MAYBE Problem: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(U11(tt(),N,X,XS)) -> splitAt#(N,XS) active#(U11(tt(),N,X,XS)) -> U12#(splitAt(N,XS),X) active#(U12(pair(YS,ZS),X)) -> cons#(X,YS) active#(U12(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS) active#(afterNth(N,XS)) -> splitAt#(N,XS) active#(afterNth(N,XS)) -> snd#(splitAt(N,XS)) active#(natsFrom(N)) -> s#(N) active#(natsFrom(N)) -> natsFrom#(s(N)) active#(natsFrom(N)) -> cons#(N,natsFrom(s(N))) active#(sel(N,XS)) -> afterNth#(N,XS) active#(sel(N,XS)) -> head#(afterNth(N,XS)) active#(splitAt(0(),XS)) -> pair#(nil(),XS) active#(splitAt(s(N),cons(X,XS))) -> U11#(tt(),N,X,XS) active#(take(N,XS)) -> splitAt#(N,XS) active#(take(N,XS)) -> fst#(splitAt(N,XS)) active#(U11(X1,X2,X3,X4)) -> active#(X1) active#(U11(X1,X2,X3,X4)) -> U11#(active(X1),X2,X3,X4) active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> U12#(active(X1),X2) 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#(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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) 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#(snd(X)) -> active#(X) active#(snd(X)) -> snd#(active(X)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(fst(X)) -> active#(X) active#(fst(X)) -> fst#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(natsFrom(X)) -> active#(X) active#(natsFrom(X)) -> natsFrom#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) 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)) 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)) U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U12#(mark(X1),X2) -> U12#(X1,X2) splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) snd#(mark(X)) -> snd#(X) and#(mark(X1),X2) -> and#(X1,X2) fst#(mark(X)) -> fst#(X) head#(mark(X)) -> head#(X) natsFrom#(mark(X)) -> natsFrom#(X) s#(mark(X)) -> s#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) tail#(mark(X)) -> tail#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) proper#(U11(X1,X2,X3,X4)) -> proper#(X4) proper#(U11(X1,X2,X3,X4)) -> proper#(X3) proper#(U11(X1,X2,X3,X4)) -> proper#(X2) proper#(U11(X1,X2,X3,X4)) -> proper#(X1) proper#(U11(X1,X2,X3,X4)) -> U11#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(splitAt(X1,X2)) -> proper#(X2) proper#(splitAt(X1,X2)) -> proper#(X1) proper#(splitAt(X1,X2)) -> splitAt#(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#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(afterNth(X1,X2)) -> proper#(X2) proper#(afterNth(X1,X2)) -> proper#(X1) proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2)) proper#(snd(X)) -> proper#(X) proper#(snd(X)) -> snd#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(fst(X)) -> proper#(X) proper#(fst(X)) -> fst#(proper(X)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(natsFrom(X)) -> proper#(X) proper#(natsFrom(X)) -> natsFrom#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) 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)) U11#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U11#(X1,X2,X3,X4) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2) pair#(ok(X1),ok(X2)) -> pair#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2) snd#(ok(X)) -> snd#(X) and#(ok(X1),ok(X2)) -> and#(X1,X2) fst#(ok(X)) -> fst#(X) head#(ok(X)) -> head#(X) natsFrom#(ok(X)) -> natsFrom#(X) s#(ok(X)) -> s#(X) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) tail#(ok(X)) -> tail#(X) take#(ok(X1),ok(X2)) -> take#(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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(U11(tt(),N,X,XS)) -> splitAt#(N,XS) active#(U11(tt(),N,X,XS)) -> U12#(splitAt(N,XS),X) active#(U12(pair(YS,ZS),X)) -> cons#(X,YS) active#(U12(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS) active#(afterNth(N,XS)) -> splitAt#(N,XS) active#(afterNth(N,XS)) -> snd#(splitAt(N,XS)) active#(natsFrom(N)) -> s#(N) active#(natsFrom(N)) -> natsFrom#(s(N)) active#(natsFrom(N)) -> cons#(N,natsFrom(s(N))) active#(sel(N,XS)) -> afterNth#(N,XS) active#(sel(N,XS)) -> head#(afterNth(N,XS)) active#(splitAt(0(),XS)) -> pair#(nil(),XS) active#(splitAt(s(N),cons(X,XS))) -> U11#(tt(),N,X,XS) active#(take(N,XS)) -> splitAt#(N,XS) active#(take(N,XS)) -> fst#(splitAt(N,XS)) active#(U11(X1,X2,X3,X4)) -> active#(X1) active#(U11(X1,X2,X3,X4)) -> U11#(active(X1),X2,X3,X4) active#(U12(X1,X2)) -> active#(X1) active#(U12(X1,X2)) -> U12#(active(X1),X2) 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#(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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) 