MAYBE Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Proof: DP Processor: DPs: a__U11#(tt(),V1) -> a__isNatList#(V1) a__U11#(tt(),V1) -> a__U12#(a__isNatList(V1)) a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__U31#(tt(),V) -> a__isNatList#(V) a__U31#(tt(),V) -> a__U32#(a__isNatList(V)) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) a__U42#(tt(),V2) -> a__U43#(a__isNatIList(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U52#(tt(),V2) -> a__U53#(a__isNatList(V2)) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__U62#(tt(),V2) -> a__U63#(a__isNatIList(V2)) a__U71#(tt(),L) -> mark#(L) a__U71#(tt(),L) -> a__length#(mark(L)) a__U91#(tt(),IL,M,N) -> mark#(N) a__and#(tt(),X) -> mark#(X) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(0(),IL) -> a__U81#(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) a__take#(s(M),cons(N,IL)) -> a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> a__U22#(mark(X)) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> a__U32#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> a__U43#(mark(X)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> a__U53#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> a__U63#(mark(X)) mark#(U71(X1,X2)) -> mark#(X1) mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> a__U81#(mark(X)) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) TDG Processor: DPs: a__U11#(tt(),V1) -> a__isNatList#(V1) a__U11#(tt(),V1) -> a__U12#(a__isNatList(V1)) a__U21#(tt(),V1) -> a__isNat#(V1) a__U21#(tt(),V1) -> a__U22#(a__isNat(V1)) a__U31#(tt(),V) -> a__isNatList#(V) a__U31#(tt(),V) -> a__U32#(a__isNatList(V)) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) a__U42#(tt(),V2) -> a__U43#(a__isNatIList(V2)) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U52#(tt(),V2) -> a__U53#(a__isNatList(V2)) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__U62#(tt(),V2) -> a__U63#(a__isNatIList(V2)) a__U71#(tt(),L) -> mark#(L) a__U71#(tt(),L) -> a__length#(mark(L)) a__U91#(tt(),IL,M,N) -> mark#(N) a__and#(tt(),X) -> mark#(X) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(0(),IL) -> a__U81#(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) a__take#(s(M),cons(N,IL)) -> a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark#(zeros()) -> a__zeros#() mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> a__U22#(mark(X)) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> a__U32#(mark(X)) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> a__U43#(mark(X)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> a__U53#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> a__U63#(mark(X)) mark#(U71(X1,X2)) -> mark#(X1) mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> a__U81#(mark(X)) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) graph: ... SCC Processor: #sccs: 1 #rules: 90 #arcs: 1552/11025 DPs: a__take#(s(M),cons(N,IL)) -> a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) mark#(U71(X1,X2)) -> mark#(X1) mark#(U71(X1,X2)) -> a__U71#(mark(X1),X2) a__U71#(tt(),L) -> mark#(L) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and( and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = x1, [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__U91#](x0, x1, x2, x3) = x3, [a__length#](x0) = x0, [mark#](x0) = x0, [a__U71#](x0, x1) = x1, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [U81](x0) = x0, [U71](x0, x1) = x0 + x1 + 1, [U63](x0) = x0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = x0, [a__take](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = x0 + 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [a__U91](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [nil] = 0, [a__U81](x0) = x0, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U71](x0, x1) = x0 + x1 + 1, [a__U63](x0) = x0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = x0, [a__U52](x0, x1) = x0, [a__U51](x0, x1, x2) = x0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = x0, [a__U31](x0, x1) = x0, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = x0, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = IL + N >= 0 = a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))) a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= 0 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) mark#(U71(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U71(X1,X2)) = X1 + X2 + 1 >= X2 = a__U71#(mark(X1),X2) a__U71#(tt(),L) = L >= L = mark#(L) mark#(length(X)) = X + 1 >= X = mark#(X) mark#(length(X)) = X + 1 >= X = a__length#(mark(X)) a__length#(cons(N,L)) = L + N >= 0 = a__isNatList#(L) a__length#(cons(N,L)) = L + N >= 0 = a__and#(a__isNatList(L),isNatIListKind(L)) a__length#(cons(N,L)) = L + N >= 0 = a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__length#(cons(N,L)) = L + N >= L = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = L >= L = a__length#(mark(L)) mark#(U81(X)) = X >= X = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X4 = a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) = N >= N = mark#(N) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X2 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = IL >= 0 = a__isNatIList#(IL) a__take#(0(),IL) = IL >= 0 = a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) = IL + N >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = IL + N >= 0 = a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) = IL + N >= N = a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind ( M)), and(isNat(N), isNatKind ( N)))), IL,M,N) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + M + N >= IL + M + N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + M + N >= IL + M + N = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 >= X1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 >= X1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 >= X1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 >= X1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 >= X1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 >= X1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 >= X1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U71(mark(X1),X2) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(U81(X)) = X >= X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X1 >= X1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 >= X1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U71(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) a__U81(X) = X >= X = U81(X) a__U91(X1,X2,X3,X4) = X1 + X2 + X3 + X4 >= X1 + X2 + X3 + X4 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__take#(s(M),cons(N,IL)) -> a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__U71#(tt(),L) -> mark#(L) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__and#(a__isNatList(L),isNatIListKind(L)) a__length#(cons(N,L)) -> a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and (and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and (and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = 0, [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__U91#](x0, x1, x2, x3) = 0, [a__length#](x0) = 1, [mark#](x0) = 0, [a__U71#](x0, x1) = 1, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = 0, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = 0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = 0 >= 0 = a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))) a__and#(tt(),X) = 0 >= 0 = mark#(X) mark#(U11(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U11(X1,X2)) = 0 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U31(X1,X2)) = 0 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U41(X1,X2,X3)) = 0 >= 0 = mark#(X1) mark#(U41(X1,X2,X3)) = 0 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U42(X1,X2)) = 0 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = 0 >= 0 = mark#(X1) mark#(U51(X1,X2,X3)) = 0 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U52(X1,X2)) = 0 >= 0 = a__U52#(mark(X1),X2) mark#(U53(X)) = 0 >= 0 = mark#(X) mark#(U61(X1,X2,X3)) = 0 >= 0 = mark#(X1) mark#(U61(X1,X2,X3)) = 0 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U62(X1,X2)) = 0 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = 0 >= 0 = mark#(X) a__U71#(tt(),L) = 1 >= 0 = mark#(L) a__length#(cons(N,L)) = 1 >= 0 = a__isNatList#(L) a__length#(cons(N,L)) = 1 >= 0 = a__and#(a__isNatList(L),isNatIListKind(L)) a__length#(cons(N,L)) = 1 >= 0 = a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(U81(X)) = 0 >= 0 = mark#(X) mark#(U91(X1,X2,X3,X4)) = 0 >= 0 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) = 0 >= 0 = mark#(N) mark#(take(X1,X2)) = 0 >= 0 = mark#(X2) mark#(take(X1,X2)) = 0 >= 0 = mark#(X1) mark#(take(X1,X2)) = 0 >= 0 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = 0 >= 0 = a__isNatIList#(IL) a__take#(0(),IL) = 0 >= 0 = a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) = 0 >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = 0 >= 0 = a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) = 0 >= 0 = a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind ( M)), and(isNat(N), isNatKind ( N)))), IL,M,N) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = 0 >= 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__take#(s(M),cons(N,IL)) -> a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and ( and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and (and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = x0 + x1, [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__U91#](x0, x1, x2, x3) = x3, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x0 + x2 + x3, [U81](x0) = x0, [U71](x0, x1) = 1, [U63](x0) = x0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = x0, [a__take](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [a__U91](x0, x1, x2, x3) = x0 + x2 + x3, [nil] = 0, [a__U81](x0) = x0, [s](x0) = x0, [a__length](x0) = 1, [mark](x0) = x0, [a__U71](x0, x1) = 1, [a__U63](x0) = x0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = x0, [a__U52](x0, x1) = x0, [a__U51](x0, x1, x2) = x0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = x0, [a__U31](x0, x1) = x0, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = x0, [tt] = 0, [cons](x0, x1) = x0, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__take#(s(M),cons(N,IL)) = M + N >= 0 = a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))) a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= 0 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(U81(X)) = X >= X = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + X3 + X4 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X3 + X4 >= X4 = a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) = N >= N = mark#(N) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take#(mark(X1),mark(X2)) a__take#(0(),IL) = IL + 1 >= 0 = a__isNatIList#(IL) a__take#(0(),IL) = IL + 1 >= 0 = a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) = M + N >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = M + N >= 0 = a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) = M + N >= N = a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind ( M)), and(isNat(N), isNatKind ( N)))), IL,M,N) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = M + N >= N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = M + N >= M + N = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = X1 >= X1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 >= X1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 >= X1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 >= X1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 >= X1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 >= X1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 >= X1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = 1 >= 1 = a__U71(mark(X1),X2) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(U81(X)) = X >= X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X3 + X4 >= X1 + X3 + X4 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X1 >= X1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 >= X1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 1 >= 1 = U71(X1,X2) a__length(X) = 1 >= 1 = length(X) a__U81(X) = X >= X = U81(X) a__U91(X1,X2,X3,X4) = X1 + X3 + X4 >= X1 + X3 + X4 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__take#(s(M),cons(N,IL)) -> a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))) a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) -> a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and ( and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = x1 + 1, [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__U91#](x0, x1, x2, x3) = x3, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = x0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = x0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = x0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = x0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = x0, [U91](x0, x1, x2, x3) = x0 + x3, [U81](x0) = x0, [U71](x0, x1) = 0, [U63](x0) = x0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = x0, [a__take](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [a__U91](x0, x1, x2, x3) = x0 + x3, [nil] = 0, [a__U81](x0) = x0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = x0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = x0, [a__U52](x0, x1) = x0, [a__U51](x0, x1, x2) = x0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = x0, [a__U31](x0, x1) = x0, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = x0, [tt] = 0, [cons](x0, x1) = x0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = N + 1 >= 0 = a__and#(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))) a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= X1 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= X1 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(U81(X)) = X >= X = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + X4 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X4 >= X4 = a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) = N >= N = mark#(N) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__take#(mark(X1),mark(X2)) a__take#(s(M),cons(N,IL)) = N + 1 >= 0 = a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) = N + 1 >= 0 = a__and#(a__isNatIList(IL),isNatIListKind(IL)) a__take#(s(M),cons(N,IL)) = N + 1 >= N = a__U91#(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind ( M)), and(isNat(N), isNatKind ( N)))), IL,M,N) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = N >= N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = M + N + 1 >= N = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 >= X1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 >= X1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 >= X1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 >= X1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 >= X1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 >= X1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 >= X1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = X >= X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X4 >= X1 + X4 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X1 >= X1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 >= X1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = X >= X = U81(X) a__U91(X1,X2,X3,X4) = X1 + X4 >= X1 + X4 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__take#](x0, x1) = 0, [a__isNatKind#](x0) = 1, [a__isNatIListKind#](x0) = 1, [a__and#](x0, x1) = 1, [a__U91#](x0, x1, x2, x3) = 1, [a__length#](x0) = 1, [mark#](x0) = 1, [a__U71#](x0, x1) = 1, [a__U62#](x0, x1) = 1, [a__U61#](x0, x1, x2) = 1, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = 1, [a__isNatIList#](x0) = 1, [a__U42#](x0, x1) = 1, [a__U41#](x0, x1, x2) = 1, [a__U31#](x0, x1) = 1, [a__isNat#](x0) = 1, [a__U21#](x0, x1) = 1, [a__isNatList#](x0) = 1, [a__U11#](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = 0, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = 0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 1 >= 1 = mark#(X) mark#(U11(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2)) = 1 >= 1 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 1 >= 1 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1 >= 1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 1 >= 1 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 1 >= 1 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 1 >= 1 = a__isNatIList#(V2) a__isNatIList#(V) = 1 >= 1 = a__isNatIListKind#(V) a__isNatIList#(V) = 1 >= 1 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 1 >= 1 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 1 >= 1 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 1 >= 1 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 1 >= 1 = a__isNatIList#(V2) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(isNatList(X)) = 1 >= 1 = a__isNatList#(X) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = a__U21#(mark(X1),X2) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = a__U31#(mark(X1),X2) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = a__U42#(mark(X1),X2) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(isNatIList(X)) = 1 >= 1 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = a__U52#(mark(X1),X2) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = a__U62#(mark(X1),X2) mark#(U63(X)) = 1 >= 1 = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(U81(X)) = 1 >= 1 = mark#(X) mark#(U91(X1,X2,X3,X4)) = 1 >= 1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = 1 >= 1 = a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) = 1 >= 1 = mark#(N) mark#(take(X1,X2)) = 1 >= 0 = a__take#(mark(X1),mark(X2)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 1 >= 1 = a__isNatKind#(X) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) mark#(s(X)) = 1 >= 1 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = 0 >= 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__U91#](x0, x1, x2, x3) = x3, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x0 + x3, [U81](x0) = x0 + 1, [U71](x0, x1) = 1, [U63](x0) = x0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = x0, [a__take](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [a__U91](x0, x1, x2, x3) = x0 + x3, [nil] = 0, [a__U81](x0) = x0 + 1, [s](x0) = x0, [a__length](x0) = 1, [mark](x0) = x0, [a__U71](x0, x1) = 1, [a__U63](x0) = x0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = x0, [a__U52](x0, x1) = x0, [a__U51](x0, x1, x2) = x0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = x0, [a__U31](x0, x1) = x0, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = x0, [tt] = 0, [cons](x0, x1) = x0, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= 0 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(U81(X)) = X + 1 >= X = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + X4 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X4 >= X4 = a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) = N >= N = mark#(N) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 0 = nil() a__U91(tt(),IL,M,N) = N >= N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 2 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = M + N + 1 >= N = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = X1 >= X1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 >= X1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 >= X1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 >= X1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 >= X1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 >= X1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 >= X1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = 1 >= 1 = a__U71(mark(X1),X2) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(U81(X)) = X + 1 >= X + 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X4 >= X1 + X4 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X1 >= X1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 >= X1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 1 >= 1 = U71(X1,X2) a__length(X) = 1 >= 1 = length(X) a__U81(X) = X + 1 >= X + 1 = U81(X) a__U91(X1,X2,X3,X4) = X1 + X4 >= X1 + X4 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__U91#](x0, x1, x2, x3) = x3, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = x0, [a__U51#](x0, x1, x2) = x0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = x0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = x0, [U91](x0, x1, x2, x3) = x0 + x3 + 1, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = x0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = x0, [a__take](x0, x1) = x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [a__U91](x0, x1, x2, x3) = x0 + x3 + 1, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = x0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = x0, [a__U52](x0, x1) = x0, [a__U51](x0, x1, x2) = x0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = x0, [a__U31](x0, x1) = x0, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = x0, [tt] = 0, [cons](x0, x1) = x0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= X1 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= X1 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(U91(X1,X2,X3,X4)) = X1 + X4 + 1 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X4 + 1 >= X4 = a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) = N >= N = mark#(N) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = N + 1 >= N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = N + 1 >= N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 >= X1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 >= X1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 >= X1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 >= X1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 >= X1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 >= X1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 >= X1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X4 + 1 >= X1 + X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X1 >= X1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 >= X1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X1 + X4 + 1 >= X1 + X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 + 1 >= X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) a__U91#(tt(),IL,M,N) -> mark#(N) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__U91#](x0, x1, x2, x3) = 1, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = 1, [U81](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = x0 + 1, [and](x0, x1) = x1, [isNatKind](x0) = 0, [isNat](x0) = x0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = x0, [a__and](x0, x1) = x1, [take](x0, x1) = x0 + 1, [a__U91](x0, x1, x2, x3) = 1, [nil] = 1, [a__U81](x0) = 1, [s](x0) = 0, [a__length](x0) = x0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = x0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 1, [a__zeros] = 0 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) mark#(U11(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U11(X1,X2)) = 0 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U31(X1,X2)) = 0 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U41(X1,X2,X3)) = 0 >= 0 = mark#(X1) mark#(U41(X1,X2,X3)) = 0 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U42(X1,X2)) = 0 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = 0 >= 0 = mark#(X1) mark#(U51(X1,X2,X3)) = 0 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U52(X1,X2)) = 0 >= 0 = a__U52#(mark(X1),X2) mark#(U53(X)) = 0 >= 0 = mark#(X) mark#(U61(X1,X2,X3)) = 0 >= 0 = mark#(X1) mark#(U61(X1,X2,X3)) = 0 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U62(X1,X2)) = 0 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) a__U91#(tt(),IL,M,N) = 1 >= 0 = mark#(N) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 1 = nil() a__U91(tt(),IL,M,N) = 1 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(length(V1)) = V1 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 2 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 1 >= 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + 1 >= X1 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = X >= X = length(X) a__U81(X) = 1 >= 1 = U81(X) a__U91(X1,X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + 1 >= X1 + 1 = take(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 1, [mark#](x0) = x0, [a__U71#](x0, x1) = 1, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = x0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x3 + 1, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = x0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [U21](x0, x1) = x0, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = x0, [a__take](x0, x1) = x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x1, [a__U91](x0, x1, x2, x3) = x3 + 1, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = x0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = x0, [a__U52](x0, x1) = x0, [a__U51](x0, x1, x2) = x0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = x0, [a__U31](x0, x1) = x0, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = x0, [tt] = 0, [cons](x0, x1) = x0 + 1, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = 0 >= 0 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 0 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = 0 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = 0 >= 0 