YES Problem: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Proof: DP Processor: DPs: active#(and(tt(),T)) -> mark#(T) active#(isNatIList(IL)) -> isNatList#(IL) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNat(0())) -> mark#(tt()) active#(isNat(s(N))) -> isNat#(N) active#(isNat(s(N))) -> mark#(isNat(N)) active#(isNat(length(L))) -> isNatList#(L) active#(isNat(length(L))) -> mark#(isNatList(L)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(N,IL))) -> isNatIList#(IL) active#(isNatIList(cons(N,IL))) -> isNat#(N) active#(isNatIList(cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(N,L))) -> isNatList#(L) active#(isNatList(cons(N,L))) -> isNat#(N) active#(isNatList(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) active#(isNatList(take(N,IL))) -> isNatIList#(IL) active#(isNatList(take(N,IL))) -> isNat#(N) active#(isNatList(take(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> uTake1#(isNatIList(IL)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) active#(uTake1(tt())) -> mark#(nil()) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),and(isNat(N),isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> uTake2#(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active#(uTake2(tt(),M,N,IL)) -> take#(M,IL) active#(uTake2(tt(),M,N,IL)) -> cons#(N,take(M,IL)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(length(cons(N,L))) -> uLength#(and(isNat(N),isNatList(L)),L) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uLength(tt(),L)) -> length#(L) active#(uLength(tt(),L)) -> s#(length(L)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(tt()) -> active#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNat(X)) -> active#(isNat(X)) mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(nil()) -> active#(nil()) 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))) mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> uTake1#(mark(X)) mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> uTake2#(mark(X1),X2,X3,X4) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> uLength#(mark(X1),X2) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) 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) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) 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) 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) uTake1#(mark(X)) -> uTake1#(X) uTake1#(active(X)) -> uTake1#(X) uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,mark(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,mark(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,mark(X4)) -> uTake2#(X1,X2,X3,X4) uTake2#(active(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,active(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,active(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,active(X4)) -> uTake2#(X1,X2,X3,X4) uLength#(mark(X1),X2) -> uLength#(X1,X2) uLength#(X1,mark(X2)) -> uLength#(X1,X2) uLength#(active(X1),X2) -> uLength#(X1,X2) uLength#(X1,active(X2)) -> uLength#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) TDG Processor: DPs: active#(and(tt(),T)) -> mark#(T) active#(isNatIList(IL)) -> isNatList#(IL) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNat(0())) -> mark#(tt()) active#(isNat(s(N))) -> isNat#(N) active#(isNat(s(N))) -> mark#(isNat(N)) active#(isNat(length(L))) -> isNatList#(L) active#(isNat(length(L))) -> mark#(isNatList(L)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(N,IL))) -> isNatIList#(IL) active#(isNatIList(cons(N,IL))) -> isNat#(N) active#(isNatIList(cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(N,L))) -> isNatList#(L) active#(isNatList(cons(N,L))) -> isNat#(N) active#(isNatList(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) active#(isNatList(take(N,IL))) -> isNatIList#(IL) active#(isNatList(take(N,IL))) -> isNat#(N) active#(isNatList(take(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> uTake1#(isNatIList(IL)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) active#(uTake1(tt())) -> mark#(nil()) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatIList(IL)) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),and(isNat(N),isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> uTake2#(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active#(uTake2(tt(),M,N,IL)) -> take#(M,IL) active#(uTake2(tt(),M,N,IL)) -> cons#(N,take(M,IL)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatList(L)) active#(length(cons(N,L))) -> uLength#(and(isNat(N),isNatList(L)),L) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uLength(tt(),L)) -> length#(L) active#(uLength(tt(),L)) -> s#(length(L)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(tt()) -> active#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNat(X)) -> active#(isNat(X)) mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(nil()) -> active#(nil()) 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))) mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> uTake1#(mark(X)) mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> uTake2#(mark(X1),X2,X3,X4) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> uLength#(mark(X1),X2) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) 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) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) 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) 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) uTake1#(mark(X)) -> uTake1#(X) uTake1#(active(X)) -> uTake1#(X) uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,mark(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,mark(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,mark(X4)) -> uTake2#(X1,X2,X3,X4) uTake2#(active(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,active(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,active(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,active(X4)) -> uTake2#(X1,X2,X3,X4) uLength#(mark(X1),X2) -> uLength#(X1,X2) uLength#(X1,mark(X2)) -> uLength#(X1,X2) uLength#(active(X1),X2) -> uLength#(X1,X2) uLength#(X1,active(X2)) -> uLength#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) graph: ... SCC Processor: #sccs: 12 #rules: 78 #arcs: 1847/13225 DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(tt()) -> active#(tt()) active#(isNatIList(IL)) -> mark#(isNatList(IL)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNat(0())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(0()) -> active#(0()) active#(isNatIList(zeros())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(nil())) -> mark#(tt()) mark#(zeros()) -> active#(zeros()) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(nil()) -> active#(nil()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> active#(uTake1(mark(X))) active#(uTake1(tt())) -> mark#(nil()) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uLength(tt(),L)) -> mark#(s(length(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) EDG Processor: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(tt()) -> active#(tt()) active#(isNatIList(IL)) -> mark#(isNatList(IL)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNat(0())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(0()) -> active#(0()) active#(isNatIList(zeros())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(nil())) -> mark#(tt()) mark#(zeros()) -> active#(zeros()) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(nil()) -> active#(nil()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> active#(uTake1(mark(X))) active#(uTake1(tt())) -> mark#(nil()) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uLength(tt(),L)) -> mark#(s(length(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) graph: mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> active#(length(mark(X))) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> active#(zeros()) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) -> active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> active#(length(mark(X))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(zeros()) -> active#(zeros()) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) -> active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(uTake1(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X2) mark#(uTake1(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(uTake1(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(uTake1(X)) -> mark#(X) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(uTake1(X)) -> mark#(X) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(uTake1(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(uTake1(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(uTake1(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(uTake1(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(length(X)) -> active#(length(mark(X))) mark#(uTake1(X)) -> mark#(X) -> mark#(zeros()) -> active#(zeros()) mark#(uTake1(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(uTake1(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(uTake1(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(uTake1(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(uTake1(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(uTake1(X)) -> active#(uTake1(mark(X))) -> active#(uTake1(tt())) -> mark#(nil()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(tt()) -> active