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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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#(U11(tt(),N,X,XS)) -> mark#(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#(U12(pair(YS,ZS),X)) -> mark#(pair(cons(X,YS),ZS)) active#(afterNth(N,XS)) -> splitAt#(N,XS) active#(afterNth(N,XS)) -> snd#(splitAt(N,XS)) 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)) -> s#(N) active#(natsFrom(N)) -> natsFrom#(s(N)) active#(natsFrom(N)) -> cons#(N,natsFrom(s(N))) active#(natsFrom(N)) -> mark#(cons(N,natsFrom(s(N)))) active#(sel(N,XS)) -> afterNth#(N,XS) active#(sel(N,XS)) -> head#(afterNth(N,XS)) active#(sel(N,XS)) -> mark#(head(afterNth(N,XS))) active#(snd(pair(X,Y))) -> mark#(Y) active#(splitAt(0(),XS)) -> pair#(nil(),XS) active#(splitAt(0(),XS)) -> mark#(pair(nil(),XS)) active#(splitAt(s(N),cons(X,XS))) -> U11#(tt(),N,X,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)) -> splitAt#(N,XS) active#(take(N,XS)) -> fst#(splitAt(N,XS)) active#(take(N,XS)) -> mark#(fst(splitAt(N,XS))) mark#(U11(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2,X3,X4)) -> U11#(mark(X1),X2,X3,X4) mark#(U11(X1,X2,X3,X4)) -> active#(U11(mark(X1),X2,X3,X4)) mark#(tt()) -> active#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> U12#(mark(X1),X2) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(splitAt(X1,X2)) -> mark#(X2) mark#(splitAt(X1,X2)) -> mark#(X1) mark#(splitAt(X1,X2)) -> splitAt#(mark(X1),mark(X2)) mark#(splitAt(X1,X2)) -> active#(splitAt(mark(X1),mark(X2))) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> pair#(mark(X1),mark(X2)) mark#(pair(X1,X2)) -> active#(pair(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(afterNth(X1,X2)) -> mark#(X2) mark#(afterNth(X1,X2)) -> mark#(X1) mark#(afterNth(X1,X2)) -> afterNth#(mark(X1),mark(X2)) mark#(afterNth(X1,X2)) -> active#(afterNth(mark(X1),mark(X2))) mark#(snd(X)) -> mark#(X) mark#(snd(X)) -> snd#(mark(X)) mark#(snd(X)) -> active#(snd(mark(X))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(fst(X)) -> mark#(X) mark#(fst(X)) -> fst#(mark(X)) mark#(fst(X)) -> active#(fst(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> active#(head(mark(X))) mark#(natsFrom(X)) -> mark#(X) mark#(natsFrom(X)) -> natsFrom#(mark(X)) mark#(natsFrom(X)) -> active#(natsFrom(mark(X))) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> sel#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> active#(sel(mark(X1),mark(X2))) mark#(0()) -> active#(0()) mark#(nil()) -> active#(nil()) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> active#(tail(mark(X))) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,mark(X2),X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,mark(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,mark(X4)) -> U11#(X1,X2,X3,X4) U11#(active(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,active(X2),X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,active(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,active(X4)) -> U11#(X1,X2,X3,X4) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U12#(active(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) splitAt#(active(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,active(X2)) -> splitAt#(X1,X2) pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) pair#(active(X1),X2) -> pair#(X1,X2) pair#(X1,active(X2)) -> pair#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) afterNth