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#(snd(X)) -> active#(X) active#(snd(X)) -> snd#(active(X)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) active#(fst(X)) -> active#(X) active#(fst(X)) -> fst#(active(X)) active#(head(X)) -> active#(X) active#(head(X)) -> head#(active(X)) active#(natsFrom(X)) -> active#(X) active#(natsFrom(X)) -> natsFrom#(active(X)) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) 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)) 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)) U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U12#(mark(X1),X2) -> U12#(X1,X2) splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) snd#(mark(X)) -> snd#(X) and#(mark(X1),X2) -> and#(X1,X2) fst#(mark(X)) -> fst#(X) head#(mark(X)) -> head#(X) natsFrom#(mark(X)) -> natsFrom#(X) s#(mark(X)) -> s#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) tail#(mark(X)) -> tail#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) proper#(U11(X1,X2,X3,X4)) -> proper#(X4) proper#(U11(X1,X2,X3,X4)) -> proper#(X3) proper#(U11(X1,X2,X3,X4)) -> proper#(X2) proper#(U11(X1,X2,X3,X4)) -> proper#(X1) proper#(U11(X1,X2,X3,X4)) -> U11#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(U12(X1,X2)) -> U12#(proper(X1),proper(X2)) proper#(splitAt(X1,X2)) -> proper#(X2) proper#(splitAt(X1,X2)) -> proper#(X1) proper#(splitAt(X1,X2)) -> splitAt#(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#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(afterNth(X1,X2)) -> proper#(X2) proper#(afterNth(X1,X2)) -> proper#(X1) proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2)) proper#(snd(X)) -> proper#(X) proper#(snd(X)) -> snd#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(fst(X)) -> proper#(X) proper#(fst(X)) -> fst#(proper(X)) proper#(head(X)) -> proper#(X) proper#(head(X)) -> head#(proper(X)) proper#(natsFrom(X)) -> proper#(X) proper#(natsFrom(X)) -> natsFrom#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2)) 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)) U11#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U11#(X1,X2,X3,X4) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2) pair#(ok(X1),ok(X2)) -> pair#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2) snd#(ok(X)) -> snd#(X) and#(ok(X1),ok(X2)) -> and#(X1,X2) fst#(ok(X)) -> fst#(X) head#(ok(X)) -> head#(X) natsFrom#(ok(X)) -> natsFrom#(X) s#(ok(X)) -> s#(X) sel#(ok(X1),ok(X2)) -> sel#(X1,X2) tail#(ok(X)) -> tail#(X) take#(ok(X1),ok(X2)) -> take#(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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 18 #rules: 83 #arcs: 2476/18225 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(U11(X1,X2,X3,X4)) -> active#(X1) active#(U12(X1,X2)) -> active#(X1) active#(splitAt(X1,X2)) -> active#(X1) active#(splitAt(X1,X2)) -> active#(X2) active#(pair(X1,X2)) -> active#(X1) active#(pair(X1,X2)) -> active#(X2) active#(cons(X1,X2)) -> active#(X1) active#(afterNth(X1,X2)) -> active#(X1) active#(afterNth(X1,X2)) -> active#(X2) active#(snd(X)) -> active#(X) active#(and(X1,X2)) -> active#(X1) active#(fst(X)) -> active#(X) active#(head(X)) -> active#(X) active#(natsFrom(X)) -> active#(X) active#(s(X)) -> active#(X) active#(sel(X1,X2)) -> active#(X1) active#(sel(X1,X2)) -> active#(X2) active#(tail(X)) -> active#(X) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(U11(X1,X2,X3,X4)) -> proper#(X4) proper#(U11(X1,X2,X3,X4)) -> proper#(X3) proper#(U11(X1,X2,X3,X4)) -> proper#(X2) proper#(U11(X1,X2,X3,X4)) -> proper#(X1) proper#(U12(X1,X2)) -> proper#(X2) proper#(U12(X1,X2)) -> proper#(X1) proper#(splitAt(X1,X2)) -> proper#(X2) proper#(splitAt(X1,X2)) -> proper#(X1) proper#(pair(X1,X2)) -> proper#(X2) proper#(pair(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(afterNth(X1,X2)) -> proper#(X2) proper#(afterNth(X1,X2)) -> proper#(X1) proper#(snd(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(fst(X)) -> proper#(X) proper#(head(X)) -> proper#(X) proper#(natsFrom(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(sel(X1,X2)) -> proper#(X2) proper#(sel(X1,X2)) -> proper#(X1) proper#(tail(X)) -> proper#(X) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed 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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(take#) = 1 problem: DPs: take#(mark(X1),X2) -> take#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(take#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: tail#(mark(X)) -> tail#(X) tail#(ok(X)) -> tail#(X) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(tail#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(sel#) = 1 problem: DPs: sel#(mark(X1),X2) -> sel#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(sel#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: natsFrom#(mark(X)) -> natsFrom#(X) natsFrom#(ok(X)) -> natsFrom#(X) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(natsFrom#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: head#(mark(X)) -> head#(X) head#(ok(X)) -> head#(X) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(head#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: fst#(mark(X)) -> fst#(X) fst#(ok(X)) -> fst#(X) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(fst#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(and#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: snd#(mark(X)) -> snd#(X) snd#(ok(X)) -> snd#(X) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(snd#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed 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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(afterNth#) = 1 problem: DPs: afterNth#(mark(X1),X2) -> afterNth#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(afterNth#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed 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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(pair#) = 1 problem: DPs: pair#(mark(X1),X2) -> pair#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(pair#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed 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(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(splitAt#) = 1 problem: DPs: splitAt#(mark(X1),X2) -> splitAt#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(splitAt#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U12#(mark(X1),X2) -> U12#(X1,X2) U12#(ok(X1),ok(X2)) -> U12#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U12#) = 1 problem: DPs: U12#(mark(X1),X2) -> U12#(X1,X2) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U12#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U11#(X1,X2,X3,X4) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 1 problem: DPs: U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: TRS: active(U11(tt(),N,X,XS)) -> mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) -> mark(snd(splitAt(N,XS))) active(and(tt(),X)) -> mark(X) active(fst(pair(X,Y))) -> mark(X) active(head(cons(N,XS))) -> mark(N) active(natsFrom(N)) -> mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) -> mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) -> mark(Y) active(splitAt(0(),XS)) -> mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) -> mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) -> mark(XS) active(take(N,XS)) -> mark(fst(splitAt(N,XS))) active(U11(X1,X2,X3,X4)) -> U11(active(X1),X2,X3,X4) active(U12(X1,X2)) -> U12(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(active(X1),X2) active(splitAt(X1,X2)) -> splitAt(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pair(X1,X2)) -> pair(X1,active(X2)) active(cons(X1,X2)) -> cons(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(active(X1),X2) active(afterNth(X1,X2)) -> afterNth(X1,active(X2)) active(snd(X)) -> snd(active(X)) active(and(X1,X2)) -> and(active(X1),X2) active(fst(X)) -> fst(active(X)) active(head(X)) -> head(active(X)) active(natsFrom(X)) -> natsFrom(active(X)) active(s(X)) -> s(active(X)) active(sel(X1,X2)) -> sel(active(X1),X2) active(sel(X1,X2)) -> sel(X1,active(X2)) active(tail(X)) -> tail(active(X)) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) U11(mark(X1),X2,X3,X4) -> mark(U11(X1,X2,X3,X4)) U12(mark(X1),X2) -> mark(U12(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) snd(mark(X)) -> mark(snd(X)) and(mark(X1),X2) -> mark(and(X1,X2)) fst(mark(X)) -> mark(fst(X)) head(mark(X)) -> mark(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) s(mark(X)) -> mark(s(X)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) tail(mark(X)) -> mark(tail(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) proper(U11(X1,X2,X3,X4)) -> U11(proper(X1),proper(X2),proper(X3),proper(X4)) proper(tt()) -> ok(tt()) proper(U12(X1,X2)) -> U12(proper(X1),proper(X2)) proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2)) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2)) proper(snd(X)) -> snd(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(fst(X)) -> fst(proper(X)) proper(head(X)) -> head(proper(X)) proper(natsFrom(X)) -> natsFrom(proper(X)) proper(s(X)) -> s(proper(X)) proper(sel(X1,X2)) -> sel(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) U11(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U11(X1,X2,X3,X4)) U12(ok(X1),ok(X2)) -> ok(U12(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) snd(ok(X)) -> ok(snd(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) fst(ok(X)) -> ok(fst(X)) head(ok(X)) -> ok(head(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) s(ok(X)) -> ok(s(X)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) tail(ok(X)) -> ok(tail(X)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed