X) proper#(isPLNat(X)) = X + 2 >= X + 1 = proper#(X) proper#(tail(X)) = X + 2 >= X + 1 = proper#(X) proper#(take(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(take(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(sel(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(sel(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) active(U101(tt(),N,XS)) = N + XS + 1 >= 1 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = N + XS + 1 >= 1 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X + 1 >= 1 = mark(X) active(U31(tt(),N)) = N + 1 >= 1 = mark(N) active(U41(tt(),N)) = N + 1 >= 1 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = N + XS + 1 >= 1 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y + 1 >= 1 = mark(Y) active(U71(tt(),XS)) = XS + 1 >= 1 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = N + X + XS + 1 >= 1 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X + YS + ZS + 2 >= 1 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = XS + 1 >= 1 = mark(XS) active(afterNth(N,XS)) = N + XS + 1 >= 1 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X + 1 >= 1 = mark(X) active(fst(pair(X,Y))) = X + Y + 2 >= 1 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = N + XS + 2 >= 1 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 1 >= 1 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + V2 + 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V1 + V2 + 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 + 1 >= 1 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 + 1 >= 1 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 + 1 >= 1 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 + 1 >= 1 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V1 + V2 + 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 1 >= 1 = mark(tt()) active(isNatural(head(V1))) = V1 + 1 >= 1 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 + 1 >= 1 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 + V2 + 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V1 + V2 + 2 >= 1 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 + V2 + 1 >= 1 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N + 1 >= 1 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N + XS + 1 >= 1 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = X + Y + 2 >= 1 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS + 1 >= 1 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N + X + XS + 2 >= 1 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = N + XS + 2 >= 1 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = N + XS + 1 >= 1 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U101(active(X1),X2,X3) active(fst(X)) = X + 1 >= X + 1 = fst(active(X)) active(splitAt(X1,X2)) = X1 + X2 >= X1 + X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 + X2 >= X1 + X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(active(X1),X2,X3) active(snd(X)) = X + 1 >= X + 1 = snd(active(X)) active(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(active(X1),X2) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U41(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U41(active(X1),X2) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(active(X)) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U51(active(X1),X2,X3) active(head(X)) = X + 1 >= X + 1 = head(active(X)) active(afterNth(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = pair(active(X1),X2) active(pair(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U82(active(X1),X2) active(U91(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U91(active(X1),X2) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) active(tail(X)) = X + 1 >= X + 1 = tail(active(X)) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(active(X1),X2) active(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,active(X2)) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X2 + X3 + 2 >= 1 = mark(U101(X1,X2,X3)) fst(mark(X)) = 2 >= 1 = mark(fst(X)) splitAt(mark(X1),X2) = X2 + 1 >= 1 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 + 1 >= 1 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X2 + X3 + 2 >= 1 = mark(U11(X1,X2,X3)) snd(mark(X)) = 2 >= 1 = mark(snd(X)) U21(mark(X1),X2) = X2 + 2 >= 1 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 + 2 >= 1 = mark(U31(X1,X2)) U41(mark(X1),X2) = X2 + 2 >= 1 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 + 2 >= 1 = mark(cons(X1,X2)) natsFrom(mark(X)) = 2 >= 1 = mark(natsFrom(X)) s(mark(X)) = 2 >= 1 = mark(s(X)) U51(mark(X1),X2,X3) = X2 + X3 + 2 >= 1 = mark(U51(X1,X2,X3)) head(mark(X)) = 2 >= 1 = mark(head(X)) afterNth(mark(X1),X2) = X2 + 2 >= 1 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 2 >= 1 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 + 2 >= 1 = mark(U61(X1,X2)) U71(mark(X1),X2) = X2 + 2 >= 1 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 + 2 >= 1 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 + 2 >= 1 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X2 + X3 + X4 + 2 >= 1 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 + 2 >= 1 = mark(U82(X1,X2)) U91(mark(X1),X2) = X2 + 2 >= 1 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 + 2 >= 1 = mark(and(X1,X2)) tail(mark(X)) = 2 >= 1 = mark(tail(X)) take(mark(X1),X2) = X2 + 2 >= 1 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 + 2 >= 1 = mark(take(X1,X2)) sel(mark(X1),X2) = X2 + 2 >= 1 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 + 2 >= 1 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X1 + X2 >= X1 + X2 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 0 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X >= X = isNatural(proper(X)) proper(isLNat(X)) = X >= X = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) proper(sel(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U101(X1,X2,X3)) fst(ok(X)) = 1 >= 0 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = 0 >= 0 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U11(X1,X2,X3)) snd(ok(X)) = 1 >= 0 = ok(snd(X)) U21(ok(X1),ok(X2)) = 1 >= 0 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = 1 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = 1 >= 0 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) natsFrom(ok(X)) = 1 >= 0 = ok(natsFrom(X)) s(ok(X)) = 1 >= 0 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U51(X1,X2,X3)) head(ok(X)) = 1 >= 0 = ok(head(X)) afterNth(ok(X1),ok(X2)) = 1 >= 0 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = 1 >= 0 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = 1 >= 0 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = 1 >= 0 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = 1 >= 0 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = 1 >= 0 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = 1 >= 0 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = 1 >= 0 = ok(and(X1,X2)) isNatural(ok(X)) = 0 >= 0 = ok(isNatural(X)) isLNat(ok(X)) = 0 >= 0 = ok(isLNat(X)) isPLNat(ok(X)) = 1 >= 0 = ok(isPLNat(X)) tail(ok(X)) = 1 >= 0 = ok(tail(X)) take(ok(X1),ok(X2)) = 1 >= 0 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = 1 >= 0 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(splitAt(X1,X2)) -> proper#(X2) proper#(splitAt(X1,X2)) -> proper#(X1) proper#(isNatural(X)) -> proper#(X) proper#(isLNat(X)) -> proper#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [sel](x0, x1) = 1, [0] = 1, [take](x0, x1) = 0, [tail](x0) = 0, [isPLNat](x0) = 0, [and](x0, x1) = 