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(cons(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = N + 1 >= N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = N + 1 >= N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = X1 >= X1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 >= X1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 >= X1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 >= X1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 >= X1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 >= X1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 >= X1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 >= X2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X1 >= X1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 >= X1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 >= X2 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatList#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) -> a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = x1, [a__U61#](x0, x1, x2) = x1 + x2, [a__U52#](x0, x1) = x1, [a__U51#](x0, x1, x2) = x1 + x2, [a__isNatIList#](x0) = x0, [a__U42#](x0, x1) = x1, [a__U41#](x0, x1, x2) = x1 + x2, [a__U31#](x0, x1) = x1, [a__isNat#](x0) = x0, [a__U21#](x0, x1) = x1, [a__isNatList#](x0) = x0, [a__U11#](x0, x1) = x1, [U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [U81](x0) = x0 + 1, [U71](x0, x1) = x1, [U63](x0) = x0, [U62](x0, x1) = x0 + x1, [U61](x0, x1, x2) = x0 + x1 + x2, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [isNatIList](x0) = x0, [U43](x0) = x0, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x0 + x1, [a__take](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = x0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = x0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [a__U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [nil] = 1, [a__U81](x0) = x0 + 1, [s](x0) = x0, [a__length](x0) = x0, [mark](x0) = x0, [a__U71](x0, x1) = x1, [a__U63](x0) = x0, [a__U62](x0, x1) = x0 + x1, [a__U61](x0, x1, x2) = x0 + x1 + x2, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1, [a__U51](x0, x1, x2) = x0 + x1 + x2, [a__U43](x0) = x0, [a__isNatIList](x0) = x0, [a__U42](x0, x1) = x0 + x1, [a__U41](x0, x1, x2) = x0 + x1 + x2, [a__U32](x0) = x0, [a__U31](x0, x1) = x0 + x1, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x0 + x1, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 + X2 >= X2 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = V1 >= V1 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = V1 + V2 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V1 + V2 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = V1 + V2 >= V1 = a__isNat#(V1) a__isNat#(length(V1)) = V1 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = V1 >= V1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = V1 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = V1 >= V1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 >= V1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = V1 + V2 >= V2 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = V2 >= V2 = a__isNatList#(V2) a__isNatList#(take(V1,V2)) = V1 + V2 + 1 >= 0 = a__isNatKind#(V1) a__isNatList#(take(V1,V2)) = V1 + V2 + 1 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(take(V1,V2)) = V1 + V2 + 1 >= V1 + V2 = a__U61#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U61#(tt(),V1,V2) = V1 + V2 >= V1 = a__isNat#(V1) a__U61#(tt(),V1,V2) = V1 + V2 >= V2 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = V2 >= V2 = a__isNatIList#(V2) a__isNatIList#(V) = V >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = V >= V = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = V >= V = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = V1 + V2 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = V1 + V2 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = V1 + V2 >= V1 = a__isNat#(V1) a__U41#(tt(),V1,V2) = V1 + V2 >= V2 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = V2 >= V2 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = X >= X = a__isNatList#(X) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X2 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = X >= X = a__isNat#(X) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X2 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X2 + X3 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 + X2 >= X2 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = X >= X = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X2 + X3 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X2 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X2 + X3 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 + X2 >= X2 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V >= V = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 >= V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = V2 >= V2 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = L >= L = s(a__length(mark(L))) a__U81(tt()) = 1 >= 1 = nil() a__U91(tt(),IL,M,N) = IL + M + N + 1 >= IL + M + N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = V1 >= V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V >= V = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = V1 + V2 + 1 >= V1 + V2 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N >= L = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= IL + 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 + X2 >= X1 + X2 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = X >= X = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 + X2 >= X1 + X2 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = X2 >= X2 = a__U71(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(U81(X)) = X + 1 >= X + 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) a__U62(X1,X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = X2 >= X2 = U71(X1,X2) a__length(X) = X >= X = length(X) a__U81(X) = X + 1 >= X + 1 = U81(X) a__U91(X1,X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U61#(tt(),V1,V2) -> a__isNat#(V1) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatIListKind#(V) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__isNat#(V1) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = 1, [a__U61#](x0, x1, x2) = x1 + 1, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 1, [a__U42#](x0, x1) = 1, [a__U41#](x0, x1, x2) = 1, [a__U31#](x0, x1) = 1, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x1 + x2 + x3, [U81](x0) = 0, [U71](x0, x1) = x1 + 1, [U63](x0) = x0, [U62](x0, x1) = x0 + x1 + 1, [U61](x0, x1, x2) = x0 + x1 + x2 + 1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1 + 1, [U51](x0, x1, x2) = x0 + x1 + x2 + 1, [isNatIList](x0) = x0 + 1, [U43](x0) = x0, [U42](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U32](x0) = x0, [U31](x0, x1) = x0 + x1 + 1, [U22](x0) = x0, [U21](x0, x1) = x0 + x1, [isNatList](x0) = x0 + 1, [U12](x0) = x0, [U11](x0, x1) = x0 + x1 + 1, [a__take](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = x0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = x0 + 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [a__U91](x0, x1, x2, x3) = x1 + x2 + x3, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U71](x0, x1) = x1 + 1, [a__U63](x0) = x0, [a__U62](x0, x1) = x0 + x1 + 1, [a__U61](x0, x1, x2) = x0 + x1 + x2 + 1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0 + x1 + 1, [a__U51](x0, x1, x2) = x0 + x1 + x2 + 1, [a__U43](x0) = x0, [a__isNatIList](x0) = x0 + 1, [a__U42](x0, x1) = x0 + x1 + 1, [a__U41](x0, x1, x2) = x0 + x1 + x2 + 1, [a__U32](x0) = x0, [a__U31](x0, x1) = x0 + x1 + 1, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x0 + x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0 + 1, [a__U11](x0, x1) = x0 + x1 + 1, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = X >= X = mark#(X) mark#(U11(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 + X2 + 1 >= 0 = a__U11#(mark(X1),X2) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__U61#(tt(),V1,V2) = V1 + 1 >= 0 = a__isNat#(V1) a__U61#(tt(),V1,V2) = V1 + 1 >= 1 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 1 >= 1 = a__isNatIList#(V2) a__isNatIList#(V) = 1 >= 0 = a__isNatIListKind#(V) a__isNatIList#(V) = 1 >= 1 = a__U31#(a__isNatIListKind(V),V) a__U31#(tt(),V) = 1 >= 0 = a__isNatList#(V) a__isNatIList#(cons(V1,V2)) = 1 >= 0 = a__isNatKind#(V1) a__isNatIList#(cons(V1,V2)) = 1 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIList#(cons(V1,V2)) = 1 >= 1 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 1 >= 0 = a__isNat#(V1) a__U41#(tt(),V1,V2) = 1 >= 1 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 1 >= 1 = a__isNatIList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(isNatList(X)) = X + 1 >= 0 = a__isNatList#(X) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = X >= 0 = a__isNat#(X) mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 + 1 >= 1 = a__U31#(mark(X1),X2) mark#(U32(X)) = X >= X = mark#(X) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= 1 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 + X2 + 1 >= 1 = a__U42#(mark(X1),X2) mark#(U43(X)) = X >= X = mark#(X) mark#(isNatIList(X)) = X + 1 >= 1 = a__isNatIList#(X) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= 0 = a__U51#(mark(X1),X2,X3) mark#(U52(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 + 1 >= 0 = a__U52#(mark(X1),X2) mark#(U53(X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + 1 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 + X2 + 1 >= 1 = a__U62#(mark(X1),X2) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = V1 + 1 >= V1 + 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V + 1 >= V + 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 + 1 >= V2 + 1 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 + 1 >= V2 + 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = V2 + 1 >= V2 + 1 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + M + N >= IL + M + N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = V1 + 1 >= V1 + 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V + 1 >= V + 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 2 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + M + N >= IL + M + N = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X + 1 >= X + 1 = a__isNatList(X) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = X + 1 >= X + 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = X2 + 1 >= X2 + 1 = a__U71(mark(X1),X2) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X + 1 >= X + 1 = isNatList(X) a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = X + 1 >= X + 1 = isNatIList(X) a__U51(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) mark#(U32(X)) -> mark#(X) mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) -> a__U42#(mark(X1),X2) mark#(U43(X)) -> mark#(X) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U53(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 1, [a__isNatIListKind#](x0) = 1, [a__and#](x0, x1) = 1, [a__length#](x0) = 0, [mark#](x0) = 1, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = 1, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 1, [a__U21#](x0, x1) = 1, [a__isNatList#](x0) = 1, [a__U11#](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = x1, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = 0, [take](x0, x1) = x1, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = 0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 1 >= 1 = mark#(X) a__U11#(tt(),V1) = 1 >= 1 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1 >= 1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) a__U61#(tt(),V1,V2) = 0 >= 0 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = a__U21#(mark(X1),X2) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(U31(X1,X2)) = 1 >= 0 = a__U31#(mark(X1),X2) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U41(X1,X2,X3)) = 1 >= 0 = a__U41#(mark(X1),X2,X3) mark#(U42(X1,X2)) = 1 >= 0 = a__U42#(mark(X1),X2) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(isNatIList(X)) = 1 >= 0 = a__isNatIList#(X) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 0 = a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) = 1 >= 0 = a__U62#(mark(X1),X2) mark#(U63(X)) = 1 >= 1 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 1 >= 1 = a__isNatKind#(X) mark#(s(X)) = 1 >= 1 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = 0 >= 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = X >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 >= X2 = take(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U62#](x0, x1) = x1 + 1, [a__U61#](x0, x1, x2) = x2 + 1, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = 0, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = 0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = x0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__U61#(tt(),V1,V2) = V2 + 1 >= V2 + 1 = a__U62#(a__isNat(V1),V2) a__U62#(tt(),V2) = V2 + 1 >= 0 = a__isNatIList#(V2) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) mark#(U63(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = 0 >= 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U61#(tt(),V1,V2) -> a__U62#(a__isNat(V1),V2) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 1, [mark#](x0) = 0, [a__U71#](x0, x1) = 1, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = x1 + 1, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = 0, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = 1, [U81](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [isNatIList](x0) = 1, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 1, [U32](x0) = x0, [U31](x0, x1) = 1, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 1, [U12](x0) = 0, [U11](x0, x1) = x1 + 1, [a__take](x0, x1) = 1, [and](x0, x1) = x1, [isNatKind](x0) = 0, [isNat](x0) = x0 + 1, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = x0, [a__and](x0, x1) = x1, [take](x0, x1) = 1, [a__U91](x0, x1, x2, x3) = 1, [nil] = 0, [a__U81](x0) = 1, [s](x0) = 0, [a__length](x0) = x0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = x0, [a__U52](x0, x1) = 1, [a__U51](x0, x1, x2) = 1, [a__U43](x0) = 0, [a__isNatIList](x0) = 1, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 1, [a__U32](x0) = x0, [a__U31](x0, x1) = 1, [a__U22](x0) = 0, [a__isNat](x0) = x0 + 1, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 1, [a__U11](x0, x1) = x1 + 1, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__U61#(tt(),V1,V2) = V1 + 1 >= 0 = a__U62#(a__isNat(V1),V2) a__isNatIList#(V) = 0 >= 0 = a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) = 0 >= 0 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 0 >= 0 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 0 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) mark#(U63(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 + 1 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 1 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 1 >= 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 1 >= 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 0 = nil() a__U91(tt(),IL,M,N) = 1 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(length(V1)) = V1 + 1 >= V1 + 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 1 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 1 >= 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X2 + 1 >= X2 + 1 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = X + 1 >= X + 1 = a__isNat(X) mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 1 >= 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 1 >= 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X2 + 1 >= X2 + 1 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = X + 1 >= X + 1 = isNat(X) a__U31(X1,X2) = 1 >= 1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = 1 >= 1 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2,X3) = 1 >= 1 = U51(X1,X2,X3) a__U52(X1,X2) = 1 >= 1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = X >= X = length(X) a__U81(X) = 1 >= 1 = U81(X) a__U91(X1,X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = 1 >= 1 = take(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatIList#(V) -> a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 1, [a__isNatIListKind#](x0) = 1, [a__and#](x0, x1) = 1, [a__length#](x0) = 1, [mark#](x0) = 1, [a__U71#](x0, x1) = 1, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = 1, [a__isNatIList#](x0) = 1, [a__U42#](x0, x1) = 1, [a__U41#](x0, x1, x2) = 1, [a__U31#](x0, x1) = 0, [a__isNat#](x0) = 1, [a__U21#](x0, x1) = 1, [a__isNatList#](x0) = 1, [a__U11#](x0, x1) = 1, [U91](x0, x1, x2, x3) = x3 + 1, [U81](x0) = x0, [U71](x0, x1) = 1, [U63](x0) = x0, [U62](x0, x1) = x1 + 1, [U61](x0, x1, x2) = x2 + 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = x0 + 1, [U43](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x1, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = x0 + 1, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = x0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [a__U91](x0, x1, x2, x3) = x3 + 1, [nil] = 0, [a__U81](x0) = x0, [s](x0) = 0, [a__length](x0) = 1, [mark](x0) = x0, [a__U71](x0, x1) = 1, [a__U63](x0) = x0, [a__U62](x0, x1) = x1 + 1, [a__U61](x0, x1, x2) = x2 + 1, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = x0 + 1, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = x1, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = x0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = x0 + 1, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = x0, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__and#(tt(),X) = 1 >= 1 = mark#(X) a__U11#(tt(),V1) = 1 >= 1 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1 >= 1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) a__isNatIList#(V) = 1 >= 0 = a__U31#(a__isNatIListKind(V),V) a__isNatIList#(cons(V1,V2)) = 1 >= 1 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = 1 >= 1 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 1 >= 1 = a__isNatIList#(V2) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = a__U21#(mark(X1),X2) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 1 >= 1 = a__isNatKind#(X) mark#(s(X)) = 1 >= 1 = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = V1 >= V1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = V2 + 1 >= V2 + 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = V2 + 1 >= V2 + 1 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 1 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = N + 1 >= N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(length(V1)) = 1 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V + 1 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 2 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + 1 >= V1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + 1 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = V1 + V2 + 2 >= V2 + 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 2 >= IL + 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = N + 1 >= N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = X + 1 >= X + 1 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X2 >= X2 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 >= X1 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = X + 1 >= X + 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X2 + 1 >= X2 + 1 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = 1 >= 1 = a__U71(mark(X1),X2) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(U81(X)) = X >= X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = X + 1 >= X + 1 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = X2 >= X2 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = X + 1 >= X + 1 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) a__U62(X1,X2) = X2 + 1 >= X2 + 1 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 1 >= 1 = U71(X1,X2) a__length(X) = 1 >= 1 = length(X) a__U81(X) = X >= X = U81(X) a__U91(X1,X2,X3,X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = x0, [mark#](x0) = 0, [a__U71#](x0, x1) = x1 + 1, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = x0 + 1, [a__U42#](x0, x1) = x1 + 1, [a__U41#](x0, x1, x2) = x2 + 1, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x1 + 1, [U81](x0) = 0, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 1, [U12](x0) = 1, [U11](x0, x1) = 0, [a__take](x0, x1) = x1, [and](x0, x1) = x1 + 1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = 1, [length](x0) = 1, [a__and](x0, x1) = x1 + 1, [take](x0, x1) = x1, [a__U91](x0, x1, x2, x3) = x1 + 1, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 1, [mark](x0) = x0 + 1, [a__U71](x0, x1) = 1, [a__U63](x0) = 1, [a__U62](x0, x1) = 1, [a__U61](x0, x1, x2) = 1, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = x0, [a__U21](x0, x1) = 0, [a__U12](x0) = 1, [a__isNatList](x0) = 1, [a__U11](x0, x1) = 1, [tt] = 0, [cons](x0, x1) = x1 + 1, [zeros] = 0, [0] = 1, [a__zeros] = 1 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__isNatIList#(cons(V1,V2)) = V2 + 2 >= V2 + 1 = a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U41#(tt(),V1,V2) = V2 + 1 >= V2 + 1 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = V2 + 1 >= V2 + 1 = a__isNatIList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) mark#(U63(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = L + 1 >= L + 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = L + 1 >= L + 1 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 1 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 1 >= 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 1 >= 1 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 1 >= 0 = tt() a__U71(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + 1 >= IL + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X + 1 >= X + 1 = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = V1 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 0 = tt() a__isNatIListKind(zeros()) = 1 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 1 >= 0 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = V1 >= V1 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + 1 >= IL + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 1 >= 1 = a__U11(mark(X1),X2) mark(U12(X)) = 2 >= 1 = a__U12(mark(X)) mark(isNatList(X)) = 2 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = 1 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 1 >= 0 = a__U22(mark(X)) mark(isNat(X)) = X + 1 >= X = a__isNat(X) mark(U31(X1,X2)) = 1 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 1 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 1 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 1 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 1 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 1 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 1 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(U63(X)) = 2 >= 1 = a__U63(mark(X)) mark(U71(X1,X2)) = 2 >= 1 = a__U71(mark(X1),X2) mark(length(X)) = 2 >= 1 = a__length(mark(X)) mark(U81(X)) = 1 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + 2 >= X2 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 + 2 >= X2 + 1 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = X + 1 >= X = a__isNatKind(X) mark(cons(X1,X2)) = X2 + 2 >= X2 + 1 = cons(mark(X1),X2) mark(0()) = 2 >= 1 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 1 >= 0 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X1,X2) = 1 >= 0 = U11(X1,X2) a__U12(X) = 1 >= 1 = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 1 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 0 = U62(X1,X2) a__U63(X) = 1 >= 1 = U63(X) a__U71(X1,X2) = 1 >= 1 = U71(X1,X2) a__length(X) = 1 >= 1 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X2 + 1 >= X2 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 >= X2 = take(X1,X2) a__and(X1,X2) = X2 + 1 >= X2 + 1 = and(X1,X2) a__isNatIListKind(X) = 1 >= 0 = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) -> a__isNatIList#(V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNatIList#](x0) = 0, [a__U42#](x0, x1) = 1, [a__U41#](x0, x1, x2) = 1, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = 0, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = 0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__U41#(tt(),V1,V2) = 1 >= 1 = a__U42#(a__isNat(V1),V2) a__U42#(tt(),V2) = 1 >= 0 = a__isNatIList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U21(X1,X2)) = 0 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) mark#(U63(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 0 >= 0 = a__isNatKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = 0 >= 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(isNatKind(X)) -> a__isNatKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__U42#](x0, x1) = x0, [a__U41#](x0, x1, x2) = 1, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x3, [U81](x0) = 0, [U71](x0, x1) = 1, [U63](x0) = x0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = 1, [isNatIList](x0) = 1, [U43](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = 0, [U32](x0) = x0, [U31](x0, x1) = 1, [U22](x0) = x0, [U21](x0, x1) = x0, [isNatList](x0) = 1, [U12](x0) = x0, [U11](x0, x1) = x0, [a__take](x0, x1) = x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 1, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 1, [length](x0) = 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [a__U91](x0, x1, x2, x3) = x3 + 1, [nil] = 0, [a__U81](x0) = 1, [s](x0) = x0, [a__length](x0) = 1, [mark](x0) = x0 + 1, [a__U71](x0, x1) = 1, [a__U63](x0) = x0, [a__U62](x0, x1) = 1, [a__U61](x0, x1, x2) = 1, [a__U53](x0) = x0, [a__U52](x0, x1) = x0, [a__U51](x0, x1, x2) = 1, [a__U43](x0) = x0, [a__isNatIList](x0) = 1, [a__U42](x0, x1) = x0, [a__U41](x0, x1, x2) = 1, [a__U32](x0) = x0, [a__U31](x0, x1) = 1, [a__U22](x0) = x0, [a__isNat](x0) = 1, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 1, [a__U11](x0, x1) = x0, [tt] = 1, [cons](x0, x1) = x0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = X >= X = mark#(X) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) a__U41#(tt(),V1,V2) = 1 >= 1 = a__U42#(a__isNat(V1),V2) mark#(U12(X)) = X >= X = mark#(X) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 0 >= 0 = a__isNat#(X) mark#(U32(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(isNatKind(X)) = 1 >= 0 = a__isNatKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 1 >= 1 = tt() a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNat(V1)) a__U22(tt()) = 1 >= 1 = tt() a__U31(tt(),V) = 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 1 >= 1 = tt() a__U41(tt(),V1,V2) = 1 >= 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 1 >= 1 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 1 >= 1 = tt() a__U51(tt(),V1,V2) = 1 >= 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 1 >= 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 1 >= 1 = tt() a__U61(tt(),V1,V2) = 1 >= 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 1 >= 1 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 1 >= 1 = tt() a__U71(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 0 = nil() a__U91(tt(),IL,M,N) = N + 1 >= N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X + 1 >= X + 1 = mark(X) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 1 = tt() a__isNatIListKind(zeros()) = 1 >= 1 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 1 >= 1 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = N + 1 >= N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 0 = a__zeros() mark(U11(X1,X2)) = X1 + 1 >= X1 + 1 = a__U11(mark(X1),X2) mark(U12(X)) = X + 1 >= X + 1 = a__U12(mark(X)) mark(isNatList(X)) = 2 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = X1 + 1 >= X1 + 1 = a__U21(mark(X1),X2) mark(U22(X)) = X + 1 >= X + 1 = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U31(X1,X2)) = 2 >= 1 = a__U31(mark(X1),X2) mark(U32(X)) = X + 1 >= X + 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 1 >= 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 + 1 >= X1 + 1 = a__U42(mark(X1),X2) mark(U43(X)) = X + 1 >= X + 1 = a__U43(mark(X)) mark(isNatIList(X)) = 2 >= 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 2 >= 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X1 + 1 >= X1 + 1 = a__U52(mark(X1),X2) mark(U53(X)) = X + 1 >= X + 1 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 2 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(U63(X)) = X + 1 >= X + 1 = a__U63(mark(X)) mark(U71(X1,X2)) = 2 >= 1 = a__U71(mark(X1),X2) mark(length(X)) = 2 >= 1 = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 + 2 >= X2 + 2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = 2 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 2 >= 1 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 1 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X1 >= X1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 1 >= 0 = isNat(X) a__U31(X1,X2) = 1 >= 1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = 1 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = X1 >= X1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2,X3) = 1 >= 1 = U51(X1,X2,X3) a__U52(X1,X2) = X1 >= X1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 0 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 1 >= 1 = U71(X1,X2) a__length(X) = 1 >= 1 = length(X) a__U81(X) = 1 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X4 + 1 >= X4 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 + 1 >= X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 1 >= 0 = isNatIListKind(X) a__isNatKind(X) = 1 >= 1 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) a__U41#(tt(),V1,V2) -> a__U42#(a__isNat(V1),V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 1, [a__isNatIListKind#](x0) = 1, [a__and#](x0, x1) = 1, [a__length#](x0) = 0, [mark#](x0) = 1, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = 1, [a__U42#](x0, x1) = 0, [a__U41#](x0, x1, x2) = x2 + 1, [a__isNat#](x0) = 1, [a__U21#](x0, x1) = 1, [a__isNatList#](x0) = 1, [a__U11#](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = 1, [U53](x0) = x0, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [U21](x0, x1) = 1, [isNatList](x0) = 1, [U12](x0) = x0, [U11](x0, x1) = 1, [a__take](x0, x1) = 0, [and](x0, x1) = x1, [isNatKind](x0) = 0, [isNat](x0) = 1, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = x1, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = 1, [a__U53](x0) = x0, [a__U52](x0, x1) = 1, [a__U51](x0, x1, x2) = 1, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = x0, [a__isNat](x0) = 1, [a__U21](x0, x1) = 1, [a__U12](x0) = x0, [a__isNatList](x0) = 1, [a__U11](x0, x1) = 1, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = 1 >= 1 = mark#(X) a__U11#(tt(),V1) = 1 >= 1 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1 >= 1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) a__U41#(tt(),V1,V2) = V2 + 1 >= 0 = a__U42#(a__isNat(V1),V2) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = a__U21#(mark(X1),X2) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(isNat(X)) = 1 >= 1 = a__isNat#(X) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(s(X)) = 1 >= 1 = mark#(X) a__zeros() = 1 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 1 >= 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 1 >= 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 1 >= 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 1 >= 1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 >= X1 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = 1 >= 1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = 1 >= 1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 1 >= 1 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 1 >= 1 = U51(X1,X2,X3) a__U52(X1,X2) = 1 >= 1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U12(X)) -> mark#(X) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2) mark#(U22(X)) -> mark#(X) mark#(isNat(X)) -> a__isNat#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = x0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = x0, [U31](x0, x1) = 0, [U22](x0) = x0, [U21](x0, x1) = x0 + 1, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 1, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = x0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = x0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = x0, [a__U31](x0, x1) = 0, [a__U22](x0) = x0, [a__isNat](x0) = 1, [a__U21](x0, x1) = x0 + 1, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = X >= X = mark#(X) a__U11#(tt(),V1) = 0 >= 0 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 0 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) mark#(U12(X)) = X >= X = mark#(X) mark#(U21(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + 1 >= 0 = a__U21#(mark(X1),X2) mark#(U22(X)) = X >= X = mark#(X) mark#(isNat(X)) = 1 >= 0 = a__isNat#(X) mark#(U32(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 1 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(length(V1)) = 1 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = X1 + 1 >= X1 + 1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = X >= X = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = X1 + 1 >= X1 + 1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 1 >= 1 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = X >= X = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatIListKind#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__isNatKind#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U12(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 1, [mark#](x0) = 0, [a__U71#](x0, x1) = 1, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = 1, [a__isNat#](x0) = 1, [a__U21#](x0, x1) = 1, [a__isNatList#](x0) = 1, [a__U11#](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = x0, [isNatList](x0) = 1, [U12](x0) = x0, [U11](x0, x1) = 1, [a__take](x0, x1) = x0, [and](x0, x1) = x1, [isNatKind](x0) = 1, [isNat](x0) = x0 + 1, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 