YES Problem: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Proof: DP Processor: DPs: a__and#(tt(),T) -> mark#(T) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__isNat#(N) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__uTake1#(a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNat#(N) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNat#(M) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2#(tt(),M,N,IL) -> mark#(N) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) -> mark#(L) a__uLength#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(zeros()) -> a__zeros#() mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) TDG Processor: DPs: a__and#(tt(),T) -> mark#(T) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__isNat#(N) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__uTake1#(a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNat#(N) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNat#(M) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2#(tt(),M,N,IL) -> mark#(N) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) -> mark#(L) a__uLength#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(zeros()) -> a__zeros#() mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) graph: a__uLength#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__uLength#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__isNat#(N) a__uLength#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__isNatList#(L) a__uLength#(tt(),L) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__uLength#(tt(),L) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__uLength#(tt(),L) -> mark#(L) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) a__uLength#(tt(),L) -> mark#(L) -> mark#(uLength(X1,X2)) -> mark#(X1) a__uLength#(tt(),L) -> mark#(L) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) a__uLength#(tt(),L) -> mark#(L) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) a__uLength#(tt(),L) -> mark#(L) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) a__uLength#(tt(),L) -> mark#(L) -> mark#(uTake1(X)) -> mark#(X) a__uLength#(tt(),L) -> mark#(L) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__uLength#(tt(),L) -> mark#(L) -> mark#(take(X1,X2)) -> mark#(X1) a__uLength#(tt(),L) -> mark#(L) -> mark#(take(X1,X2)) -> mark#(X2) a__uLength#(tt(),L) -> mark#(L) -> mark#(zeros()) -> a__zeros#() a__uLength#(tt(),L) -> mark#(L) -> mark#(length(X)) -> a__length#(mark(X)) a__uLength#(tt(),L) -> mark#(L) -> mark#(length(X)) -> mark#(X) a__uLength#(tt(),L) -> mark#(L) -> mark#(isNat(X)) -> a__isNat#(X) a__uLength#(tt(),L) -> mark#(L) -> mark#(isNatList(X)) -> a__isNatList#(X) a__uLength#(tt(),L) -> mark#(L) -> mark#(isNatIList(X)) -> a__isNatIList#(X) a__uLength#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) a__uLength#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> mark#(X1) a__uLength#(tt(),L) -> mark#(L) -> mark#(and(X1,X2)) -> mark#(X2) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) -> a__uLength#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) -> a__uLength#(tt(),L) -> mark#(L) a__length#(cons(N,L)) -> a__isNat#(N) -> a__isNat#(length(L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__isNat#(N) -> a__isNat#(s(N)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) -> a__and#(tt(),T) -> mark#(T) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(uLength(X1,X2)) -> mark#(X1) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(uTake1(X)) -> mark#(X) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X1) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X2) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(zeros()) -> a__zeros#() a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(length(X)) -> a__length#(mark(X)) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(length(X)) -> mark#(X) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(isNat(X)) -> a__isNat#(X) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(isNatList(X)) -> a__isNatList#(X) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(isNatIList(X)) -> a__isNatIList#(X) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__uTake2#(tt(),M,N,IL) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X2) a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) -> a__uTake2#(tt(),M,N,IL) -> mark#(N) a__take#(s(M),cons(N,IL)) -> a__isNat#(M) -> a__isNat#(length(L)) -> a__isNatList#(L) a__take#(s(M),cons(N,IL)) -> a__isNat#(M) -> a__isNat#(s(N)) -> a__isNat#(N) a__take#(s(M),cons(N,IL)) -> a__isNat#(N) -> a__isNat#(length(L)) -> a__isNatList#(L) a__take#(s(M),cons(N,IL)) -> a__isNat#(N) -> a__isNat#(s(N)) -> a__isNat#(N) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(IL) -> a__isNatList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) -> a__and#(tt(),T) -> mark#(T) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) -> a__and#(tt(),T) -> mark#(T) a__take#(0(),IL) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(0(),IL) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__take#(0(),IL) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__isNatIList#(IL) -> a__isNatIList#(IL) -> a__isNatList#(IL) a__isNat#(length(L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNat#(length(L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNat#(length(L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__isNat#(length(L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNat#(s(N)) -> a__isNat#(N) -> a__isNat#(length(L)) -> a__isNatList#(L) a__isNat#(s(N)) -> a__isNat#(N) -> a__isNat#(s(N)) -> a__isNat#(N) a__isNatList#(take(N,IL)) -> a__isNat#(N) -> a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(take(N,IL)) -> a__isNat#(N) -> a__isNat#(s(N)) -> a__isNat#(N) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) -> a__and#(tt(),T) -> mark#(T) a__isNatList#(cons(N,L)) -> a__isNat#(N) -> a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) -> a__isNat#(s(N)) -> a__isNat#(N) a__isNatList#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__isNat#(N) a__isNatList#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNatList#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) -> a__and#(tt(),T) -> mark#(T) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) -> a__isNat#(length(L)) -> a__isNatList#(L) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) -> a__isNat#(s(N)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) -> a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) -> a__and#(tt(),T) -> mark#(T) a__isNatIList#(IL) -> a__isNatList#(IL) -> a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatIList#(IL) -> a__isNatList#(IL) -> a__isNatList#(take(N,IL)) -> a__isNat#(N) a__isNatIList#(IL) -> a__isNatList#(IL) -> a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) -> a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__isNatIList#(IL) -> a__isNatList#(IL) -> a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNatIList#(IL) -> a__isNatList#(IL) -> a__isNatList#(cons(N,L)) -> a__isNatList#(L) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) -> a__uLength#(tt(),L) -> a__length#(mark(L)) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) -> a__uLength#(tt(),L) -> mark#(L) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) -> a__uTake2#(tt(),M,N,IL) -> mark#(N) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(uTake1(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(uTake1(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(uTake1(X)) -> mark#(X) -> mark#(uTake1(X)) -> mark#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(uTake1(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(uTake1(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(uTake1(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(uTake1(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(uTake1(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(uTake1(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(uTake1(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X2) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(length(L)) -> a__isNatList#(L) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(N)) -> a__isNat#(N) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(take(N,IL)) -> a__isNat#(N) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