#(active(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,active(X2)) -> afterNth#(X1,X2) snd#(mark(X)) -> snd#(X) snd#(active(X)) -> snd#(X) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) fst#(mark(X)) -> fst#(X) fst#(active(X)) -> fst#(X) head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) natsFrom#(mark(X)) -> natsFrom#(X) natsFrom#(active(X)) -> natsFrom#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(active(X1),X2) -> sel#(X1,X2) sel#(X1,active(X2)) -> sel#(X1,X2) tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) TDG Processor: DPs: active#(U11(tt(),N,X,XS)) -> splitAt#(N,XS) active#(U11(tt(),N,X,XS)) -> U12#(splitAt(N,XS),X) active#(U11(tt(),N,X,XS)) -> mark#(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#(U12(pair(YS,ZS),X)) -> mark#(pair(cons(X,YS),ZS)) active#(afterNth(N,XS)) -> splitAt#(N,XS) active#(afterNth(N,XS)) -> snd#(splitAt(N,XS)) 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)) -> s#(N) active#(natsFrom(N)) -> natsFrom#(s(N)) active#(natsFrom(N)) -> cons#(N,natsFrom(s(N))) active#(natsFrom(N)) -> mark#(cons(N,natsFrom(s(N)))) active#(sel(N,XS)) -> afterNth#(N,XS) active#(sel(N,XS)) -> head#(afterNth(N,XS)) active#(sel(N,XS)) -> mark#(head(afterNth(N,XS))) active#(snd(pair(X,Y))) -> mark#(Y) active#(splitAt(0(),XS)) -> pair#(nil(),XS) active#(splitAt(0(),XS)) -> mark#(pair(nil(),XS)) active#(splitAt(s(N),cons(X,XS))) -> U11#(tt(),N,X,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)) -> splitAt#(N,XS) active#(take(N,XS)) -> fst#(splitAt(N,XS)) active#(take(N,XS)) -> mark#(fst(splitAt(N,XS))) mark#(U11(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2,X3,X4)) -> U11#(mark(X1),X2,X3,X4) mark#(U11(X1,X2,X3,X4)) -> active#(U11(mark(X1),X2,X3,X4)) mark#(tt()) -> active#(tt()) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> U12#(mark(X1),X2) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(splitAt(X1,X2)) -> mark#(X2) mark#(splitAt(X1,X2)) -> mark#(X1) mark#(splitAt(X1,X2)) -> splitAt#(mark(X1),mark(X2)) mark#(splitAt(X1,X2)) -> active#(splitAt(mark(X1),mark(X2))) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> pair#(mark(X1),mark(X2)) mark#(pair(X1,X2)) -> active#(pair(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(afterNth(X1,X2)) -> mark#(X2) mark#(afterNth(X1,X2)) -> mark#(X1) mark#(afterNth(X1,X2)) -> afterNth#(mark(X1),mark(X2)) mark#(afterNth(X1,X2)) -> active#(afterNth(mark(X1),mark(X2))) mark#(snd(X)) -> mark#(X) mark#(snd(X)) -> snd#(mark(X)) mark#(snd(X)) -> active#(snd(mark(X))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(fst(X)) -> mark#(X) mark#(fst(X)) -> fst#(mark(X)) mark#(fst(X)) -> active#(fst(mark(X))) mark#(head(X)) -> mark#(X) mark#(head(X)) -> head#(mark(X)) mark#(head(X)) -> active#(head(mark(X))) mark#(natsFrom(X)) -> mark#(X) mark#(natsFrom(X)) -> natsFrom#(mark(X)) mark#(natsFrom(X)) -> active#(natsFrom(mark(X))) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> sel#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> active#(sel(mark(X1),mark(X2))) mark#(0()) -> active#(0()) mark#(nil()) -> active#(nil()) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> tail#(mark(X)) mark#(tail(X)) -> active#(tail(mark(X))) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,mark(X2),X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,mark(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,mark(X4)) -> U11#(X1,X2,X3,X4) U11#(active(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,active(X2),X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,active(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,active(X4)) -> U11#(X1,X2,X3,X4) U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U12#(active(X1),X2) -> U12#(X1,X2) U12#(X1,active(X2)) -> U12#(X1,X2) splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) splitAt#(active(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,active(X2)) -> splitAt#(X1,X2) pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) pair#(active(X1),X2) -> pair#(X1,X2) pair#(X1,active(X2)) -> pair#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) afterNth#(active(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,active(X2)) -> afterNth#(X1,X2) snd#(mark(X)) -> snd#(X) snd#(active(X)) -> snd#(X) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) fst#(mark(X)) -> fst#(X) fst#(active(X)) -> fst#(X) head#(mark(X)) -> head#(X) head#(active(X)) -> head#(X) natsFrom#(mark(X)) -> natsFrom#(X) natsFrom#(active(X)) -> natsFrom#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(active(X1),X2) -> sel#(X1,X2) sel#(X1,active(X2)) -> sel#(X1,X2) tail#(mark(X)) -> tail#(X) tail#(active(X)) -> tail#(X) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) graph: ... SCC Processor: #sccs: 16 #rules: 103 #arcs: 2575/17689 DPs: mark#(take(X1,X2)) -> mark#(X2) mark#(U11(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2,X3,X4)) -> active#(U11(mark(X1),X2,X3,X4)) active#(U11(tt(),N,X,XS)) -> mark#(U12(splitAt(N,XS),X)) mark#(tt()) -> active#(tt()) active#(U12(pair(YS,ZS),X)) -> mark#(pair(cons(X,YS),ZS)) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(afterNth(N,XS)) -> mark#(snd(splitAt(N,XS))) mark#(splitAt(X1,X2)) -> mark#(X2) mark#(splitAt(X1,X2)) -> mark#(X1) mark#(splitAt(X1,X2)) -> active#(splitAt(mark(X1),mark(X2))) active#(and(tt(),X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> active#(pair(mark(X1),mark(X2))) active#(fst(pair(X,Y))) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(head(cons(N,XS))) -> mark#(N) mark#(afterNth(X1,X2)) -> mark#(X2) mark#(afterNth(X1,X2)) -> mark#(X1) mark#(afterNth(X1,X2)) -> active#(afterNth(mark(X1),mark(X2))) active#(natsFrom(N)) -> mark#(cons(N,natsFrom(s(N)))) mark#(snd(X)) -> mark#(X) mark#(snd(X)) -> active#(snd(mark(X))) active#(sel(N,XS)) -> mark#(head(afterNth(N,XS))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(snd(pair(X,Y))) -> mark#(Y) mark#(fst(X)) -> mark#(X) mark#(fst(X)) -> active#(fst(mark(X))) active#(splitAt(0(),XS)) -> mark#(pair(nil(),XS)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) active#(splitAt(s(N),cons(X,XS))) -> mark#(U11(tt(),N,X,XS)) mark#(natsFrom(X)) -> mark#(X) mark#(natsFrom(X)) -> active#(natsFrom(mark(X))) active#(tail(cons(N,XS))) -> mark#(XS) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(take(N,XS)) -> mark#(fst(splitAt(N,XS))) mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> active#(sel(mark(X1),mark(X2))) mark#(0()) -> active#(0()) mark#(nil()) -> active#(nil()) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) EDG Processor: DPs: mark#(take(X1,X2)) -> mark#(X2) mark#(U11(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2,X3,X4)) -> active#(U11(mark(X1),X2,X3,X4)) active#(U11(tt(),N,X,XS)) -> mark#(U12(splitAt(N,XS),X)) mark#(tt()) -> active#(tt()) active#(U12(pair(YS,ZS),X)) -> mark#(pair(cons(X,YS),ZS)) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) active#(afterNth(N,XS)) -> mark#(snd(splitAt(N,XS))) mark#(splitAt(X1,X2)) -> mark#(X2) mark#(splitAt(X1,X2)) -> mark#(X1) mark#(splitAt(X1,X2)) -> active#(splitAt(mark(X1),mark(X2))) active#(and(tt(),X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> active#(pair(mark(X1),mark(X2))) active#(fst(pair(X,Y))) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(head(cons(N,XS))) -> mark#(N) mark#(afterNth(X1,X2)) -> mark#(X2) mark#(afterNth(X1,X2)) -> mark#(X1) mark#(afterNth(X1,X2)) -> active#(afterNth(mark(X1),mark(X2))) active#(natsFrom(N)) -> mark#(cons(N,natsFrom(s(N)))) mark#(snd(X)) -> mark#(X) mark#(snd(X)) -> active#(snd(mark(X))) active#(sel(N,XS)) -> mark#(head(afterNth(N,XS))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(snd(pair(X,Y))) -> mark#(Y) mark#(fst(X)) -> mark#(X) mark#(fst(X)) -> active#(fst(mark(X))) active#(splitAt(0(),XS)) -> mark#(pair(nil(),XS)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) active#(splitAt(s(N),cons(X,XS))) -> mark#(U11(tt(),N,X,XS)) mark#(natsFrom(X)) -> mark#(X) mark#(natsFrom(X)) -> active#(natsFrom(mark(X))) active#(tail(cons(N,XS))) -> mark#(XS) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(take(N,XS)) -> mark#(fst(splitAt(N,XS))) mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> active#(sel(mark(X1),mark(X2))) mark#(0()) -> active#(0()) mark#(nil()) -> active#(nil()) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> active#(tail(mark(X))) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) graph: ... SCC Processor: #sccs: 1 #rules: 48 #arcs: 1425/2601 DPs: mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(N,XS)) -> mark#(fst(splitAt(N,XS))) mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> active#(tail(mark(X))) active#(tail(cons(N,XS))) -> mark#(XS) mark#(tail(X)) -> mark#(X) mark#(sel(X1,X2)) -> active#(sel(mark(X1),mark(X2))) active#(splitAt(s(N),cons(X,XS))) -> mark#(U11(tt(),N,X,XS)) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X2) mark#(s(X)) -> active#(s(mark(X))) active#(splitAt(0(),XS)) -> mark#(pair(nil(),XS)) mark#(s(X)) -> mark#(X) mark#(natsFrom(X)) -> active#(natsFrom(mark(X))) active#(snd(pair(X,Y))) -> mark#(Y) mark#(natsFrom(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) active#(sel(N,XS)) -> mark#(head(afterNth(N,XS))) mark#(head(X)) -> mark#(X) mark#(fst(X)) -> active#(fst(mark(X))) active#(natsFrom(N)) -> mark#(cons(N,natsFrom(s(N)))) mark#(fst(X)) -> mark#(X) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(head(cons(N,XS))) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(snd(X)) -> active#(snd(mark(X))) active#(fst(pair(X,Y))) -> mark#(X) mark#(snd(X)) -> mark#(X) mark#(afterNth(X1,X2)) -> active#(afterNth(mark(X1),mark(X2))) active#(and(tt(),X)) -> mark#(X) mark#(afterNth(X1,X2)) -> mark#(X1) mark#(afterNth(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(afterNth(N,XS)) -> mark#(snd(splitAt(N,XS))) mark#(cons(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> active#(pair(mark(X1),mark(X2))) active#(U12(pair(YS,ZS),X)) -> mark#(pair(cons(X,YS),ZS)) mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) mark#(splitAt(X1,X2)) -> active#(splitAt(mark(X1),mark(X2))) active#(U11(tt(),N,X,XS)) -> mark#(U12(splitAt(N,XS),X)) mark#(splitAt(X1,X2)) -> mark#(X1) mark#(splitAt(X1,X2)) -> mark#(X2) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(U12(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3,X4)) -> active#(U11(mark(X1),X2,X3,X4)) mark#(U11(X1,X2,X3,X4)) -> mark#(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [mark#](x0) = 