MAYBE 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) Matrix Interpretation Processor: dimension: 1 interpretation: [a__uLength#](x0, x1) = x1, [a__length#](x0) = x0, [a__uTake2#](x0, x1, x2, x3) = x2 + x3 + 1, [a__take#](x0, x1) = x1 + 1, [a__isNat#](x0) = 0, [a__isNatList#](x0) = 0, [a__isNatIList#](x0) = 0, [mark#](x0) = x0, [a__and#](x0, x1) = x1, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [uTake1](x0) = x0, [isNat](x0) = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [and](x0, x1) = x0 + x1, [a__uLength](x0, x1) = x0 + x1, [a__length](x0) = x0, [a__uTake2](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [a__uTake1](x0) = x0, [a__take](x0, x1) = x0 + x1 + 1, [a__zeros] = 0, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [a__isNat](x0) = 0, [0] = 0, [a__isNatList](x0) = 0, [a__isNatIList](x0) = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0 orientation: a__uLength#(tt(),L) = L >= L = a__length#(mark(L)) a__length#(cons(N,L)) = L + N >= 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 >= 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 >= X2 = 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__isNat#(N) 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)) = X >= X = mark#(X) mark#(length(X)) = X >= X = a__length#(mark(X)) a__length#(cons(N,L)) = L + N >= 0 = a__isNat#(N) a__length#(cons(N,L)) = L + N >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = L + N >= L = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = L >= L = mark#(L) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = IL + 1 >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = IL + N + 1 >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = IL + N + 1 >= 0 = a__isNat#(N) a__take#(s(M),cons(N,IL)) = IL + N + 1 >= 0 = a__and#(a__isNat(N),a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) = IL + N + 1 >= 0 = a__isNat#(M) a__take#(s(M),cons(N,IL)) = IL + N + 1 >= 0 = a__and#(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))) a__take#(s(M),cons(N,IL)) = IL + N + 1 >= IL + N + 1 = 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 + N + 1 >= N = mark#(N) mark#(uTake1(X)) = X >= X = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 = mark#(X1) mark#(uTake2(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X3 + X4 + 1 = a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 >= X2 = 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 >= 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) = IL + 1 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = 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 + 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N >= L = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L >= L = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + 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)) = X >= X = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = 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 + 1 >= X1 + X2 + X3 + X4 + 1 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X1 + X2 >= X1 + X2 = 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)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 >= X1 + 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) = X >= X = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__uTake1(X) = X >= X = uTake1(X) a__uTake2(X1,X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 + X2 >= 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__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)) -> 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#(uTake1(X)) -> mark#(X) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [a__uLength#](x0, x1) = 1, [a__length#](x0) = 1, [a__uTake2#](x0, x1, x2, x3) = 0, [a__take#](x0, x1) = 0, [a__isNat#](x0) = 1, [a__isNatList#](x0) = 1, [a__isNatIList#](x0) = 1, [mark#](x0) = 1, [a__and#](x0, x1) = 1, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = 1, [uTake1](x0) = 1, [isNat](x0) = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [and](x0, x1) = x1, [a__uLength](x0, x1) = 0, [a__length](x0) = x0, [a__uTake2](x0, x1, x2, x3) = 1, [a__uTake1](x0) = 1, [a__take](x0, x1) = 1, [a__zeros] = 0, [take](x0, x1) = 1, [nil] = 1, [cons](x0, x1) = x1, [zeros] = 0, [length](x0) = x0, [s](x0) = 0, [a__isNat](x0) = 0, [0] = 0, [a__isNatList](x0) = 0, [a__isNatIList](x0) = 0, [mark](x0) = x0, [a__and](x0, x1) = x1, [tt] = 0 orientation: a__uLength#(tt(),L) = 1 >= 1 = a__length#(mark(L)) a__length#(cons(N,L)) = 1 >= 1 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 1 >= 1 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 1 >= 1 = a__isNat#(N) a__isNat#(s(N)) = 1 >= 1 = a__isNat#(N) a__isNat#(length(L)) = 1 >= 1 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 1 >= 1 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = 1 >= 1 = mark#(T) mark#(and(X1,X2)) = 1 >= 1 = mark#(X2) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 1 >= 1 = a__isNatIList#(X) a__isNatIList#(IL) = 1 >= 1 = a__isNatList#(IL) a__isNatList#(take(N,IL)) = 1 >= 1 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 1 >= 1 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 1 >= 1 = a__isNat#(N) a__isNatIList#(cons(N,IL)) = 1 >= 1 = a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) = 1 >= 1 = a__isNat#(N) a__isNatList#(take(N,IL)) = 1 >= 1 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = 1 >= 1 = a__isNatList#(X) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(length(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = a__length#(mark(X)) a__length#(cons(N,L)) = 1 >= 1 = a__isNat#(N) a__length#(cons(N,L)) = 1 >= 1 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = 1 >= 1 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = 1 >= 1 = mark#(L) mark#(take(X1,X2)) = 1 >= 0 = a__take#(mark(X1),mark(X2)) a__take#(s(M),cons(N,IL)) = 0 >= 0 = a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) mark#(uTake1(X)) = 1 >= 1 = mark#(X) mark#(uTake2(X1,X2,X3,X4)) = 1 >= 0 = a__uTake2#(mark(X1),X2,X3,X4) mark#(uLength(X1,X2)) = 1 >= 1 = mark#(X1) mark#(uLength(X1,X2)) = 1 >= 1 = a__uLength#(mark(X1),X2) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) 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) = 1 >= 1 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 1 >= 1 = nil() a__take(s(M),cons(N,IL)) = 1 >= 1 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 1 >= 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L >= 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)) = X2 >= 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)) = X >= X = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = 1 >= 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 1 >= 1 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 1 >= 1 = 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)) = 0 >= 0 = s(mark(X)) mark(cons(X1,X2)) = X2 >= X2 = cons(mark(X1),X2) mark(nil()) = 1 >= 1 = nil() a__and(X1,X2) = X2 >= 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) = X >= X = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = 1 >= 1 = take(X1,X2) a__uTake1(X) = 1 >= 1 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 0 >= 0 = 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__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) 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#(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) Matrix Interpretation Processor: dimension: 1 interpretation: [a__uLength#](x0, x1) = 0, [a__length#](x0) = 0, [a__uTake2#](x0, x1, x2, x3) = 0, [a__take#](x0, x1) = 1, [a__isNat#](x0) = 0, [a__isNatList#](x0) = 0, [a__isNatIList#](x0) = 0, [mark#](x0) = 0, [a__and#](x0, x1) = 0, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = 1, [uTake1](x0) = 1, [isNat](x0) = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [and](x0, x1) = x1, [a__uLength](x0, x1) = 0, [a__length](x0) = x0, [a__uTake2](x0, x1, x2, x3) = 1, [a__uTake1](x0) = 1, [a__take](x0, x1) = 1, [a__zeros] = 0, [take](x0, x1) = 1, [nil] = 1, [cons](x0, x1) = x1, [zeros] = 0, [length](x0) = x0, [s](x0) = 0, [a__isNat](x0) = 0, [0] = 0, [a__isNatList](x0) = 0, [a__isNatIList](x0) = 0, [mark](x0) = x0, [a__and](x0, x1) = x1, [tt] = 0 orientation: a__uLength#(tt(),L) = 0 >= 0 = a__length#(mark(L)) a__length#(cons(N,L)) = 0 >= 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) = 0 >= 0 = mark#(T) mark#(and(X1,X2)) = 0 >= 0 = mark#(X2) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 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__isNat#(N) 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)) = 0 >= 0 = mark#(X) mark#(length(X)) = 0 >= 0 = a__length#(mark(X)) a__length#(cons(N,L)) = 0 >= 0 = a__isNat#(N) a__length#(cons(N,L)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = 0 >= 0 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = 0 >= 0 = mark#(L) a__take#(s(M),cons(N,IL)) = 