0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = 0, [U82](x0, x1) = 0, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = 1, [nil] = 1, [U71](x0, x1) = 0, [U61](x0, x1) = 1, [head](x0) = 0, [afterNth](x0, x1) = 0, [U51](x0, x1, x2) = x1 + x2, [cons](x0, x1) = 1, [natsFrom](x0) = 0, [s](x0) = 0, [U41](x0, x1) = 0, [U31](x0, x1) = 0, [U21](x0, x1) = 1, [snd](x0) = x0 + 1, [U11](x0, x1, x2) = x1 + x2, [mark](x0) = 0, [fst](x0) = 0, [splitAt](x0, x1) = x0 + x1 + 1, [active](x0) = x0, [U101](x0, x1, x2) = 0, [tt] = 0 orientation: proper#(splitAt(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(splitAt(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(isNatural(X)) = X + 1 >= X + 1 = proper#(X) proper#(isLNat(X)) = X + 1 >= X + 1 = proper#(X) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = N + XS >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = 1 >= 0 = mark(X) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = N + XS >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = 1 >= 0 = mark(Y) active(U71(tt(),XS)) = 0 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = 0 >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = 0 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = 0 >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = 0 >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 1 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = 0 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = 0 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 + 1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = 0 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 1 >= 0 = mark(tt()) active(isNatural(head(V1))) = 0 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = 0 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = 0 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = 0 >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = 1 >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = 2 >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS + 2 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = 2 >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = 0 >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = 0 >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = 0 >= 0 = U101(active(X1),X2,X3) active(fst(X)) = 0 >= 0 = fst(active(X)) active(splitAt(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = U11(active(X1),X2,X3) active(snd(X)) = X + 1 >= X + 1 = snd(active(X)) active(U21(X1,X2)) = 1 >= 1 = U21(active(X1),X2) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) active(natsFrom(X)) = 0 >= 0 = natsFrom(active(X)) active(s(X)) = 0 >= 0 = s(active(X)) active(U51(X1,X2,X3)) = X2 + X3 >= X2 + X3 = U51(active(X1),X2,X3) active(head(X)) = 0 >= 0 = head(active(X)) active(afterNth(X1,X2)) = 0 >= 0 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = 0 >= 0 = afterNth(X1,active(X2)) active(U61(X1,X2)) = 1 >= 1 = U61(active(X1),X2) active(U71(X1,X2)) = 0 >= 0 = U71(active(X1),X2) active(pair(X1,X2)) = 1 >= 1 = pair(active(X1),X2) active(pair(X1,X2)) = 1 >= 1 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = 0 >= 0 = U82(active(X1),X2) active(U91(X1,X2)) = 0 >= 0 = U91(active(X1),X2) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) active(tail(X)) = 0 >= 0 = tail(active(X)) active(take(X1,X2)) = 0 >= 0 = take(active(X1),X2) active(take(X1,X2)) = 0 >= 0 = take(X1,active(X2)) active(sel(X1,X2)) = 1 >= 1 = sel(active(X1),X2) active(sel(X1,X2)) = 1 >= 1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 + 1 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 + 1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 1 >= 0 = mark(snd(X)) U21(mark(X1),X2) = 1 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 0 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = 0 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 0 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = 1 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 1 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = 0 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 1 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = 0 >= 0 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(fst(X)) = 0 >= 0 = fst(proper(X)) proper(splitAt(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = 0 >= 0 = natsFrom(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(U51(X1,X2,X3)) = X2 + X3 >= X2 + X3 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(afterNth(X1,X2)) = 0 >= 0 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = 0 >= 0 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = 1 >= 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 0 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = 0 >= 0 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = 0 >= 0 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X >= X = isNatural(proper(X)) proper(isLNat(X)) = X >= X = isLNat(proper(X)) proper(isPLNat(X)) = 0 >= 0 = isPLNat(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 0 = ok(0()) proper(sel(X1,X2)) = 1 >= 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U101(X1,X2,X3)) fst(ok(X)) = 0 >= 0 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = 1 >= 0 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) snd(ok(X)) = 1 >= 0 = ok(snd(X)) U21(ok(X1),ok(X2)) = 1 >= 0 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) natsFrom(ok(X)) = 0 >= 0 = ok(natsFrom(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) head(ok(X)) = 0 >= 0 = ok(head(X)) afterNth(ok(X1),ok(X2)) = 0 >= 0 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = 1 >= 0 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = 0 >= 0 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = 1 >= 0 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = 0 >= 0 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = 0 >= 0 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatural(ok(X)) = 0 >= 0 = ok(isNatural(X)) isLNat(ok(X)) = 0 >= 0 = ok(isLNat(X)) isPLNat(ok(X)) = 0 >= 0 = ok(isPLNat(X)) tail(ok(X)) = 0 >= 0 = ok(tail(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = 1 >= 0 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(isNatural(X)) -> proper#(X) proper#(isLNat(X)) -> proper#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [sel](x0, x1) = 0, [0] = 0, [take](x0, x1) = 0, [tail](x0) = 0, [isPLNat](x0) = 0, [and](x0, x1) = x0, [isLNat](x0) = x0 + 1, [isNatural](x0) = x0 + 1, [U91](x0, x1) = 0, [U82](x0, x1) = 0, [U81](x0, x1, x2, x3) = x0, [pair](x0, x1) = 0, [nil] = 1, [U71](x0, x1) = 0, [U61](x0, x1) = x0, [head](x0) = 0, [afterNth](x0, x1) = 0, [U51](x0, x1, x2) = 0, [cons](x0, x1) = 0, [natsFrom](x0) = 0, [s](x0) = 0, [U41](x0, x1) = 0, [U31](x0, x1) = 0, [U21](x0, x1) = 0, [snd](x0) = 0, [U11](x0, x1, x2) = 0, [mark](x0) = 0, [fst](x0) = 0, [splitAt](x0, x1) = 0, [active](x0) = 0, [U101](x0, x1, x2) = 0, [tt] = 0 orientation: proper#(isNatural(X)) = X + 2 >= X + 1 = proper#(X) proper#(isLNat(X)) = X + 2 >= X + 1 = proper#(X) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = 0 >= 0 = mark(X) active(U31(tt(),N)) = 0 >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 0 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = 0 >= 0 = mark(Y) active(U71(tt(),XS)) = 0 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = 0 >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = 0 >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = 0 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = 0 >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = 0 >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = 0 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = 0 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = 0 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = 0 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = 0 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = 0 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = 0 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = 0 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = 0 >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = 0 >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = 0 >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = 0 >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = 0 >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = 0 >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = 0 >= 0 = U101(active(X1),X2,X3) active(fst(X)) = 0 >= 0 = fst(active(X)) active(splitAt(X1,X2)) = 0 >= 0 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = 0 >= 0 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = 0 >= 0 = U11(active(X1),X2,X3) active(snd(X)) = 0 >= 0 = snd(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(natsFrom(X)) = 0 >= 0 = natsFrom(active(X)) active(s(X)) = 0 >= 0 = s(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(head(X)) = 0 >= 0 = head(active(X)) active(afterNth(X1,X2)) = 0 >= 0 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = 0 >= 0 = afterNth(X1,active(X2)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(U71(X1,X2)) = 0 >= 0 = U71(active(X1),X2) active(pair(X1,X2)) = 0 >= 0 = pair(active(X1),X2) active(pair(X1,X2)) = 0 >= 0 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = 0 >= 0 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = 0 >= 0 = U82(active(X1),X2) active(U91(X1,X2)) = 0 >= 0 = U91(active(X1),X2) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) active(tail(X)) = 0 >= 0 = tail(active(X)) active(take(X1,X2)) = 0 >= 0 = take(active(X1),X2) active(take(X1,X2)) = 0 >= 0 = take(X1,active(X2)) active(sel(X1,X2)) = 0 >= 0 = sel(active(X1),X2) active(sel(X1,X2)) = 0 >= 0 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 0 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = 0 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 0 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = 0 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = 0 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = 0 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = 0 >= 0 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 0 >= 0 = ok(tt()) proper(fst(X)) = 0 >= 0 = fst(proper(X)) proper(splitAt(X1,X2)) = 0 >= 0 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = 0 >= 0 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = 0 >= 0 = snd(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = 0 >= 0 = natsFrom(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = 0 >= 0 = head(proper(X)) proper(afterNth(X1,X2)) = 0 >= 0 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X1 >= X1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = 0 >= 0 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = 0 >= 0 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 0 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X1 >= X1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = 0 >= 0 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = 0 >= 0 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 >= X1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = 0 >= 0 = isPLNat(proper(X)) proper(tail(X)) = 0 >= 0 = tail(proper(X)) proper(take(X1,X2)) = 0 >= 0 = take(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(sel(X1,X2)) = 0 >= 0 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U101(X1,X2,X3)) fst(ok(X)) = 0 >= 0 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = 0 >= 0 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U11(X1,X2,X3)) snd(ok(X)) = 0 >= 0 = ok(snd(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) natsFrom(ok(X)) = 0 >= 0 = ok(natsFrom(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) head(ok(X)) = 0 >= 0 = ok(head(X)) afterNth(ok(X1),ok(X2)) = 0 >= 0 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = 0 >= 0 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = 0 >= 0 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = 0 >= 0 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = 0 >= 0 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = 0 >= 0 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatural(ok(X)) = 1 >= 0 = ok(isNatural(X)) isLNat(ok(X)) = 1 >= 0 = ok(isLNat(X)) isPLNat(ok(X)) = 0 >= 0 = ok(isPLNat(X)) tail(ok(X)) = 0 >= 0 = ok(tail(X)) take(ok(X1),ok(X2)) = 0 >= 0 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = 0 >= 0 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(ok(X1),ok(X2)) -> take#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: tail#(mark(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isPLNat#(ok(X)) -> isPLNat#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isPLNat#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x1, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x1, [U82](x0, x1) = x0, [U81](x0, x1, x2, x3) = x2 + 1, [pair](x0, x1) = x0, [nil] = 0, [U71](x0, x1) = x1, [U61](x0, x1) = x0 + 1, [head](x0) = x0, [afterNth](x0, x1) = x0, [U51](x0, x1, x2) = x2, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1 + 1, [U21](x0, x1) = x0, [snd](x0) = x0, [U11](x0, x1, x2) = x2, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x0, [active](x0) = x0, [U101](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: isPLNat#(ok(X)) = X + 2 >= X + 1 = isPLNat#(X) active(U101(tt(),N,XS)) = XS + 1 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = XS >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = 0 >= 0 = mark(X) active(U31(tt(),N)) = N + 1 >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = XS >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = 1 >= 0 = mark(Y) active(U71(tt(),XS)) = XS >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X + 1 >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = YS >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = XS >= 0 = mark(XS) active(afterNth(N,XS)) = N >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = X >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V1 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = XS >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = X >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X1 >= X1 = U21(active(X1),X2) active(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X3 >= X3 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 >= X1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 >= X1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(active(X1),X2) active(U71(X1,X2)) = X2 >= X2 = U71(active(X1),X2) active(pair(X1,X2)) = X1 >= X1 = pair(active(X1),X2) active(pair(X1,X2)) = X1 >= X1 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X1 >= X1 = U82(active(X1),X2) active(U91(X1,X2)) = X2 >= X2 = U91(active(X1),X2) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X2 >= X2 = sel(active(X1),X2) active(sel(X1,X2)) = X2 >= X2 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 + 1 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = X3 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 0 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = X2 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = 0 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 + 1 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = 0 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = X2 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = X2 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X1 + 1 >= X1 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 2 >= X2 + 2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X1 + 2 >= X1 + 2 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X2 + 1 >= X2 + 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + 1 >= X1 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 2 >= X3 + 2 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X1 + 1 >= X1 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X2 + 1 >= X2 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X2 + 1 >= X2 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 2 >= X3 + 2 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isLNat#(ok(X)) -> isLNat#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isLNat#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x1, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x1, [U82](x0, x1) = x0, [U81](x0, x1, x2, x3) = x2 + 1, [pair](x0, x1) = x0, [nil] = 0, [U71](x0, x1) = x1, [U61](x0, x1) = x0 + 1, [head](x0) = x0, [afterNth](x0, x1) = x0, [U51](x0, x1, x2) = x2, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1 + 1, [U21](x0, x1) = x0, [snd](x0) = x0, [U11](x0, x1, x2) = x2, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x0, [active](x0) = x0, [U101](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: isLNat#(ok(X)) = X + 2 >= X + 1 = isLNat#(X) active(U101(tt(),N,XS)) = XS + 1 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = XS >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = 0 >= 0 = mark(X) active(U31(tt(),N)) = N + 1 >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = XS >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = 1 >= 0 = mark(Y) active(U71(tt(),XS)) = XS >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X + 1 >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = YS >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = XS >= 0 = mark(XS) active(afterNth(N,XS)) = N >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = X >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V1 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = XS >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = X >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X1 >= X1 = U21(active(X1),X2) active(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X3 >= X3 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 >= X1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 >= X1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(active(X1),X2) active(U71(X1,X2)) = X2 >= X2 = U71(active(X1),X2) active(pair(X1,X2)) = X1 >= X1 = pair(active(X1),X2) active(pair(X1,X2)) = X1 >= X1 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X1 >= X1 = U82(active(X1),X2) active(U91(X1,X2)) = X2 >= X2 = U91(active(X1),X2) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X2 >= X2 = sel(active(X1),X2) active(sel(X1,X2)) = X2 >= X2 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 + 1 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = X3 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 0 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = X2 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = 0 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 + 1 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = 0 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = X2 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = X2 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X1 + 1 >= X1 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 2 >= X2 + 2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X1 + 2 >= X1 + 2 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X2 + 1 >= X2 + 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + 1 >= X1 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 2 >= X3 + 2 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X1 + 1 >= X1 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X2 + 1 >= X2 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X2 + 1 >= X2 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 2 >= X3 + 2 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatural#(ok(X)) -> isNatural#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatural#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x1, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x1, [U82](x0, x1) = x0, [U81](x0, x1, x2, x3) = x2 + 1, [pair](x0, x1) = x0, [nil] = 0, [U71](x0, x1) = x1, [U61](x0, x1) = x0 + 1, [head](x0) = x0, [afterNth](x0, x1) = x0, [U51](x0, x1, x2) = x2, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1 + 1, [U21](x0, x1) = x0, [snd](x0) = x0, [U11](x0, x1, x2) = x2, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x0, [active](x0) = x0, [U101](x0, x1, x2) = x2 + 1, [tt] = 0 orientation: isNatural#(ok(X)) = X + 2 >= X + 1 = isNatural#(X) active(U101(tt(),N,XS)) = XS + 1 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = XS >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = 0 >= 0 = mark(X) active(U31(tt(),N)) = N + 1 >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = XS >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = 1 >= 0 = mark(Y) active(U71(tt(),XS)) = XS >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X + 1 >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = YS >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = XS >= 0 = mark(XS) active(afterNth(N,XS)) = N >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = X >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V1 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = XS >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = X >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X1 >= X1 = U21(active(X1),X2) active(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X3 >= X3 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 >= X1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 >= X1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(active(X1),X2) active(U71(X1,X2)) = X2 >= X2 = U71(active(X1),X2) active(pair(X1,X2)) = X1 >= X1 = pair(active(X1),X2) active(pair(X1,X2)) = X1 >= X1 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X1 >= X1 = U82(active(X1),X2) active(U91(X1,X2)) = X2 >= X2 = U91(active(X1),X2) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X2 >= X2 = sel(active(X1),X2) active(sel(X1,X2)) = X2 >= X2 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X3 + 1 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 + 1 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = X3 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 0 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = X2 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = 0 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = X1 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 + 1 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = 0 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = X2 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = X2 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X3 + 2 >= X3 + 2 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X1 + 1 >= X1 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 2 >= X2 + 2 