4, [active#](x0) = 4x0, [take](x0, x1) = 1, [tail](x0) = 1, [nil] = 0, [0] = 0, [sel](x0, x1) = 1, [s](x0) = 0, [natsFrom](x0) = 1, [head](x0) = 1, [fst](x0) = 1, [and](x0, x1) = 1, [snd](x0) = 1, [afterNth](x0, x1) = 1, [cons](x0, x1) = 0, [pair](x0, x1) = 0, [mark](x0) = 3, [U12](x0, x1) = 1, [splitAt](x0, x1) = 1, [active](x0) = 3, [U11](x0, x1, x2, x3) = 1, [tt] = 0 orientation: mark#(take(X1,X2)) = 4 >= 4 = mark#(X2) mark#(take(X1,X2)) = 4 >= 4 = active#(take(mark(X1),mark(X2))) active#(take(N,XS)) = 4 >= 4 = mark#(fst(splitAt(N,XS))) mark#(take(X1,X2)) = 4 >= 4 = mark#(X1) mark#(tail(X)) = 4 >= 4 = active#(tail(mark(X))) active#(tail(cons(N,XS))) = 4 >= 4 = mark#(XS) mark#(tail(X)) = 4 >= 4 = mark#(X) mark#(sel(X1,X2)) = 4 >= 4 = active#(sel(mark(X1),mark(X2))) active#(splitAt(s(N),cons(X,XS))) = 4 >= 4 = mark#(U11(tt(),N,X,XS)) mark#(sel(X1,X2)) = 4 >= 4 = mark#(X1) mark#(sel(X1,X2)) = 4 >= 4 = mark#(X2) mark#(s(X)) = 4 >= 0 = active#(s(mark(X))) active#(splitAt(0(),XS)) = 4 >= 4 = mark#(pair(nil(),XS)) mark#(s(X)) = 4 >= 4 = mark#(X) mark#(natsFrom(X)) = 4 >= 4 = active#(natsFrom(mark(X))) active#(snd(pair(X,Y))) = 4 >= 4 = mark#(Y) mark#(natsFrom(X)) = 4 >= 4 = mark#(X) mark#(head(X)) = 4 >= 4 = active#(head(mark(X))) active#(sel(N,XS)) = 4 >= 4 = mark#(head(afterNth(N,XS))) mark#(head(X)) = 4 >= 4 = mark#(X) mark#(fst(X)) = 4 >= 4 = active#(fst(mark(X))) active#(natsFrom(N)) = 4 >= 4 = mark#(cons(N,natsFrom(s(N)))) mark#(fst(X)) = 4 >= 4 = mark#(X) mark#(and(X1,X2)) = 4 >= 4 = active#(and(mark(X1),X2)) active#(head(cons(N,XS))) = 4 >= 4 = mark#(N) mark#(and(X1,X2)) = 4 >= 4 = mark#(X1) mark#(snd(X)) = 4 >= 4 = active#(snd(mark(X))) active#(fst(pair(X,Y))) = 4 >= 4 = mark#(X) mark#(snd(X)) = 4 >= 4 = mark#(X) mark#(afterNth(X1,X2)) = 4 >= 4 = active#(afterNth(mark(X1),mark(X2))) active#(and(tt(),X)) = 4 >= 4 = mark#(X) mark#(afterNth(X1,X2)) = 4 >= 4 = mark#(X1) mark#(afterNth(X1,X2)) = 4 >= 4 = mark#(X2) mark#(cons(X1,X2)) = 4 >= 0 = active#(cons(mark(X1),X2)) active#(afterNth(N,XS)) = 4 >= 4 = mark#(snd(splitAt(N,XS))) mark#(cons(X1,X2)) = 4 >= 4 = mark#(X1) mark#(pair(X1,X2)) = 4 >= 0 = active#(pair(mark(X1),mark(X2))) active#(U12(pair(YS,ZS),X)) = 4 >= 4 = mark#(pair(cons(X,YS),ZS)) mark#(pair(X1,X2)) = 4 >= 4 = mark#(X1) mark#(pair(X1,X2)) = 4 >= 4 = mark#(X2) mark#(splitAt(X1,X2)) = 4 >= 4 = active#(splitAt(mark(X1),mark(X2))) active#(U11(tt(),N,X,XS)) = 4 >= 4 = mark#(U12(splitAt(N,XS),X)) mark#(splitAt(X1,X2)) = 4 >= 4 = mark#(X1) mark#(splitAt(X1,X2)) = 4 >= 4 = mark#(X2) mark#(U12(X1,X2)) = 4 >= 4 = active#(U12(mark(X1),X2)) mark#(U12(X1,X2)) = 4 >= 4 = mark#(X1) mark#(U11(X1,X2,X3,X4)) = 4 >= 4 = active#(U11(mark(X1),X2,X3,X4)) mark#(U11(X1,X2,X3,X4)) = 4 >= 4 = mark#(X1) active(U11(tt(),N,X,XS)) = 3 >= 3 = mark(U12(splitAt(N,XS),X)) active(U12(pair(YS,ZS),X)) = 3 >= 3 = mark(pair(cons(X,YS),ZS)) active(afterNth(N,XS)) = 3 >= 3 = mark(snd(splitAt(N,XS))) active(and(tt(),X)) = 3 >= 3 = mark(X) active(fst(pair(X,Y))) = 3 >= 3 = mark(X) active(head(cons(N,XS))) = 3 >= 3 = mark(N) active(natsFrom(N)) = 3 >= 3 = mark(cons(N,natsFrom(s(N)))) active(sel(N,XS)) = 3 >= 3 = mark(head(afterNth(N,XS))) active(snd(pair(X,Y))) = 3 >= 3 = mark(Y) active(splitAt(0(),XS)) = 3 >= 3 = mark(pair(nil(),XS)) active(splitAt(s(N),cons(X,XS))) = 3 >= 3 = mark(U11(tt(),N,X,XS)) active(tail(cons(N,XS))) = 