1 >= 0 = a__uTake2#(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) mark#(uTake1(X)) = 0 >= 0 = mark#(X) mark#(uLength(X1,X2)) = 0 >= 0 = mark#(X1) mark#(uLength(X1,X2)) = 0 >= 0 = a__uLength#(mark(X1),X2) mark#(s(X)) = 0 >= 0 = mark#(X) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) 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) = 1 >= 1 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 1 >= 1 = nil() a__take(s(M),cons(N,IL)) = 1 >= 1 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = 1 >= 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L >= 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)) = X2 >= 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)) = X >= X = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = 1 >= 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 1 >= 1 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 1 >= 1 = 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)) = 0 >= 0 = s(mark(X)) mark(cons(X1,X2)) = X2 >= X2 = cons(mark(X1),X2) mark(nil()) = 1 >= 1 = nil() a__and(X1,X2) = X2 >= 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) = X >= X = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = 1 >= 1 = take(X1,X2) a__uTake1(X) = 1 >= 1 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 1 >= 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 0 >= 0 = 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__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#(uTake1(X)) -> mark#(X) 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) Matrix Interpretation Processor: dimension: 1 interpretation: [a__uLength#](x0, x1) = x1, [a__length#](x0) = x0, [a__isNat#](x0) = 0, [a__isNatList#](x0) = 0, [a__isNatIList#](x0) = 0, [mark#](x0) = x0, [a__and#](x0, x1) = x1, [uLength](x0, x1) = x0 + x1, [uTake2](x0, x1, x2, x3) = x2 + x3 + 1, [uTake1](x0) = x0 + 1, [isNat](x0) = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [and](x0, x1) = x0 + x1, [a__uLength](x0, x1) = x0 + x1, [a__length](x0) = x0, [a__uTake2](x0, x1, x2, x3) = x2 + x3 + 1, [a__uTake1](x0) = x0 + 1, [a__take](x0, x1) = x1 + 1, [a__zeros] = 0, [take](x0, x1) = x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [a__isNat](x0) = 0, [0] = 0, [a__isNatList](x0) = 0, [a__isNatIList](x0) = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0 orientation: a__uLength#(tt(),L) = L >= L = a__length#(mark(L)) a__length#(cons(N,L)) = L + N >= 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 >= 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 >= X2 = 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__isNat#(N) 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)) = X >= X = mark#(X) mark#(length(X)) = X >= X = a__length#(mark(X)) a__length#(cons(N,L)) = L + N >= 0 = a__isNat#(N) a__length#(cons(N,L)) = L + N >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = L + N >= L = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = L >= L = mark#(L) mark#(uTake1(X)) = X + 1 >= X = mark#(X) mark#(uLength(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 >= X2 = 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 >= 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) = IL + 1 >= 1 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 1 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + N + 1 >= IL + N + 1 = 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 + N + 1 >= IL + N + 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N >= L = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L >= L = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + 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)) = X >= X = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = X + 1 >= X + 1 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X3 + X4 + 1 >= X3 + X4 + 1 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X1 + X2 >= X1 + X2 = 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)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 >= X1 + 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) = X >= X = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X2 + 1 >= X2 + 1 = take(X1,X2) a__uTake1(X) = X + 1 >= X + 1 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X3 + X4 + 1 >= X3 + X4 + 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 + X2 >= 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__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#(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) Matrix