#(tt()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> active#(s(mark(X))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> active#(length(mark(X))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zeros()) -> active#(zeros()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(nil()) -> active#(nil()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(take(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(take(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(take(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> active#(length(mark(X))) mark#(take(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> active#(zeros()) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) -> active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) -> active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> active#(length(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(zeros()) -> active#(zeros()) -> active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X2) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(length(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(length(X)) -> mark#(X) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(length(X)) -> mark#(X) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(length(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(length(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> active#(length(mark(X))) mark#(length(X)) -> mark#(X) -> mark#(zeros()) -> active#(zeros()) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(length(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(length(X)) -> mark#(X) -> mark#(uTake1(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(length(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(length(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(length(X)) -> active#(length(mark(X))) -> active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(s(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(s(X)) -> mark#(X) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> active#(length(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> active#(zeros()) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(s(X)) -> mark#(X) -> mark#(uTake1(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(s(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(s(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(0())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(s(N))) -> mark#(isNat(N)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) -> active#(isNatList(nil())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) -> active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(isNatList(X)) -> active#(isNatList(X)) -> active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(isNatIList(X)) -> active#(isNatIList(X)) -> active#(isNatIList(IL)) -> mark#(isNatList(IL)) mark#(isNatIList(X)) -> active#(isNatIList(X)) -> active#(isNatIList(zeros())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) -> active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X2) -> mark#(tt()) -> active#(tt()) mark#(and(X1,X2)) -> mark#(X2) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(and(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> active#(s(mark(X))) mark#(and(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> active#(length(mark(X))) mark#(and(X1,X2)) -> mark#(X2) -> mark#(zeros()) -> active#(zeros()) mark#(and(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(nil()) -> active#(nil()) mark#(and(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> active#(s(mark(X))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> active#(length(mark(X))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> active#(zeros()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(nil()) -> active#(nil()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) -> active#(and(tt(),T)) -> mark#(T) active#(uLength(tt(),L)) -> mark#(s(length(L))) -> mark#(s(X)) -> mark#(X) active#(uLength(tt(),L)) -> mark#(s(length(L))) -> mark#(s(X)) -> active#(s(mark(X))) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(uTake1(tt())) -> mark#(nil()) -> mark#(nil()) -> active#(nil()) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) -> mark#(uTake1(X)) -> mark#(X) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) active#(zeros()) -> mark#(cons(0(),zeros())) -> mark#(cons(X1,X2)) -> mark#(X1) active#(zeros()) -> mark#(cons(0(),zeros())) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) -> mark#(uLength(X1,X2)) -> mark#(X1) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(isNat(length(L))) -> mark#(isNatList(L)) -> mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(s(N))) -> mark#(isNat(N)) -> mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(0())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) -> mark#(and(X1,X2)) -> mark#(X2) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) -> mark#(and(X1,X2)) -> mark#(X1) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNatList(nil())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) -> mark#(and(X1,X2)) -> mark#(X2) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) -> mark#(and(X1,X2)) -> mark#(X1) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) -> mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) -> mark#(and(X1,X2)) -> mark#(X1) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNatIList(zeros())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNatIList(IL)) -> mark#(isNatList(IL)) -> mark#(isNatList(X)) -> active#(isNatList(X)) active#(and(tt(),T)) -> mark#(T) -> mark#(and(X1,X2)) -> mark#(X2) active#(and(tt(),T)) -> mark#(T) -> mark#(and(X1,X2)) -> mark#(X1) active#(and(tt(),T)) -> mark#(T) -> mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) -> mark#(tt()) -> active#(tt()) active#(and(tt(),T)) -> mark#(T) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(and(tt(),T)) -> mark#(T) -> mark#(isNatList(X)) -> active#(isNatList(X)) active#(and(tt(),T)) -> mark#(T) -> mark#(isNat(X)) -> active#(isNat(X)) active#(and(tt(),T)) -> mark#(T) -> mark#(0()) -> active#(0()) active#(and(tt(),T)) -> mark#(T) -> mark#(s(X)) -> mark#(X) active#(and(tt(),T)) -> mark#(T) -> mark#(s(X)) -> active#(s(mark(X))) active#(and(tt(),T)) -> mark#(T) -> mark#(length(X)) -> mark#(X) active#(and(tt(),T)) -> mark#(T) -> mark#(length(X)) -> active#(length(mark(X))) active#(and(tt(),T)) -> mark#(T) -> mark#(zeros()) -> active#(zeros()) active#(and(tt(),T)) -> mark#(T) -> mark#(cons(X1,X2)) -> mark#(X1) active#(and(tt(),T)) -> mark#(T) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(and(tt(),T)) -> mark#(T) -> mark#(nil()) -> active#(nil()) active#(and(tt(),T)) -> mark#(T) -> mark#(take(X1,X2)) -> mark#(X2) active#(and(tt(),T)) -> mark#(T) -> mark#(take(X1,X2)) -> mark#(X1) active#(and(tt(),T)) -> mark#(T) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) -> mark#(uTake1(X)) -> mark#(X) active#(and(tt(),T)) -> mark#(T) -> mark#(uTake1(X)) -> active#(uTake1(mark(X))) active#(and(tt(),T)) -> mark#(T) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) active#(and(tt(),T)) -> mark#(T) -> mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(and(tt(),T)) -> mark#(T) -> mark#(uLength(X1,X2)) -> mark#(X1) active#(and(tt(),T)) -> mark#(T) -> mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) SCC Processor: #sccs: 1 #rules: 32 #arcs: 320/1764 DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) active#(isNat(s(N))) -> mark#(isNat(N)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = x0 + x1 + 0, [uTake2](x0, x1, x2, x3) = x0 + 2x1 + x2 + x3 + 0, [uTake1](x0) = x0 + 0, [take](x0, x1) = 2x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1 + 0, [zeros] = 0, [length](x0) = x0 + 0, [s](x0) = x0, [isNat](x0) = 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 0, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 0 >= L + 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) = IL + 2M + N + 0 >= IL + 2M + N + 0 = mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(uTake2(X1,X2,X3,X4)) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 0 = mark#(X1) mark#(uTake1(X)) = X + 0 >= X + 0 = mark#(X) mark#(take(X1,X2)) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = IL + 2M + N + 0 >= IL + 2M + N + 0 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active#(take(0(),IL)) = IL + 2 >= 0 = mark#(uTake1(isNatIList(IL))) mark#(take(X1,X2)) = 2X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = 2X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(length(X)) = X + 0 >= X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = L + N + 0 >= L + 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(length(L))) = 0 >= 0 = mark#(isNatList(L)) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(isNatList(take(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T + 0 = mark#(T) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) active#(isNatIList(IL)) = 0 >= 0 = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = 0 >= 0 = mark#(and(isNat(N),isNatList(L))) active#(isNat(s(N))) = 0 >= 0 = mark#(isNat(N)) active(and(tt(),T)) = T + 0 >= T = mark(T) active(isNatIList(IL)) = 0 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = 0 >= 0 = mark(isNat(N)) active(isNat(length(L))) = 0 >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 0 