3 >= 3 = mark(XS) active(take(N,XS)) = 3 >= 3 = mark(fst(splitAt(N,XS))) mark(U11(X1,X2,X3,X4)) = 3 >= 3 = active(U11(mark(X1),X2,X3,X4)) mark(tt()) = 3 >= 3 = active(tt()) mark(U12(X1,X2)) = 3 >= 3 = active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) = 3 >= 3 = active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) = 3 >= 3 = active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) = 3 >= 3 = active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) = 3 >= 3 = active(afterNth(mark(X1),mark(X2))) mark(snd(X)) = 3 >= 3 = active(snd(mark(X))) mark(and(X1,X2)) = 3 >= 3 = active(and(mark(X1),X2)) mark(fst(X)) = 3 >= 3 = active(fst(mark(X))) mark(head(X)) = 3 >= 3 = active(head(mark(X))) mark(natsFrom(X)) = 3 >= 3 = active(natsFrom(mark(X))) mark(s(X)) = 3 >= 3 = active(s(mark(X))) mark(sel(X1,X2)) = 3 >= 3 = active(sel(mark(X1),mark(X2))) mark(0()) = 3 >= 3 = active(0()) mark(nil()) = 3 >= 3 = active(nil()) mark(tail(X)) = 3 >= 3 = active(tail(mark(X))) mark(take(X1,X2)) = 3 >= 3 = active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) = 1 >= 1 = U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) = 1 >= 1 = U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) = 1 >= 1 = U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) = 1 >= 1 = U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) = 1 >= 1 = U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) = 1 >= 1 = U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) = 1 >= 1 = U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) = 1 >= 1 = U11(X1,X2,X3,X4) U12(mark(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,mark(X2)) = 1 >= 1 = U12(X1,X2) U12(active(X1),X2) = 1 >= 1 = U12(X1,X2) U12(X1,active(X2)) = 1 >= 1 = U12(X1,X2) splitAt(mark(X1),X2) = 1 >= 1 = splitAt(X1,X2) splitAt(X1,mark(X2)) = 1 >= 1 = splitAt(X1,X2) splitAt(active(X1),X2) = 1 >= 1 = splitAt(X1,X2) splitAt(X1,active(X2)) = 1 >= 1 = splitAt(X1,X2) pair(mark(X1),X2) = 0 >= 0 = pair(X1,X2) pair(X1,mark(X2)) = 0 >= 0 = pair(X1,X2) pair(active(X1),X2) = 0 >= 0 = pair(X1,X2) pair(X1,active(X2)) = 0 >= 0 = pair(X1,X2) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) afterNth(mark(X1),X2) = 1 >= 1 = afterNth(X1,X2) afterNth(X1,mark(X2)) = 1 >= 1 = afterNth(X1,X2) afterNth(active(X1),X2) = 1 >= 1 = afterNth(X1,X2) afterNth(X1,active(X2)) = 1 >= 1 = afterNth(X1,X2) snd(mark(X)) = 1 >= 1 = snd(X) snd(active(X)) = 1 >= 1 = snd(X) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) fst(mark(X)) = 1 >= 1 = fst(X) fst(active(X)) = 1 >= 1 = fst(X) head(mark(X)) = 1 >= 1 = head(X) head(active(X)) = 1 >= 1 = head(X) natsFrom(mark(X)) = 1 >= 1 = natsFrom(X) natsFrom(active(X)) = 1 >= 1 = natsFrom(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) sel(mark(X1),X2) = 1 >= 1 = sel(X1,X2) sel(X1,mark(X2)) = 1 >= 1 = sel(X1,X2) sel(active(X1),X2) = 1 >= 1 = sel(X1,X2) sel(X1,active(X2)) = 1 >= 1 = sel(X1,X2) tail(mark(X)) = 1 >= 1 = tail(X) tail(active(X)) = 1 >= 1 = tail(X) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) problem: DPs: mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(N,XS)) -> mark#(fst(splitAt(N,XS))) mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> active#(tail(mark(X))) active#(tail(cons(N,XS))) -> mark#(XS) mark#(tail(X)) -> mark#(X) mark#(sel(X1,X2)) -> active#(sel(mark(X1),mark(X2))) active#(splitAt(s(N),cons(X,XS))) -> mark#(U11(tt(),N,X,XS)) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X2) active#(splitAt(0(),XS)) -> mark#(pair(nil(),XS)) mark#(s(X)) -> mark#(X) mark#(natsFrom(X)) -> active#(natsFrom(mark(X))) active#(snd(pair(X,Y))) -> mark#(Y) mark#(natsFrom(X)) -> mark#(X) mark#(head(X)) -> active#(head(mark(X))) active#(sel(N,XS)) -> mark#(head(afterNth(N,XS))) mark#(head(X)) -> mark#(X) mark#(fst(X)) -> active#(fst(mark(X))) active#(natsFrom(N)) -> mark#(cons(N,natsFrom(s(N)))) mark#(fst(X)) -> mark#(X) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(head(cons(N,XS))) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(snd(X)) -> active#(snd(mark(X))) active#(fst(pair(X,Y))) -> mark#(X) mark#(snd(X)) -> mark#(X) mark#(afterNth(X1,X2)) -> active#(afterNth(mark(X1),mark(X2))) active#(and(tt(),X)) -> mark#(X) mark#(afterNth(X1,X2)) -> mark#(X1) mark#(afterNth(X1,X2)) -> mark#(X2) active#(afterNth(N,XS)) -> mark#(snd(splitAt(N,XS))) mark#(cons(X1,X2)) -> mark#(X1) active#(U12(pair(YS,ZS),X)) -> mark#(pair(cons(X,YS),ZS)) mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) mark#(splitAt(X1,X2)) -> active#(splitAt(mark(X1),mark(X2))) active#(U11(tt(),N,X,XS)) -> mark#(U12(splitAt(N,XS),X)) mark#(splitAt(X1,X2)) -> mark#(X1) mark#(splitAt(X1,X2)) -> mark#(X2) mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2)) mark#(U12(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3,X4)) -> active#(U11(mark(X1),X2,X3,X4)) mark#(U11(X1,X2,X3,X4)) -> mark#(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Open DPs: U12#(mark(X1),X2) -> U12#(X1,X2) U12#(X1,mark(X2)) -> U12#(X1,X2) U12#(active(X1),X2) -> U12#(X1,X2) U12#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(U12#) = 1 problem: DPs: U12#(mark(X1),X2) -> U12#(X1,X2) U12#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,mark(X2)) -> splitAt#(X1,X2) splitAt#(active(X1),X2) -> splitAt#(X1,X2) splitAt#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(splitAt#) = 1 problem: DPs: splitAt#(mark(X1),X2) -> splitAt#(X1,X2) splitAt#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: pair#(mark(X1),X2) -> pair#(X1,X2) pair#(X1,mark(X2)) -> pair#(X1,X2) pair#(active(X1),X2) -> pair#(X1,X2) pair#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(pair#) = 1 problem: DPs: pair#(mark(X1),X2) -> pair#(X1,X2) pair#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: snd#(mark(X)) -> snd#(X) snd#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: s#(mark(X)) -> s#(X) s#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: natsFrom#(mark(X)) -> natsFrom#(X) natsFrom#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,mark(X2)) -> afterNth#(X1,X2) afterNth#(active(X1),X2) -> afterNth#(X1,X2) afterNth#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(afterNth#) = 1 problem: DPs: afterNth#(mark(X1),X2) -> afterNth#(X1,X2) afterNth#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: head#(mark(X)) -> head#(X) head#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,mark(X2),X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,mark(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,mark(X4)) -> U11#(X1,X2,X3,X4) U11#(active(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,active(X2),X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,active(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(U11#) = 1 problem: DPs: U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,mark(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,mark(X4)) -> U11#(X1,X2,X3,X4) U11#(active(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,active(X3),X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(U11#) = 