(N,L)) -> a__isNat#(N) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(N,L)) -> a__isNatList#(L) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(cons(N,IL)) -> a__isNat#(N) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(IL) -> a__isNatList#(IL) mark#(and(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X2) -> mark#(zeros()) -> a__zeros#() mark#(and(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> a__isNat#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(and(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) -> a__and#(tt(),T) -> mark#(T) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> a__isNat#(M) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> a__isNat#(N) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(0(),IL) -> a__uTake1#(a__isNatIList(IL)) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(0(),IL) -> a__isNatIList#(IL) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zeros()) -> a__zeros#() mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> a__length#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> a__isNat#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(take(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(uTake1(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X2) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__isNat#(N) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__isNatList#(L) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(length(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(uTake1(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(length(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(length(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(uLength(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(s(X)) -> mark#(X) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(uTake1(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X2) a__and#(tt(),T) -> mark#(T) -> mark#(cons(X1,X2)) -> mark#(X1) a__and#(tt(),T) -> mark#(T) -> mark#(s(X)) -> mark#(X) a__and#(tt(),T) -> mark#(T) -> mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) a__and#(tt(),T) -> mark#(T) -> mark#(uLength(X1,X2)) -> mark#(X1) a__and#(tt(),T) -> mark#(T) -> mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) a__and#(tt(),T) -> mark#(T) -> mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) a__and#(tt(),T) -> mark#(T) -> mark#(uTake1(X)) -> a__uTake1#(mark(X)) a__and#(tt(),T) -> mark#(T) -> mark#(uTake1(X)) -> mark#(X) a__and#(tt(),T) -> mark#(T) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__and#(tt(),T) -> mark#(T) -> mark#(take(X1,X2)) -> mark#(X1) a__and#(tt(),T) -> mark#(T) -> mark#(take(X1,X2)) -> mark#(X2) a__and#(tt(),T) -> mark#(T) -> mark#(zeros()) -> a__zeros#() a__and#(tt(),T) -> mark#(T) -> mark#(length(X)) -> a__length#(mark(X)) a__and#(tt(),T) -> mark#(T) -> mark#(length(X)) -> mark#(X) a__and#(tt(),T) -> mark#(T) -> mark#(isNat(X)) -> a__isNat#(X) a__and#(tt(),T) -> mark#(T) -> mark#(isNatList(X)) -> a__isNatList#(X) a__and#(tt(),T) -> mark#(T) -> mark#(isNatIList(X)) -> a__isNatIList#(X) a__and#(tt(),T) -> mark#(T) -> mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) a__and#(tt(),T) -> mark#(T) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),T) -> mark#(T) -> mark#(and(X1,X2)) -> mark#(X2) SCC Processor: #sccs: 1 #rules: 45 #arcs: 355/2304 DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) -> a__isNat#(N) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) -> mark#(L) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNat#(N) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNat#(M) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2#(tt(),M,N,IL) -> mark#(N) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = x0 + x1 + 0, [a__isNatList](x0) = x0 + 0, [a__uTake1](x0) = x0 + 0, [length](x0) = x0 + 0, [a__and](x0, x1) = x0 + x1 + 0, [a__isNatIList](x0) = x0 + 7, [a__uTake2](x0, x1, x2, x3) = x0 + 7x1 + 7x2 + 7x3 + 7, [isNatList](x0) = x0, [a__uLength#](x0, x1) = x1 + 0, [a__and#](x0, x1) = x0 + x1, [a__take#](x0, x1) = 7x0 + 7x1 + 0, [and](x0, x1) = x0 + x1, [a__length](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__uTake2#](x0, x1, x2, x3) = x0 + 7x2 + x3 + 1, [isNat](x0) = x0, [a__isNatIList#](x0) = x0 + 7, [0] = 0, [cons](x0, x1) = x0 + x1, [tt] = 0, [s](x0) = x0 + 0, [isNatIList](x0) = x0 + 7, [uLength](x0, x1) = x0 + x1, [a__isNat#](x0) = x0 + 0, [a__take](x0, x1) = 7x0 + 7x1 + 7, [take](x0, x1) = 7x0 + 7x1 + 7, [uTake1](x0) = x0 + 0, [uTake2](x0, x1, x2, x3) = x0 + 7x1 + 7x2 + 7x3 + 7, [nil] = 0, [zeros] = 5, [a__zeros] = 5, [a__isNat](x0) = x0, [a__length#](x0) = x0 + 0, [a__isNatList#](x0) = x0 + 0, [mark](x0) = x0 + 0 orientation: a__uLength#(tt(),L) = L + 0 >= L + 0 = a__length#(mark(L)) a__length#(cons(N,L)) = L + N + 0 >= L + 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 0 >= L + 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 0 >= N + 0 = a__isNat#(N) a__isNat#(s(N)) = N + 0 >= N + 0 = a__isNat#(N) a__isNat#(length(L)) = L + 0 >= L + 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 0 >= L + N + 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 0 >= T + 0 = mark#(T) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X + 7 >= X + 7 = a__isNatIList#(X) a__isNatIList#(IL) = IL + 7 >= IL + 0 = a__isNatList#(IL) a__isNatList#(take(N,IL)) = 7IL + 7N + 7 >= IL + 7 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 7 >= IL + 7 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 7 >= N + 0 = a__isNat#(N) a__isNatIList#(cons(N,IL)) = IL + N + 7 >= IL + N + 7 = a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) = 7IL + 7N + 7 >= N + 0 = a__isNat#(N) a__isNatList#(take(N,IL)) = 7IL + 7N + 7 >= IL + N + 7 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X + 0 >= X + 0 = a__isNatList#(X) mark#(isNat(X)) = X + 0 >= X + 0 = a__isNat#(X) mark#(length(X)) = X + 0 >= X + 0 = mark#(X) mark#(length(X)) = X + 0 >= X + 0 = a__length#(mark(X)) a__length#(cons(N,L)) = L + N + 0 >= N + 0 = a__isNat#(N) a__length#(cons(N,L)) = L + N + 0 >= L + N + 0 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = L + N + 0 >= L + 0 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = L + 0 >= L + 0 = mark#(L) mark#(take(X1,X2)) = 7X1 + 7X2 + 7 >= X2 + 0 = mark#(X2) mark#(take(X1,X2)) = 7X1 + 7X2 + 7 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = 7X1 + 7X2 + 7 >= 7X1 + 7X2 + 7 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = 7IL + 7 >= IL + 7 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = 7IL + 7M + 7N + 7 >= IL + 7 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = 7IL + 7M + 7N + 7 >= N + 0 = a__isNat#(N) a__take#(s(M),cons(N,IL)) = 7IL + 7M + 7N + 7 >= IL + N + 7 = a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) = 7IL + 7M + 7N + 7 >= M + 0 = a__isNat#(M) a__take#(s(M),cons(N,IL)) = 7IL + 7M + 7N + 7 >= IL + M + N + 7 = a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) a__take#(s(M),cons(N,IL)) = 7IL + 7M + 7N + 7 >= IL + M + 7N + 7 = a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2#(tt(),M,N,IL) = IL + 7N + 1 >= N + 0 = mark#(N) mark#(uTake1(X)) = X + 0 >= X + 0 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + 7X2 + 7X3 + 7X4 + 7 >= X1 + 0 = mark#(X1) mark#(uTake2(X1,X2,X3,X4)) = X1 + 7X2 + 7X3 + 7X4 + 7 >= X1 + 7X3 + X4 + 1 = a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = a__uLength#(mark(X1),X2) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) a__and(tt(),T) = T + 0 >= T + 0 = mark(T) a__isNatIList(IL) = IL + 7 >= IL + 0 = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = N + 0 >= N = a__isNat(N) a__isNat(length(L)) = L + 0 >= L + 0 = a__isNatList(L) a__isNatIList(zeros()) = 7 >= 0 = tt() a__isNatIList(cons(N,IL)) = IL + N + 7 >= IL + N + 7 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = L + N + 0 >= L + N + 0 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 7IL + 7N + 7 >= IL + N + 7 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 5 >= 5 = cons(0(),zeros()) a__take(0(),IL) = 7IL + 7 >= IL + 7 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 7IL + 7M + 7N + 7 >= 7IL + 7M + 7N + 7 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 7IL + 7M + 7N + 7 >= 7IL + 7M + N + 7 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N + 0 >= L + N + 0 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L + 0 >= L + 0 = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = X + 7 >= X + 7 = a__isNatIList(X) mark(isNatList(X)) = X + 0 >= X + 0 = a__isNatList(X) mark(isNat(X)) = X + 0 >= X = a__isNat(X) mark(length(X)) = X + 0 >= X + 0 = a__length(mark(X)) mark(zeros()) = 5 >= 5 = a__zeros() mark(take(X1,X2)) = 7X1 + 7X2 + 7 >= 7X1 + 7X2 + 7 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = X + 0 >= X + 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X1 + 7X2 + 7X3 + 7X4 + 7 >= X1 + 7X2 + 7X3 + 7X4 + 7 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 = and(X1,X2) a__isNatIList(X) = X + 7 >= X + 7 = isNatIList(X) a__isNatList(X) = X + 0 >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 0 >= X + 0 = length(X) a__zeros() = 5 >= 5 = zeros() a__take(X1,X2) = 7X1 + 7X2 + 7 >= 7X1 + 7X2 + 7 = take(X1,X2) a__uTake1(X) = X + 0 >= X + 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X1 + 7X2 + 7X3 + 7X4 + 7 >= X1 + 7X2 + 7X3 + 7X4 + 7 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 + X2 + 0 >= X1 + X2 = uLength(X1,X2) problem: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) -> mark#(L) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) -> mark#(L) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) a__take#(s(M),cons(N,IL)) -> a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uTake2(X1,X2,X3,X4)) -> a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 37 #arcs: 328/1521 DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNat#(N) a__length#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) -> mark#(L) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(uLength(X1,X2)) -> mark#(X1) mark#(uLength(X1,X2)) -> a__uLength#(mark(X1),X2) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 7x0 + 7x1 + 7, [a__isNatList](x0) = 0, [a__uTake1](x0) = x0 + 0, [length](x0) = 7x0 + 7, [a__and](x0, x1) = x0 + x1 + 0, [a__isNatIList](x0) = 0, [a__uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 0, [isNatList](x0) = 0, [a__uLength#](x0, x1) = 6x0 + 6x1 + 0, [a__and#](x0, x1) = x0 + x1 + 0, [a__take#](x0, x1) = x0 + 0, [and](x0, x1) = x0 + x1 + 0, [a__length](x0) = 7x0 + 7, [mark#](x0) = x0, [isNat](x0) = 0, [a__isNatIList#](x0) = 0, [0] = 0, [cons](x0, x1) = x0 + x1, [tt] = 0, [s](x0) = x0, [isNatIList](x0) = 0, [uLength](x0, x1) = 7x0 + 7x1 + 7, [a__isNat#](x0) = 0, [a__take](x0, x1) = x0 + x1 + 0, [take](x0, x1) = x0 + x1 + 0, [uTake1](x0) = x0, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [zeros] = 0, [a__zeros] = 0, [a__isNat](x0) = 0, [a__length#](x0) = 6x0 + 6, [a__isNatList#](x0) = 0, [mark](x0) = x0 + 0 orientation: a__uLength#(tt(),L) = 6L + 6 >= 6L + 6 = a__length#(mark(L)) a__length#(cons(N,L)) = 6L + 6N + 6 >= 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 0 >= 0 = a__isNat#(N) a__isNat#(s(N)) = 0 >= 0 = a__isNat#(N) a__isNat#(length(L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 0 >= T = mark#(T) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) a__isNatIList#(IL) = 0 >= 0 = a__isNatList#(IL) a__isNatList#(take(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__isNat#(N) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(length(X)) = 7X + 7 >= X = mark#(X) mark#(length(X)) = 7X + 7 >= 6X + 6 = a__length#(mark(X)) a__length#(cons(N,L)) = 6L + 6N + 6 >= 0 = a__isNat#(N) a__length#(cons(N,L)) = 6L + 6N + 6 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = 6L + 6N + 6 >= 6L + 6 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = 6L + 6 >= L = mark#(L) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = 0 >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = M + 0 >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = M + 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) = M + 0 >= 0 = a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) = X >= X = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 = mark#(X1) mark#(uLength(X1,X2)) = 7X1 + 7X2 + 7 >= X1 = mark#(X1) mark#(uLength(X1,X2)) = 7X1 + 7X2 + 7 >= 6X1 + 6X2 + 6 = a__uLength#(mark(X1),X2) mark#(s(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) a__and(tt(),T) = T + 0 >= T + 0 = mark(T) a__isNatIList(IL) = 0 >= 0 = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = 0 >= 0 = a__isNat(N) a__isNat(length(L)) = 0 >= 0 = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = 0 >= 0 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__take(0(),IL) = IL + 0 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + M + N + 0 >= IL + M + N + 0 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = IL + M + N + 0 >= IL + M + N + 0 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 7L + 7N + 7 >= 7L + 7 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 7L + 7 >= 7L + 7 = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = 7X + 7 >= 7X + 7 = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = X + 0 >= X + 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 0 >= X1 + X2 + X3 + X4 + 0 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 7X1 + 7X2 + 7 >= 7X1 + 7X2 + 7 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 7X + 7 >= 7X + 7 = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) a__uTake1(X) = X + 0 >= X = uTake1(X) a__uTake2(X1,X2,X3,X4) = X1 + X2 + X3 + X4 + 0 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 7X1 + 7X2 + 7 >= 7X1 + 7X2 + 7 = uLength(X1,X2) problem: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(isNat(X)) -> a__isNat#(X) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 2 #rules: 29 #arcs: 228/841 DPs: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNat#(length(L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNat(X)) -> a__isNat#(X) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 1x0 + 1x1, [a__isNatList](x0) = x0, [a__uTake1](x0) = x0, [length](x0) = 1x0 + 1, [a__and](x0, x1) = x0 + x1, [a__isNatIList](x0) = x0, [a__uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [isNatList](x0) = x0, [a__and#](x0, x1) = x0 + x1 + 0, [and](x0, x1) = x0 + x1, [a__take#](x0, x1) = x0 + x1 + 0, [a__length](x0) = 1x0 + 1, [mark#](x0) = x0 + 0, [isNat](x0) = x0, [a__isNatIList#](x0) = x0 + 0, [0] = 1, [cons](x0, x1) = x0 + x1, [tt] = 1, [s](x0) = x0 + 2, [isNatIList](x0) = x0, [uLength](x0, x1) = 1x0 + 1x1, [a__isNat#](x0) = x0 + 0, [a__take](x0, x1) = x0 + x1 + 0, [take](x0, x1) = x0 + x1 + 0, [uTake1](x0) = x0, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 1, [zeros] = 1, [a__zeros] = 1, [a__isNat](x0) = x0, [a__isNatList#](x0) = x0 + 0, [mark](x0) = x0 orientation: a__take#(s(M),cons(N,IL)) = IL + M + N + 2 >= IL + 0 = a__isNatIList#(IL) a__isNatIList#(IL) = IL + 0 >= IL + 0 = a__isNatList#(IL) a__isNatList#(cons(N,L)) = L + N + 0 >= L + 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 0 >= N + 0 = a__isNat#(N) a__isNat#(s(N)) = N + 2 >= N + 0 = a__isNat#(N) a__isNat#(length(L)) = 1L + 1 >= L + 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 0 >= L + N + 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 1 >= T + 0 = mark#(T) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X + 0 >= X + 0 = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = IL + N + 0 >= IL + 0 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 0 >= N + 0 = a__isNat#(N) a__isNatIList#(cons(N,IL)) = IL + N + 0 >= IL + N + 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X + 0 >= X + 0 = a__isNatList#(X) a__isNatList#(take(N,IL)) = IL + N + 0 >= IL + 0 = a__isNatIList#(IL) a__isNatList#(take(N,IL)) = IL + N + 0 >= IL + N + 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNat(X)) = X + 0 >= X + 0 = a__isNat#(X) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = IL + 1 >= IL + 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = IL + M + N + 2 >= IL + N + 0 = a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) = IL + M + N + 2 >= IL + M + N + 0 = a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) = X + 0 >= X + 0 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 2 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) a__and(tt(),T) = T + 1 >= T = mark(T) a__isNatIList(IL) = IL >= IL = a__isNatList(IL) a__isNat(0()) = 1 >= 1 = tt() a__isNat(s(N)) = N + 2 >= N = a__isNat(N) a__isNat(length(L)) = 