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 2 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + 2M + N + 0 >= IL + 2M + N + 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + 2M + N + 0 >= IL + 2M + N + 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 0 >= L + 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 0 >= L + 0 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 0 >= X + 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X + 0 >= X + 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 0 >= X + 0 = length(X) length(active(X)) = X + 0 >= X + 0 = length(X) cons(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) take(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = take(X1,X2) take(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = take(X1,X2) take(active(X1),X2) = 2X1 + X2 >= 2X1 + X2 = take(X1,X2) take(X1,active(X2)) = 2X1 + X2 >= 2X1 + X2 = take(X1,X2) uTake1(mark(X)) = X + 0 >= X + 0 = uTake1(X) uTake1(active(X)) = X + 0 >= X + 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + 2X2 + X3 + X4 + 0 >= X1 + 2X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) active#(isNat(s(N))) -> mark#(isNat(N)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) active#(isNat(s(N))) -> mark#(isNat(N)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = x0 + x1 + 0, [uTake2](x0, x1, x2, x3) = x0 + 4x1 + 4x2 + 2x3 + 0, [uTake1](x0) = 2x0, [take](x0, x1) = 4x0 + 2x1 + 0, [nil] = 2, [cons](x0, x1) = 4x0 + x1, [zeros] = 4, [length](x0) = x0 + 0, [s](x0) = x0, [isNat](x0) = 4x0 + 0, [0] = 0, [isNatList](x0) = x0 + 0, [isNatIList](x0) = x0 + 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1 + 0, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 0 >= L + 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) = 2IL + 4M + 4N + 0 >= 2IL + 4M + 4N + 0 = mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) = 4X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(uTake2(X1,X2,X3,X4)) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 0 = mark#(X1) mark#(uTake1(X)) = 2X + 0 >= X + 0 = mark#(X) mark#(take(X1,X2)) = 4X1 + 2X2 + 0 >= 4X1 + 2X2 + 0 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = 2IL + 4M + 6N + 0 >= 2IL + 4M + 4N + 0 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = 4X1 + 2X2 + 0 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = 4X1 + 2X2 + 0 >= X2 + 0 = mark#(X2) mark#(zeros()) = 4 >= 4 = active#(zeros()) active#(zeros()) = 4 >= 4 = mark#(cons(0(),zeros())) mark#(length(X)) = X + 0 >= X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = L + 4N + 0 >= L + 4N + 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNat(X)) = 4X + 0 >= 4X + 0 = active#(isNat(X)) active#(isNat(length(L))) = 4L + 4 >= L + 0 = mark#(isNatList(L)) mark#(isNatList(X)) = X + 0 >= X + 0 = active#(isNatList(X)) active#(isNatList(take(N,IL))) = 2IL + 4N + 0 >= IL + 4N + 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T + 0 = mark#(T) mark#(isNatIList(X)) = X + 0 >= X + 0 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + 4N + 0 >= IL + 4N + 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) active#(isNatIList(IL)) = IL + 0 >= IL + 0 = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = L + 4N + 0 >= L + 4N + 0 = mark#(and(isNat(N),isNatList(L))) active#(isNat(s(N))) = 4N + 0 >= 4N + 0 = mark#(isNat(N)) active(and(tt(),T)) = T + 0 >= T = mark(T) active(isNatIList(IL)) = IL + 0 >= IL + 0 = mark(isNatList(IL)) active(isNat(0())) = 4 >= 0 = mark(tt()) active(isNat(s(N))) = 4N + 0 >= 4N + 0 = mark(isNat(N)) active(isNat(length(L))) = 4L + 4 >= L + 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 4 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + 4N + 0 >= IL + 4N + 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + 4N + 0 >= L + 4N + 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 2IL + 4N + 0 >= IL + 4N + 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 4 >= 4 = mark(cons(0(),zeros())) active(take(0(),IL)) = 2IL + 4 >= 2IL + 2 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 2 >= 2 = mark(nil()) active(take(s(M),cons(N,IL))) = 2IL + 4M + 6N + 0 >= 2IL + 4M + 4N + 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 2IL + 4M + 4N + 0 >= 2IL + 4M + 4N + 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + 4N + 0 >= L + 4N + 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 0 >= L + 0 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = X + 0 >= X + 0 = active(isNatIList(X)) mark(isNatList(X)) = X + 0 >= X + 0 = active(isNatList(X)) mark(isNat(X)) = 4X + 0 >= 4X + 0 = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 0 >= X + 0 = active(length(mark(X))) mark(zeros()) = 4 >= 4 = active(zeros()) mark(cons(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = active(cons(mark(X1),X2)) mark(nil()) = 2 >= 2 = active(nil()) mark(take(X1,X2)) = 4X1 + 2X2 + 0 >= 4X1 + 2X2 + 0 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 2X >= 2X = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) isNatIList(mark(X)) = X + 0 >= X + 0 = isNatIList(X) isNatIList(active(X)) = X + 0 >= X + 0 = isNatIList(X) isNatList(mark(X)) = X + 0 >= X + 0 = isNatList(X) isNatList(active(X)) = X + 0 >= X + 0 = isNatList(X) isNat(mark(X)) = 4X + 0 >= 4X + 0 = isNat(X) isNat(active(X)) = 4X + 0 >= 4X + 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 0 >= X + 0 = length(X) length(active(X)) = X + 0 >= X + 0 = length(X) cons(mark(X1),X2) = 4X1 + X2 >= 4X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = 4X1 + X2 >= 4X1 + X2 = cons(X1,X2) cons(active(X1),X2) = 4X1 + X2 >= 4X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = 4X1 + X2 >= 4X1 + X2 = cons(X1,X2) take(mark(X1),X2) = 4X1 + 2X2 + 0 >= 4X1 + 2X2 + 0 = take(X1,X2) take(X1,mark(X2)) = 4X1 + 2X2 + 0 >= 4X1 + 2X2 + 0 = take(X1,X2) take(active(X1),X2) = 4X1 + 2X2 + 0 >= 4X1 + 2X2 + 0 = take(X1,X2) take(X1,active(X2)) = 4X1 + 2X2 + 0 >= 4X1 + 2X2 + 0 = take(X1,X2) uTake1(mark(X)) = 2X >= 2X = uTake1(X) uTake1(active(X)) = 2X >= 2X = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + 4X2 + 4X3 + 2X4 + 0 >= X1 + 4X2 + 4X3 + 2X4 + 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) active#(isNat(s(N))) -> mark#(isNat(N)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) active#(isNat(s(N))) -> mark#(isNat(N)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) SCC Processor: #sccs: 2 #rules: 30 #arcs: 242/900 DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + 7x1 + x2 + x3 + 4, [uTake1](x0) = x0 + 0, [take](x0, x1) = 7x0 + x1 + 0, [nil] = 0, [cons](x0, x1) = x0 + x1 + 0, [zeros] = 0, [length](x0) = x0, [s](x0) = x0 + 0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 >= X1 + X2 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 0 >= L + 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) = IL + 7M + N + 4 >= IL + 7M + N + 0 = mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(uTake2(X1,X2,X3,X4)) = X1 + 7X2 + X3 + X4 + 4 >= X1 = mark#(X1) mark#(uTake1(X)) = X + 0 >= X = mark#(X) mark#(take(X1,X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = IL + 7M + N + 7 >= IL + 7M + N + 4 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = 7X1 + X2 + 0 >= X1 = mark#(X1) mark#(take(X1,X2)) = 7X1 + X2 + 0 >= X2 = mark#(X2) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(length(cons(N,L))) = L + N + 0 >= L + N = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = X >= X = mark#(X) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(isNatList(take(N,IL))) = IL + 7N + 0 >= IL + N + 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T = mark#(T) mark#(isNatIList(X)) = X + 0 >= X + 0 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + N + 0 >= IL + N + 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = IL + 0 >= IL = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = L + N + 0 >= L + N = mark#(and(isNat(N),isNatList(L))) active(and(tt(),T)) = T + 0 >= T = mark(T) active(isNatIList(IL)) = IL + 0 >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N + 0 >= N = mark(isNat(N)) active(isNat(length(L))) = L >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + N + 0 >= IL + N + 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N + 0 >= L + N = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + 7N + 0 >= IL + N + 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 7 >= IL + 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + 7M + N + 7 >= IL + 7M + N + 4 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + 7M + N + 4 >= IL + 7M + N + 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 0 >= L + N = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 0 >= L + 0 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = X + 0 >= X + 0 = active(isNatIList(X)) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X + 0 >= X + 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + X2 >= X1 + X2 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIList(mark(X)) = X + 0 >= X + 0 = isNatIList(X) isNatIList(active(X)) = X + 0 >= X + 0 = isNatIList(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) cons(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) take(mark(X1),X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) take(X1,mark(X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) take(active(X1),X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) take(X1,active(X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) uTake1(mark(X)) = X + 0 >= X + 0 = uTake1(X) uTake1(active(X)) = X + 0 >= X + 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + 7X2 + X3 + X4 + 4 >= X1 + 7X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + X2 >= X1 + X2 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + X2 >= X1 + X2 = uLength(X1,X2) uLength(active(X1),X2) = X1 + X2 >= X1 + X2 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + X2 >= X1 + X2 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = x0 + x1 + 0, [uTake2](x0, x1, x2, x3) = x0 + 5x1 + 7x2 + 7x3, [uTake1](x0) = 2x0 + 5, [take](x0, x1) = 5x0 + 7x1, [nil] = 4, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0 + 0, [s](x0) = x0 + 0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0 + 0, [isNatIList](x0) = x0 + 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 0 >= L + 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + 5X2 + 7X3 + 7X4 + 0 >= X1 + 5X2 + 7X3 + 7X4 + 0 = active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) = 7IL + 5M + 7N + 0 >= 7IL + 5M + N + 0 = mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(uTake2(X1,X2,X3,X4)) = X1 + 5X2 + 7X3 + 7X4 + 0 >= X1 + 0 = mark#(X1) mark#(uTake1(X)) = 2X + 5 >= X + 0 = mark#(X) mark#(take(X1,X2)) = 5X1 + 7X2 + 0 >= 5X1 + 7X2 + 0 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = 7IL + 5M + 7N + 5 >= 7IL + 5M + 7N + 0 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = 5X1 + 7X2 + 0 >= X2 + 0 = mark#(X2) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(length(X)) = X + 0 >= X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = L + N + 0 >= L + N + 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNatList(X)) = X + 0 >= X + 0 = active#(isNatList(X)) active#(isNatList(take(N,IL))) = 7IL + 5N + 0 >= IL + N + 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T + 0 = mark#(T) mark#(isNatIList(X)) = X + 0 >= X + 0 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + N + 0 >= IL + N + 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) active#(isNatIList(IL)) = IL + 0 >= IL + 0 = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = L + N + 0 >= L + N + 0 = mark#(and(isNat(N),isNatList(L))) active(and(tt(),T)) = T + 0 >= T = mark(T) active(isNatIList(IL)) = IL + 0 >= IL + 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N + 0 >= N = mark(isNat(N)) active(isNat(length(L))) = L + 0 >= L + 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + N + 0 >= IL + N + 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 4 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N + 0 >= L + N + 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 7IL + 5N + 0 >= IL + N + 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 7IL + 5 >= 2IL + 5 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 5 >= 4 = mark(nil()) active(take(s(M),cons(N,IL))) = 7IL + 5M + 7N + 5 >= 7IL + 5M + 7N + 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 7IL + 5M + 7N + 0 >= 7IL + 5M + N = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 0 >= L + N + 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 0 >= L + 0 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = X + 0 >= X + 0 = active(isNatIList(X)) mark(isNatList(X)) = X + 0 >= X + 0 = active(isNatList(X)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(length(X)) = X + 0 >= X + 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(nil()) = 4 >= 4 = active(nil()) mark(take(X1,X2)) = 5X1 + 7X2 >= 5X1 + 7X2 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 2X + 5 >= 2X + 5 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIList(mark(X)) = X + 0 >= X + 0 = isNatIList(X) isNatIList(active(X)) = X + 0 >= X + 0 = isNatIList(X) isNatList(mark(X)) = X + 0 >= X + 0 = isNatList(X) isNatList(active(X)) = X + 0 >= X + 0 = isNatList(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) length(mark(X)) = X + 0 >= X + 0 = length(X) length(active(X)) = X + 0 >= X + 0 = length(X) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) take(mark(X1),X2) = 5X1 + 7X2 >= 5X1 + 7X2 = take(X1,X2) take(X1,mark(X2)) = 5X1 + 7X2 >= 5X1 + 7X2 = take(X1,X2) take(active(X1),X2) = 5X1 + 7X2 >= 5X1 + 7X2 = take(X1,X2) take(X1,active(X2)) = 5X1 + 7X2 >= 5X1 + 7X2 = take(X1,X2) uTake1(mark(X)) = 2X + 5 >= 2X + 5 = uTake1(X) uTake1(active(X)) = 2X + 5 >= 2X + 5 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + 5X2 + 7X3 + 7X4 >= X1 + 5X2 + 7X3 + 7X4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = 4x0 + x1 + 4, [uTake2](x0, x1, x2, x3) = 4x0 + x1 + x2 + x3 + 4, [uTake1](x0) = x0 + 0, [take](x0, x1) = x0 + x1 + 0, [nil] = 0, [cons](x0, x1) = x0 + x1 + 4, [zeros] = 4, [length](x0) = x0 + 0, [s](x0) = x0 + 4, [isNat](x0) = 0, [0] = 4, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 0, [active](x0) = x0 + 0, [and](x0, x1) = x0 + x1 + 0, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 4X1 + X2 + 4 >= X1 + 0 = mark#(X1) mark#(uLength(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 4 >= L + 4 = mark#(s(length(L))) mark#(s(X)) = X + 4 >= X + 0 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) = IL + M + N + 4 >= IL + M + N + 4 = mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) = X1 + X2 + 4 >= X1 + 0 = mark#(X1) mark#(uTake2(X1,X2,X3,X4)) = 4X1 + X2 + X3 + X4 + 4 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = IL + M + N + 4 >= IL + M + N + 4 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(zeros()) = 4 >= 4 = active#(zeros()) active#(zeros()) = 4 >= 4 = mark#(cons(0(),zeros())) mark#(length(X)) = X + 0 >= X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = L + N + 4 >= L + 4 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = X + 0 >= X + 0 = mark#(X) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(isNatList(take(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T + 0 = mark#(T) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) active#(isNatIList(IL)) = 0 >= 0 = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = 0 >= 0 = mark#(and(isNat(N),isNatList(L))) active(and(tt(),T)) = T + 0 >= T + 0 = mark(T) active(isNatIList(IL)) = 0 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = 0 >= 0 = mark(isNat(N)) active(isNat(length(L))) = 0 >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 0 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 4 >= 4 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 4 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 4 >= IL + M + N + 4 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 4 >= IL + M + N + 4 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 4 >= L + 4 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 4 >= L + 4 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(0()) = 4 >= 4 = active(0()) mark(s(X)) = X + 4 >= X + 4 = active(s(mark(X))) mark(length(X)) = X + 0 >= X + 0 = active(length(mark(X))) mark(zeros()) = 4 >= 4 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X + 0 >= X + 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X + 4 >= X + 4 = s(X) s(active(X)) = X + 4 >= X + 4 = s(X) length(mark(X)) = X + 0 >= X + 0 = length(X) length(active(X)) = X + 0 >= X + 0 = length(X) cons(mark(X1),X2) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(X1,X2) take(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) uTake1(mark(X)) = X + 0 >= X + 0 = uTake1(X) uTake1(active(X)) = X + 0 >= X + 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 4X1 + X2 + X3 + X4 + 4 >= 4X1 + X2 + X3 + X4 + 4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = uLength(X1,X2) uLength(X1,mark(X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = uLength(X1,X2) uLength(active(X1),X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = uLength(X1,X2) uLength(X1,active(X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> mark#(X) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = x0 + 2x1, [uTake2](x0, x1, x2, x3) = x1 + x2 + x3 + 0, [uTake1](x0) = 0, [take](x0, x1) = x0 + x1 + 0, [nil] = 0, [cons](x0, x1) = x0 + x1 + 0, [zeros] = 0, [length](x0) = 2x0 + 0, [s](x0) = x0 + 0, [isNat](x0) = x0 + 0, [0] = 0, [isNatList](x0) = 2x0, [isNatIList](x0) = 2x0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = 2x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 2L + 0 >= 2L + 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) = IL + M + N + 0 >= IL + M + N + 0 = mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = IL + M + N + 0 >= IL + M + N + 0 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(length(X)) = 2X + 0 >= 2X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 2L + 2N + 2 >= 2L + 2N + 2 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = 2X + 0 >= X = mark#(X) mark#(isNatList(X)) = 2X >= 2X = active#(isNatList(X)) active#(isNatList(take(N,IL))) = 2IL + 2N + 2 >= 2IL + 2N + 2 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 2 >= T = mark#(T) mark#(isNatIList(X)) = 2X >= 2X = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 2IL + 2N + 2 >= 2IL + 2N + 2 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = 2X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = 2X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = 2IL >= 2IL = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = 2L + 2N + 2 >= 2L + 2N + 2 = mark#(and(isNat(N),isNatList(L))) active(and(tt(),T)) = T + 2 >= T = mark(T) active(isNatIList(IL)) = 