2 problem: DPs: U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,mark(X4)) -> U11#(X1,X2,X3,X4) U11#(active(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(X1,X2,X3,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(U11#) = 3 problem: DPs: U11#(mark(X1),X2,X3,X4) -> U11#(X1,X2,X3,X4) U11#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: fst#(mark(X)) -> fst#(X) fst#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: sel#(mark(X1),X2) -> sel#(X1,X2) sel#(X1,mark(X2)) -> sel#(X1,X2) sel#(active(X1),X2) -> sel#(X1,X2) sel#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(sel#) = 1 problem: DPs: sel#(mark(X1),X2) -> sel#(X1,X2) sel#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: tail#(mark(X)) -> tail#(X) tail#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(take#) = 1 problem: DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(active(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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) 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))) mark(U11(X1,X2,X3,X4)) -> active(U11(mark(X1),X2,X3,X4)) mark(tt()) -> active(tt()) mark(U12(X1,X2)) -> active(U12(mark(X1),X2)) mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2))) mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2))) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2))) mark(snd(X)) -> active(snd(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(fst(X)) -> active(fst(mark(X))) mark(head(X)) -> active(head(mark(X))) mark(natsFrom(X)) -> active(natsFrom(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2))) mark(0()) -> active(0()) mark(nil()) -> active(nil()) mark(tail(X)) -> active(tail(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) U11(mark(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,mark(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,mark(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,mark(X4)) -> U11(X1,X2,X3,X4) U11(active(X1),X2,X3,X4) -> U11(X1,X2,X3,X4) U11(X1,active(X2),X3,X4) -> U11(X1,X2,X3,X4) U11(X1,X2,active(X3),X4) -> U11(X1,X2,X3,X4) U11(X1,X2,X3,active(X4)) -> U11(X1,X2,X3,X4) U12(mark(X1),X2) -> U12(X1,X2) U12(X1,mark(X2)) -> U12(X1,X2) U12(active(X1),X2) -> U12(X1,X2) U12(X1,active(X2)) -> U12(X1,X2) splitAt(mark(X1),X2) -> splitAt(X1,X2) splitAt(X1,mark(X2)) -> splitAt(X1,X2) splitAt(active(X1),X2) -> splitAt(X1,X2) splitAt(X1,active(X2)) -> splitAt(X1,X2) pair(mark(X1),X2) -> pair(X1,X2) pair(X1,mark(X2)) -> pair(X1,X2) pair(active(X1),X2) -> pair(X1,X2) pair(X1,active(X2)) -> pair(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) afterNth(mark(X1),X2) -> afterNth(X1,X2) afterNth(X1,mark(X2)) -> afterNth(X1,X2) afterNth(active(X1),X2) -> afterNth(X1,X2) afterNth(X1,active(X2)) -> afterNth(X1,X2) snd(mark(X)) -> snd(X) snd(active(X)) -> snd(X) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) fst(mark(X)) -> fst(X) fst(active(X)) -> fst(X) head(mark(X)) -> head(X) head(active(X)) -> head(X) natsFrom(mark(X)) -> natsFrom(X) natsFrom(active(X)) -> natsFrom(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) sel(mark(X1),X2) -> sel(X1,X2) sel(X1,mark(X2)) -> sel(X1,X2) sel(active(X1),X2) -> sel(X1,X2) sel(X1,active(X2)) -> sel(X1,X2) tail(mark(X)) -> tail(X) tail(active(X)) -> tail(X) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) Qed