/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(U11(tt(),L)) -> length#(L) active#(U11(tt(),L)) -> s#(length(L)) active#(U11(tt(),L)) -> mark#(s(length(L))) active#(U21(tt())) -> mark#(nil()) active#(U31(tt(),IL,M,N)) -> take#(M,IL) active#(U31(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> mark#(isNatList(V1)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> mark#(isNatList(V)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNatIList#(V2) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatIList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNatList#(V2) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNat(V1),isNatList(V2)) active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) active#(isNatList(take(V1,V2))) -> isNatIList#(V2) active#(isNatList(take(V1,V2))) -> isNat#(V1) active#(isNatList(take(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNat(N)) active#(length(cons(N,L))) -> U11#(and(isNatList(L),isNat(N)),L) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> U21#(isNatIList(IL)) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNat(N)) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),and(isNat(M),isNat(N))) active#(take(s(M),cons(N,IL))) -> U31#(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(nil()) -> active#(nil()) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(U31(X1,X2,X3,X4)) -> U31#(mark(X1),X2,X3,X4) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,mark(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(active(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,active(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) TDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(U11(tt(),L)) -> length#(L) active#(U11(tt(),L)) -> s#(length(L)) active#(U11(tt(),L)) -> mark#(s(length(L))) active#(U21(tt())) -> mark#(nil()) active#(U31(tt(),IL,M,N)) -> take#(M,IL) active#(U31(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> mark#(isNatList(V1)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> mark#(isNat(V1)) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> mark#(isNatList(V)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNatIList#(V2) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatIList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNatList#(V2) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNat(V1),isNatList(V2)) active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) active#(isNatList(take(V1,V2))) -> isNatIList#(V2) active#(isNatList(take(V1,V2))) -> isNat#(V1) active#(isNatList(take(V1,V2))) -> and#(isNat(V1),isNatIList(V2)) active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNat(N)) active#(length(cons(N,L))) -> U11#(and(isNatList(L),isNat(N)),L) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> U21#(isNatIList(IL)) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNat(N)) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),and(isNat(M),isNat(N))) active#(take(s(M),cons(N,IL))) -> U31#(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(nil()) -> active#(nil()) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(U31(X1,X2,X3,X4)) -> U31#(mark(X1),X2,X3,X4) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(isNatIList(X)) -> active#(isNatIList(X)) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,mark(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(active(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,active(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) graph: ... SCC Processor: #sccs: 12 #rules: 78 #arcs: 1835/13225 DPs: mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),L)) -> mark#(s(length(L))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U21(tt())) -> mark#(nil()) mark#(0()) -> active#(0()) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(tt()) -> active#(tt()) active#(isNat(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(length(V1))) -> mark#(isNatList(V1)) mark#(length(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> active#(U21(mark(X))) active#(isNatIList(V)) -> mark#(isNatList(V)) mark#(nil()) -> active#(nil()) active#(isNatIList(zeros())) -> mark#(tt()) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) active#(isNatIList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatList(nil())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) mark#(isNat(X)) -> active#(isNat(X)) active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) EDG Processor: DPs: mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),L)) -> mark#(s(length(L))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U21(tt())) -> mark#(nil()) mark#(0()) -> active#(0()) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(tt()) -> active#(tt()) active#(isNat(0())) -> mark#(tt()) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNat(length(V1))) -> mark#(isNatList(V1)) mark#(length(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> active#(U21(mark(X))) active#(isNatIList(V)) -> mark#(isNatList(V)) mark#(nil()) -> active#(nil()) active#(isNatIList(zeros())) -> mark#(tt()) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) active#(isNatIList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatList(nil())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) mark#(isNat(X)) -> active#(isNat(X)) active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) graph: mark#(isNatList(X)) -> active#(isNatList(X)) -> active#(isNatList(nil())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) -> active#(isNatList(cons(V1,V2))) -> mark#(and(isNat(V1),isNatList(V2))) mark#(isNatList(X)) -> active#(isNatList(X)) -> active#(isNatList(take(V1,V2))) -> mark#(and(isNat(V1),isNatIList(V2))) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(0())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(length(V1))) -> mark#(isNatList(V1)) mark#(isNat(X)) -> active#(isNat(X)) -> active#(isNat(s(V1))) -> mark#(isNat(V1)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> active#(zeros()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(and(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> active#(length(mark(X))) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) -> active#(and(tt(),X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zeros()) -> active#(zeros()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(0()) -> active#(0()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(tt()) -> active#(tt()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> active#(s(mark(X))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(length(X)) -> active#(length(mark(X))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U21(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U21(X)) -> active#(U21(mark(X))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(nil()) -> active#(nil()) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> active#(zeros()) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(0()) -> active#(0()) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> active#(isNatList(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) -> active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) -> active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) -> active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U21(X)) -> mark#(X) -> mark#(zeros()) -> active#(zeros()) mark#(U21(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(U21(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(U21(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(U21(X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) mark#(U21(X)) -> active#(U21(mark(X))) -> active#(U21(tt())) -> mark#(nil()) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> active#(zeros()) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(s(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(length(X)) -> mark#(X) -> mark#(zeros()) -> active#(zeros()) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(length(X)) -> mark#(X) -> mark#(0()) -> active#(0()) mark#(length(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(length(X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> active#(length(mark(X))) mark#(length(X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(length(X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) mark#(length(X)) -> active#(length(mark(X))) -> active#(length(nil())) -> mark#(0()) mark#(length(X)) -> active#(length(mark(X))) -> active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(U11(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) -> active#(U11(tt(),L)) -> mark#(s(length(L))) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tt()) -> active#(tt()) mark#(zeros()) -> active#(zeros()) -> active#(zeros()) -> mark#(cons(0(),zeros())) active#(isNatIList(zeros())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNatList(nil())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(isNat(0())) -> mark#(tt()) -> mark#(tt()) -> active#(tt()) active#(and(tt(),X)) -> mark#(X) -> mark#(zeros()) -> active#(zeros()) active#(and(tt(),X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(0()) -> active#(0()) active#(and(tt(),X)) -> mark#(X) -> mark#(U11(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(tt()) -> active#(tt()) active#(and(tt(),X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) active#(and(tt(),X)) -> mark#(X) -> mark#(s(X)) -> active#(s(mark(X))) active#(and(tt(),X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) active#(and(tt(),X)) -> mark#(X) -> mark#(length(X)) -> active#(length(mark(X))) active#(and(tt(),X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) active#(and(tt(),X)) -> mark#(X) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(and(tt(),X)) -> mark#(X) -> mark#(nil()) -> active#(nil()) active#(and(tt(),X)) -> mark#(X) -> mark#(U31(X1,X2,X3,X4)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) active#(and(tt(),X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) active#(and(tt(),X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(and(tt(),X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) active#(and(tt(),X)) -> mark#(X) -> mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) -> mark#(isNat(X)) -> active#(isNat(X)) active#(and(tt(),X)) -> mark#(X) -> mark#(isNatList(X)) -> active#(isNatList(X)) active#(and(tt(),X)) -> mark#(X) -> mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) -> mark#(U31(X1,X2,X3,X4)) -> mark#(X1) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) -> mark#(U31(X1,X2,X3,X4)) -> active#(U31(mark(X1),X2,X3,X4)) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) -> mark#(U21(X)) -> mark#(X) active#(take(0(),IL)) -> mark#(U21(isNatIList(IL))) -> mark#(U21(X)) -> active#(U21(mark(X))) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) -> mark#(cons(X1,X2)) -> mark#(X1) active#(U31(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U21(tt())) -> mark#(nil()) -> mark#(nil()) -> active#(nil()) active#(length(nil())) -> mark#(0()) -> mark#(0()) -> active#(0()) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) -> mark#(U11(X1,X2)) -> mark#(X1) active#(length(cons(N,L))) -> mark#(U11(and(isNatList(L),isNat(N)),L)) -> mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),L)) -> mark#(s(length(L))) -> mark#(s(X)) -> mark#(X) active#(U11(tt(),L)) -> mark#(s(length(L))) -> mark#(s(X)) -> active#(s(mark(X))) active#(zeros()) -> mark#(cons(0(),zeros())) -> mark#(cons(X1,X2)) -> mark#(X1) active#(zeros()) -> mark#(cons(0(),zeros())) -> mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) SCC Processor: #sccs: 2 #rules: 11 #arcs: 127/1764 DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(length(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Matrix Interpretation Processor: dim=1 usable rules: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) interpretation: [mark#](x0) = 4x0, [active#](x0) = 4x0, [isNatIList](x0) = 0, [isNatList](x0) = 0, [isNat](x0) = 0, [and](x0, x1) = 2x1, [take](x0, x1) = 2x1, [U31](x0, x1, x2, x3) = x0, [nil] = 0, [U21](x0) = 0, [s](x0) = 0, [length](x0) = 2x0 + 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(and(X1,X2)) = 8X2 >= 8X2 = active#(and(mark(X1),X2)) active#(and(tt(),X)) = 8X >= 4X = mark#(X) mark#(take(X1,X2)) = 8X2 >= 8X2 = active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) = 0 >= 0 = mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(U31(X1,X2,X3,X4)) = 4X1 >= 4X1 = mark#(X1) mark#(length(X)) = 8X + 4 >= 4X = mark#(X) mark#(take(X1,X2)) = 8X2 >= 4X2 = mark#(X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(U21(tt())) = 0 >= 0 = mark(nil()) active(U31(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 2X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(isNatList(V1)) active(isNat(s(V1))) = 0 >= 0 = mark(isNat(V1)) active(isNatIList(V)) = 0 >= 0 = mark(isNatList(V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) = 2IL >= 0 = mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 2X + 1 >= 2X + 1 = active(length(mark(X))) mark(U21(X)) = 0 >= 0 = active(U21(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U31(X1,X2,X3,X4)) = X1 >= X1 = active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2X2 >= 2X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 2X2 >= 2X2 = active(and(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 2X + 1 >= 2X + 1 = length(X) length(active(X)) = 2X + 1 >= 2X + 1 = length(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X1),X2,X3,X4) = X1 >= X1 = U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) = X1 >= X1 = U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) = X1 >= X1 = U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) = X1 >= X1 = U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) = X1 >= X1 = U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) = X1 >= X1 = U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) = X1 >= X1 = U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) = X1 >= X1 = U31(X1,X2,X3,X4) take(mark(X1),X2) = 2X2 >= 2X2 = take(X1,X2) take(X1,mark(X2)) = 2X2 >= 2X2 = take(X1,X2) take(active(X1),X2) = 2X2 >= 2X2 = take(X1,X2) take(X1,active(X2)) = 2X2 >= 2X2 = take(X1,X2) and(mark(X1),X2) = 2X2 >= 2X2 = and(X1,X2) and(X1,mark(X2)) = 2X2 >= 2X2 = and(X1,X2) and(active(X1),X2) = 2X2 >= 2X2 = and(X1,X2) and(X1,active(X2)) = 2X2 >= 2X2 = and(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Restore Modifier: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(take(s(M),cons(N,IL))) -> mark#(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark#(U31(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) SCC Processor: #sccs: 1 #rules: 3 #arcs: 15/36 DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(and) = 1 pi(active#) = 0 pi(mark#) = 0 problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(and) = [1,1] pi(isNat) = [0,0] pi(active#) = 0 pi(mark#) = 0 problem: DPs: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/1 DPs: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U11(tt(),L)) -> mark#(s(length(L))) mark#(s(X)) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) KBO Processor: argument filtering: pi(zeros) = [] pi(active) = 0 pi(0) = [] pi(cons) = 1 pi(mark) = [0] pi(tt) = [] pi(U11) = [0] pi(length) = [] pi(s) = 0 pi(U21) = [] pi(nil) = [] pi(U31) = [0,1] pi(take) = 0 pi(and) = [] pi(isNat) = 0 pi(isNatList) = [] pi(isNatIList) = [] pi(active#) = [] pi(mark#) = 0 usable rules: s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) weight function: w0 = 1 w(mark#) = w(active#) = w(isNatIList) = w(isNatList) = w(and) = w( U31) = w(nil) = w(U21) = w(length) = w(U11) = w(tt) = w(mark) = w( 0) = w(zeros) = 1 w(isNat) = w(take) = w(s) = w(cons) = w(active) = 0 precedence: active# > mark# ~ isNatIList ~ isNatList ~ isNat ~ and ~ take ~ U31 ~ nil ~ U21 ~ s ~ length ~ U11 ~ tt ~ mark ~ cons ~ 0 ~ active ~ zeros problem: DPs: mark#(s(X)) -> mark#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Restore Modifier: DPs: mark#(s(X)) -> mark#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/1 DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(length#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(take#) = 0 problem: DPs: take#(X1,mark(X2)) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(take#) = 1 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(isNatList#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(isNatIList#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(and#) = 0 problem: DPs: and#(X1,mark(X2)) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(and#) = 1 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(U11#) = 1 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(U21#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed DPs: U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,mark(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(active(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,active(X3),X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(U31#) = 2 problem: DPs: U31#(mark(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(active(X1),X2,X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(U31#) = 0 problem: DPs: U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,mark(X4)) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,X2,X3,active(X4)) -> U31#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(U31#) = 3 problem: DPs: U31#(X1,mark(X2),X3,X4) -> U31#(X1,X2,X3,X4) U31#(X1,active(X2),X3,X4) -> U31#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Subterm Criterion Processor: simple projection: pi(U31#) = 1 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),L)) -> mark(s(length(L))) active(U21(tt())) -> mark(nil()) active(U31(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(and(isNat(V1),isNatList(V2))) active(isNatList(take(V1,V2))) -> mark(and(isNat(V1),isNatIList(V2))) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U11(and(isNatList(L),isNat(N)),L)) active(take(0(),IL)) -> mark(U21(isNatIList(IL))) active(take(s(M),cons(N,IL))) -> mark(U31(and(isNatIList(IL),and(isNat(M),isNat(N))),IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U21(X)) -> active(U21(mark(X))) mark(nil()) -> active(nil()) mark(U31(X1,X2,X3,X4)) -> active(U31(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(isNatList(X)) -> active(isNatList(X)) mark(isNatIList(X)) -> active(isNatIList(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,mark(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,mark(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,mark(X4)) -> U31(X1,X2,X3,X4) U31(active(X1),X2,X3,X4) -> U31(X1,X2,X3,X4) U31(X1,active(X2),X3,X4) -> U31(X1,X2,X3,X4) U31(X1,X2,active(X3),X4) -> U31(X1,X2,X3,X4) U31(X1,X2,X3,active(X4)) -> U31(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) Qed