2IL >= 2IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N + 0 >= N + 0 = mark(isNat(N)) active(isNat(length(L))) = 2L + 0 >= 2L = mark(isNatList(L)) active(isNatIList(zeros())) = 2 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 2IL + 2N + 2 >= 2IL + 2N + 2 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 2L + 2N + 2 >= 2L + 2N + 2 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 2IL + 2N + 2 >= 2IL + 2N + 2 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 0 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 0 >= IL + M + N + 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 0 >= IL + M + N + 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 2L + 2N + 2 >= 2L + 2N + 2 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 2L + 0 >= 2L + 0 = mark(s(length(L))) mark(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = 2X >= 2X = active(isNatIList(X)) mark(isNatList(X)) = 2X >= 2X = active(isNatList(X)) mark(isNat(X)) = X + 0 >= X + 0 = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(length(X)) = 2X + 0 >= 2X + 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) and(X1,mark(X2)) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) and(active(X1),X2) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) and(X1,active(X2)) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) isNatIList(mark(X)) = 2X >= 2X = isNatIList(X) isNatIList(active(X)) = 2X >= 2X = isNatIList(X) isNatList(mark(X)) = 2X >= 2X = isNatList(X) isNatList(active(X)) = 2X >= 2X = isNatList(X) isNat(mark(X)) = X + 0 >= X + 0 = isNat(X) isNat(active(X)) = X + 0 >= X + 0 = isNat(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) length(mark(X)) = 2X + 0 >= 2X + 0 = length(X) length(active(X)) = 2X + 0 >= 2X + 0 = length(X) cons(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) take(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) uTake1(mark(X)) = 0 >= 0 = uTake1(X) uTake1(active(X)) = 0 >= 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X2 + X3 + X4 + 0 >= X2 + X3 + X4 + 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + 2X2 >= X1 + 2X2 = uLength(X1,X2) uLength(active(X1),X2) = X1 + 2X2 >= X1 + 2X2 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + 2X2 >= X1 + 2X2 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) -> mark#(X2) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = x0 + 0, [uTake2](x0, x1, x2, x3) = 4x0 + 4x1 + 4x2 + 4, [uTake1](x0) = 4x0 + 4, [take](x0, x1) = 4x0 + 4x1 + 4, [nil] = 0, [cons](x0, x1) = x0 + 0, [zeros] = 0, [length](x0) = 0, [s](x0) = x0, [isNat](x0) = 1x0 + 0, [0] = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0 + 0, [active](x0) = x0 + 0, [and](x0, x1) = x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X1 + 0 >= X1 + 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = active#(uTake2(mark(X1),X2,X3,X4)) active#(uTake2(tt(),M,N,IL)) = 4M + 4N + 4 >= N + 0 = mark#(cons(N,take(M,IL))) mark#(cons(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = 4M + 4N + 4 >= 4M + 4N + 4 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = 4X1 + 4X2 + 4 >= X2 + 0 = mark#(X2) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 0 >= 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(isNatList(take(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X2 + 0 >= X2 + 0 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T + 0 = mark#(T) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X2 + 0 >= X2 + 0 = mark#(X2) active#(isNatIList(IL)) = 0 >= 0 = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = 0 >= 0 = mark#(and(isNat(N),isNatList(L))) active(and(tt(),T)) = T + 0 >= T + 0 = mark(T) active(isNatIList(IL)) = 0 >= 0 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(s(N))) = 1N + 0 >= 1N + 0 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = 0 >= 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 0 >= 0 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 4IL + 4 >= 4 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 4 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 4M + 4N + 4 >= 4M + 4N + 4 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 4M + 4N + 4 >= N + 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 0 >= 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 0 >= 0 = mark(s(length(L))) mark(and(X1,X2)) = X2 + 0 >= X2 + 0 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(isNat(X)) = 1X + 0 >= 1X + 0 = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + 0 >= X1 + 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 4X + 4 >= 4X + 4 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + 0 >= X1 + 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 0 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 0 >= X2 = and(X1,X2) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) isNat(mark(X)) = 1X + 1 >= 1X + 0 = isNat(X) isNat(active(X)) = 1X + 1 >= 1X + 0 = isNat(X) s(mark(X)) = X + 0 >= X = s(X) s(active(X)) = X + 0 >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) cons(mark(X1),X2) = X1 + 0 >= X1 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 0 >= X1 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + 0 >= X1 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + 0 >= X1 + 0 = cons(X1,X2) take(mark(X1),X2) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = take(X1,X2) take(X1,mark(X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = take(X1,X2) take(active(X1),X2) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = take(X1,X2) take(X1,active(X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = take(X1,X2) uTake1(mark(X)) = 4X + 4 >= 4X + 4 = uTake1(X) uTake1(active(X)) = 4X + 4 >= 4X + 4 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4X2 + 4X3 + 4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + 0 >= X1 + 0 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + 0 >= X1 + 0 = uLength(X1,X2) uLength(active(X1),X2) = X1 + 0 >= X1 + 0 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + 0 >= X1 + 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 17 #arcs: 225/400 DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x0 + 6x1 + 1x2 + 1x3 + 6, [uTake1](x0) = x0 + 0, [take](x0, x1) = 6x0 + 1x1 + 6, [nil] = 0, [cons](x0, x1) = x0 + x1 + 0, [zeros] = 6, [length](x0) = x0 + 0, [s](x0) = x0 + 0, [isNat](x0) = x0 + 2, [0] = 0, [isNatList](x0) = x0 + 2, [isNatIList](x0) = x0 + 4, [mark](x0) = x0 + 0, [active](x0) = x0 + 0, [and](x0, x1) = x0 + x1 + 0, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X2 + 0 >= X2 + 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 0 >= L + 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(zeros()) = 6 >= 6 = active#(zeros()) active#(zeros()) = 6 >= 6 = mark#(cons(0(),zeros())) mark#(length(X)) = X + 0 >= X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = L + N + 0 >= L + 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) = X + 2 >= X + 2 = active#(isNatList(X)) active#(isNatList(take(N,IL))) = 1IL + 6N + 6 >= IL + N + 4 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T + 0 = mark#(T) mark#(isNatIList(X)) = X + 4 >= X + 4 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + N + 4 >= IL + N + 4 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) active#(isNatIList(IL)) = IL + 4 >= IL + 2 = mark#(isNatList(IL)) active#(isNatList(cons(N,L))) = L + N + 2 >= L + N + 2 = mark#(and(isNat(N),isNatList(L))) active(and(tt(),T)) = T + 0 >= T + 0 = mark(T) active(isNatIList(IL)) = IL + 4 >= IL + 2 = mark(isNatList(IL)) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(s(N))) = N + 2 >= N + 2 = mark(isNat(N)) active(isNat(length(L))) = L + 2 >= L + 2 = mark(isNatList(L)) active(isNatIList(zeros())) = 6 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + N + 4 >= IL + N + 4 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N + 2 >= L + N + 2 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1IL + 6N + 6 >= IL + N + 4 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 6 >= 6 = mark(cons(0(),zeros())) active(take(0(),IL)) = 1IL + 6 >= IL + 4 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 1IL + 6M + 1N + 6 >= 1IL + 6M + 1N + 6 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1IL + 6M + 1N + 6 >= 1IL + 6M + N + 6 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 0 >= L + 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 0 >= L + 0 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = X + 4 >= X + 4 = active(isNatIList(X)) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(length(X)) = X + 0 >= X + 0 = active(length(mark(X))) mark(zeros()) = 6 >= 6 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = 6X1 + 1X2 + 6 >= 6X1 + 1X2 + 6 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X + 0 >= X + 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X2 + 0 >= X2 + 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) and(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) isNatIList(mark(X)) = X + 4 >= X + 4 = isNatIList(X) isNatIList(active(X)) = X + 4 >= X + 4 = isNatIList(X) isNatList(mark(X)) = X + 2 >= X + 2 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 2 = isNatList(X) isNat(mark(X)) = X + 2 >= X + 2 = isNat(X) isNat(active(X)) = X + 2 >= X + 2 = isNat(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) length(mark(X)) = X + 0 >= X + 0 = length(X) length(active(X)) = X + 0 >= X + 0 = length(X) cons(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) take(mark(X1),X2) = 6X1 + 1X2 + 6 >= 6X1 + 1X2 + 6 = take(X1,X2) take(X1,mark(X2)) = 6X1 + 1X2 + 6 >= 6X1 + 1X2 + 6 = take(X1,X2) take(active(X1),X2) = 6X1 + 1X2 + 6 >= 6X1 + 1X2 + 6 = take(X1,X2) take(X1,active(X2)) = 6X1 + 1X2 + 6 >= 6X1 + 1X2 + 6 = take(X1,X2) uTake1(mark(X)) = X + 0 >= X + 0 = uTake1(X) uTake1(active(X)) = X + 0 >= X + 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + 6X2 + 1X3 + 1X4 + 6 >= X1 + 6X2 + 1X3 + 1X4 + 6 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X2 >= X2 = uLength(X1,X2) uLength(X1,mark(X2)) = X2 + 0 >= X2 = uLength(X1,X2) uLength(active(X1),X2) = X2 >= X2 = uLength(X1,X2) uLength(X1,active(X2)) = X2 + 0 >= X2 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {39} transitions: cons0(41,40) -> 42* zeros0() -> 40* mark{#,0}(42) -> 39* mark{#,0}(41) -> 39* 00() -> 41* problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 14 #arcs: 54/225 DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = x1 + 0, [uTake2](x0, x1, x2, x3) = x0 + 7x1 + 1x2 + x3, [uTake1](x0) = 0, [take](x0, x1) = 7x0 + x1 + 0, [nil] = 0, [cons](x0, x1) = 1x0 + x1, [zeros] = 7, [length](x0) = x0 + 0, [s](x0) = x0 + 0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X2 + 0 >= X2 + 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 0 >= L + 0 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(cons(X1,X2)) = 1X1 + X2 >= X1 = mark#(X1) mark#(length(X)) = X + 0 >= X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = L + 1N + 0 >= L + 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(isNatList(cons(N,L))) = L + 1N >= L + N = mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T = mark#(T) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + 1N >= IL + N = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = IL >= IL = mark#(isNatList(IL)) active(and(tt(),T)) = T + 0 >= T = mark(T) active(isNatIList(IL)) = IL >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N + 0 >= N = mark(isNat(N)) active(isNat(length(L))) = L + 0 >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 7 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + 1N >= IL + N = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + 1N >= L + N = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + 7N + 0 >= IL + N = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 7 >= 7 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 7 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + 7M + 1N + 7 >= IL + 7M + 1N = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + 7M + 1N + 0 >= IL + 7M + 1N + 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + 1N + 0 >= L + 0 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 0 >= L + 0 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = X >= X = active(isNatIList(X)) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 0 >= X + 0 = active(s(mark(X))) mark(length(X)) = X + 0 >= X + 0 = active(length(mark(X))) mark(zeros()) = 7 >= 7 = active(zeros()) mark(cons(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X2 + 0 >= X2 + 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIList(mark(X)) = X >= X = isNatIList(X) isNatIList(active(X)) = X >= X = isNatIList(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) s(mark(X)) = X + 0 >= X + 0 = s(X) s(active(X)) = X + 0 >= X + 0 = s(X) length(mark(X)) = X + 0 >= X + 0 = length(X) length(active(X)) = X + 0 >= X + 0 = length(X) cons(mark(X1),X2) = 1X1 + X2 >= 1X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = 1X1 + X2 >= 1X1 + X2 = cons(X1,X2) cons(active(X1),X2) = 1X1 + X2 >= 1X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = 1X1 + X2 >= 1X1 + X2 = cons(X1,X2) take(mark(X1),X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) take(X1,mark(X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) take(active(X1),X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) take(X1,active(X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = take(X1,X2) uTake1(mark(X)) = 0 >= 0 = uTake1(X) uTake1(active(X)) = 0 >= 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + 7X2 + 1X3 + X4 >= X1 + 7X2 + 1X3 + X4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X2 + 0 >= X2 + 0 = uLength(X1,X2) uLength(X1,mark(X2)) = X2 + 0 >= X2 + 0 = uLength(X1,X2) uLength(active(X1),X2) = X2 + 0 >= X2 + 0 = uLength(X1,X2) uLength(X1,active(X2)) = X2 + 0 >= X2 + 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = 4x0 + 4x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + 4x3 + 4, [uTake1](x0) = x0 + 3, [take](x0, x1) = x0 + 4x1 + 4, [nil] = 0, [cons](x0, x1) = x0 + x1 + 0, [zeros] = 1, [length](x0) = 4x0 + 0, [s](x0) = x0 + 4, [isNat](x0) = x0 + 0, [0] = 0, [isNatList](x0) = x0 + 0, [isNatIList](x0) = 4x0 + 4, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 4X1 + 4X2 + 0 >= 4X1 + 4X2 + 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 4L + 4 >= 4L + 4 = mark#(s(length(L))) mark#(s(X)) = X + 4 >= X + 0 = mark#(X) mark#(length(X)) = 4X + 0 >= 4X + 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 4L + 4N + 4 >= 4L + 4N + 4 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) = X + 0 >= X + 0 = active#(isNatList(X)) active#(isNatList(cons(N,L))) = L + N + 0 >= L + N + 0 = mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 0 >= T + 0 = mark#(T) mark#(isNatIList(X)) = 4X + 4 >= 4X + 4 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 4IL + 4N + 4 >= 4IL + N + 4 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) active#(isNatIList(IL)) = 4IL + 4 >= IL + 0 = mark#(isNatList(IL)) active(and(tt(),T)) = T + 0 >= T = mark(T) active(isNatIList(IL)) = 4IL + 4 >= IL + 0 = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N + 4 >= N + 0 = mark(isNat(N)) active(isNat(length(L))) = 4L + 0 >= L + 0 = mark(isNatList(L)) active(isNatIList(zeros())) = 5 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = 4IL + 4N + 4 >= 4IL + N + 4 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N + 0 >= L + N + 0 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 4IL + N + 4 >= 4IL + N + 4 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = 4IL + 4 >= 4IL + 4 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 3 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 4IL + M + 4N + 4 >= 4IL + M + N + 4 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 4IL + M + N + 4 >= 4IL + M + N + 4 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 4L + 4N + 4 >= 4L + 4N + 4 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 4L + 4 >= 4L + 4 = mark(s(length(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),mark(X2))) mark(tt()) = 0 >= 0 = active(tt()) mark(isNatIList(X)) = 4X + 4 >= 4X + 4 = active(isNatIList(X)) mark(isNatList(X)) = X + 0 >= X + 0 = active(isNatList(X)) mark(isNat(X)) = X + 0 >= X + 0 = active(isNat(X)) mark(0()) = 0 >= 0 = active(0()) mark(s(X)) = X + 4 >= X + 4 = active(s(mark(X))) mark(length(X)) = 4X + 0 >= 4X + 0 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X + 3 >= X + 3 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIList(mark(X)) = 4X + 4 >= 4X + 4 = isNatIList(X) isNatIList(active(X)) = 4X + 4 >= 4X + 4 = isNatIList(X) isNatList(mark(X)) = X + 0 >= X + 0 = isNatList(X) isNatList(active(X)) = X + 0 >= X + 0 = isNatList(X) isNat(mark(X)) = X + 0 >= X + 0 = isNat(X) isNat(active(X)) = X + 0 >= X + 0 = isNat(X) s(mark(X)) = X + 4 >= X + 4 = s(X) s(active(X)) = X + 4 >= X + 4 = s(X) length(mark(X)) = 4X + 0 >= 4X + 0 = length(X) length(active(X)) = 4X + 0 >= 4X + 0 = length(X) cons(mark(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(X1,X2) take(mark(X1),X2) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = take(X1,X2) take(X1,mark(X2)) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = take(X1,X2) take(active(X1),X2) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = take(X1,X2) take(X1,active(X2)) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = take(X1,X2) uTake1(mark(X)) = X + 3 >= X + 3 = uTake1(X) uTake1(active(X)) = X + 3 >= X + 3 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + X2 + X3 + 4X4 + 4 >= X1 + X2 + X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = uLength(X1,X2) uLength(X1,mark(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = uLength(X1,X2) uLength(active(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = uLength(X1,X2) uLength(X1,active(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> mark#(X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 0, [active#](x0) = x0 + 0, [uLength](x0, x1) = x0 + 4x1 + 1, [uTake2](x0, x1, x2, x3) = x2 + 1x3 + 0, [uTake1](x0) = 1, [take](x0, x1) = x1 + 0, [nil] = 0, [cons](x0, x1) = x0 + 1x1 + 1, [zeros] = 0, [length](x0) = 4x0 + 5, [s](x0) = x0, [isNat](x0) = 1x0 + 6, [0] = 1, [isNatList](x0) = 5x0 + 0, [isNatIList](x0) = 5x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x1 + 5, [tt] = 5 orientation: mark#(uLength(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 4L + 5 >= 4L + 5 = mark#(s(length(L))) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(length(X)) = 4X + 5 >= 4X + 5 = active#(length(mark(X))) active#(length(cons(N,L))) = 5L + 4N + 5 >= 5L + 5 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) = 5X + 0 >= 5X + 0 = active#(isNatList(X)) active#(isNatList(cons(N,L))) = 6L + 5N + 6 >= 5L + 5 = mark#(and(isNat(N),isNatList(L))) mark#(and(X1,X2)) = X2 + 5 >= X2 + 5 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 5 >= T + 0 = mark#(T) mark#(isNatIList(X)) = 5X + 0 >= 5X + 0 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 6IL + 5N + 6 >= 5IL + 5 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X2 + 5 >= X2 + 0 = mark#(X2) active(and(tt(),T)) = T + 5 >= T + 1 = mark(T) active(isNatIList(IL)) = 5IL + 1 >= 5IL + 1 = mark(isNatList(IL)) active(isNat(0())) = 6 >= 5 = mark(tt()) active(isNat(s(N))) = 1N + 6 >= 1N + 6 = mark(isNat(N)) active(isNat(length(L))) = 5L + 6 >= 5L + 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 5 >= 5 = mark(tt()) active(isNatIList(cons(N,IL))) = 6IL + 5N + 6 >= 5IL + 5 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 5 >= 5 = mark(tt()) active(isNatList(cons(N,L))) = 6L + 5N + 6 >= 5L + 5 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 5IL + 5 >= 5IL + 5 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = 1IL + N + 1 >= 1IL + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1IL + N + 1 >= 1IL + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 5L + 4N + 5 >= 5L + 5 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 4L + 5 >= 4L + 5 = mark(s(length(L))) mark(and(X1,X2)) = X2 + 5 >= X2 + 5 = active(and(mark(X1),mark(X2))) mark(tt()) = 5 >= 5 = active(tt()) mark(isNatIList(X)) = 5X + 1 >= 5X + 1 = active(isNatIList(X)) mark(isNatList(X)) = 5X + 1 >= 5X + 1 = active(isNatList(X)) mark(isNat(X)) = 1X + 6 >= 1X + 6 = active(isNat(X)) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = X + 1 >= X + 1 = active(s(mark(X))) mark(length(X)) = 4X + 5 >= 4X + 5 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = active(cons(mark(X1),X2)) mark(nil()) = 1 >= 1 = active(nil()) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 1 >= 1 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X3 + 1X4 + 1 >= X3 + 1X4 + 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X2 + 5 >= X2 + 5 = and(X1,X2) and(X1,mark(X2)) = X2 + 5 >= X2 + 5 = and(X1,X2) and(active(X1),X2) = X2 + 5 >= X2 + 5 = and(X1,X2) and(X1,active(X2)) = X2 + 5 >= X2 + 5 = and(X1,X2) isNatIList(mark(X)) = 5X + 6 >= 5X = isNatIList(X) isNatIList(active(X)) = 5X + 6 >= 5X = isNatIList(X) isNatList(mark(X)) = 5X + 6 >= 5X + 0 = isNatList(X) isNatList(active(X)) = 5X + 6 >= 5X + 0 = isNatList(X) isNat(mark(X)) = 1X + 6 >= 1X + 6 = isNat(X) isNat(active(X)) = 1X + 6 >= 1X + 6 = isNat(X) s(mark(X)) = X + 1 >= X = s(X) s(active(X)) = X + 1 >= X = s(X) length(mark(X)) = 4X + 5 >= 4X + 5 = length(X) length(active(X)) = 4X + 5 >= 4X + 5 = length(X) cons(mark(X1),X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 1X2 + 2 >= X1 + 1X2 + 1 = cons(X1,X2) cons(active(X1),X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = cons(X1,X2) cons(X1,active(X2)) = X1 + 1X2 + 2 >= X1 + 1X2 + 1 = cons(X1,X2) take(mark(X1),X2) = X2 + 0 >= X2 + 0 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 0 = take(X1,X2) take(active(X1),X2) = X2 + 0 >= X2 + 0 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 0 = take(X1,X2) uTake1(mark(X)) = 1 >= 1 = uTake1(X) uTake1(active(X)) = 1 >= 1 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X3 + 1X4 + 0 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X3 + 1X4 + 0 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X3 + 1X4 + 1 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X3 + 1X4 + 2 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X3 + 1X4 + 0 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X3 + 1X4 + 0 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X3 + 1X4 + 1 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X3 + 1X4 + 2 >= X3 + 1X4 + 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + 4X2 + 5 >= X1 + 4X2 + 1 = uLength(X1,X2) uLength(active(X1),X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + 4X2 + 5 >= X1 + 4X2 + 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(and(X1,X2)) -> mark#(X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(and(X1,X2)) -> mark#(X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 8 #arcs: 45/100 DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = x0 + 1, [active#](x0) = x0 + 1, [uLength](x0, x1) = x0 + 1x1 + 0, [uTake2](x0, x1, x2, x3) = x0 + 4x1 + 5x3 + 0, [uTake1](x0) = 6, [take](x0, x1) = 3x0 + 4x1 + 6, [nil] = 6, [cons](x0, x1) = 1x1, [zeros] = 0, [length](x0) = x0 + 0, [s](x0) = 1x0 + 4, [isNat](x0) = 1x0 + 7, [0] = 0, [isNatList](x0) = 1x0 + 0, [isNatIList](x0) = 2x0 + 7, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = x1 + 1, [tt] = 7 orientation: mark#(uLength(X1,X2)) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1L + 7 >= 1L + 4 = mark#(s(length(L))) mark#(s(X)) = 1X + 4 >= X + 1 = mark#(X) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(length(cons(N,L))) = 1L + 1 >= 1L + 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(and(X1,X2)) = X2 + 1 >= X2 + 1 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = T + 1 >= T + 1 = mark#(T) mark#(and(X1,X2)) = X2 + 1 >= X2 + 1 = mark#(X2) active(and(tt(),T)) = T + 1 >= T + 1 = mark(T) active(isNatIList(IL)) = 2IL + 7 >= 1IL + 1 = mark(isNatList(IL)) active(isNat(0())) = 7 >= 7 = mark(tt()) active(isNat(s(N))) = 2N + 7 >= 1N + 7 = mark(isNat(N)) active(isNat(length(L))) = 1L + 7 >= 1L + 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 7 >= 7 = mark(tt()) active(isNatIList(cons(N,IL))) = 3IL + 7 >= 2IL + 7 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 7 >= 7 = mark(tt()) active(isNatList(cons(N,L))) = 2L + 1 >= 1L + 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 5IL + 4N + 7 >= 2IL + 7 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = 4IL + 6 >= 6 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 6 >= 6 = mark(nil()) active(take(s(M),cons(N,IL))) = 5IL + 4M + 7 >= 5IL + 4M + 7 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 5IL + 4M + 7 >= 5IL + 4M + 7 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 1L + 1 >= 1L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1L + 7 >= 1L + 4 = mark(s(length(L))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),mark(X2))) mark(tt()) = 7 >= 7 = active(tt()) mark(isNatIList(X)) = 2X + 7 >= 2X + 7 = active(isNatIList(X)) mark(isNatList(X)) = 1X + 1 >= 1X + 1 = active(isNatList(X)) mark(isNat(X)) = 1X + 7 >= 1X + 7 = active(isNat(X)) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1X + 4 >= 1X + 4 = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1X2 + 1 >= 1X2 + 1 = active(cons(mark(X1),X2)) mark(nil()) = 6 >= 6 = active(nil()) mark(take(X1,X2)) = 3X1 + 4X2 + 6 >= 3X1 + 4X2 + 6 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 6 >= 6 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + 4X2 + 5X4 + 1 >= X1 + 4X2 + 5X4 + 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = X2 + 1 >= X2 + 1 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 + 1 = and(X1,X2) and(active(X1),X2) = X2 + 1 >= X2 + 1 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 + 1 = and(X1,X2) isNatIList(mark(X)) = 2X + 7 >= 2X + 7 = isNatIList(X) isNatIList(active(X)) = 2X + 7 >= 2X + 7 = isNatIList(X) isNatList(mark(X)) = 1X + 2 >= 1X + 0 = isNatList(X) isNatList(active(X)) = 1X + 2 >= 1X + 0 = isNatList(X) isNat(mark(X)) = 1X + 7 >= 1X + 7 = isNat(X) isNat(active(X)) = 1X + 7 >= 1X + 7 = isNat(X) s(mark(X)) = 1X + 4 >= 1X + 4 = s(X) s(active(X)) = 1X + 4 >= 1X + 4 = s(X) length(mark(X)) = X + 1 >= X + 0 = length(X) length(active(X)) = X + 1 >= X + 0 = length(X) cons(mark(X1),X2) = 1X2 >= 1X2 = cons(X1,X2) cons(X1,mark(X2)) = 1X2 + 2 >= 1X2 = cons(X1,X2) cons(active(X1),X2) = 1X2 >= 1X2 = cons(X1,X2) cons(X1,active(X2)) = 1X2 + 2 >= 1X2 = cons(X1,X2) take(mark(X1),X2) = 3X1 + 4X2 + 6 >= 3X1 + 4X2 + 6 = take(X1,X2) take(X1,mark(X2)) = 3X1 + 4X2 + 6 >= 3X1 + 4X2 + 6 = take(X1,X2) take(active(X1),X2) = 3X1 + 4X2 + 6 >= 3X1 + 4X2 + 6 = take(X1,X2) take(X1,active(X2)) = 3X1 + 4X2 + 6 >= 3X1 + 4X2 + 6 = take(X1,X2) uTake1(mark(X)) = 6 >= 6 = uTake1(X) uTake1(active(X)) = 6 >= 6 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + 4X2 + 5X4 + 1 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + 4X2 + 5X4 + 5 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + 4X2 + 5X4 + 0 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + 4X2 + 5X4 + 6 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + 4X2 + 5X4 + 1 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + 4X2 + 5X4 + 5 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + 4X2 + 5X4 + 0 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + 4X2 + 5X4 + 6 >= X1 + 4X2 + 5X4 + 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 0 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + 1X2 + 2 >= X1 + 1X2 + 0 = uLength(X1,X2) uLength(active(X1),X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 0 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + 1X2 + 2 >= X1 + 1X2 + 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 3 #arcs: 20/49 DPs: mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) -> mark#(T) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) interpretation: [mark#](x0) = 4x0 + 0, [active#](x0) = 4x0 + 0, [uLength](x0, x1) = 2x0 + 5x1 + 4, [uTake2](x0, x1, x2, x3) = 5x1 + 3x2 + 5x3 + 6, [uTake1](x0) = 4, [take](x0, x1) = 4x0 + 4x1 + 5, [nil] = 2, [cons](x0, x1) = x0 + 1x1 + 1, [zeros] = 0, [length](x0) = 4x0 + 5, [s](x0) = 