MAYBE 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) EDG 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: 71 #arcs: 1270/13225 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#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> active#(uTake1(mark(X))) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) 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#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(length(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [active#](x0) = x0 + 1, [uLength](x0, x1) = x0 + x1 + 1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0 + 1, [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, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 2 >= L + 2 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = L + N + 2 >= L + 2 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + 1 = mark#(X1) mark#(uTake1(X)) = X + 1 >= X + 1 = active#(uTake1(mark(X))) active#(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark#(cons(N,take(M,IL))) mark#(uTake1(X)) = X + 1 >= X + 1 = mark#(X) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(cons(mark(X1),X2)) active#(take(0(),IL)) = IL + 1 >= 1 = mark#(uTake1(isNatIList(IL))) mark#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(length(X)) = X + 2 >= X + 2 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(length(X)) = X + 2 >= X + 1 = mark#(X) mark#(s(X)) = X + 1 >= X + 1 = active#(s(mark(X))) active#(isNatList(cons(N,L))) = 1 >= 1 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(length(L))) = 1 >= 1 = mark#(isNatList(L)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 1 >= 1 = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) active#(isNatIList(IL)) = 1 >= 1 = mark#(isNatList(IL)) active#(and(tt(),T)) = T + 1 >= T + 1 = mark#(T) active(and(tt(),T)) = T >= 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 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N >= IL + M + N = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N >= IL + M + N = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = 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)) = 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 + 1 >= X + 1 = 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()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X >= X = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = 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)) = 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 + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = 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) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) uTake1(mark(X)) = X >= X = uTake1(X) uTake1(active(X)) = X >= X = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(X1,X2) uLength(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(X1,X2) uLength(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(X1,X2) uLength(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> active#(uTake1(mark(X))) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) 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#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(s(X)) -> active#(s(mark(X))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [uLength](x0, x1) = 1, [uTake2](x0, x1, x2, x3) = 1, [uTake1](x0) = 0, [take](x0, x1) = 1, [nil] = 0, [cons](x0, x1) = 1, [zeros] = 1, [length](x0) = 1, [s](x0) = 0, [isNat](x0) = 1, [0] = 0, [isNatList](x0) = 1, [isNatIList](x0) = 1, [mark](x0) = 1, [active](x0) = 1, [and](x0, x1) = 1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 1 >= 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 1 >= 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 1 = mark#(X1) mark#(uTake1(X)) = 1 >= 0 = active#(uTake1(mark(X))) active#(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark#(cons(N,take(M,IL))) mark#(uTake1(X)) = 1 >= 1 = mark#(X) mark#(take(X1,X2)) = 1 >= 1 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = 1 >= 1 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = 1 >= 1 = mark#(X1) mark#(take(X1,X2)) = 1 >= 1 = mark#(X2) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(take(0(),IL)) = 1 >= 1 = mark#(uTake1(isNatIList(IL))) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(s(X)) = 1 >= 0 = active#(s(mark(X))) active#(isNatList(cons(N,L))) = 1 >= 1 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(length(L))) = 1 >= 1 = mark#(isNatList(L)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 1 >= 1 = mark#(isNat(N)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = mark#(X2) active#(isNatIList(IL)) = 1 >= 1 = mark#(isNatList(IL)) active#(and(tt(),T)) = 1 >= 1 = mark#(T) active(and(tt(),T)) = 1 >= 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 1 >= 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 1 = mark(s(length(L))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),mark(X2))) mark(tt()) = 1 >= 1 = active(tt()) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(nil()) = 1 >= 1 = active(nil()) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 1 >= 1 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 1 >= 1 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) uTake1(mark(X)) = 0 >= 0 = uTake1(X) uTake1(active(X)) = 0 >= 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,mark(X2)) = 1 >= 1 = uLength(X1,X2) uLength(active(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,active(X2)) = 1 >= 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) 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#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = x0 + x2 + 1, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + 1, [zeros] = 1, [length](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, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 0 >= 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = X1 + X3 + 1 >= X1 + X3 + 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 0 >= 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(uTake2(X1,X2,X3,X4)) = X1 + X3 + 1 >= X1 = mark#(X1) active#(uTake2(tt(),M,N,IL)) = N + 1 >= N + 1 = mark#(cons(N,take(M,IL))) mark#(uTake1(X)) = X >= X = mark#(X) mark#(take(X1,X2)) = X1 + X2 >= X1 + X2 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = M + N + 1 >= N + 1 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active#(cons(mark(X1),X2)) active#(take(0(),IL)) = IL >= 0 = mark#(uTake1(isNatIList(IL))) mark#(cons(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = 0 >= 0 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X >= X = 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)) 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 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 0 >= 0 = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = 0 >= 0 = mark#(isNatList(IL)) active#(and(tt(),T)) = T >= T = mark#(T) active(and(tt(),T)) = T >= 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()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + N + 1 >= N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = N + 1 >= N + 1 = 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)) = X1 + X2 >= X1 + X2 = 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)) = 0 >= 0 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X >= X = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X1 + X3 + 1 >= X1 + X3 + 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 0 >= 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)) = 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)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) cons(mark(X1),X2) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(active(X1),X2) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(X1,active(X2)) = X1 + 1 >= X1 + 1 = cons(X1,X2) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) uTake1(mark(X)) = X >= X = uTake1(X) uTake1(active(X)) = X >= X = uTake1(X) uTake2(mark(X1),X2,X3,X4) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X1 + X3 + 1 >= X1 + X3 + 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,mark(X2)) = 0 >= 0 = uLength(X1,X2) uLength(active(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,active(X2)) = 0 >= 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) 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#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = 1, [uTake2](x0, x1, x2, x3) = 0, [uTake1](x0) = x0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 1, [length](x0) = 1, [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, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 1 >= 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 0 >= 0 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 1 >= 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) = 0 >= 0 = mark#(cons(N,take(M,IL))) mark#(uTake1(X)) = X >= X = mark#(X) mark#(take(X1,X2)) = X1 + X2 >= X1 + X2 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = M >= 0 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(cons(X1,X2)) = 0 >= 0 = active#(cons(mark(X1),X2)) active#(take(0(),IL)) = IL >= 0 = mark#(uTake1(isNatIList(IL))) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(zeros()) = 1 >= 0 = mark#(cons(0(),zeros())) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = 0 >= 0 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X >= X = 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)) 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 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 0 >= 0 = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = 0 >= 0 = mark#(isNatList(IL)) active#(and(tt(),T)) = T >= T = mark#(T) active(and(tt(),T)) = T >= 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()) = 1 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 1 >= 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 1 = 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)) = 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)) = 1 >= 1 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X >= X = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 0 >= 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 1 >= 1 = 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)) = 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)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) uTake1(mark(X)) = X >= X = uTake1(X) uTake1(active(X)) = X >= X = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,mark(X2)) = 1 >= 1 = uLength(X1,X2) uLength(active(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,active(X2)) = 1 >= 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) 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#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(zeros()) -> active#(zeros()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [active#](x0) = x0 + 1, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = 0, [uTake1](x0) = x0 + 1, [take](x0, x1) = x0 + x1, [nil] = 1, [cons](x0, x1) = 0, [zeros] = 0, [length](x0) = 0, [s](x0) = x0, [isNat](x0) = 0, [0] = 1, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 1 >= 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 1 >= 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark#(cons(N,take(M,IL))) mark#(uTake1(X)) = X + 2 >= X + 1 = mark#(X) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = M + 1 >= 1 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(take(0(),IL)) = IL + 2 >= 2 = mark#(uTake1(isNatIList(IL))) mark#(zeros()) = 1 >= 1 = active#(zeros()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = 1 >= 1 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(length(L))) = 1 >= 1 = mark#(isNatList(L)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 1 >= 1 = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) active#(isNatIList(IL)) = 1 >= 1 = mark#(isNatList(IL)) active#(and(tt(),T)) = T + 1 >= T + 1 = mark#(T) active(and(tt(),T)) = T >= 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 + 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = M >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 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)) = X1 + X2 >= X1 + X2 = 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()) = 1 >= 1 = active(0()) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(nil()) = 1 >= 1 = active(nil()) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = X + 1 >= X + 1 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 0 >= 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 0 >= 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)) = 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)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) uTake1(mark(X)) = X + 1 >= X + 1 = uTake1(X) uTake1(active(X)) = X + 1 >= X + 1 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,mark(X2)) = 0 >= 0 = uLength(X1,X2) uLength(active(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,active(X2)) = 0 >= 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,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#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(zeros()) -> active#(zeros()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [uLength](x0, x1) = 1, [uTake2](x0, x1, x2, x3) = 1, [uTake1](x0) = 1, [take](x0, x1) = 1, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 1, [length](x0) = 1, [s](x0) = 0, [isNat](x0) = 1, [0] = 0, [isNatList](x0) = 1, [isNatIList](x0) = 1, [mark](x0) = 1, [active](x0) = 1, [and](x0, x1) = 1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 1 >= 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 1 >= 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark#(cons(N,take(M,IL))) mark#(take(X1,X2)) = 1 >= 1 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = 1 >= 1 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = 1 >= 1 = mark#(X1) mark#(take(X1,X2)) = 1 >= 1 = mark#(X2) mark#(cons(X1,X2)) = 1 >= 0 = active#(cons(mark(X1),X2)) active#(take(0(),IL)) = 1 >= 1 = mark#(uTake1(isNatIList(IL))) mark#(zeros()) = 1 >= 1 = active#(zeros()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = 1 >= 1 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(length(L))) = 1 >= 1 = mark#(isNatList(L)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 1 >= 1 = mark#(isNat(N)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = mark#(X2) active#(isNatIList(IL)) = 1 >= 1 = mark#(isNatList(IL)) active#(and(tt(),T)) = 1 >= 1 = mark#(T) active(and(tt(),T)) = 1 >= 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 1 >= 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 1 = mark(s(length(L))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),mark(X2))) mark(tt()) = 1 >= 1 = active(tt()) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(nil()) = 1 >= 1 = active(nil()) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 1 >= 1 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 1 >= 1 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) uTake1(mark(X)) = 1 >= 1 = uTake1(X) uTake1(active(X)) = 1 >= 1 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,mark(X2)) = 1 >= 1 = uLength(X1,X2) uLength(active(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,active(X2)) = 1 >= 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,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#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(zeros()) -> active#(zeros()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [uLength](x0, x1) = 1, [uTake2](x0, x1, x2, x3) = 1, [uTake1](x0) = 0, [take](x0, x1) = 1, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 0, [length](x0) = 1, [s](x0) = 0, [isNat](x0) = 1, [0] = 0, [isNatList](x0) = 1, [isNatIList](x0) = 1, [mark](x0) = 1, [active](x0) = 1, [and](x0, x1) = 1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 1 >= 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 1 >= 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark#(cons(N,take(M,IL))) mark#(take(X1,X2)) = 1 >= 1 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = 1 >= 1 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = 1 >= 1 = mark#(X1) mark#(take(X1,X2)) = 1 >= 1 = mark#(X2) active#(take(0(),IL)) = 1 >= 1 = mark#(uTake1(isNatIList(IL))) mark#(zeros()) = 1 >= 0 = active#(zeros()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = 1 >= 1 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(length(L))) = 1 >= 1 = mark#(isNatList(L)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 1 >= 1 = mark#(isNat(N)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = mark#(X2) active#(isNatIList(IL)) = 1 >= 1 = mark#(isNatList(IL)) active#(and(tt(),T)) = 1 >= 1 = mark#(T) active(and(tt(),T)) = 1 >= 1 = mark(T) active(isNatIList(IL)) = 1 >= 1 = mark(isNatList(IL)) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(s(N))) = 1 >= 1 = mark(isNat(N)) active(isNat(length(L))) = 1 >= 1 = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(N,L))) = 1 >= 1 = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = 1 >= 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(take(0(),IL)) = 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 1 = mark(nil()) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = 1 >= 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = 1 >= 1 = mark(s(length(L))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),mark(X2))) mark(tt()) = 1 >= 1 = active(tt()) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(0()) = 1 >= 1 = active(0()) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(nil()) = 1 >= 1 = active(nil()) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 1 >= 1 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 1 >= 1 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) uTake1(mark(X)) = 0 >= 0 = uTake1(X) uTake1(active(X)) = 0 >= 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,mark(X2)) = 1 >= 1 = uLength(X1,X2) uLength(active(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,active(X2)) = 1 >= 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,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#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) active#(take(0(),IL)) -> mark#(uTake1(isNatIList(IL))) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = 0, [uTake1](x0) = 0, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 0, [length](x0) = 0, [s](x0) = x0, [isNat](x0) = 0, [0] = 1, [isNatList](x0) = 0, [isNatIList](x0) = 0, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 0 >= 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 0 >= 0 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 0 >= 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) = 0 >= 0 = mark#(cons(N,take(M,IL))) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = M + 1 >= 0 = mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) active#(take(0(),IL)) = IL + 2 >= 0 = mark#(uTake1(isNatIList(IL))) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = 0 >= 0 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X >= X = 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)) 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 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 0 >= 0 = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = 0 >= 0 = mark#(isNatList(IL)) active#(and(tt(),T)) = T >= T = mark#(T) active(and(tt(),T)) = T >= 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))) = M + 1 >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 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)) = X1 + X2 >= X1 + X2 = 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()) = 1 >= 1 = active(0()) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 0 >= 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 0 >= 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)) = 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)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) uTake1(mark(X)) = 0 >= 0 = uTake1(X) uTake1(active(X)) = 0 >= 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,mark(X2)) = 0 >= 0 = uLength(X1,X2) uLength(active(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,active(X2)) = 0 >= 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) -> mark#(cons(N,take(M,IL))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = 1, [uTake1](x0) = 1, [take](x0, x1) = x0 + 1, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 1, [length](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, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 0 >= 0 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 0 >= 0 = mark#(uLength(and(isNat(N),isNatList(L)),L)) active#(uTake2(tt(),M,N,IL)) = 1 >= 0 = mark#(cons(N,take(M,IL))) mark#(take(X1,X2)) = X1 + 1 >= X1 + 1 = active#(take(mark(X1),mark(X2))) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatList(take(N,IL))) = 0 >= 0 = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = 0 >= 0 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X >= X = 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)) 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 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 0 >= 0 = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = 0 >= 0 = mark#(isNatList(IL)) active#(and(tt(),T)) = T >= T = mark#(T) active(and(tt(),T)) = T >= 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()) = 1 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 1 >= 1 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 1 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = M + 1 >= 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 1 >= 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)) = X1 + X2 >= X1 + X2 = 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)) = 0 >= 0 = active(length(mark(X))) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + 1 >= X1 + 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 1 >= 1 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 0 >= 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)) = 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)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) take(mark(X1),X2) = X1 + 1 >= X1 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + 1 >= X1 + 1 = take(X1,X2) take(active(X1),X2) = X1 + 1 >= X1 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + 1 >= X1 + 1 = take(X1,X2) uTake1(mark(X)) = 1 >= 1 = uTake1(X) uTake1(active(X)) = 1 >= 1 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,mark(X2)) = 0 >= 0 = uLength(X1,X2) uLength(active(X1),X2) = 0 >= 0 = uLength(X1,X2) uLength(X1,active(X2)) = 0 >= 0 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(take(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [uTake1](x0) = 0, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [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 >= X2 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L >= L = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = L + N >= L = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(take(mark(X1),mark(X2))) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatList(take(N,IL))) = IL + N + 1 >= IL + N = mark#(and(isNat(N),isNatIList(IL))) active#(isNatList(cons(N,L))) = L + N >= L + N = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X >= X = mark#(X) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(isNat(length(L))) = L >= L = mark#(isNatList(L)) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + N >= IL + N = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = N >= N = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = IL >= IL = mark#(isNatList(IL)) active#(and(tt(),T)) = T >= T = mark#(T) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= 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 >= IL + N = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N >= L + N = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + N + 1 >= IL + N = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N >= L = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= L = 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 >= X = 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 >= X1 + X2 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X2 >= 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 >= 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 >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = 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) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = 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 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X2 >= X2 = uLength(X1,X2) uLength(X1,mark(X2)) = X2 >= X2 = uLength(X1,X2) uLength(active(X1),X2) = X2 >= X2 = uLength(X1,X2) uLength(X1,active(X2)) = X2 >= X2 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [uLength](x0, x1) = 1, [uTake2](x0, x1, x2, x3) = 1, [uTake1](x0) = 0, [take](x0, x1) = 0, [nil] = 0, [cons](x0, x1) = 1, [zeros] = 0, [length](x0) = 1, [s](x0) = 0, [isNat](x0) = 1, [0] = 0, [isNatList](x0) = 1, [isNatIList](x0) = 1, [mark](x0) = 0, [active](x0) = 0, [and](x0, x1) = 1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 1 >= 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 1 >= 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(take(X1,X2)) = 1 >= 0 = active#(take(mark(X1),mark(X2))) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatList(cons(N,L))) = 1 >= 1 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(length(L))) = 1 >= 1 = mark#(isNatList(L)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 1 >= 1 = mark#(isNat(N)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = mark#(X2) active#(isNatIList(IL)) = 1 >= 1 = mark#(isNatList(IL)) active#(and(tt(),T)) = 1 >= 1 = mark#(T) active(and(tt(),T)) = 0 >= 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()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 0 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 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)) = 0 >= 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)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 0 >= 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 0 >= 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) uTake1(mark(X)) = 0 >= 0 = uTake1(X) uTake1(active(X)) = 0 >= 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 1 >= 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,mark(X2)) = 1 >= 1 = uLength(X1,X2) uLength(active(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,active(X2)) = 1 >= 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) active#(isNatIList(IL)) -> mark#(isNatList(IL)) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [uTake1](x0) = 0, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [isNat](x0) = x0, [0] = 0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [mark](x0) = x0, [active](x0) = x0, [and](x0, x1) = x0 + x1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = X2 >= X2 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L >= L = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = L + N >= L = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatList(cons(N,L))) = L + N >= L + N = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X >= X = mark#(X) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(isNat(length(L))) = L >= L = mark#(isNatList(L)) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) mark#(isNatIList(X)) = X + 1 >= X + 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = N >= N = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(isNatIList(IL)) = IL + 1 >= IL = mark#(isNatList(IL)) active#(and(tt(),T)) = T >= T = mark#(T) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL + 1 >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= N = mark(isNat(N)) active(isNat(length(L))) = L >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N >= L + N = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL + 1 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N >= L = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L >= L = 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 + 1 >= X + 1 = 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 >= X = 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 >= X1 + X2 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X2 >= 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 + 1 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 1 >= X + 1 = 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 >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = 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) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = 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 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X2 >= X2 = uLength(X1,X2) uLength(X1,mark(X2)) = X2 >= X2 = uLength(X1,X2) uLength(active(X1),X2) = X2 >= X2 = uLength(X1,X2) uLength(X1,active(X2)) = X2 >= X2 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(length(L))) -> mark#(isNatList(L)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [uLength](x0, x1) = x1 + 1, [uTake2](x0, x1, x2, x3) = x1 + x2 + x3, [uTake1](x0) = 0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0 + 1, [s](x0) = x0, [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 + 1 >= X2 + 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = L + N + 1 >= L + 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(isNatList(cons(N,L))) = L + N >= L + N = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = X >= X = mark#(X) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(isNat(length(L))) = L + 1 >= L = mark#(isNatList(L)) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = IL + N >= IL + N = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = N >= N = mark#(isNat(N)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) active#(and(tt(),T)) = T >= T = mark#(T) active(and(tt(),T)) = T >= T = mark(T) active(isNatIList(IL)) = IL >= IL = mark(isNatList(IL)) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(s(N))) = N >= N = mark(isNat(N)) active(isNat(length(L))) = L + 1 >= L = mark(isNatList(L)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(N,IL))) = IL + N >= IL + N = mark(and(isNat(N),isNatIList(IL))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(N,L))) = L + N >= L + N = mark(and(isNat(N),isNatList(L))) active(isNatList(take(N,IL))) = IL + N >= IL + N = mark(and(isNat(N),isNatIList(IL))) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = IL >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = IL + M + N >= IL + M + N = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = IL + M + N >= IL + M + N = mark(cons(N,take(M,IL))) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(uLength(and(isNat(N),isNatList(L)),L)) active(uLength(tt(),L)) = L + 1 >= L + 1 = 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 >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = 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()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = X2 + 1 >= X2 + 1 = 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 >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = 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) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = 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 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = X2 + 1 >= X2 + 1 = uLength(X1,X2) uLength(X1,mark(X2)) = X2 + 1 >= X2 + 1 = uLength(X1,X2) uLength(active(X1),X2) = X2 + 1 >= X2 + 1 = uLength(X1,X2) uLength(X1,active(X2)) = X2 + 1 >= X2 + 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) -> active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> 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) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [uLength](x0, x1) = 1, [uTake2](x0, x1, x2, x3) = 0, [uTake1](x0) = 0, [take](x0, x1) = 0, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 0, [length](x0) = 1, [s](x0) = 0, [isNat](x0) = 1, [0] = 0, [isNatList](x0) = 1, [isNatIList](x0) = 1, [mark](x0) = 0, [active](x0) = 0, [and](x0, x1) = 1, [tt] = 0 orientation: mark#(uLength(X1,X2)) = 1 >= 1 = active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 0 = active#(uTake2(mark(X1),X2,X3,X4)) active#(length(cons(N,L))) = 1 >= 1 = mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatList(cons(N,L))) = 1 >= 1 = mark#(and(isNat(N),isNatList(L))) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) = 1 >= 1 = mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) = 1 >= 1 = mark#(isNat(N)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = mark#(X2) active#(and(tt(),T)) = 1 >= 1 = mark#(T) active(and(tt(),T)) = 0 >= 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()) = 0 >= 0 = mark(cons(0(),zeros())) active(take(0(),IL)) = 0 >= 0 = mark(uTake1(isNatIList(IL))) active(uTake1(tt())) = 0 >= 0 = mark(nil()) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) active(uTake2(tt(),M,N,IL)) = 0 >= 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)) = 0 >= 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)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(nil()) = 0 >= 0 = active(nil()) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(uTake1(X)) = 0 >= 0 = active(uTake1(mark(X))) mark(uTake2(X1,X2,X3,X4)) = 0 >= 0 = active(uTake2(mark(X1),X2,X3,X4)) mark(uLength(X1,X2)) = 0 >= 0 = active(uLength(mark(X1),X2)) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) uTake1(mark(X)) = 0 >= 0 = uTake1(X) uTake1(active(X)) = 0 >= 0 = uTake1(X) uTake2(mark(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,mark(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,mark(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,mark(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(active(X1),X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,active(X2),X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,active(X3),X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) uTake2(X1,X2,X3,active(X4)) = 0 >= 0 = uTake2(X1,X2,X3,X4) uLength(mark(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,mark(X2)) = 1 >= 1 = uLength(X1,X2) uLength(active(X1),X2) = 1 >= 1 = uLength(X1,X2) uLength(X1,active(X2)) = 1 >= 1 = uLength(X1,X2) problem: DPs: mark#(uLength(X1,X2)) -> active#(uLength(mark(X1),X2)) active#(uLength(tt(),L)) -> mark#(s(length(L))) active#(length(cons(N,L))) -> mark#(uLength(and(isNat(N),isNatList(L)),L)) mark#(length(X)) -> active#(length(mark(X))) active#(isNatList(cons(N,L))) -> mark#(and(isNat(N),isNatList(L))) mark#(s(X)) -> mark#(X) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNatIList(cons(N,IL))) -> mark#(and(isNat(N),isNatIList(IL))) mark#(and(X1,X2)) -> active#(and(mark(X1),mark(X2))) active#(isNat(s(N))) -> mark#(isNat(N)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> 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) Open 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) Open 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) Open 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) Open DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,mark(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) Open DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,mark(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) Open 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) Open DPs: uTake2#(mark(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,active(X4)) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,active(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,active(X2),X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(active(X1),X2,X3,X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,X3,mark(X4)) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,X2,mark(X3),X4) -> uTake2#(X1,X2,X3,X4) uTake2#(X1,mark(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) Open DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,mark(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) Open DPs: uLength#(mark(X1),X2) -> uLength#(X1,X2) uLength#(X1,active(X2)) -> uLength#(X1,X2) uLength#(active(X1),X2) -> uLength#(X1,X2) uLength#(X1,mark(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) Open 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) Open 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) Open