MAYBE Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U61(tt(),V1,V2)) -> isNat#(V1) active#(U61(tt(),V1,V2)) -> U62#(isNat(V1),V2) active#(U62(tt(),V2)) -> isNatIList#(V2) active#(U62(tt(),V2)) -> U63#(isNatIList(V2)) active#(U71(tt(),L)) -> length#(L) active#(U71(tt(),L)) -> s#(length(L)) active#(U91(tt(),IL,M,N)) -> take#(M,IL) active#(U91(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(take(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(take(V1,V2))) -> isNatKind#(V1) active#(isNatList(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(take(V1,V2))) -> U61#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(length(cons(N,L))) -> isNatKind#(N) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatKind(N)) active#(length(cons(N,L))) -> isNatIListKind#(L) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNatIListKind(L)) active#(length(cons(N,L))) -> and#(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) active#(length(cons(N,L))) -> U71#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(take(0(),IL)) -> isNatIListKind#(IL) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(0(),IL)) -> U81#(and(isNatIList(IL),isNatIListKind(IL))) active#(take(s(M),cons(N,IL))) -> isNatKind#(N) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatKind(N)) active#(take(s(M),cons(N,IL))) -> isNatKind#(M) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNatKind(M)) active#(take(s(M),cons(N,IL))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(take(s(M),cons(N,IL))) -> isNatIListKind#(IL) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(s(M),cons(N,IL))) -> and#(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) active#(take(s(M),cons(N,IL))) -> U91#(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X)) -> active#(X) active#(U12(X)) -> U12#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U32(X)) -> active#(X) active#(U32(X)) -> U32#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(U42(X1,X2)) -> active#(X1) active#(U42(X1,X2)) -> U42#(active(X1),X2) active#(U43(X)) -> active#(X) active#(U43(X)) -> U43#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(U52(X1,X2)) -> active#(X1) active#(U52(X1,X2)) -> U52#(active(X1),X2) active#(U53(X)) -> active#(X) active#(U53(X)) -> U53#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(U63(X)) -> active#(X) active#(U63(X)) -> U63#(active(X)) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(U81(X)) -> active#(X) active#(U81(X)) -> U81#(active(X)) active#(U91(X1,X2,X3,X4)) -> active#(X1) active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U63#(mark(X)) -> U63#(X) U71#(mark(X1),X2) -> U71#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) U81#(mark(X)) -> U81#(X) U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X)) -> proper#(X) proper#(U12(X)) -> U12#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U32(X)) -> proper#(X) proper#(U32(X)) -> U32#(proper(X)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(U42(X1,X2)) -> proper#(X2) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2)) proper#(U43(X)) -> proper#(X) proper#(U43(X)) -> U43#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(U52(X1,X2)) -> proper#(X2) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2)) proper#(U53(X)) -> proper#(X) proper#(U53(X)) -> U53#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(U63(X)) -> proper#(X) proper#(U63(X)) -> U63#(proper(X)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(U81(X)) -> proper#(X) proper#(U81(X)) -> U81#(proper(X)) proper#(U91(X1,X2,X3,X4)) -> proper#(X4) proper#(U91(X1,X2,X3,X4)) -> proper#(X3) proper#(U91(X1,X2,X3,X4)) -> proper#(X2) proper#(U91(X1,X2,X3,X4)) -> proper#(X1) proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIListKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> isNatIListKind#(proper(X)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X)) -> U12#(X) isNatList#(ok(X)) -> isNatList#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) isNat#(ok(X)) -> isNat#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U32#(ok(X)) -> U32#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U43#(ok(X)) -> U43#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U53#(ok(X)) -> U53#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) U63#(ok(X)) -> U63#(X) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) U81#(ok(X)) -> U81#(X) U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4) take#(ok(X1),ok(X2)) -> take#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIListKind#(ok(X)) -> isNatIListKind#(X) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Usable