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X1 + 2 >= X1 + 2 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X2 + 1 >= X2 + 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X1 + 1 >= X1 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 2 >= X3 + 2 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X1 + 1 >= X1 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X2 + 1 >= X2 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X2 + 1 >= X2 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 2 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 2 >= X3 + 2 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: and#(mark(X1),X2) = X2 + 1 >= X2 + 1 = and#(X1,X2) and#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = and#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U91#(mark(X1),X2) -> U91#(X1,X2) U91#(ok(X1),ok(X2)) -> U91#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U91#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: U91#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U91#(X1,X2) U91#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U91#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U91#(mark(X1),X2) -> U91#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U82#(mark(X1),X2) -> U82#(X1,X2) U82#(ok(X1),ok(X2)) -> U82#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U82#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: U82#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U82#(X1,X2) U82#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U82#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U82#(mark(X1),X2) -> U82#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U81#(mark(X1),X2,X3,X4) -> U81#(X1,X2,X3,X4) U81#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U81#(X1,X2,X3,X4) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U81#](x0, x1, x2, x3) = x3 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [sel](x0, x1) = x1, [0] = 0, [take](x0, x1) = x0, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x1, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x0, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x1, [U51](x0, x1, x2) = x2, [cons](x0, x1) = x0, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x2, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x0, [active](x0) = x0, [U101](x0, x1, x2) = x1, [tt] = 0 orientation: U81#(mark(X1),X2,X3,X4) = X4 + 1 >= X4 + 1 = U81#(X1,X2,X3,X4) U81#(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 2 >= X4 + 1 = U81#(X1,X2,X3,X4) active(U101(tt(),N,XS)) = N >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = XS >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = XS >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 0 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = 0 >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = XS >= 0 = mark(XS) active(afterNth(N,XS)) = XS >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = N >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = XS >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = N >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = N >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X2 >= X2 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X3 >= X3 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X3 >= X3 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 >= X1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X1 >= X1 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X2 >= X2 = U91(active(X1),X2) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(sel(X1,X2)) = X2 >= X2 = sel(active(X1),X2) active(sel(X1,X2)) = X2 >= X2 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X2 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X3 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = X3 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = X2 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = 0 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 0 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = 0 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = X2 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = X2 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = 0 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = 1 >= 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = 1 >= 1 = fst(proper(X)) proper(splitAt(X1,X2)) = 1 >= 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = 1 >= 1 = snd(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = 1 >= 1 = natsFrom(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = 1 >= 1 = head(proper(X)) proper(afterNth(X1,X2)) = 1 >= 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = 1 >= 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = 1 >= 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = 1 >= 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = 1 >= 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = 1 >= 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = 1 >= 1 = isNatural(proper(X)) proper(isLNat(X)) = 1 >= 1 = isLNat(proper(X)) proper(isPLNat(X)) = 1 >= 1 = isPLNat(proper(X)) proper(tail(X)) = 1 >= 1 = tail(proper(X)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = 1 >= 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X1 + 1 >= X1 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U81#(mark(X1),X2,X3,X4) -> U81#(X1,X2,X3,X4) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) pair#(ok(X1),ok(X2)) -> pair#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U71#(mark(X1),X2) -> U71#(X1,X2) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U71#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: U71#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U71#(X1,X2) U71#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U71#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U71#(mark(X1),X2) -> U71#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U61#(mark(X1),X2) -> U61#(X1,X2) U61#(ok(X1),ok(X2)) -> U61#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: U61#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U61#(X1,X2) U61#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U61#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U61#(mark(X1),X2) -> U61#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: head#(mark(X)) -> head#(X) head#(ok(X)) -> head#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1, x2) = x1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x0, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x3, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x1, [U51](x0, x1, x2) = x0, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x1, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x1, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x0, [active](x0) = x0, [U101](x0, x1, x2) = x2, [tt] = 0 orientation: U51#(mark(X1),X2,X3) = X2 >= X2 = U51#(X1,X2,X3) U51#(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 = U51#(X1,X2,X3) active(U101(tt(),N,XS)) = XS >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = N >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = N >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 0 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = XS >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = XS >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = XS >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = N >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X3 >= X3 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X2 >= X2 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X2 >= X2 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X2 >= X2 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X4 >= X4 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X3 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X2 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = X2 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = X2 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = 0 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = X2 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = 1 >= 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = 1 >= 1 = fst(proper(X)) proper(splitAt(X1,X2)) = 1 >= 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = 1 >= 1 = snd(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = 1 >= 1 = natsFrom(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = 1 >= 1 = head(proper(X)) proper(afterNth(X1,X2)) = 1 >= 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = 1 >= 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = 1 >= 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = 1 >= 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = 1 >= 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = 1 >= 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = 1 >= 1 = isNatural(proper(X)) proper(isLNat(X)) = 1 >= 1 = isLNat(proper(X)) proper(isPLNat(X)) = 1 >= 1 = isPLNat(proper(X)) proper(tail(X)) = 1 >= 1 = tail(proper(X)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = 1 >= 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: natsFrom#(mark(X)) -> natsFrom#(X) natsFrom#(ok(X)) -> natsFrom#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: cons#(mark(X1),X2) = X2 + 1 >= X2 + 1 = cons#(X1,X2) cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U41#(mark(X1),X2) -> U41#(X1,X2) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: U41#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U41#(X1,X2) U41#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U41#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U41#(mark(X1),X2) -> U41#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U31#(mark(X1),X2) -> U31#(X1,X2) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: U31#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U31#(X1,X2) U31#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U31#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U31#(mark(X1),X2) -> U31#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U21#(mark(X1),X2) -> U21#(X1,X2) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x1, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x1, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x2, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x0 + 1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x0, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x0, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x1, [active](x0) = x0, [U101](x0, x1, x2) = x0, [tt] = 0 orientation: U21#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U21#(X1,X2) U21#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U21#(X1,X2) active(U101(tt(),N,XS)) = 0 >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = 0 >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = 0 >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 1 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = 1 >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = X >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = N + 1 >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = X >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V1 + 1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = XS >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = XS >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = XS >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X1 >= X1 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X2 >= X2 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X1 >= X1 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X1 >= X1 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X1 + 1 >= X1 + 1 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X1 + 1 >= X1 + 1 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X3 >= X3 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X2 >= X2 = take(active(X1),X2) active(take(X1,X2)) = X2 >= X2 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = 0 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = X2 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = 0 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = 0 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = 1 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = X1 + 1 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = 1 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X3 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = X2 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = 0 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = X + 1 >= X + 1 = fst(proper(X)) proper(splitAt(X1,X2)) = X2 + 1 >= X2 + 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = X + 1 >= X + 1 = snd(proper(X)) proper(U21(X1,X2)) = X2 + 1 >= X2 + 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = X + 1 >= X + 1 = natsFrom(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = X + 1 >= X + 1 = head(proper(X)) proper(afterNth(X1,X2)) = X1 + 2 >= X1 + 2 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = X1 + 2 >= X1 + 2 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = X2 + 1 >= X2 + 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = X2 + 1 >= X2 + 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = X1 + 1 >= X1 + 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = X2 + 1 >= X2 + 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = X + 1 >= X + 1 = isNatural(proper(X)) proper(isLNat(X)) = X + 1 >= X + 1 = isLNat(proper(X)) proper(isPLNat(X)) = X + 1 >= X + 1 = isPLNat(proper(X)) proper(tail(X)) = X + 1 >= X + 1 = tail(proper(X)) proper(take(X1,X2)) = X2 + 1 >= X2 + 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = X1 + 1 >= X1 + 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 2 >= X1 + 2 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X1 + 2 >= X1 + 2 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X3 + 1 >= X3 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U21#(mark(X1),X2) -> U21#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: snd#(mark(X)) -> snd#(X) snd#(ok(X)) -> snd#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1, x2) = x1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x0, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x3, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x1, [U51](x0, x1, x2) = x0, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x1, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x1, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x0, [active](x0) = x0, [U101](x0, x1, x2) = x2, [tt] = 0 orientation: U11#(mark(X1),X2,X3) = X2 >= X2 = U11#(X1,X2,X3) U11#(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 = U11#(X1,X2,X3) active(U101(tt(),N,XS)) = XS >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = N >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = N >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 0 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = XS >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = XS >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = XS >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = N >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X3 >= X3 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X2 >= X2 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X2 >= X2 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X2 >= X2 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X4 >= X4 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X3 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X2 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = X2 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = X2 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = 0 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = X2 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = 1 >= 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = 1 >= 1 = fst(proper(X)) proper(splitAt(X1,X2)) = 1 >= 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = 1 >= 1 = snd(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = 1 >= 1 = natsFrom(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = 1 >= 1 = head(proper(X)) proper(afterNth(X1,X2)) = 1 >= 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = 1 >= 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = 1 >= 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = 1 >= 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = 1 >= 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = 1 >= 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = 1 >= 1 = isNatural(proper(X)) proper(isLNat(X)) = 1 >= 1 = isLNat(proper(X)) proper(isPLNat(X)) = 1 >= 1 = isPLNat(proper(X)) proper(tail(X)) = 1 >= 1 = tail(proper(X)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = 1 >= 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: fst#(mark(X)) -> fst#(X) fst#(ok(X)) -> fst#(X) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3) U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U101#](x0, x1, x2) = x1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [sel](x0, x1) = x0, [0] = 0, [take](x0, x1) = x0, [tail](x0) = x0, [isPLNat](x0) = x0, [and](x0, x1) = x0, [isLNat](x0) = x0, [isNatural](x0) = x0, [U91](x0, x1) = x0, [U82](x0, x1) = x1, [U81](x0, x1, x2, x3) = x3, [pair](x0, x1) = x1, [nil] = 0, [U71](x0, x1) = x1, [U61](x0, x1) = x1, [head](x0) = x0, [afterNth](x0, x1) = x1, [U51](x0, x1, x2) = x0, [cons](x0, x1) = x1, [natsFrom](x0) = x0, [s](x0) = x0, [U41](x0, x1) = x1, [U31](x0, x1) = x1, [U21](x0, x1) = x1, [snd](x0) = x0, [U11](x0, x1, x2) = x1, [mark](x0) = 0, [fst](x0) = x0, [splitAt](x0, x1) = x0, [active](x0) = x0, [U101](x0, x1, x2) = x2, [tt] = 0 orientation: U101#(mark(X1),X2,X3) = X2 >= X2 = U101#(X1,X2,X3) U101#(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 = U101#(X1,X2,X3) active(U101(tt(),N,XS)) = XS >= 0 = mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) = N >= 0 = mark(snd(splitAt(N,XS))) active(U21(tt(),X)) = X >= 0 = mark(X) active(U31(tt(),N)) = N >= 0 = mark(N) active(U41(tt(),N)) = N >= 0 = mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) = 0 >= 0 = mark(head(afterNth(N,XS))) active(U61(tt(),Y)) = Y >= 0 = mark(Y) active(U71(tt(),XS)) = XS >= 0 = mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) = XS >= 0 = mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) = X >= 0 = mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) = 0 >= 0 = mark(XS) active(afterNth(N,XS)) = XS >= 0 = mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) = 0 >= 0 = mark(X) active(fst(pair(X,Y))) = Y >= 0 = mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) = XS >= 0 = mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) = 0 >= 0 = mark(tt()) active(isLNat(afterNth(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) = V2 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isLNat(snd(V1))) = V1 >= 0 = mark(isPLNat(V1)) active(isLNat(tail(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isLNat(take(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) = 0 >= 0 = mark(tt()) active(isNatural(head(V1))) = V1 >= 0 = mark(isLNat(V1)) active(isNatural(s(V1))) = V1 >= 0 = mark(isNatural(V1)) active(isNatural(sel(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) = V2 >= 0 = mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) = V1 >= 0 = mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) = N >= 0 = mark(U41(isNatural(N),N)) active(sel(N,XS)) = N >= 0 = mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) = Y >= 0 = mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) = 0 >= 0 = mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) = N >= 0 = mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) = XS >= 0 = mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) = N >= 0 = mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) = X3 >= X3 = U101(active(X1),X2,X3) active(fst(X)) = X >= X = fst(active(X)) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(active(X1),X2) active(splitAt(X1,X2)) = X1 >= X1 = splitAt(X1,active(X2)) active(U11(X1,X2,X3)) = X2 >= X2 = U11(active(X1),X2,X3) active(snd(X)) = X >= X = snd(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U31(X1,X2)) = X2 >= X2 = U31(active(X1),X2) active(U41(X1,X2)) = X2 >= X2 = U41(active(X1),X2) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(natsFrom(X)) = X >= X = natsFrom(active(X)) active(s(X)) = X >= X = s(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(head(X)) = X >= X = head(active(X)) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(active(X1),X2) active(afterNth(X1,X2)) = X2 >= X2 = afterNth(X1,active(X2)) active(U61(X1,X2)) = X2 >= X2 = U61(active(X1),X2) active(U71(X1,X2)) = X2 >= X2 = U71(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(active(X1),X2) active(pair(X1,X2)) = X2 >= X2 = pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) = X4 >= X4 = U81(active(X1),X2,X3,X4) active(U82(X1,X2)) = X2 >= X2 = U82(active(X1),X2) active(U91(X1,X2)) = X1 >= X1 = U91(active(X1),X2) active(and(X1,X2)) = X1 >= X1 = and(active(X1),X2) active(tail(X)) = X >= X = tail(active(X)) active(take(X1,X2)) = X1 >= X1 = take(active(X1),X2) active(take(X1,X2)) = X1 >= X1 = take(X1,active(X2)) active(sel(X1,X2)) = X1 >= X1 = sel(active(X1),X2) active(sel(X1,X2)) = X1 >= X1 = sel(X1,active(X2)) U101(mark(X1),X2,X3) = X3 >= 0 = mark(U101(X1,X2,X3)) fst(mark(X)) = 0 >= 0 = mark(fst(X)) splitAt(mark(X1),X2) = 0 >= 0 = mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) = X1 >= 0 = mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) = X2 >= 0 = mark(U11(X1,X2,X3)) snd(mark(X)) = 0 >= 0 = mark(snd(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U41(mark(X1),X2) = X2 >= 0 = mark(U41(X1,X2)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) natsFrom(mark(X)) = 0 >= 0 = mark(natsFrom(X)) s(mark(X)) = 0 >= 0 = mark(s(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) head(mark(X)) = 0 >= 0 = mark(head(X)) afterNth(mark(X1),X2) = X2 >= 0 = mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) = 0 >= 0 = mark(afterNth(X1,X2)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) U71(mark(X1),X2) = X2 >= 0 = mark(U71(X1,X2)) pair(mark(X1),X2) = X2 >= 0 = mark(pair(X1,X2)) pair(X1,mark(X2)) = 0 >= 0 = mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) = X4 >= 0 = mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) = X2 >= 0 = mark(U82(X1,X2)) U91(mark(X1),X2) = 0 >= 0 = mark(U91(X1,X2)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) tail(mark(X)) = 