Interpretation Processor: dimension: 1 interpretation: [a__uLength#](x0, x1) = x1, [a__length#](x0) = x0, [a__isNat#](x0) = 0, [a__isNatList#](x0) = 0, [a__isNatIList#](x0) = 0, [mark#](x0) = x0, [a__and#](x0, x1) = x1, [uLength](x0, x1) = x0 + x1 + 1, [uTake2](x0, x1, x2, x3) = x2 + x3 + 1, [uTake1](x0) = 0, [isNat](x0) = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [and](x0, x1) = x0 + x1, [a__uLength](x0, x1) = x0 + x1 + 1, [a__length](x0) = x0 + 1, [a__uTake2](x0, x1, x2, x3) = x2 + x3 + 1, [a__uTake1](x0) = 0, [a__take](x0, x1) = x1 + 1, [a__zeros] = 0, [take](x0, x1) = x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0 + 1, [s](x0) = x0, [a__isNat](x0) = 0, [0] = 0, [a__isNatList](x0) = 0, [a__isNatIList](x0) = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0 orientation: a__uLength#(tt(),L) = L >= L = a__length#(mark(L)) a__length#(cons(N,L)) = L + N >= 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 >= 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 >= X2 = 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__isNat#(N) 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)) = X + 1 >= X = mark#(X) mark#(length(X)) = X + 1 >= X = a__length#(mark(X)) a__length#(cons(N,L)) = L + N >= 0 = a__isNat#(N) a__length#(cons(N,L)) = L + N >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = L + N >= L = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = L >= L = mark#(L) mark#(uLength(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(uLength(X1,X2)) = X1 + X2 + 1 >= X2 = 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 >= 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) = IL + 1 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + N + 1 >= IL + N + 1 = 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 + N + 1 >= IL + N + 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N + 1 >= L + 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 >= X1 + 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)) = X + 1 >= X + 1 = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 0 >= 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X3 + X4 + 1 >= X3 + X4 + 1 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = 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)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 >= X1 + 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) = X + 1 >= X + 1 = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X2 + 1 >= X2 + 1 = take(X1,X2) a__uTake1(X) = 0 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X3 + X4 + 1 >= X3 + X4 + 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = 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__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) 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#(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: dimension: 1 interpretation: [a__uLength#](x0, x1) = 1, [a__length#](x0) = 1, [a__isNat#](x0) = 0, [a__isNatList#](x0) = 0, [a__isNatIList#](x0) = 0, [mark#](x0) = 0, [a__and#](x0, x1) = 0, [uLength](x0, x1) = x0, [uTake2](x0, x1, x2, x3) = 0, [uTake1](x0) = x0, [isNat](x0) = 1, [isNatList](x0) = 1, [isNatIList](x0) = 1, [and](x0, x1) = 1, [a__uLength](x0, x1) = x0, [a__length](x0) = 1, [a__uTake2](x0, x1, x2, x3) = 0, [a__uTake1](x0) = x0, [a__take](x0, x1) = x0, [a__zeros] = 0, [take](x0, x1) = x0, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 0, [length](x0) = 1, [s](x0) = 0, [a__isNat](x0) = 1, [0] = 1, [a__isNatList](x0) = 1, [a__isNatIList](x0) = 1, [mark](x0) = 1, [a__and](x0, x1) = 1, [tt] = 0 orientation: a__uLength#(tt(),L) = 1 >= 1 = a__length#(mark(L)) a__length#(cons(N,L)) = 1 >= 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) = 0 >= 0 = mark#(T) mark#(and(X1,X2)) = 0 >= 0 = mark#(X2) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 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__isNat#(N) 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) a__length#(cons(N,L)) = 1 >= 0 = a__isNat#(N) a__length#(cons(N,L)) = 1 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__length#(cons(N,L)) = 1 >= 1 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength#(tt(),L) = 1 >= 0 = mark#(L) mark#(s(X)) = 0 >= 0 = mark#(X) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) a__and(tt(),T) = 1 >= 1 = mark(T) a__isNatIList(IL) = 1 >= 1 = a__isNatList(IL) a__isNat(0()) = 1 >= 0 = tt() a__isNat(s(N)) = 1 >= 1 = a__isNat(N) a__isNat(length(L)) = 1 >= 1 = a__isNatList(L) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(N,IL)) = 1 >= 1 = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(N,L)) = 1 >= 1 = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = 1 >= 1 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__take(0(),IL) = 1 >= 1 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 0 >= 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) = 0 >= 0 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = 1 >= 1 = 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)) = 1 >= 1 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(zeros()) = 1 >= 0 = a__zeros() mark(take(X1,X2)) = 1 >= 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 1 >= 1 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 1 >= 0 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = 1 >= 1 = a__uLength(mark(X1),X2) mark(tt()) = 1 >= 0 = tt() mark(0()) = 1 >= 1 = 0() mark(s(X)) = 1 >= 0 = s(mark(X)) mark(cons(X1,X2)) = 1 >= 0 = cons(mark(X1),X2) mark(nil()) = 1 >= 0 = nil() a__and(X1,X2) = 1 >= 1 = and(X1,X2) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__isNat(X) = 1 >= 1 = isNat(X) a__length(X) = 1 >= 1 = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X1 >= X1 = take(X1,X2) a__uTake1(X) = X >= X = uTake1(X) a__uTake2(X1,X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X1 >= X1 = 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__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) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) 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: dimension: 1 interpretation: [a__uLength#](x0, x1) = x1, [a__length#](x0) = x0, [a__isNat#](x0) = x0, [a__isNatList#](x0) = x0 + 1, [a__isNatIList#](x0) = x0 + 1, [mark#](x0) = x0 + 1, [a__and#](x0, x1) = x1 + 1, [uLength](x0, x1) = x1 + 1, [uTake2](x0, x1, x2, x3) = x1 + x2 + x3, [uTake1](x0) = 0, [isNat](x0) = x0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [and](x0, x1) = x0 + x1, [a__uLength](x0, x1) = x1 + 1, [a__length](x0) = x0 + 1, [a__uTake2](x0, x1, x2, x3) = x1 + x2 + x3, [a__uTake1](x0) = 0, [a__take](x0, x1) = x0 + x1, [a__zeros] = 0, [take](x0, x1) = x0 + x1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0 + 1, [s](x0) = x0, [a__isNat](x0) = x0, [0] = 0, [a__isNatList](x0) = x0, [a__isNatIList](x0) = x0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0 orientation: a__uLength#(tt(),L) = L >= L = a__length#(mark(L)) a__isNatList#(cons(N,L)) = L + N + 1 >= L + 1 = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 1 >= N = a__isNat#(N) a__isNat#(s(N)) = N >= N = a__isNat#(N) a__isNat#(length(L)) = L + 1 >= L + 1 = a__isNatList#(L) a__isNatList#(cons(N,L)) = L + N + 1 >= L + 1 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 1 >= T + 1 = mark#(T) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X + 1 >= X + 1 = a__isNatIList#(X) a__isNatIList#(IL) = IL + 1 >= IL + 1 = a__isNatList#(IL) a__isNatList#(take(N,IL)) = IL + N + 1 >= IL + 1 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 1 >= IL + 1 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 1 >= N = a__isNat#(N) a__isNatIList#(cons(N,IL)) = IL + N + 1 >= IL + 1 = a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) = IL + N + 1 >= N = a__isNat#(N) a__isNatList#(take(N,IL)) = IL + N + 1 >= IL + 1 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X + 1 >= X + 1 = a__isNatList#(X) mark#(isNat(X)) = X + 1 >= X = a__isNat#(X) a__length#(cons(N,L)) = L + N >= L = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) a__and(tt(),T) = T >= T = mark(T) a__isNatIList(IL) = IL >= IL = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = N >= N = a__isNat(N) a__isNat(length(L)) = L + 1 >= L = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = IL + N >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = 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 >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__take(0(),IL) = IL >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + M + N >= 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 >= IL + M + N = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N + 1 >= L + 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 >= 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)) = X + 1 >= X + 1 = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 0 >= 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X2 + 1 >= X2 + 1 = 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)) = X1 + X2 >= X1 + X2 = 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) = X + 1 >= X + 1 = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__uTake1(X) = 0 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X2 + 1 >= X2 + 1 = uLength(X1,X2) problem: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) 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__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) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) 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: dimension: 1 interpretation: [a__uLength#](x0, x1) = 1, [a__length#](x0) = 1, [a__isNat#](x0) = 1, [a__isNatList#](x0) = 0, [a__isNatIList#](x0) = 0, [mark#](x0) = 0, [a__and#](x0, x1) = 0, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = 0, [uTake1](x0) = 0, [isNat](x0) = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [and](x0, x1) = 0, [a__uLength](x0, x1) = 0, [a__length](x0) = 0, [a__uTake2](x0, x1, x2, x3) = 0, [a__uTake1](x0) = 0, [a__take](x0, x1) = 0, [a__zeros] = 0, [take](x0, x1) = 0, [nil] = 0, [cons](x0, x1) = 0, [zeros] = 0, [length](x0) = 0, [s](x0) = 0, [a__isNat](x0) = 0, [0] = 0, [a__isNatList](x0) = 0, [a__isNatIList](x0) = 0, [mark](x0) = 0, [a__and](x0, x1) = 0, [tt] = 0 orientation: a__uLength#(tt(),L) = 1 >= 1 = a__length#(mark(L)) a__isNatList#(cons(N,L)) = 0 >= 0 = a__isNatList#(L) a__isNat#(s(N)) = 1 >= 1 = a__isNat#(N) a__isNat#(length(L)) = 1 >= 0 = a__isNatList#(L) a__isNatList#(cons(N,L)) = 0 >= 0 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = 0 >= 0 = mark#(T) mark#(and(X1,X2)) = 0 >= 0 = mark#(X2) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 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__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) a__length#(cons(N,L)) = 1 >= 1 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(s(X)) = 0 >= 0 = mark#(X) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) a__and(tt(),T) = 0 >= 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) = 0 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = 0 >= 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) = 0 >= 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)) = 0 >= 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)) = 0 >= 0 = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 0 >= 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = 0 >= 0 = 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)) = 0 >= 0 = s(mark(X)) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = 0 >= 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) = 0 >= 0 = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__uTake1(X) = 0 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = 0 >= 0 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 0 >= 0 = uLength(X1,X2) problem: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) 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#(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__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) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) 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: dimension: 1 interpretation: [a__uLength#](x0, x1) = 1, [a__length#](x0) = 1, [a__isNat#](x0) = 0, [a__isNatList#](x0) = 1, [a__isNatIList#](x0) = 1, [mark#](x0) = x0 + 1, [a__and#](x0, x1) = x1 + 1, [uLength](x0, x1) = 0, [uTake2](x0, x1, x2, x3) = x2 + 1, [uTake1](x0) = 0, [isNat](x0) = 0, [isNatList](x0) = 0, [isNatIList](x0) = 0, [and](x0, x1) = x0 + x1, [a__uLength](x0, x1) = 0, [a__length](x0) = 0, [a__uTake2](x0, x1, x2, x3) = x2 + 1, [a__uTake1](x0) = 0, [a__take](x0, x1) = x1, [a__zeros] = 1, [take](x0, x1) = x1, [nil] = 0, [cons](x0, x1) = x0 + 1, [zeros] = 1, [length](x0) = 0, [s](x0) = x0, [a__isNat](x0) = 0, [0] = 0, [a__isNatList](x0) = 0, [a__isNatIList](x0) = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0 orientation: a__uLength#(tt(),L) = 1 >= 1 = a__length#(mark(L)) a__isNatList#(cons(N,L)) = 1 >= 1 = a__isNatList#(L) a__isNat#(s(N)) = 0 >= 0 = a__isNat#(N) a__isNatList#(cons(N,L)) = 1 >= 1 = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T + 1 >= T + 1 = mark#(T) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = 1 >= 1 = a__isNatIList#(X) a__isNatIList#(IL) = 1 >= 1 = a__isNatList#(IL) a__isNatList#(take(N,IL)) = 1 >= 1 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 