1L + 1 >= L = a__isNatList(L) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(N,IL)) = IL + N >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(N,L)) = L + N >= L + N = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = IL + N + 0 >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__take(0(),IL) = IL + 1 >= IL = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 1 >= 1 = nil() a__take(s(M),cons(N,IL)) = IL + M + N + 2 >= IL + M + N = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = IL + M + N + 1 >= IL + M + N + 0 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 1L + 1N + 1 >= 1L + 1N = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 1L + 2 >= 1L + 2 = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = X >= X = a__isNatIList(X) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(isNat(X)) = X >= X = a__isNat(X) mark(length(X)) = 1X + 1 >= 1X + 1 = a__length(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = X >= X = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 1X1 + 1X2 >= 1X1 + 1X2 = a__uLength(mark(X1),X2) mark(tt()) = 1 >= 1 = tt() mark(0()) = 1 >= 1 = 0() mark(s(X)) = X + 2 >= X + 2 = s(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(nil()) = 1 >= 1 = nil() a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIList(X) = X >= X = isNatIList(X) a__isNatList(X) = X >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = 1X + 1 >= 1X + 1 = length(X) a__zeros() = 1 >= 1 = zeros() a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) a__uTake1(X) = X >= X = uTake1(X) a__uTake2(X1,X2,X3,X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 1X1 + 1X2 >= 1X1 + 1X2 = uLength(X1,X2) problem: DPs: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNat(X)) -> a__isNat#(X) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__isNat#(N) a__isNat#(s(N)) -> a__isNat#(N) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__isNat#(N) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNat(X)) -> a__isNat#(X) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 2 #rules: 23 #arcs: 135/676 DPs: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 0, [a__isNatList](x0) = 0, [a__uTake1](x0) = x0, [length](x0) = 0, [a__and](x0, x1) = x0 + 4x1, [a__isNatIList](x0) = 0, [a__uTake2](x0, x1, x2, x3) = 4x0 + 4x2, [isNatList](x0) = 0, [a__and#](x0, x1) = x1, [and](x0, x1) = x0 + 4x1, [a__take#](x0, x1) = x0, [a__length](x0) = 0, [mark#](x0) = x0, [isNat](x0) = 0, [a__isNatIList#](x0) = 0, [0] = 2, [cons](x0, x1) = 2x0, [tt] = 0, [s](x0) = 4x0, [isNatIList](x0) = 0, [uLength](x0, x1) = 0, [a__take](x0, x1) = 2x0 + 2x1, [take](x0, x1) = 2x0 + 2x1, [uTake1](x0) = x0, [uTake2](x0, x1, x2, x3) = 4x0 + 2x2, [nil] = 0, [zeros] = 3, [a__zeros] = 4, [a__isNat](x0) = 0, [a__isNatList#](x0) = 0, [mark](x0) = 2x0 orientation: a__take#(s(M),cons(N,IL)) = 4M >= 0 = a__isNatIList#(IL) a__isNatIList#(IL) = 0 >= 0 = a__isNatList#(IL) a__isNatList#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T >= T = mark#(T) mark#(and(X1,X2)) = X1 + 4X2 >= X2 = mark#(X2) mark#(and(X1,X2)) = X1 + 4X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + 4X2 >= 2X2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) a__isNatList#(take(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatList#(take(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) = 2X1 + 2X2 >= 2X1 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = 2 >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = 4M >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) = 4M >= 0 = a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) = X >= X = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = 4X1 + 2X3 >= X1 = mark#(X1) mark#(s(X)) = 4X >= X = mark#(X) mark#(cons(X1,X2)) = 2X1 >= X1 = mark#(X1) a__and(tt(),T) = 4T >= 2T = mark(T) a__isNatIList(IL) = 0 >= 0 = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = 0 >= 0 = a__isNat(N) a__isNat(length(L)) = 0 >= 0 = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = 0 >= 0 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 4 >= 4 = cons(0(),zeros()) a__take(0(),IL) = 2IL + 4 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 8M + 4N >= 4N = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 4N >= 4N = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 0 >= 0 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 0 >= 0 = s(a__length(mark(L))) mark(and(X1,X2)) = 2X1 + 8X2 >= 2X1 + 8X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(zeros()) = 6 >= 4 = a__zeros() mark(take(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 2X >= 2X = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 8X1 + 4X3 >= 8X1 + 4X3 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 0 >= 0 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 4 >= 2 = 0() mark(s(X)) = 8X >= 8X = s(mark(X)) mark(cons(X1,X2)) = 4X1 >= 4X1 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + 4X2 >= X1 + 4X2 = and(X1,X2) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 0 >= 0 = length(X) a__zeros() = 4 >= 3 = zeros() a__take(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = take(X1,X2) a__uTake1(X) = X >= X = uTake1(X) a__uTake2(X1,X2,X3,X4) = 4X1 + 4X3 >= 4X1 + 2X3 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 0 >= 0 = uLength(X1,X2) problem: DPs: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) -> mark#(X) mark#(uTake2(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 0, [a__isNatList](x0) = 0, [a__uTake1](x0) = x0, [length](x0) = 0, [a__and](x0, x1) = 2x0 + 2x1, [a__isNatIList](x0) = 0, [a__uTake2](x0, x1, x2, x3) = x0 + 2x2 + 5, [isNatList](x0) = 0, [a__and#](x0, x1) = 2x0 + 2x1, [and](x0, x1) = 2x0 + 2x1, [a__take#](x0, x1) = 4, [a__length](x0) = 0, [mark#](x0) = 2x0, [isNat](x0) = 0, [a__isNatIList#](x0) = 0, [0] = 0, [cons](x0, x1) = x0 + 1, [tt] = 0, [s](x0) = 3x0, [isNatIList](x0) = 0, [uLength](x0, x1) = 0, [a__take](x0, x1) = 3x1 + 4, [take](x0, x1) = 3x1 + 2, [uTake1](x0) = x0, [uTake2](x0, x1, x2, x3) = x0 + x2 + 4, [nil] = 0, [zeros] = 1, [a__zeros] = 1, [a__isNat](x0) = 0, [a__isNatList#](x0) = 0, [mark](x0) = 2x0 orientation: a__take#(s(M),cons(N,IL)) = 4 >= 0 = a__isNatIList#(IL) a__isNatIList#(IL) = 0 >= 0 = a__isNatList#(IL) a__isNatList#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = 2T >= 2T = mark#(T) mark#(and(X1,X2)) = 4X1 + 4X2 >= 2X2 = mark#(X2) mark#(and(X1,X2)) = 4X1 + 4X2 >= 2X1 = mark#(X1) mark#(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) a__isNatList#(take(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatList#(take(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) = 6X2 + 4 >= 4 = a__take#(mark(X1),mark(X2)) a__take#(s(M),cons(N,IL)) = 4 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) = 4 >= 0 = a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) mark#(uTake1(X)) = 2X >= 2X = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = 2X1 + 2X3 + 8 >= 2X1 = mark#(X1) mark#(s(X)) = 6X >= 2X = mark#(X) mark#(cons(X1,X2)) = 2X1 + 2 >= 2X1 = mark#(X1) a__and(tt(),T) = 2T >= 2T = mark(T) a__isNatIList(IL) = 0 >= 0 = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = 0 >= 0 = a__isNat(N) a__isNat(length(L)) = 0 >= 0 = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = 0 >= 0 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__take(0(),IL) = 3IL + 4 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 3N + 7 >= 2N + 5 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 2N + 5 >= 2N + 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 0 >= 0 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 0 >= 0 = s(a__length(mark(L))) mark(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(zeros()) = 2 >= 1 = a__zeros() mark(take(X1,X2)) = 6X2 + 4 >= 6X2 + 4 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 2X >= 2X = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 2X1 + 2X3 + 8 >= 2X1 + 2X3 + 5 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 0 >= 0 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = 6X >= 6X = s(mark(X)) mark(cons(X1,X2)) = 2X1 + 2 >= 2X1 + 1 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = and(X1,X2) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 0 >= 0 = length(X) a__zeros() = 1 >= 1 = zeros() a__take(X1,X2) = 3X2 + 4 >= 3X2 + 2 = take(X1,X2) a__uTake1(X) = X >= X = uTake1(X) a__uTake2(X1,X2,X3,X4) = X1 + 2X3 + 5 >= X1 + X3 + 4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 0 >= 0 = uLength(X1,X2) problem: DPs: a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(uTake1(X)) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(uTake1(X)) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 15 #arcs: 107/256 DPs: a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(uTake1(X)) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 0, [a__isNatList](x0) = 0, [a__uTake1](x0) = 2x0 + 1, [length](x0) = 0, [a__and](x0, x1) = 2x0 + x1, [a__isNatIList](x0) = 0, [a__uTake2](x0, x1, x2, x3) = 2x0 + x1, [isNatList](x0) = 0, [a__and#](x0, x1) = 1/2x0 + 1/2x1, [and](x0, x1) = 2x0 + x1, [a__length](x0) = 0, [mark#](x0) = 1/2x0, [isNat](x0) = 0, [a__isNatIList#](x0) = 0, [0] = 3, [cons](x0, x1) = 0, [tt] = 0, [s](x0) = x0, [isNatIList](x0) = 0, [uLength](x0, x1) = 0, [a__take](x0, x1) = x0 + 1, [take](x0, x1) = x0 + 1, [uTake1](x0) = 2x0 + 1, [uTake2](x0, x1, x2, x3) = 2x0 + x1, [nil] = 0, [zeros] = 0, [a__zeros] = 0, [a__isNat](x0) = 0, [a__isNatList#](x0) = 0, [mark](x0) = x0 orientation: a__isNatList#(take(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatIList#(IL) = 0 >= 0 = a__isNatList#(IL) a__isNatList#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = 1/2T >= 1/2T = mark#(T) mark#(and(X1,X2)) = X1 + 1/2X2 >= 1/2X2 = mark#(X2) mark#(and(X1,X2)) = X1 + 1/2X2 >= 1/2X1 = mark#(X1) mark#(and(X1,X2)) = X1 + 1/2X2 >= 1/2X1 + 1/2X2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) a__isNatList#(take(N,IL)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(uTake1(X)) = X + 1/2 >= 1/2X = mark#(X) mark#(s(X)) = 1/2X >= 1/2X = mark#(X) a__and(tt(),T) = T >= T = mark(T) a__isNatIList(IL) = 0 >= 0 = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = 0 >= 0 = a__isNat(N) a__isNat(length(L)) = 0 >= 0 = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = 0 >= 0 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 0 >= 0 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__take(0(),IL) = 4 >= 1 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 1 >= 0 = nil() a__take(s(M),cons(N,IL)) = M + 1 >= M = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = M >= 0 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 0 >= 0 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 0 >= 0 = s(a__length(mark(L))) mark(and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X1 + 1 >= X1 + 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 2X + 1 >= 2X + 1 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 2X1 + X2 >= 2X1 + X2 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 0 >= 0 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 3 >= 3 = 0() mark(s(X)) = X >= X = s(mark(X)) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 0 >= 0 = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X1 + 1 >= X1 + 1 = take(X1,X2) a__uTake1(X) = 2X + 1 >= 2X + 1 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 2X1 + X2 >= 2X1 + X2 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 0 >= 0 = uLength(X1,X2) problem: DPs: a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__isNatList#(take(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = x0 + x1 + 1, [a__isNatList](x0) = x0 + 0, [a__uTake1](x0) = 0, [length](x0) = x0 + 0, [a__and](x0, x1) = x0 + x1 + 1, [a__isNatIList](x0) = x0 + 0, [a__uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + 1x3 + 4, [isNatList](x0) = x0 + 0, [a__and#](x0, x1) = x0 + x1 + 0, [and](x0, x1) = x0 + x1 + 1, [a__length](x0) = x0 + 1, [mark#](x0) = x0, [isNat](x0) = x0, [a__isNatIList#](x0) = x0, [0] = 1, [cons](x0, x1) = x0 + x1 + 1, [tt] = 0, [s](x0) = x0, [isNatIList](x0) = x0 + 0, [uLength](x0, x1) = x0 + x1, [a__take](x0, x1) = x0 + 1x1 + 4, [take](x0, x1) = x0 + 1x1 + 4, [uTake1](x0) = 0, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + 1x3 + 4, [nil] = 0, [zeros] = 0, [a__zeros] = 1, [a__isNat](x0) = x0, [a__isNatList#](x0) = x0, [mark](x0) = x0 + 1 orientation: a__isNatList#(take(N,IL)) = 1IL + N + 4 >= IL = a__isNatIList#(IL) a__isNatIList#(IL) = IL >= IL = a__isNatList#(IL) a__isNatList#(cons(N,L)) = L + N + 1 >= L = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 1 >= L + N + 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 0 >= T = mark#(T) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X + 0 >= X = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = IL + N + 1 >= IL = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 1 >= IL + N + 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X + 0 >= X = a__isNatList#(X) a__isNatList#(take(N,IL)) = 1IL + N + 4 >= IL + N + 0 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(s(X)) = X >= X = mark#(X) a__and(tt(),T) = T + 1 >= T + 1 = mark(T) a__isNatIList(IL) = IL + 0 >= IL + 0 = a__isNatList(IL) a__isNat(0()) = 1 >= 0 = tt() a__isNat(s(N)) = N >= N = a__isNat(N) a__isNat(length(L)) = L + 0 >= L + 0 = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = IL + N + 1 >= IL + N + 1 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = L + N + 1 >= L + N + 1 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 1IL + N + 4 >= IL + N + 1 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__take(0(),IL) = 1IL + 4 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 1IL + M + 1N + 4 >= 1IL + M + N + 4 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 1IL + M + N + 4 >= 1IL + M + N + 4 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N + 1 >= L + N + 1 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = X + 1 >= X + 0 = a__isNatIList(X) mark(isNatList(X)) = X + 1 >= X + 0 = a__isNatList(X) mark(isNat(X)) = X + 1 >= X = a__isNat(X) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() mark(take(X1,X2)) = X1 + 1X2 + 4 >= X1 + 1X2 + 4 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 1 >= 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + 1X4 + 4 >= X1 + X2 + X3 + 1X4 + 4 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__uLength(mark(X1),X2) mark(tt()) = 1 >= 0 = tt() mark(0()) = 1 >= 1 = 0() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(mark(X1),X2) mark(nil()) = 1 >= 0 = nil() a__and(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) a__isNatIList(X) = X + 0 >= X + 0 = isNatIList(X) a__isNatList(X) = X + 0 >= X + 0 = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 1 >= X + 0 = length(X) a__zeros() = 1 >= 0 = zeros() a__take(X1,X2) = X1 + 1X2 + 4 >= X1 + 1X2 + 4 = take(X1,X2) a__uTake1(X) = 0 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X1 + X2 + X3 + 1X4 + 4 >= X1 + X2 + X3 + 1X4 + 4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 + X2 + 1 >= X1 + X2 = uLength(X1,X2) problem: DPs: a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__isNatList#(take(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 2x1 + 1, [a__isNatList](x0) = x0, [a__uTake1](x0) = x0 + 0, [length](x0) = 2x0, [a__and](x0, x1) = x0 + x1, [a__isNatIList](x0) = x0, [a__uTake2](x0, x1, x2, x3) = x0 + 2x1 + 2x2 + 2x3, [isNatList](x0) = x0, [a__and#](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [a__length](x0) = 2x0, [mark#](x0) = x0, [isNat](x0) = x0, [a__isNatIList#](x0) = x0, [0] = 0, [cons](x0, x1) = x0 + x1 + 0, [tt] = 0, [s](x0) = x0 + 0, [isNatIList](x0) = x0, [uLength](x0, x1) = 2x1 + 1, [a__take](x0, x1) = 2x0 + 2x1 + 0, [take](x0, x1) = 2x0 + 2x1 + 0, [uTake1](x0) = x0 + 0, [uTake2](x0, x1, x2, x3) = x0 + 2x1 + 2x2 + 2x3, [nil] = 0, [zeros] = 0, [a__zeros] = 0, [a__isNat](x0) = x0, [a__isNatList#](x0) = x0, [mark](x0) = x0 orientation: a__isNatIList#(IL) = IL >= IL = a__isNatList#(IL) a__isNatList#(cons(N,L)) = L + N + 0 >= L = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 0 >= L + N = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 0 >= T = mark#(T) mark#(and(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X >= X = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = IL + N + 0 >= IL = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 0 >= IL + N = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X >= X = a__isNatList#(X) a__isNatList#(take(N,IL)) = 2IL + 2N + 0 >= IL + N = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(s(X)) = X + 0 >= X = mark#(X) a__and(tt(),T) = T + 0 >= T = mark(T) a__isNatIList(IL) = IL >= IL = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = N + 0 >= N = a__isNat(N) a__isNat(length(L)) = 2L >= L = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = IL + N + 0 >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = L + N + 0 >= L + N = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 2IL + 2N + 0 >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__take(0(),IL) = 2IL + 2 >= IL + 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 2IL + 2M + 2N + 2 >= 2IL + 2M + 2N = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 2IL + 2M + 2N + 0 >= 2IL + 2M + N + 0 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 2L + 2N + 2 >= 2L + 1 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 2L + 1 >= 2L + 0 = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = X >= X = a__isNatIList(X) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(isNat(X)) = X >= X = a__isNat(X) mark(length(X)) = 2X >= 2X = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = X + 0 >= X + 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X1 + 2X2 + 2X3 + 2X4 >= X1 + 2X2 + 2X3 + 2X4 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 2X2 + 1 >= 2X2 + 1 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIList(X) = X >= X = isNatIList(X) a__isNatList(X) = X >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = 2X >= 2X = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = take(X1,X2) a__uTake1(X) = X + 0 >= X + 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X1 + 2X2 + 2X3 + 2X4 >= X1 + 2X2 + 2X3 + 2X4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 2X2 + 1 >= 2X2 + 1 = uLength(X1,X2) problem: DPs: a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__isNatIList#(IL) -> a__isNatList#(IL) a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = x0 + x1 + 0, [a__isNatList](x0) = x0, [a__uTake1](x0) = 5, [length](x0) = x0, [a__and](x0, x1) = x0 + x1 + 0, [a__isNatIList](x0) = 1x0 + 4, [a__uTake2](x0, x1, x2, x3) = 4x1 + 4x2 + 4x3 + 4, [isNatList](x0) = x0, [a__and#](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [a__length](x0) = x0 + 0, [mark#](x0) = x0 + 0, [isNat](x0) = x0, [a__isNatIList#](x0) = 1x0 + 4, [0] = 2, [cons](x0, x1) = x0 + x1 + 0, [tt] = 0, [s](x0) = x0 + 0, [isNatIList](x0) = 1x0 + 4, [uLength](x0, x1) = x0 + x1, [a__take](x0, x1) = 4x0 + 4x1 + 4, [take](x0, x1) = 4x0 + 4x1 + 4, [uTake1](x0) = 5, [uTake2](x0, x1, x2, x3) = 4x1 + 4x2 + 4x3 + 4, [nil] = 4, [zeros] = 2, [a__zeros] = 2, [a__isNat](x0) = x0, [a__isNatList#](x0) = x0, [mark](x0) = x0 + 0 orientation: a__isNatIList#(IL) = 1IL + 4 >= IL = a__isNatList#(IL) a__isNatList#(cons(N,L)) = L + N + 0 >= L = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 0 >= L + N = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 0 >= T + 0 = mark#(T) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 1X + 4 >= 1X + 4 = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = 1IL + 1N + 4 >= 1IL + 4 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 1IL + 1N + 4 >= 1IL + N + 4 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X + 0 >= X = a__isNatList#(X) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__and(tt(),T) = T + 0 >= T + 0 = mark(T) a__isNatIList(IL) = 1IL + 4 >= IL = a__isNatList(IL) a__isNat(0()) = 2 >= 0 = tt() a__isNat(s(N)) = N + 0 >= N = a__isNat(N) a__isNat(length(L)) = L >= L = a__isNatList(L) a__isNatIList(zeros()) = 4 >= 0 = tt() a__isNatIList(cons(N,IL)) = 1IL + 1N + 4 >= 1IL + N + 4 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 4 >= 0 = tt() a__isNatList(cons(N,L)) = L + N + 0 >= L + N + 0 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 4IL + 4N + 4 >= 1IL + N + 4 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 2 >= 2 = cons(0(),zeros()) a__take(0(),IL) = 4IL + 6 >= 5 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 5 >= 4 = nil() a__take(s(M),cons(N,IL)) = 4IL + 4M + 4N + 4 >= 4IL + 4M + 4N + 4 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 4IL + 4M + 4N + 4 >= 4IL + 4M + N + 4 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N + 0 >= L + N + 0 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L + 0 >= L + 0 = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 1X + 4 >= 1X + 4 = a__isNatIList(X) mark(isNatList(X)) = X + 0 >= X = a__isNatList(X) mark(isNat(X)) = X + 0 >= X = a__isNat(X) mark(length(X)) = X + 0 >= X + 0 = a__length(mark(X)) mark(zeros()) = 2 >= 2 = a__zeros() mark(take(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 5 >= 5 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 4X2 + 4X3 + 4X4 + 4 >= 4X2 + 4X3 + 4X4 + 4 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 2 >= 2 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(nil()) = 4 >= 4 = nil() a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 = and(X1,X2) a__isNatIList(X) = 1X + 4 >= 1X + 4 = isNatIList(X) a__isNatList(X) = X >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 0 >= X = length(X) a__zeros() = 2 >= 2 = zeros() a__take(X1,X2) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = take(X1,X2) a__uTake1(X) = 5 >= 5 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 4X2 + 4X3 + 4X4 + 4 >= 4X2 + 4X3 + 4X4 + 4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 + X2 + 0 >= X1 + X2 = uLength(X1,X2) problem: DPs: a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 4x1, [a__isNatList](x0) = 3x0, [a__uTake1](x0) = x0, [length](x0) = 4x0, [a__and](x0, x1) = 4x0 + x1, [a__isNatIList](x0) = 3x0, [a__uTake2](x0, x1, x2, x3) = 4x1 + 4x2 + 3x3, [isNatList](x0) = 3x0, [a__and#](x0, x1) = x1, [and](x0, x1) = 4x0 + x1, [a__length](x0) = 4x0, [mark#](x0) = x0, [isNat](x0) = 1x0, [a__isNatIList#](x0) = 3x0, [0] = 0, [cons](x0, x1) = 2x0 + x1, [tt] = 0, [s](x0) = x0, [isNatIList](x0) = 3x0, [uLength](x0, x1) = 4x1, [a__take](x0, x1) = 4x0 + 3x1, [take](x0, x1) = 4x0 + 3x1, [uTake1](x0) = x0, [uTake2](x0, x1, x2, x3) = 4x1 + 4x2 + 3x3, [nil] = 0, [zeros] = 2, [a__zeros] = 2, [a__isNat](x0) = 1x0, [a__isNatList#](x0) = 3x0, [mark](x0) = x0 orientation: a__isNatList#(cons(N,L)) = 3L + 5N >= 3L = a__isNatList#(L) a__isNatList#(cons(N,L)) = 3L + 5N >= 3L = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T >= T = mark#(T) mark#(and(X1,X2)) = 4X1 + X2 >= X2 = mark#(X2) mark#(and(X1,X2)) = 4X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = 4X1 + X2 >= X2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 3X >= 3X = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = 3IL + 5N >= 3IL = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 3IL + 5N >= 3IL = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = 3X >= 3X = a__isNatList#(X) mark#(s(X)) = X >= X = mark#(X) a__and(tt(),T) = T + 4 >= T = mark(T) a__isNatIList(IL) = 3IL >= 3IL = a__isNatList(IL) a__isNat(0()) = 1 >= 0 = tt() a__isNat(s(N)) = 1N >= 1N = a__isNat(N) a__isNat(length(L)) = 5L >= 3L = a__isNatList(L) a__isNatIList(zeros()) = 5 >= 0 = tt() a__isNatIList(cons(N,IL)) = 3IL + 5N >= 3IL + 5N = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 3 >= 0 = tt() a__isNatList(cons(N,L)) = 3L + 5N >= 3L + 5N = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 6IL + 7N >= 3IL + 5N = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 2 >= 2 = cons(0(),zeros()) a__take(0(),IL) = 3IL + 4 >= 3IL = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 3IL + 4M + 5N >= 3IL + 4M + 4N = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 3IL + 4M + 4N >= 3IL + 4M + 2N = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 4L + 6N >= 4L = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 4L >= 4L = s(a__length(mark(L))) mark(and(X1,X2)) = 4X1 + X2 >= 4X1 + X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 3X >= 3X = a__isNatIList(X) mark(isNatList(X)) = 3X >= 3X = a__isNatList(X) mark(isNat(X)) = 1X >= 1X = a__isNat(X) mark(length(X)) = 4X >= 4X = a__length(mark(X)) mark(zeros()) = 2 >= 2 = a__zeros() mark(take(X1,X2)) = 4X1 + 3X2 >= 4X1 + 3X2 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = X >= X = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 4X2 + 4X3 + 3X4 >= 4X2 + 4X3 + 3X4 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 4X2 >= 4X2 = a__uLength(mark(X1),X2) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = 4X1 + X2 >= 4X1 + X2 = and(X1,X2) a__isNatIList(X) = 3X >= 3X = isNatIList(X) a__isNatList(X) = 3X >= 3X = isNatList(X) a__isNat(X) = 1X >= 1X = isNat(X) a__length(X) = 4X >= 4X = length(X) a__zeros() = 2 >= 2 = zeros() a__take(X1,X2) = 4X1 + 3X2 >= 4X1 + 3X2 = take(X1,X2) a__uTake1(X) = X >= X = uTake1(X) a__uTake2(X1,X2,X3,X4) = 4X2 + 4X3 + 3X4 >= 4X2 + 4X3 + 3X4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 4X2 >= 4X2 = uLength(X1,X2) problem: DPs: a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__isNatList#(cons(N,L)) -> a__isNatList#(L) a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) a__isNatIList#(cons(N,IL)) -> a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) -> a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = x1 + 5/2, [a__isNatList](x0) = x0 + 1/2, [a__uTake1](x0) = 2, [length](x0) = x0 + 2, [a__and](x0, x1) = x1 + 1/2, [a__isNatIList](x0) = x0 + 2, [a__uTake2](x0, x1, x2, x3) = x3 + 5/2, [isNatList](x0) = x0, [a__and#](x0, x1) = x1, [and](x0, x1) = x1 + 1/2, [a__length](x0) = x0 + 2, [mark#](x0) = x0, [isNat](x0) = x0, [a__isNatIList#](x0) = x0 + 2, [0] = 0, [cons](x0, x1) = x1 + 1/2, [tt] = 0, [s](x0) = x0, [isNatIList](x0) = x0 + 2, [uLength](x0, x1) = x1 + 2, [a__take](x0, x1) = x1 + 2, [take](x0, x1) = x1 + 2, [uTake1](x0) = 3/2, [uTake2](x0, x1, x2, x3) = x3 + 5/2, [nil] = 0, [zeros] = 2, [a__zeros] = 5/2, [a__isNat](x0) = x0, [a__isNatList#](x0) = x0, [mark](x0) = x0 + 1/2 orientation: a__isNatList#(cons(N,L)) = L + 1/2 >= L = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + 1/2 >= L + 1/2 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T >= T = mark#(T) mark#(and(X1,X2)) = X2 + 1/2 >= X2 = mark#(X2) mark#(and(X1,X2)) = X2 + 1/2 >= X2 + 1/2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X + 2 >= X + 2 = a__isNatIList#(X) a__isNatIList#(cons(N,IL)) = IL + 5/2 >= IL + 2 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + 5/2 >= IL + 2 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X >= X = a__isNatList#(X) mark#(s(X)) = X >= X = mark#(X) a__and(tt(),T) = T + 1/2 >= T + 1/2 = mark(T) a__isNatIList(IL) = IL + 2 >= IL + 1/2 = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = N >= N = a__isNat(N) a__isNat(length(L)) = L + 2 >= L + 1/2 = a__isNatList(L) a__isNatIList(zeros()) = 4 >= 0 = tt() a__isNatIList(cons(N,IL)) = IL + 5/2 >= IL + 5/2 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 1/2 >= 0 = tt() a__isNatList(cons(N,L)) = L + 1 >= L + 1 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = IL + 5/2 >= IL + 5/2 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 5/2 >= 5/2 = cons(0(),zeros()) a__take(0(),IL) = IL + 2 >= 2 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 2 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + 5/2 >= IL + 5/2 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = IL + 5/2 >= IL + 5/2 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + 5/2 >= L + 5/2 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L + 5/2 >= L + 5/2 = s(a__length(mark(L))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = X + 5/2 >= X + 2 = a__isNatIList(X) mark(isNatList(X)) = X + 1/2 >= X + 1/2 = a__isNatList(X) mark(isNat(X)) = X + 1/2 >= X = a__isNat(X) mark(length(X)) = X + 5/2 >= X + 5/2 = a__length(mark(X)) mark(zeros()) = 5/2 >= 5/2 = a__zeros() mark(take(X1,X2)) = X2 + 5/2 >= X2 + 5/2 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 2 >= 2 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X4 + 3 >= X4 + 5/2 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X2 + 5/2 >= X2 + 5/2 = a__uLength(mark(X1),X2) mark(tt()) = 1/2 >= 0 = tt() mark(0()) = 1/2 >= 0 = 0() mark(s(X)) = X + 1/2 >= X + 1/2 = s(mark(X)) mark(cons(X1,X2)) = X2 + 1 >= X2 + 1/2 = cons(mark(X1),X2) mark(nil()) = 1/2 >= 0 = nil() a__and(X1,X2) = X2 + 1/2 >= X2 + 1/2 = and(X1,X2) a__isNatIList(X) = X + 2 >= X + 2 = isNatIList(X) a__isNatList(X) = X + 1/2 >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 2 >= X + 2 = length(X) a__zeros() = 5/2 >= 2 = zeros() a__take(X1,X2) = X2 + 2 >= X2 + 2 = take(X1,X2) a__uTake1(X) = 2 >= 3/2 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X4 + 5/2 >= X4 + 5/2 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X2 + 5/2 >= X2 + 2 = uLength(X1,X2) problem: DPs: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 5 #arcs: 60/36 DPs: a__isNatList#(cons(N,L)) -> a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = x0 + 1x1, [a__isNatList](x0) = x0, [a__uTake1](x0) = 4, [length](x0) = x0 + 0, [a__and](x0, x1) = 1x1, [a__isNatIList](x0) = x0, [a__uTake2](x0, x1, x2, x3) = 5x3 + 4, [isNatList](x0) = x0, [a__and#](x0, x1) = x1, [and](x0, x1) = 1x1, [a__length](x0) = x0 + 0, [mark#](x0) = x0, [isNat](x0) = 2x0 + 5, [0] = 2, [cons](x0, x1) = 1x1, [tt] = 0, [s](x0) = x0 + 0, [isNatIList](x0) = x0, [uLength](x0, x1) = x0 + 1x1, [a__take](x0, x1) = 4x1 + 4, [take](x0, x1) = 4x1 + 3, [uTake1](x0) = 4, [uTake2](x0, x1, x2, x3) = 4x3 + 4, [nil] = 0, [zeros] = 0, [a__zeros] = 1, [a__isNat](x0) = 2x0 + 5, [a__isNatList#](x0) = x0, [mark](x0) = 1x0 orientation: a__isNatList#(cons(N,L)) = 1L >= L = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T >= T = mark#(T) mark#(and(X1,X2)) = 1X2 >= 1X2 = a__and#(mark(X1),mark(X2)) mark#(isNatList(X)) = X >= X = a__isNatList#(X) mark#(s(X)) = X + 0 >= X = mark#(X) a__and(tt(),T) = 1T >= 1T = mark(T) a__isNatIList(IL) = IL >= IL = a__isNatList(IL) a__isNat(0()) = 5 >= 0 = tt() a__isNat(s(N)) = 2N + 5 >= 2N + 5 = a__isNat(N) a__isNat(length(L)) = 2L + 5 >= L = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = 1IL >= 1IL = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = 1L >= 1L = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 4IL + 3 >= 1IL = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__take(0(),IL) = 4IL + 4 >= 4 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 4 >= 0 = nil() a__take(s(M),cons(N,IL)) = 5IL + 4 >= 5IL + 4 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 5IL + 4 >= 5IL + 4 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 1L + 0 >= 1L = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 1L + 0 >= 1L + 0 = s(a__length(mark(L))) mark(and(X1,X2)) = 2X2 >= 2X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 1X >= X = a__isNatIList(X) mark(isNatList(X)) = 1X >= X = a__isNatList(X) mark(isNat(X)) = 3X + 6 >= 2X + 5 = a__isNat(X) mark(length(X)) = 1X + 1 >= 1X + 0 = a__length(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() mark(take(X1,X2)) = 5X2 + 4 >= 5X2 + 4 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 5 >= 4 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 5X4 + 5 >= 5X4 + 4 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 1X1 + 2X2 >= 1X1 + 1X2 = a__uLength(mark(X1),X2) mark(tt()) = 1 >= 0 = tt() mark(0()) = 3 >= 2 = 0() mark(s(X)) = 1X + 1 >= 1X + 0 = s(mark(X)) mark(cons(X1,X2)) = 2X2 >= 1X2 = cons(mark(X1),X2) mark(nil()) = 1 >= 0 = nil() a__and(X1,X2) = 1X2 >= 1X2 = and(X1,X2) a__isNatIList(X) = X >= X = isNatIList(X) a__isNatList(X) = X >= X = isNatList(X) a__isNat(X) = 2X + 5 >= 2X + 5 = isNat(X) a__length(X) = X + 0 >= X + 0 = length(X) a__zeros() = 1 >= 0 = zeros() a__take(X1,X2) = 4X2 + 4 >= 4X2 + 3 = take(X1,X2) a__uTake1(X) = 4 >= 4 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 5X4 + 4 >= 4X4 + 4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 + 1X2 >= X1 + 1X2 = uLength(X1,X2) problem: DPs: a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and (a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__and#(tt(),T) -> mark#(T) mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and (a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 3 #arcs: 9/16 DPs: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) a__and#(tt(),T) -> mark#(T) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and (a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Matrix Interpretation Processor: dim=1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and (a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 2x1 + 5, [a__isNatList](x0) = 3x0, [a__uTake1](x0) = 1, [length](x0) = x0 + 2, [a__and](x0, x1) = 2x1 + 2, [a__isNatIList](x0) = 4x0, [a__uTake2](x0, x1, x2, x3) = 6x3 + 7, [isNatList](x0) = 3x0, [a__and#](x0, x1) = 4x1 + 4, [and](x0, x1) = 2x1 + 2, [a__length](x0) = x0 + 