1x0 + 2, [isNat](x0) = x0 + 5, [0] = 0, [isNatList](x0) = 2x0 + 1, [isNatIList](x0) = 4x0, [mark](x0) = x0 + 1, [active](x0) = x0 + 1, [and](x0, x1) = 1x1 + 2, [tt] = 4 orientation: mark#(and(X1,X2)) = 5X2 + 6 >= 4X2 + 0 = mark#(X2) mark#(and(X1,X2)) = 5X2 + 6 >= 5X2 + 6 = active#(and(mark(X1),mark(X2))) active#(and(tt(),T)) = 5T + 6 >= 4T + 0 = mark#(T) active(and(tt(),T)) = 1T + 2 >= T + 1 = mark(T) active(isNatIList(IL)) = 4IL + 1 >= 2IL + 1 = mark(isNatList(IL)) active(isNat(0())) = 5 >= 4 = mark(tt()) active(isNat(s(N))) = 1N + 5 >= N + 5 = mark(isNat(N)) active(isNat(length(L))) = 4L + 5 >= 2L + 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 4 >= 4 = mark(tt()) active(isNatIList(cons(N,IL))) = 5IL + 4N + 5 >= 5IL + 2 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 4 >= 4 = mark(tt()) active(isNatList(cons(N,L))) = 3L + 2N + 3 >= 3L + 2 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 6IL + 6N + 7 >= 5IL + 2 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = 4IL + 5 >= 4 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 4 >= 2 = mark(nil()) active(take(s(M),cons(N,IL))) = 5IL + 5M + 4N + 6 >= 5IL + 5M + 3N + 6 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 5IL + 5M + 3N + 6 >= 5IL + 5M + N + 6 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 5L + 4N + 5 >= 5L + 4 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 5L + 6 >= 5L + 6 = mark(s(length(L))) mark(and(X1,X2)) = 1X2 + 2 >= 1X2 + 2 = active(and(mark(X1),mark(X2))) mark(tt()) = 4 >= 4 = active(tt()) mark(isNatIList(X)) = 4X + 1 >= 4X + 1 = active(isNatIList(X)) mark(isNatList(X)) = 2X + 1 >= 2X + 1 = active(isNatList(X)) mark(isNat(X)) = X + 5 >= X + 5 = active(isNat(X)) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1X + 2 >= 1X + 2 = active(s(mark(X))) mark(length(X)) = 4X + 5 >= 4X + 5 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = active(cons(mark(X1),X2)) mark(nil()) = 2 >= 2 = active(nil()) mark(take(X1,X2)) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 4 >= 4 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 2X1 + 5X2 + 4 >= 2X1 + 5X2 + 4 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = 1X2 + 2 >= 1X2 + 2 = and(X1,X2) and(X1,mark(X2)) = 1X2 + 2 >= 1X2 + 2 = and(X1,X2) and(active(X1),X2) = 1X2 + 2 >= 1X2 + 2 = and(X1,X2) and(X1,active(X2)) = 1X2 + 2 >= 1X2 + 2 = and(X1,X2) isNatIList(mark(X)) = 4X + 5 >= 4X = isNatIList(X) isNatIList(active(X)) = 4X + 5 >= 4X = isNatIList(X) isNatList(mark(X)) = 2X + 3 >= 2X + 1 = isNatList(X) isNatList(active(X)) = 2X + 3 >= 2X + 1 = isNatList(X) isNat(mark(X)) = X + 5 >= X + 5 = isNat(X) isNat(active(X)) = X + 5 >= X + 5 = isNat(X) s(mark(X)) = 1X + 2 >= 1X + 2 = s(X) s(active(X)) = 1X + 2 >= 1X + 2 = s(X) length(mark(X)) = 4X + 5 >= 4X + 5 = length(X) length(active(X)) = 4X + 5 >= 4X + 5 = length(X) cons(mark(X1),X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 1X2 + 2 >= X1 + 1X2 + 1 = cons(X1,X2) cons(active(X1),X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = cons(X1,X2) cons(X1,active(X2)) = X1 + 1X2 + 2 >= X1 + 1X2 + 1 = cons(X1,X2) take(mark(X1),X2) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = take(X1,X2) take(X1,mark(X2)) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = take(X1,X2) take(active(X1),X2) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = take(X1,X2) take(X1,active(X2)) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = take(X1,X2) uTake1(mark(X)) = 4 >= 4 = uTake1(X) uTake1(active(X)) = 4 >= 4 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 5X2 + 3X3 + 5X4 + 6 >= 5X2 + 3X3 + 5X4 + 6 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 2X1 + 5X2 + 4 >= 2X1 + 5X2 + 4 = uLength(X1,X2) uLength(X1,mark(X2)) = 2X1 + 5X2 + 6 >= 2X1 + 5X2 + 4 = uLength(X1,X2) uLength(active(X1),X2) = 2X1 + 5X2 + 4 >= 2X1 + 5X2 + 4 = uLength(X1,X2) uLength(X1,active(X2)) = 2X1 + 5X2 + 6 >= 2X1 + 5X2 + 4 = uLength(X1,X2) problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))), M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Restore Modifier: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))), M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/1 DPs: mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(N))) -> mark#(isNat(N)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(and) = 0 pi(isNat) = 0 pi(active#) = 0 pi(mark#) = 0 problem: DPs: mark#(isNat(X)) -> active#(isNat(X)) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1 DPs: isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(isNatList#) = 0 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed DPs: isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(isNatIList#) = 0 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(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(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(and#) = 0 problem: DPs: and#(X1,mark(X2)) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(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(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed DPs: uTake1#(mark(X)) -> uTake1#(X) uTake1#(active(X)) -> uTake1#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(uTake1#) = 0 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed DPs: uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,mark(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,mark(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,mark(X4)) -> uTake2#(X1,X2,X3,X4) uTake2#(active(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,active(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,active(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,active(X4)) -> uTake2#(X1,X2,X3,X4) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(uTake2#) = 2 problem: DPs: uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,mark(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,mark(X4)) -> uTake2#(X1,X2,X3,X4) uTake2#(active(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,active(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,active(X4)) -> uTake2#(X1,X2,X3,X4) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(uTake2#) = 0 problem: DPs: uTake2#(X1,mark(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,mark(X4)) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,active(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,active(X4)) -> uTake2#(X1,X2,X3,X4) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(uTake2#) = 3 problem: DPs: uTake2#(X1,mark(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,active(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(uTake2#) = 1 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(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(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(take#) = 0 problem: DPs: take#(X1,mark(X2)) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(take#) = 1 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed DPs: uLength#(mark(X1),X2) -> uLength#(X1,X2) uLength#(X1,mark(X2)) -> uLength#(X1,X2) uLength#(active(X1),X2) -> uLength#(X1,X2) uLength#(X1,active(X2)) -> uLength#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(uLength#) = 0 problem: DPs: uLength#(X1,mark(X2)) -> uLength#(X1,X2) uLength#(X1,active(X2)) -> uLength#(X1,X2) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(uLength#) = 1 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed DPs: length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(length#) = 0 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(and(tt(),T)) -> mark(T) active(isNatIList(IL)) -> mark(isNatList(IL)) active(isNat(0())) -> mark(tt()) active(isNat(s(N))) -> mark(isNat(N)) active(isNat(length(L))) -> mark(isNatList(L)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(N,L))) -> mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) -> mark(and(isNat(N),isNatIList(IL))) active(zeros()) -> mark(cons(0(),zeros())) active(take(0(),IL)) -> mark(uTake1(isNatIList(IL))) active(uTake1(tt())) -> mark(nil()) active(take(s(M),cons(N,IL))) -> mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) -> mark(cons(N,take(M,IL))) active(length(cons(N,L))) -> mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) -> mark(s(length(L))) mark(and(X1,X2)) -> active(and(mark(X1),mark(X2))) mark(tt()) -> active(tt()) mark(isNatIList(X)) -> active(isNatIList(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNat(X)) -> active(isNat(X)) mark(0()) -> active(0()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(nil()) -> active(nil()) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(uTake1(X)) -> active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) -> active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) -> active(uLength(mark(X1),X2)) 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) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) 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) uTake1(mark(X)) -> uTake1(X) uTake1(active(X)) -> uTake1(X) uTake2(mark(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) -> uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) -> uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) -> uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) -> uLength(X1,X2) uLength(X1,mark(X2)) -> uLength(X1,X2) uLength(active(X1),X2) -> uLength(X1,X2) uLength(X1,active(X2)) -> uLength(X1,X2) Qed