1 >= 1 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = 1 >= 1 = a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) = 1 >= 1 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = 1 >= 1 = a__isNatList#(X) a__length#(cons(N,L)) = 1 >= 1 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) mark#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = mark#(X1) 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() = 1 >= 1 = cons(0(),zeros()) a__take(0(),IL) = IL >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = N + 1 >= N + 1 = a__uTake2(a__and(a__isNat(M),a__and(a__isNat(N),a__isNatIList(IL))),M,N,IL) a__uTake2(tt(),M,N,IL) = N + 1 >= N + 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)) = X1 + X2 >= X1 + 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()) = 1 >= 1 = a__zeros() mark(take(X1,X2)) = X2 >= X2 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 0 >= 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X3 + 1 >= X3 + 1 = 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)) = X >= X = s(mark(X)) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 >= X1 + 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() = 1 >= 1 = zeros() a__take(X1,X2) = X2 >= X2 = take(X1,X2) a__uTake1(X) = 0 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X3 + 1 >= X3 + 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = 0 >= 0 = uLength(X1,X2) problem: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) 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#(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__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) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) 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: dimension: 1 interpretation: [a__uLength#](x0, x1) = 0, [a__length#](x0) = 0, [a__isNat#](x0) = 1, [a__isNatList#](x0) = x0, [a__isNatIList#](x0) = x0, [mark#](x0) = x0, [a__and#](x0, x1) = x1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [uTake1](x0) = 0, [isNat](x0) = x0, [isNatList](x0) = x0, [isNatIList](x0) = x0, [and](x0, x1) = x0 + x1, [a__uLength](x0, x1) = x1, [a__length](x0) = x0, [a__uTake2](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [a__uTake1](x0) = 0, [a__take](x0, x1) = x0 + x1 + 1, [a__zeros] = 0, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [a__isNat](x0) = x0, [0] = 0, [a__isNatList](x0) = x0, [a__isNatIList](x0) = x0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0 orientation: a__uLength#(tt(),L) = 0 >= 0 = a__length#(mark(L)) a__isNatList#(cons(N,L)) = L + N >= L = a__isNatList#(L) a__isNat#(s(N)) = 1 >= 1 = a__isNat#(N) a__isNatList#(cons(N,L)) = L + N >= L = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T >= 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 >= X2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X >= X = a__isNatIList#(X) a__isNatIList#(IL) = IL >= IL = a__isNatList#(IL) a__isNatList#(take(N,IL)) = IL + N + 1 >= IL = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N >= IL = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N >= IL = a__and#(a__isNat(N),a__isNatIList(IL)) a__isNatList#(take(N,IL)) = IL + N + 1 >= IL = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X >= X = a__isNatList#(X) a__length#(cons(N,L)) = 0 >= 0 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(s(X)) = X >= X = mark#(X) a__and(tt(),T) = T >= T = mark(T) a__isNatIList(IL) = IL >= IL = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = N >= N = a__isNat(N) a__isNat(length(L)) = L >= L = a__isNatList(L) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(N,IL)) = IL + N >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__isNatList(nil()) = 0 >= 0 = 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 + 1 >= IL + N = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__take(0(),IL) = IL + 1 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = 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 + 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N >= L = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L >= L = 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)) = X >= X = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 0 >= 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X2 >= X2 = 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)) = X1 + X2 >= X1 + X2 = 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) = X >= X = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__uTake1(X) = 0 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X2 >= X2 = uLength(X1,X2) problem: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) 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#(IL) -> a__isNatList#(IL) 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__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) 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: dimension: 1 interpretation: [a__uLength#](x0, x1) = 0, [a__length#](x0) = 0, [a__isNat#](x0) = 0, [a__isNatList#](x0) = x0, [a__isNatIList#](x0) = x0 + 1, [mark#](x0) = x0, [a__and#](x0, x1) = x1, [uLength](x0, x1) = x1, [uTake2](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [uTake1](x0) = 0, [isNat](x0) = x0, [isNatList](x0) = x0, [isNatIList](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [a__uLength](x0, x1) = x1, [a__length](x0) = x0, [a__uTake2](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [a__uTake1](x0) = 0, [a__take](x0, x1) = x0 + x1 + 1, [a__zeros] = 0, [take](x0, x1) = x0 + x1 + 1, [nil] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [length](x0) = x0, [s](x0) = x0, [a__isNat](x0) = x0, [0] = 0, [a__isNatList](x0) = x0, [a__isNatIList](x0) = x0 + 1, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 0 orientation: a__uLength#(tt(),L) = 0 >= 0 = a__length#(mark(L)) a__isNatList#(cons(N,L)) = L + N >= L = a__isNatList#(L) a__isNat#(s(N)) = 0 >= 0 = a__isNat#(N) a__isNatList#(cons(N,L)) = L + N >= L = a__and#(a__isNat(N),a__isNatList(L)) a__and#(tt(),T) = T >= 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 >= X2 = a__and#(mark(X1),mark(X2)) mark#(isNatIList(X)) = X + 1 >= X + 1 = a__isNatIList#(X) a__isNatIList#(IL) = IL + 1 >= IL = a__isNatList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 1 >= IL + 1 = a__isNatIList#(IL) a__isNatIList#(cons(N,IL)) = IL + N + 1 >= IL + 1 = a__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) = X >= X = a__isNatList#(X) a__length#(cons(N,L)) = 0 >= 0 = a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) mark#(s(X)) = X >= X = mark#(X) a__and(tt(),T) = T >= T = mark(T) a__isNatIList(IL) = IL + 1 >= IL = a__isNatList(IL) a__isNat(0()) = 0 >= 0 = tt() a__isNat(s(N)) = N >= N = a__isNat(N) a__isNat(length(L)) = L >= L = a__isNatList(L) a__isNatIList(zeros()) = 1 >= 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 >= L + N = a__and(a__isNat(N),a__isNatList(L)) a__isNatList(take(N,IL)) = IL + N + 1 >= IL + N + 1 = a__and(a__isNat(N),a__isNatIList(IL)) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__take(0(),IL) = IL + 1 >= 0 = a__uTake1(a__isNatIList(IL)) a__uTake1(tt()) = 0 >= 0 = nil() a__take(s(M),cons(N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = 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 + 1 = cons(mark(N),take(M,IL)) a__length(cons(N,L)) = L + N >= L = a__uLength(a__and(a__isNat(N),a__isNatList(L)),L) a__uLength(tt(),L) = L >= L = s(a__length(mark(L))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),mark(X2)) mark(isNatIList(X)) = X + 1 >= X + 1 = a__isNatIList(X) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(isNat(X)) = X >= X = a__isNat(X) mark(length(X)) = X >= X = a__length(mark(X)) mark(zeros()) = 0 >= 0 = a__zeros() mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(uTake1(X)) = 0 >= 0 = a__uTake1(mark(X)) mark(uTake2(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = a__uTake2(mark(X1),X2,X3,X4) mark(uLength(X1,X2)) = X2 >= X2 = 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)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(nil()) = 0 >= 0 = nil() a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIList(X) = X + 1 >= X + 1 = isNatIList(X) a__isNatList(X) = X >= X = isNatList(X) a__isNat(X) = X >= X = isNat(X) a__length(X) = X >= X = length(X) a__zeros() = 0 >= 0 = zeros() a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__uTake1(X) = 0 >= 0 = uTake1(X) a__uTake2(X1,X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = uTake2(X1,X2,X3,X4) a__uLength(X1,X2) = X2 >= X2 = uLength(X1,X2) problem: DPs: a__uLength#(tt(),L) -> a__length#(mark(L)) a__isNatList#(cons(N,L)) -> a__isNatList#(L) 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__and#(a__isNat(N),a__isNatIList(IL)) mark#(isNatList(X)) -> a__isNatList#(X) a__length#(cons(N,L)) -> a__uLength#(a__and(a__isNat(N),a__isNatList(L)),L) 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) Open