Rule Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U61(tt(),V1,V2)) -> isNat#(V1) active#(U61(tt(),V1,V2)) -> U62#(isNat(V1),V2) active#(U62(tt(),V2)) -> isNatIList#(V2) active#(U62(tt(),V2)) -> U63#(isNatIList(V2)) active#(U71(tt(),L)) -> length#(L) active#(U71(tt(),L)) -> s#(length(L)) active#(U91(tt(),IL,M,N)) -> take#(M,IL) active#(U91(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(take(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(take(V1,V2))) -> isNatKind#(V1) active#(isNatList(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(take(V1,V2))) -> U61#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(length(cons(N,L))) -> isNatKind#(N) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatKind(N)) active#(length(cons(N,L))) -> isNatIListKind#(L) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNatIListKind(L)) active#(length(cons(N,L))) -> and#(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) active#(length(cons(N,L))) -> U71#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(take(0(),IL)) -> isNatIListKind#(IL) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(0(),IL)) -> U81#(and(isNatIList(IL),isNatIListKind(IL))) active#(take(s(M),cons(N,IL))) -> isNatKind#(N) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatKind(N)) active#(take(s(M),cons(N,IL))) -> isNatKind#(M) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNatKind(M)) active#(take(s(M),cons(N,IL))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(take(s(M),cons(N,IL))) -> isNatIListKind#(IL) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(s(M),cons(N,IL))) -> and#(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) active#(take(s(M),cons(N,IL))) -> U91#(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X)) -> active#(X) active#(U12(X)) -> U12#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U32(X)) -> active#(X) active#(U32(X)) -> U32#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(U42(X1,X2)) -> active#(X1) active#(U42(X1,X2)) -> U42#(active(X1),X2) active#(U43(X)) -> active#(X) active#(U43(X)) -> U43#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(U52(X1,X2)) -> active#(X1) active#(U52(X1,X2)) -> U52#(active(X1),X2) active#(U53(X)) -> active#(X) active#(U53(X)) -> U53#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(U63(X)) -> active#(X) active#(U63(X)) -> U63#(active(X)) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(U81(X)) -> active#(X) active#(U81(X)) -> U81#(active(X)) active#(U91(X1,X2,X3,X4)) -> active#(X1) active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U63#(mark(X)) -> U63#(X) U71#(mark(X1),X2) -> U71#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) U81#(mark(X)) -> U81#(X) U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X)) -> proper#(X) proper#(U12(X)) -> U12#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U32(X)) -> proper#(X) proper#(U32(X)) -> U32#(proper(X)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(U42(X1,X2)) -> proper#(X2) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2)) proper#(U43(X)) -> proper#(X) proper#(U43(X)) -> U43#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(U52(X1,X2)) -> proper#(X2) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2)) proper#(U53(X)) -> proper#(X) proper#(U53(X)) -> U53#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(U63(X)) -> proper#(X) proper#(U63(X)) -> U63#(proper(X)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(U81(X)) -> proper#(X) proper#(U81(X)) -> U81#(proper(X)) proper#(U91(X1,X2,X3,X4)) -> proper#(X4) proper#(U91(X1,X2,X3,X4)) -> proper#(X3) proper#(U91(X1,X2,X3,X4)) -> proper#(X2) proper#(U91(X1,X2,X3,X4)) -> proper#(X1) proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIListKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> isNatIListKind#(proper(X)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X)) -> U12#(X) isNatList#(ok(X)) -> isNatList#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) isNat#(ok(X)) -> isNat#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U32#(ok(X)) -> U32#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U43#(ok(X)) -> U43#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U53#(ok(X)) -> U53#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) U63#(ok(X)) -> U63#(X) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) U81#(ok(X)) -> U81#(X) U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4) take#(ok(X1),ok(X2)) -> take#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIListKind#(ok(X)) -> isNatIListKind#(X) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: f68(x,y) -> x f68(x,y) -> y isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U12(mark(X)) -> mark(U12(X)) U12(ok(X)) -> ok(U12(X)) U22(mark(X)) -> mark(U22(X)) U22(ok(X)) -> ok(U22(X)) U32(mark(X)) -> mark(U32(X)) U32(ok(X)) -> ok(U32(X)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U43(ok(X)) -> ok(U43(X)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U53(ok(X)) -> ok(U53(X)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U63(ok(X)) -> ok(U63(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U81(mark(X)) -> mark(U81(X)) U81(ok(X)) -> ok(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) EDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U61(tt(),V1,V2)) -> isNat#(V1) active#(U61(tt(),V1,V2)) -> U62#(isNat(V1),V2) active#(U62(tt(),V2)) -> isNatIList#(V2) active#(U62(tt(),V2)) -> U63#(isNatIList(V2)) active#(U71(tt(),L)) -> length#(L) active#(U71(tt(),L)) -> s#(length(L)) active#(U91(tt(),IL,M,N)) -> take#(M,IL) active#(U91(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(take(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(take(V1,V2))) -> isNatKind#(V1) active#(isNatList(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(take(V1,V2))) -> U61#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(length(cons(N,L))) -> isNatKind#(N) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatKind(N)) active#(length(cons(N,L))) -> isNatIListKind#(L) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNatIListKind(L)) active#(length(cons(N,L))) -> and#(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) active#(length(cons(N,L))) -> U71#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(take(0(),IL)) -> isNatIListKind#(IL) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(0(),IL)) -> U81#(and(isNatIList(IL),isNatIListKind(IL))) active#(take(s(M),cons(N,IL))) -> isNatKind#(N) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatKind(N)) active#(take(s(M),cons(N,IL))) -> isNatKind#(M) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNatKind(M)) active#(take(s(M),cons(N,IL))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(take(s(M),cons(N,IL))) -> isNatIListKind#(IL) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(s(M),cons(N,IL))) -> and#(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) active#(take(s(M),cons(N,IL))) -> U91#(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X)) -> active#(X) active#(U12(X)) -> U12#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U32(X)) -> active#(X) active#(U32(X)) -> U32#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(U42(X1,X2)) -> active#(X1) active#(U42(X1,X2)) -> U42#(active(X1),X2) active#(U43(X)) -> active#(X) active#(U43(X)) -> U43#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(U52(X1,X2)) -> active#(X1) active#(U52(X1,X2)) -> U52#(active(X1),X2) active#(U53(X)) -> active#(X) active#(U53(X)) -> U53#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(U63(X)) -> active#(X) active#(U63(X)) -> U63#(active(X)) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(U81(X)) -> active#(X) active#(U81(X)) -> U81#(active(X)) active#(U91(X1,X2,X3,X4)) -> active#(X1) active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U63#(mark(X)) -> U63#(X) U71#(mark(X1),X2) -> U71#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) U81#(mark(X)) -> U81#(X) U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X)) -> proper#(X) proper#(U12(X)) -> U12#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U32(X)) -> proper#(X) proper#(U32(X)) -> U32#(proper(X)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(U42(X1,X2)) -> proper#(X2) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2)) proper#(U43(X)) -> proper#(X) proper#(U43(X)) -> U43#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(U52(X1,X2)) -> proper#(X2) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2)) proper#(U53(X)) -> proper#(X) proper#(U53(X)) -> U53#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(U63(X)) -> proper#(X) proper#(U63(X)) -> U63#(proper(X)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(U81(X)) -> proper#(X) proper#(U81(X)) -> U81#(proper(X)) proper#(U91(X1,X2,X3,X4)) -> proper#(X4) proper#(U91(X1,X2,X3,X4)) -> proper#(X3) proper#(U91(X1,X2,X3,X4)) -> proper#(X2) proper#(U91(X1,X2,X3,X4)) -> proper#(X1) proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIListKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> isNatIListKind#(proper(X)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X)) -> U12#(X) isNatList#(ok(X)) -> isNatList#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) isNat#(ok(X)) -> isNat#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U32#(ok(X)) -> U32#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U43#(ok(X)) -> U43#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U53#(ok(X)) -> U53#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) U63#(ok(X)) -> U63#(X) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) U81#(ok(X)) -> U81#(X) U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4) take#(ok(X1),ok(X2)) -> take#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIListKind#(ok(X)) -> isNatIListKind#(X) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: f68(x,y) -> x f68(x,y) -> y isNatList(ok(X)) -> ok(isNatList(X)) isNat(ok(X)) -> ok(isNat(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) length(mark(X)) -> mark(length(X)) length(ok(X)) -> ok(length(X)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) and(mark(X1),X2) -> mark(and(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) U12(mark(X)) -> mark(U12(X)) U12(ok(X)) -> ok(U12(X)) U22(mark(X)) -> mark(U22(X)) U22(ok(X)) -> ok(U22(X)) U32(mark(X)) -> mark(U32(X)) U32(ok(X)) -> ok(U32(X)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U43(ok(X)) -> ok(U43(X)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U53(ok(X)) -> ok(U53(X)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U63(ok(X)) -> ok(U63(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U71(mark(X1),X2) -> mark(U71(X1,X2)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) U81(mark(X)) -> mark(U81(X)) U81(ok(X)) -> ok(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) graph: ... Restore Modifier: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U61(tt(),V1,V2)) -> isNat#(V1) active#(U61(tt(),V1,V2)) -> U62#(isNat(V1),V2) active#(U62(tt(),V2)) -> isNatIList#(V2) active#(U62(tt(),V2)) -> U63#(isNatIList(V2)) active#(U71(tt(),L)) -> length#(L) active#(U71(tt(),L)) -> s#(length(L)) active#(U91(tt(),IL,M,N)) -> take#(M,IL) active#(U91(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(take(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(take(V1,V2))) -> isNatKind#(V1) active#(isNatList(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(take(V1,V2))) -> U61#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(length(cons(N,L))) -> isNatKind#(N) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatKind(N)) active#(length(cons(N,L))) -> isNatIListKind#(L) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNatIListKind(L)) active#(length(cons(N,L))) -> and#(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) active#(length(cons(N,L))) -> U71#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(take(0(),IL)) -> isNatIListKind#(IL) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(0(),IL)) -> U81#(and(isNatIList(IL),isNatIListKind(IL))) active#(take(s(M),cons(N,IL))) -> isNatKind#(N) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatKind(N)) active#(take(s(M),cons(N,IL))) -> isNatKind#(M) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNatKind(M)) active#(take(s(M),cons(N,IL))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(take(s(M),cons(N,IL))) -> isNatIListKind#(IL) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(s(M),cons(N,IL))) -> and#(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) active#(take(s(M),cons(N,IL))) -> U91#(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X)) -> active#(X) active#(U12(X)) -> U12#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U32(X)) -> active#(X) active#(U32(X)) -> U32#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(U42(X1,X2)) -> active#(X1) active#(U42(X1,X2)) -> U42#(active(X1),X2) active#(U43(X)) -> active#(X) active#(U43(X)) -> U43#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(U52(X1,X2)) -> active#(X1) active#(U52(X1,X2)) -> U52#(active(X1),X2) active#(U53(X)) -> active#(X) active#(U53(X)) -> U53#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(U63(X)) -> active#(X) active#(U63(X)) -> U63#(active(X)) active#(U71(X1,X2)) -> active#(X1) active#(U71(X1,X2)) -> U71#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(U81(X)) -> active#(X) active#(U81(X)) -> U81#(active(X)) active#(U91(X1,X2,X3,X4)) -> active#(X1) active#(U91(X1,X2,X3,X4)) -> U91#(active(X1),X2,X3,X4) active#(take(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> take#(active(X1),X2) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> take#(X1,active(X2)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U63#(mark(X)) -> U63#(X) U71#(mark(X1),X2) -> U71#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) U81#(mark(X)) -> U81#(X) U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X)) -> proper#(X) proper#(U12(X)) -> U12#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U32(X)) -> proper#(X) proper#(U32(X)) -> U32#(proper(X)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(U42(X1,X2)) -> proper#(X2) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2)) proper#(U43(X)) -> proper#(X) proper#(U43(X)) -> U43#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(U52(X1,X2)) -> proper#(X2) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2)) proper#(U53(X)) -> proper#(X) proper#(U53(X)) -> U53#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(U63(X)) -> proper#(X) proper#(U63(X)) -> U63#(proper(X)) proper#(U71(X1,X2)) -> proper#(X2) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(U81(X)) -> proper#(X) proper#(U81(X)) -> U81#(proper(X)) proper#(U91(X1,X2,X3,X4)) -> proper#(X4) proper#(U91(X1,X2,X3,X4)) -> proper#(X3) proper#(U91(X1,X2,X3,X4)) -> proper#(X2) proper#(U91(X1,X2,X3,X4)) -> proper#(X1) proper#(U91(X1,X2,X3,X4)) -> U91#(proper(X1),proper(X2),proper(X3),proper(X4)) proper#(take(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> take#(proper(X1),proper(X2)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIListKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> isNatIListKind#(proper(X)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X)) -> U12#(X) isNatList#(ok(X)) -> isNatList#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) isNat#(ok(X)) -> isNat#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U32#(ok(X)) -> U32#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U43#(ok(X)) -> U43#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U53#(ok(X)) -> U53#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) U63#(ok(X)) -> U63#(X) U71#(ok(X1),ok(X2)) -> U71#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) U81#(ok(X)) -> U81#(X) U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4) take#(ok(X1),ok(X2)) -> take#(X1,X2) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIListKind#(ok(X)) -> isNatIListKind#(X) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) SCC