0 >= 0 = mark(tail(X)) take(mark(X1),X2) = 0 >= 0 = mark(take(X1,X2)) take(X1,mark(X2)) = X1 >= 0 = mark(take(X1,X2)) sel(mark(X1),X2) = 0 >= 0 = mark(sel(X1,X2)) sel(X1,mark(X2)) = X1 >= 0 = mark(sel(X1,X2)) proper(U101(X1,X2,X3)) = 1 >= 1 = U101(proper(X1),proper(X2),proper(X3)) proper(tt()) = 1 >= 1 = ok(tt()) proper(fst(X)) = 1 >= 1 = fst(proper(X)) proper(splitAt(X1,X2)) = 1 >= 1 = splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) = 1 >= 1 = U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) = 1 >= 1 = snd(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(natsFrom(X)) = 1 >= 1 = natsFrom(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) = 1 >= 1 = head(proper(X)) proper(afterNth(X1,X2)) = 1 >= 1 = afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(U71(X1,X2)) = 1 >= 1 = U71(proper(X1),proper(X2)) proper(pair(X1,X2)) = 1 >= 1 = pair(proper(X1),proper(X2)) proper(nil()) = 1 >= 1 = ok(nil()) proper(U81(X1,X2,X3,X4)) = 1 >= 1 = U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) = 1 >= 1 = U82(proper(X1),proper(X2)) proper(U91(X1,X2)) = 1 >= 1 = U91(proper(X1),proper(X2)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatural(X)) = 1 >= 1 = isNatural(proper(X)) proper(isLNat(X)) = 1 >= 1 = isLNat(proper(X)) proper(isPLNat(X)) = 1 >= 1 = isPLNat(proper(X)) proper(tail(X)) = 1 >= 1 = tail(proper(X)) proper(take(X1,X2)) = 1 >= 1 = take(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(sel(X1,X2)) = 1 >= 1 = sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U101(X1,X2,X3)) fst(ok(X)) = X + 1 >= X + 1 = ok(fst(X)) splitAt(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) = X2 + 1 >= X2 + 1 = ok(U11(X1,X2,X3)) snd(ok(X)) = X + 1 >= X + 1 = ok(snd(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U31(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U31(X1,X2)) U41(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U41(X1,X2)) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) natsFrom(ok(X)) = X + 1 >= X + 1 = ok(natsFrom(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) head(ok(X)) = X + 1 >= X + 1 = ok(head(X)) afterNth(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U61(X1,X2)) U71(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U71(X1,X2)) pair(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) = X4 + 1 >= X4 + 1 = ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U82(X1,X2)) U91(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U91(X1,X2)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatural(ok(X)) = X + 1 >= X + 1 = ok(isNatural(X)) isLNat(ok(X)) = X + 1 >= X + 1 = ok(isLNat(X)) isPLNat(ok(X)) = X + 1 >= X + 1 = ok(isPLNat(X)) tail(ok(X)) = X + 1 >= X + 1 = ok(tail(X)) take(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(take(X1,X2)) sel(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(sel(X1,X2)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3) TRS: active(U101(tt(),N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS))) active(U21(tt(),X)) -> mark(X) active(U31(tt(),N)) -> mark(N) active(U41(tt(),N)) -> mark(cons(N,natsFrom(s(N)))) active(U51(tt(),N,XS)) -> mark(head(afterNth(N,XS))) active(U61(tt(),Y)) -> mark(Y) active(U71(tt(),XS)) -> mark(pair(nil(),XS)) active(U81(tt(),N,X,XS)) -> mark(U82(splitAt(N,XS),X)) active(U82(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(U91(tt(),XS)) -> mark(XS) active(afterNth(N,XS)) -> mark(U11(and(isNatural(N),isLNat(XS)),N,XS)) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(U21(and(isLNat(X),isLNat(Y)),X)) active(head(cons(N,XS))) -> mark(U31(and(isNatural(N),isLNat(XS)),N)) active(isLNat(nil())) -> mark(tt()) active(isLNat(afterNth(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(cons(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isLNat(fst(V1))) -> mark(isPLNat(V1)) active(isLNat(natsFrom(V1))) -> mark(isNatural(V1)) active(isLNat(snd(V1))) -> mark(isPLNat(V1)) active(isLNat(tail(V1))) -> mark(isLNat(V1)) active(isLNat(take(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isNatural(0())) -> mark(tt()) active(isNatural(head(V1))) -> mark(isLNat(V1)) active(isNatural(s(V1))) -> mark(isNatural(V1)) active(isNatural(sel(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(isPLNat(pair(V1,V2))) -> mark(and(isLNat(V1),isLNat(V2))) active(isPLNat(splitAt(V1,V2))) -> mark(and(isNatural(V1),isLNat(V2))) active(natsFrom(N)) -> mark(U41(isNatural(N),N)) active(sel(N,XS)) -> mark(U51(and(isNatural(N),isLNat(XS)),N,XS)) active(snd(pair(X,Y))) -> mark(U61(and(isLNat(X),isLNat(Y)),Y)) active(splitAt(0(),XS)) -> mark(U71(isLNat(XS),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U81(and(isNatural(N),and(isNatural(X),isLNat(XS))),N,X,XS)) active(tail(cons(N,XS))) -> mark(U91(and(isNatural(N),isLNat(XS)),XS)) active(take(N,XS)) -> mark(U101(and(isNatural(N),isLNat(XS)),N,XS)) active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3) active(fst(X)) -> fst(active(X)) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) active(snd(X)) -> snd(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U31(X1,X2)) -> U31(active(X1),X2) active(U41(X1,X2)) -> U41(active(X1),X2) active(cons(X1,X2)) -> cons(active(X1),X2) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(head(X)) -> head(active(X)) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(U61(X1,X2)) -> U61(active(X1),X2) active(U71(X1,X2)) -> U71(active(X1),X2) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(U81(X1,X2,X3,X4)) -> U81(active(X1),X2,X3,X4) active(U82(X1,X2)) -> U82(active(X1),X2) active(U91(X1,X2)) -> U91(active(X1),X2) active(and(X1,X2)) -> and(active(X1),X2) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3)) fst(mark(X)) -> mark(fst(X)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) snd(mark(X)) -> mark(snd(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U41(mark(X1),X2) -> mark(U41(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) head(mark(X)) -> mark(head(X)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) U61(mark(X1),X2) -> mark(U61(X1,X2)) U71(mark(X1),X2) -> mark(U71(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) U81(mark(X1),X2,X3,X4) -> mark(U81(X1,X2,X3,X4)) U82(mark(X1),X2) -> mark(U82(X1,X2)) U91(mark(X1),X2) -> mark(U91(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3)) proper(tt()) -> ok(tt()) proper(fst(X)) -> fst(proper(X)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) proper(snd(X)) -> snd(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(head(X)) -> head(proper(X)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(U81(X1,X2,X3,X4)) -> U81(proper(X1),proper(X2),proper(X3),proper(X4)) proper(U82(X1,X2)) -> U82(proper(X1),proper(X2)) proper(U91(X1,X2)) -> U91(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatural(X)) -> isNatural(proper(X)) proper(isLNat(X)) -> isLNat(proper(X)) proper(isPLNat(X)) -> isPLNat(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3)) fst(ok(X)) -> ok(fst(X)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) snd(ok(X)) -> ok(snd(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) head(ok(X)) -> ok(head(X)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) U81(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U81(X1,X2,X3,X4)) U82(ok(X1),ok(X2)) -> ok(U82(X1,X2)) U91(ok(X1),ok(X2)) -> ok(U91(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatural(ok(X)) -> ok(isNatural(X)) isLNat(ok(X)) -> ok(isLNat(X)) isPLNat(ok(X)) -> ok(isPLNat(X)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open