4, [mark#](x0) = 4x0, [isNat](x0) = 2x0, [0] = 0, [cons](x0, x1) = 2x1 + 1, [tt] = 0, [s](x0) = x0, [isNatIList](x0) = 2x0, [uLength](x0, x1) = x1 + 2, [a__take](x0, x1) = 3x1 + 4, [take](x0, x1) = 3x1 + 3, [uTake1](x0) = 0, [uTake2](x0, x1, x2, x3) = 4x3 + 3, [nil] = 0, [zeros] = 0, [a__zeros] = 1, [a__isNat](x0) = 4x0, [mark](x0) = 2x0 + 1 orientation: mark#(and(X1,X2)) = 8X2 + 8 >= 8X2 + 8 = a__and#(mark(X1),mark(X2)) a__and#(tt(),T) = 4T + 4 >= 4T = mark#(T) mark#(s(X)) = 4X >= 4X = mark#(X) a__and(tt(),T) = 2T + 2 >= 2T + 1 = mark(T) a__isNatIList(IL) = 4IL >= 3IL = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = 4N >= 4N = a__isNat(N) a__isNat(length(L)) = 4L + 8 >= 3L = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = 8IL + 4 >= 8IL + 2 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(N,L)) = 6L + 3 >= 6L + 2 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 9IL + 9 >= 8IL + 2 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__take(0(),IL) = 3IL + 4 >= 1 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 1 >= 0 = nil() a__take(s(M),cons(N,IL)) = 6IL + 7 >= 6IL + 7 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 6IL + 7 >= 6IL + 7 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 2L + 5 >= 2L + 5 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 2L + 5 >= 2L + 5 = s(a__length(mark(L))) mark(and(X1,X2)) = 4X2 + 5 >= 4X2 + 4 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 4X + 1 >= 4X = a__isNatIList(X) mark(isNatList(X)) = 6X + 1 >= 3X = a__isNatList(X) mark(isNat(X)) = 4X + 1 >= 4X = a__isNat(X) mark(length(X)) = 2X + 5 >= 2X + 5 = a__length(mark(X)) mark(zeros()) = 1 >= 1 = a__zeros() mark(take(X1,X2)) = 6X2 + 7 >= 6X2 + 7 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 1 >= 1 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 8X4 + 7 >= 6X4 + 7 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 2X2 + 5 >= 2X2 + 5 = a__uLength(mark(X1),X2) mark(tt()) = 1 >= 0 = tt() mark(0()) = 1 >= 0 = 0() mark(s(X)) = 2X + 1 >= 2X + 1 = s(mark(X)) mark(cons(X1,X2)) = 4X2 + 3 >= 2X2 + 1 = cons(mark(X1),X2) mark(nil()) = 1 >= 0 = nil() a__and(X1,X2) = 2X2 + 2 >= 2X2 + 2 = and(X1,X2) a__isNatIList(X) = 4X >= 2X = isNatIList(X) a__isNatList(X) = 3X >= 3X = isNatList(X) a__isNat(X) = 4X >= 2X = isNat(X) a__length(X) = X + 4 >= X + 2 = length(X) a__zeros() = 1 >= 0 = zeros() a__take(X1,X2) = 3X2 + 4 >= 3X2 + 3 = take(X1,X2) a__uTake1(X) = 1 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 6X4 + 7 >= 4X4 + 3 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 2X2 + 5 >= X2 + 2 = uLength(X1,X2) problem: DPs: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and (a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: mark#(and(X1,X2)) -> a__and#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and (a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 1 #rules: 1 #arcs: 5/4 DPs: mark#(s(X)) -> mark#(X) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and (a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Size-Change Termination Processor: DPs: TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and ( a__isNat(N),a__isNatIList(IL))), M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) The DP: mark#(s(X)) -> mark#(X) has the edges: 0 > 0 Qed DPs: a__isNat#(s(N)) -> a__isNat#(N) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Size-Change Termination Processor: DPs: TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) The DP: a__isNat#(s(N)) -> a__isNat#(N) has the edges: 0 > 0 Qed DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) interpretation: [a__uLength](x0, x1) = 1x0 + 3x1 + 6, [a__isNatList](x0) = 2x0 + 0, [a__uTake1](x0) = 4, [length](x0) = 2x0 + 6, [a__and](x0, x1) = x1 + 4, [a__isNatIList](x0) = 3x0 + 6, [a__uTake2](x0, x1, x2, x3) = 2x1 + 1x2 + 2x3 + 6, [isNatList](x0) = 2x0, [a__uLength#](x0, x1) = x0 + 2x1 + 0, [and](x0, x1) = x1, [a__length](x0) = 2x0 + 6, [isNat](x0) = x0 + 6, [0] = 4, [cons](x0, x1) = x0 + 1x1 + 2, [tt] = 6, [s](x0) = 1x0 + 6, [isNatIList](x0) = 3x0 + 6, [uLength](x0, x1) = 1x0 + 3x1 + 6, [a__take](x0, x1) = 1x0 + 1x1 + 5, [take](x0, x1) = 1x0 + 1x1 + 5, [uTake1](x0) = 4, [uTake2](x0, x1, x2, x3) = 2x1 + 1x2 + 2x3 + 6, [nil] = 4, [zeros] = 1, [a__zeros] = 4, [a__isNat](x0) = x0 + 6, [a__length#](x0) = 1x0 + 4, [mark](x0) = x0 + 4 orientation: a__uLength#(tt(),L) = 2L + 6 >= 1L + 5 = a__length#(mark(L)) a__length#(cons(N,L)) = 2L + 1N + 4 >= 2L + 4 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__and(tt(),T) = T + 4 >= T + 4 = mark(T) a__isNatIList(IL) = 3IL + 6 >= 2IL + 0 = a__isNatList(IL) a__isNat(0()) = 6 >= 6 = tt() a__isNat(s(N)) = 1N + 6 >= N + 6 = a__isNat(N) a__isNat(length(L)) = 2L + 6 >= 2L + 0 = a__isNatList(L) a__isNatIList(zeros()) = 6 >= 6 = tt() a__isNatIList(cons(N,IL)) = 4IL + 3N + 6 >= 3IL + 6 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 6 >= 6 = tt() a__isNatList(cons(N,L)) = 3L + 2N + 4 >= 2L + 4 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 3IL + 3N + 7 >= 3IL + 6 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 4 >= 4 = cons(0(),zeros()) a__take(0(),IL) = 1IL + 5 >= 4 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 4 >= 4 = nil() a__take(s(M),cons(N,IL)) = 2IL + 2M + 1N + 7 >= 2IL + 2M + 1N + 6 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 2IL + 2M + 1N + 6 >= 2IL + 2M + N + 6 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 3L + 2N + 6 >= 3L + 6 = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = 3L + 7 >= 3L + 7 = s(a__length(mark(L))) mark(and(X1,X2)) = X2 + 4 >= X2 + 4 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 3X + 6 >= 3X + 6 = a__isNatIList(X) mark(isNatList(X)) = 2X + 4 >= 2X + 0 = a__isNatList(X) mark(isNat(X)) = X + 6 >= X + 6 = a__isNat(X) mark(length(X)) = 2X + 6 >= 2X + 6 = a__length(mark(X)) mark(zeros()) = 4 >= 4 = a__zeros() mark(take(X1,X2)) = 1X1 + 1X2 + 5 >= 1X1 + 1X2 + 5 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 4 >= 4 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 2X2 + 1X3 + 2X4 + 6 >= 2X2 + 1X3 + 2X4 + 6 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 1X1 + 3X2 + 6 >= 1X1 + 3X2 + 6 = a__uLength(mark(X1),X2) mark(tt()) = 6 >= 6 = tt() mark(0()) = 4 >= 4 = 0() mark(s(X)) = 1X + 6 >= 1X + 6 = s(mark(X)) mark(cons(X1,X2)) = X1 + 1X2 + 4 >= X1 + 1X2 + 4 = cons(mark(X1),X2) mark(nil()) = 4 >= 4 = nil() a__and(X1,X2) = X2 + 4 >= X2 = and(X1,X2) a__isNatIList(X) = 3X + 6 >= 3X + 6 = isNatIList(X) a__isNatList(X) = 2X + 0 >= 2X = isNatList(X) a__isNat(X) = X + 6 >= X + 6 = isNat(X) a__length(X) = 2X + 6 >= 2X + 6 = length(X) a__zeros() = 4 >= 1 = zeros() a__take(X1,X2) = 1X1 + 1X2 + 5 >= 1X1 + 1X2 + 5 = take(X1,X2) a__uTake1(X) = 4 >= 4 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 2X2 + 1X3 + 2X4 + 6 >= 2X2 + 1X3 + 2X4 + 6 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 1X1 + 3X2 + 6 >= 1X1 + 3X2 + 6 = uLength(X1,X2) problem: DPs: a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) Restore Modifier: DPs: a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) TRS: a__and(tt(),T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0()) -> tt() a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) -> tt() a__isNatList(cons(N,L)) -> a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) -> a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() -> cons(0(),zeros()) a__take(0(),IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) -> nil() a__take(s(M),cons(N,IL)) -> a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) -> cons(mark(N),take(M,IL)) a__length(cons(N,L)) -> a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) -> s(a__length(mark(L))) mark(and(X1,X2)) -> a__and(mark(X1),mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros()) -> a__zeros() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) -> a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) -> a__uLength(mark(X1),X2) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(nil()) -> nil() a__and(X1,X2) -> and(X1,X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros() -> zeros() a__take(X1,X2) -> take(X1,X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1,X2,X3,X4) -> uTake2(X1,X2,X3,X4) a__uLength(X1,X2) -> uLength(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1