Processor: #sccs: 31 #rules: 125 #arcs: 6946/63504 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(isNatKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(take(X1,X2)) -> proper#(X1) proper#(take(X1,X2)) -> proper#(X2) proper#(U91(X1,X2,X3,X4)) -> proper#(X1) proper#(U91(X1,X2,X3,X4)) -> proper#(X2) proper#(U91(X1,X2,X3,X4)) -> proper#(X3) proper#(U91(X1,X2,X3,X4)) -> proper#(X4) proper#(U81(X)) -> proper#(X) proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(U71(X1,X2)) -> proper#(X1) proper#(U71(X1,X2)) -> proper#(X2) proper#(U63(X)) -> proper#(X) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U53(X)) -> proper#(X) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(isNatIList(X)) -> proper#(X) proper#(U43(X)) -> proper#(X) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(isNat(X)) -> proper#(X) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(isNatList(X)) -> proper#(X) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(and(X1,X2)) -> active#(X1) active#(take(X1,X2)) -> active#(X2) active#(take(X1,X2)) -> active#(X1) active#(U91(X1,X2,X3,X4)) -> active#(X1) active#(U81(X)) -> active#(X) active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) active#(U71(X1,X2)) -> active#(X1) active#(U63(X)) -> active#(X) active#(U62(X1,X2)) -> active#(X1) active#(U61(X1,X2,X3)) -> active#(X1) active#(U53(X)) -> active#(X) active#(U52(X1,X2)) -> active#(X1) active#(U51(X1,X2,X3)) -> active#(X1) active#(U43(X)) -> active#(X) active#(U42(X1,X2)) -> active#(X1) active#(U41(X1,X2,X3)) -> active#(X1) active#(U32(X)) -> active#(X) active#(U31(X1,X2)) -> active#(X1) active#(U22(X)) -> active#(X) active#(U21(X1,X2)) -> active#(X1) active#(U12(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isNatList#(ok(X)) -> isNatList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isNatKind#(ok(X)) -> isNatKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isNat#(ok(X)) -> isNat#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isNatIListKind#(ok(X)) -> isNatIListKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: isNatIList#(ok(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U12#(ok(X)) -> U12#(X) U12#(mark(X)) -> U12#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U22#(ok(X)) -> U22#(X) U22#(mark(X)) -> U22#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U32#(ok(X)) -> U32#(X) U32#(mark(X)) -> U32#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U42#(mark(X1),X2) -> U42#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U43#(ok(X)) -> U43#(X) U43#(mark(X)) -> U43#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U53#(ok(X)) -> U53#(X) U53#(mark(X)) -> U53#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U62#(ok(X1),ok(X2)) -> U62#(X1,X2) U62#(mark(X1),X2) -> U62#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U63#(ok(X)) -> U63#(X) U63#(mark(X)) -> U63#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U71#(ok(X1),ok(X2)) -> U71#(X1,X2) U71#(mark(X1),X2) -> U71#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: length#(ok(X)) -> length#(X) length#(mark(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U81#(ok(X)) -> U81#(X) U81#(mark(X)) -> U81#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: U91#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U91#(X1,X2,X3,X4) U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: take#(ok(X1),ok(X2)) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(mark(X1),X2) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(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(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(U63(X)) -> U63(active(X)) active(U71(X1,X2)) -> U71(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U81(X)) -> U81(active(X)) active(U91(X1,X2,X3,X4)) -> U91(active(X1),X2,X3,X4) active(take(X1,X2)) -> take(active(X1),X2) active(take(X1,X2)) -> take(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) U63(mark(X)) -> mark(U63(X)) U71(mark(X1),X2) -> mark(U71(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U81(mark(X)) -> mark(U81(X)) U91(mark(X1),X2,X3,X4) -> mark(U91(X1,X2,X3,X4)) take(mark(X1),X2) -> mark(take(X1,X2)) take(X1,mark(X2)) -> mark(take(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(U63(X)) -> U63(proper(X)) proper(U71(X1,X2)) -> U71(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U81(X)) -> U81(proper(X)) proper(nil()) -> ok(nil()) proper(U91(X1,X2,X3,X4)) -> U91(proper(X1),proper(X2),proper(X3),proper(X4)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) U63(ok(X)) -> ok(U63(X)) U71(ok(X1),ok(X2)) -> ok(U71(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U81(ok(X)) -> ok(U81(X)) U91(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U91(X1,X2,X3,X4)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open