MAYBE Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Proof: DP Processor: DPs: a__U11#(tt(),L) -> a__U12#(tt(),L) a__U12#(tt(),L) -> mark#(L) a__U12#(tt(),L) -> a__length#(mark(L)) a__U21#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) a__U22#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) a__U23#(tt(),IL,M,N) -> mark#(N) a__length#(cons(N,L)) -> a__U11#(tt(),L) a__take#(s(M),cons(N,IL)) -> a__U21#(tt(),IL,M,N) mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) TDG Processor: DPs: a__U11#(tt(),L) -> a__U12#(tt(),L) a__U12#(tt(),L) -> mark#(L) a__U12#(tt(),L) -> a__length#(mark(L)) a__U21#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) a__U22#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) a__U23#(tt(),IL,M,N) -> mark#(N) a__length#(cons(N,L)) -> a__U11#(tt(),L) a__take#(s(M),cons(N,IL)) -> a__U21#(tt(),IL,M,N) mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) graph: a__take#(s(M),cons(N,IL)) -> a__U21#(tt(),IL,M,N) -> a__U21#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X1) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X2) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(length(X)) -> a__length#(mark(X)) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(length(X)) -> mark#(X) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U12(X1,X2)) -> mark#(X1) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(U11(X1,X2)) -> mark#(X1) a__U23#(tt(),IL,M,N) -> mark#(N) -> mark#(zeros()) -> a__zeros#() a__U22#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) -> mark#(N) a__U21#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) a__length#(cons(N,L)) -> a__U11#(tt(),L) -> a__U11#(tt(),L) -> a__U12#(tt(),L) mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) -> a__U23#(tt(),IL,M,N) -> mark#(N) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) -> a__U22#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) -> a__U21#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__U11#(tt(),L) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) 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#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(length(X)) -> mark#(X) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(length(X)) -> mark#(X) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(length(X)) -> mark#(X) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) 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#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U12(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) -> a__U12#(tt(),L) -> a__length#(mark(L)) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) -> a__U12#(tt(),L) -> mark#(L) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) -> a__U11#(tt(),L) -> a__U12#(tt(),L) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(M),cons(N,IL)) -> a__U21#(tt(),IL,M,N) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) 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#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) 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#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zeros()) -> a__zeros#() mark#(take(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) 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#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) 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#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) 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#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(s(X)) -> mark#(X) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(s(X)) -> mark#(X) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(s(X)) -> mark#(X) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) 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#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) 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#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) 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#(U12(X1,X2)) -> a__U12#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() a__U12#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__U11#(tt(),L) a__U12#(tt(),L) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__U12#(tt(),L) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__U12#(tt(),L) -> mark#(L) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__U12#(tt(),L) -> mark#(L) -> mark#(take(X1,X2)) -> mark#(X1) a__U12#(tt(),L) -> mark#(L) -> mark#(take(X1,X2)) -> mark#(X2) a__U12#(tt(),L) -> mark#(L) -> mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) a__U12#(tt(),L) -> mark#(L) -> mark#(U23(X1,X2,X3,X4)) -> mark#(X1) a__U12#(tt(),L) -> mark#(L) -> mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) a__U12#(tt(),L) -> mark#(L) -> mark#(U22(X1,X2,X3,X4)) -> mark#(X1) a__U12#(tt(),L) -> mark#(L) -> mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) a__U12#(tt(),L) -> mark#(L) -> mark#(U21(X1,X2,X3,X4)) -> mark#(X1) a__U12#(tt(),L) -> mark#(L) -> mark#(length(X)) -> a__length#(mark(X)) a__U12#(tt(),L) -> mark#(L) -> mark#(length(X)) -> mark#(X) a__U12#(tt(),L) -> mark#(L) -> mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__U12#(tt(),L) -> mark#(L) -> mark#(U12(X1,X2)) -> mark#(X1) a__U12#(tt(),L) -> mark#(L) -> mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U12#(tt(),L) -> mark#(L) -> mark#(U11(X1,X2)) -> mark#(X1) a__U12#(tt(),L) -> mark#(L) -> mark#(zeros()) -> a__zeros#() a__U11#(tt(),L) -> a__U12#(tt(),L) -> a__U12#(tt(),L) -> a__length#(mark(L)) a__U11#(tt(),L) -> a__U12#(tt(),L) -> a__U12#(tt(),L) -> mark#(L) SCC Processor: #sccs: 1 #rules: 25 #arcs: 231/676 DPs: a__take#(s(M),cons(N,IL)) -> a__U21#(tt(),IL,M,N) a__U21#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) a__U22#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) a__U23#(tt(),IL,M,N) -> mark#(N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),L) -> a__U12#(tt(),L) a__U12#(tt(),L) -> mark#(L) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__U12#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U11#(tt(),L) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U21(X1,X2,X3,X4)) -> mark#(X1) mark#(U21(X1,X2,X3,X4)) -> a__U21#(mark(X1),X2,X3,X4) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [a__take#](x0, x1) = x0 + x1 + 6, [a__U23#](x0, x1, x2, x3) = 3x0 + x3, [a__U22#](x0, x1, x2, x3) = 3x0 + x2 + x3, [a__U21#](x0, x1, x2, x3) = 2x0 + x2 + x3 + 1, [a__length#](x0) = x0 + 1, [mark#](x0) = x0, [a__U12#](x0, x1) = x0 + x1, [a__U11#](x0, x1) = x0 + x1, [U23](x0, x1, x2, x3) = 6x0 + 2x1 + x2 + x3, [U22](x0, x1, x2, x3) = 6x0 + 2x1 + x2 + x3, [U21](x0, x1, x2, x3) = 4x0 + 2x1 + x2 + 2x3 + 2, [length](x0) = x0 + 1, [U12](x0, x1) = x0 + x1, [U11](x0, x1) = x0 + x1, [a__take](x0, x1) = x0 + 2x1 + 6, [nil] = 2, [take](x0, x1) = x0 + 2x1 + 6, [a__U23](x0, x1, x2, x3) = 6x0 + 2x1 + x2 + x3, [a__U22](x0, x1, x2, x3) = 6x0 + 2x1 + x2 + x3, [a__U21](x0, x1, x2, x3) = 4x0 + 2x1 + x2 + 2x3 + 2, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U12](x0, x1) = x0 + x1, [a__U11](x0, x1) = x0 + x1, [tt] = 1, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = IL + M + N + 6 >= M + N + 3 = a__U21#(tt(),IL,M,N) a__U21#(tt(),IL,M,N) = M + N + 3 >= M + N + 3 = a__U22#(tt(),IL,M,N) a__U22#(tt(),IL,M,N) = M + N + 3 >= N + 3 = a__U23#(tt(),IL,M,N) a__U23#(tt(),IL,M,N) = N + 3 >= N = mark#(N) mark#(U11(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11#(mark(X1),X2) a__U11#(tt(),L) = L + 1 >= L + 1 = a__U12#(tt(),L) a__U12#(tt(),L) = L + 1 >= L = mark#(L) mark#(U12(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U12(X1,X2)) = X1 + X2 >= X1 + X2 = a__U12#(mark(X1),X2) a__U12#(tt(),L) = L + 1 >= L + 1 = a__length#(mark(L)) a__length#(cons(N,L)) = L + N + 1 >= L + 1 = a__U11#(tt(),L) mark#(length(X)) = X + 1 >= X = mark#(X) mark#(length(X)) = X + 1 >= X + 1 = a__length#(mark(X)) mark#(U21(X1,X2,X3,X4)) = 4X1 + 2X2 + X3 + 2X4 + 2 >= X1 = mark#(X1) mark#(U21(X1,X2,X3,X4)) = 4X1 + 2X2 + X3 + 2X4 + 2 >= 2X1 + X3 + X4 + 1 = a__U21#(mark(X1),X2,X3,X4) mark#(U22(X1,X2,X3,X4)) = 6X1 + 2X2 + X3 + X4 >= X1 = mark#(X1) mark#(U22(X1,X2,X3,X4)) = 6X1 + 2X2 + X3 + X4 >= 3X1 + X3 + X4 = a__U22#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) = 6X1 + 2X2 + X3 + X4 >= X1 = mark#(X1) mark#(U23(X1,X2,X3,X4)) = 6X1 + 2X2 + X3 + X4 >= 3X1 + X4 = a__U23#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) = X1 + 2X2 + 6 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + 2X2 + 6 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + 2X2 + 6 >= X1 + X2 + 6 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),L) = L + 1 >= L + 1 = a__U12(tt(),L) a__U12(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__U21(tt(),IL,M,N) = 2IL + M + 2N + 6 >= 2IL + M + N + 6 = a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) = 2IL + M + N + 6 >= 2IL + M + N + 6 = a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) = 2IL + M + N + 6 >= 2IL + M + N + 6 = cons(mark(N),take(M,IL)) a__length(nil()) = 3 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + 1 = a__U11(tt(),L) a__take(0(),IL) = 2IL + 6 >= 2 = nil() a__take(s(M),cons(N,IL)) = 2IL + M + 2N + 6 >= 2IL + M + 2N + 6 = a__U21(tt(),IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(U12(X1,X2)) = X1 + X2 >= X1 + X2 = a__U12(mark(X1),X2) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(U21(X1,X2,X3,X4)) = 4X1 + 2X2 + X3 + 2X4 + 2 >= 4X1 + 2X2 + X3 + 2X4 + 2 = a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) = 6X1 + 2X2 + X3 + X4 >= 6X1 + 2X2 + X3 + X4 = a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) = 6X1 + 2X2 + X3 + X4 >= 6X1 + 2X2 + X3 + X4 = a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + 2X2 + 6 >= X1 + 2X2 + 6 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 2 >= 2 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X1,X2) = X1 + X2 >= X1 + X2 = U12(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) a__U21(X1,X2,X3,X4) = 4X1 + 2X2 + X3 + 2X4 + 2 >= 4X1 + 2X2 + X3 + 2X4 + 2 = U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) = 6X1 + 2X2 + X3 + X4 >= 6X1 + 2X2 + X3 + X4 = U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) = 6X1 + 2X2 + X3 + X4 >= 6X1 + 2X2 + X3 + X4 = U23(X1,X2,X3,X4) a__take(X1,X2) = X1 + 2X2 + 6 >= X1 + 2X2 + 6 = take(X1,X2) problem: DPs: a__U21#(tt(),IL,M,N) -> a__U22#(tt(),IL,M,N) a__U22#(tt(),IL,M,N) -> a__U23#(tt(),IL,M,N) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),L) -> a__U12#(tt(),L) mark#(U12(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2) a__U12#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U11#(tt(),L) mark#(length(X)) -> a__length#(mark(X)) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> a__U22#(mark(X1),X2,X3,X4) mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U23(X1,X2,X3,X4)) -> a__U23#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) SCC Processor: #sccs: 2 #rules: 9 #arcs: 219/289 DPs: mark#(U23(X1,X2,X3,X4)) -> mark#(X1) mark#(U11(X1,X2)) -> mark#(X1) mark#(U12(X1,X2)) -> mark#(X1) mark#(U22(X1,X2,X3,X4)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Qed DPs: a__length#(cons(N,L)) -> a__U11#(tt(),L) a__U11#(tt(),L) -> a__U12#(tt(),L) a__U12#(tt(),L) -> a__length#(mark(L)) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),L) -> a__U12(tt(),L) a__U12(tt(),L) -> s(a__length(mark(L))) a__U21(tt(),IL,M,N) -> a__U22(tt(),IL,M,N) a__U22(tt(),IL,M,N) -> a__U23(tt(),IL,M,N) a__U23(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U11(tt(),L) a__take(0(),IL) -> nil() a__take(s(M),cons(N,IL)) -> a__U21(tt(),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X1,X2)) -> a__U12(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U21(X1,X2,X3,X4)) -> a__U21(mark(X1),X2,X3,X4) mark(U22(X1,X2,X3,X4)) -> a__U22(mark(X1),X2,X3,X4) mark(U23(X1,X2,X3,X4)) -> a__U23(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X1,X2) -> U12(X1,X2) a__length(X) -> length(X) a__U21(X1,X2,X3,X4) -> U21(X1,X2,X3,X4) a__U22(X1,X2,X3,X4) -> U22(X1,X2,X3,X4) a__U23(X1,X2,X3,X4) -> U23(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Open