0, [length](x0) = x0, [a__and](x0, x1) = x1, [take](x0, x1) = x0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 1, [a__U81](x0) = 1, [s](x0) = 0, [a__length](x0) = x0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = x0, [a__U52](x0, x1) = 1, [a__U51](x0, x1, x2) = 1, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = x0 + 1, [a__U21](x0, x1) = x0, [a__U12](x0) = x0, [a__isNatList](x0) = 1, [a__U11](x0, x1) = 1, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__U11#(tt(),V1) = 1 >= 1 = a__isNatList#(V1) a__isNatList#(cons(V1,V2)) = 1 >= 0 = a__isNatKind#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = 1 >= 1 = a__isNat#(V1) a__isNat#(length(V1)) = 1 >= 0 = a__isNatIListKind#(V1) a__isNat#(length(V1)) = 1 >= 1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 1 >= 0 = a__isNatKind#(V1) a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 1 >= 1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) mark#(U63(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 1 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 1 >= 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 1 >= 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 1 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 2 >= 0 = tt() a__isNat(length(V1)) = V1 + 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 1 >= 0 = tt() a__isNatKind(length(V1)) = 1 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 1 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 1 >= 1 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = X + 1 >= X + 1 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 >= X1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = 1 >= 1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = X1 >= X1 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = X + 1 >= X + 1 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 1 >= 1 = U51(X1,X2,X3) a__U52(X1,X2) = 1 >= 1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = X >= X = length(X) a__U81(X) = 1 >= 1 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 >= X1 = take(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 1 >= 1 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U12(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) mark#(U63(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 1, [a__isNatIListKind#](x0) = 1, [a__and#](x0, x1) = x1 + 1, [a__length#](x0) = 0, [mark#](x0) = x0 + 1, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = x1, [a__U51#](x0, x1, x2) = x1 + x2, [a__isNat#](x0) = x0, [a__U21#](x0, x1) = x1, [a__isNatList#](x0) = x0, [a__U11#](x0, x1) = x1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [U81](x0) = 0, [U71](x0, x1) = x1, [U63](x0) = x0 + 1, [U62](x0, x1) = x1 + 1, [U61](x0, x1, x2) = x2 + 1, [U53](x0) = x0, [U52](x0, x1) = x1, [U51](x0, x1, x2) = x2, [isNatIList](x0) = x0, [U43](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x2, [U32](x0) = x0, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x1, [isNatList](x0) = x0, [U12](x0) = x0, [U11](x0, x1) = x1, [a__take](x0, x1) = x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = x0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = x0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [a__U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = x0, [mark](x0) = x0, [a__U71](x0, x1) = x1, [a__U63](x0) = x0 + 1, [a__U62](x0, x1) = x1 + 1, [a__U61](x0, x1, x2) = x2 + 1, [a__U53](x0) = x0, [a__U52](x0, x1) = x1, [a__U51](x0, x1, x2) = x2, [a__U43](x0) = x0, [a__isNatIList](x0) = x0, [a__U42](x0, x1) = x1, [a__U41](x0, x1, x2) = x2, [a__U32](x0) = x0, [a__U31](x0, x1) = x1, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x1, [a__U12](x0) = x0, [a__isNatList](x0) = x0, [a__U11](x0, x1) = x1, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = X + 1 >= X + 1 = mark#(X) a__U11#(tt(),V1) = V1 >= V1 = a__isNatList#(V1) a__isNatKind#(length(V1)) = 1 >= 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = V1 + V2 >= V1 = a__isNat#(V1) a__isNat#(length(V1)) = V1 >= V1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = V1 >= V1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 >= V1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = V1 + V2 >= V2 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = V2 >= V2 = a__isNatList#(V2) mark#(U12(X)) = X + 1 >= X + 1 = mark#(X) mark#(U22(X)) = X + 1 >= X + 1 = mark#(X) mark#(U32(X)) = X + 1 >= X + 1 = mark#(X) mark#(U43(X)) = X + 1 >= X + 1 = mark#(X) mark#(U53(X)) = X + 1 >= X + 1 = mark#(X) mark#(U63(X)) = X + 2 >= X + 1 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = V1 >= V1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V >= V = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = V2 >= V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 >= V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = V2 >= V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = V2 + 1 >= V2 + 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = V2 + 1 >= V2 + 1 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 1 >= 0 = tt() a__U71(tt(),L) = L >= L = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + N + 1 >= IL + N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = V1 >= V1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V >= V = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 >= V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 >= V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = V2 + 1 >= V2 + 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = L + N >= L = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + N + 1 >= IL + N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = X2 >= X2 = a__U11(mark(X1),X2) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X2 >= X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = X2 >= X2 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X3 >= X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X2 >= X2 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = X >= X = a__isNatIList(X) mark(U51(X1,X2,X3)) = X3 >= X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X2 + 1 >= X2 + 1 = a__U62(mark(X1),X2) mark(U63(X)) = X + 1 >= X + 1 = a__U63(mark(X)) mark(U71(X1,X2)) = X2 >= X2 = a__U71(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = X2 >= X2 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X2 >= X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = X2 >= X2 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X3 >= X3 = U41(X1,X2,X3) a__U42(X1,X2) = X2 >= X2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2,X3) = X3 >= X3 = U51(X1,X2,X3) a__U52(X1,X2) = X2 >= X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) a__U62(X1,X2) = X2 + 1 >= X2 + 1 = U62(X1,X2) a__U63(X) = X + 1 >= X + 1 = U63(X) a__U71(X1,X2) = X2 >= X2 = U71(X1,X2) a__length(X) = X >= X = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 + 1 >= X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__U11#(tt(),V1) -> a__isNatList#(V1) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U12(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = x1, [a__U51#](x0, x1, x2) = x1 + x2, [a__isNat#](x0) = x0, [a__U21#](x0, x1) = x1, [a__isNatList#](x0) = x0, [a__U11#](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [U81](x0) = 0, [U71](x0, x1) = x1 + 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = x0 + 1, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = x0, [U32](x0) = 0, [U31](x0, x1) = x0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = 0, [isNatIListKind](x0) = x0 + 1, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0 + 1, [length](x0) = x0 + 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [a__U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U71](x0, x1) = x1 + 1, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = x0 + 1, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = 0, [a__U31](x0, x1) = x0, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__U11#(tt(),V1) = V1 + 1 >= V1 = a__isNatList#(V1) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = V1 + V2 >= V1 = a__isNat#(V1) a__isNat#(length(V1)) = V1 + 1 >= V1 + 1 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = V1 >= V1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 >= V1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = V1 + V2 >= V2 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = V2 >= V2 = a__isNatList#(V2) mark#(U12(X)) = 0 >= 0 = mark#(X) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + M + N + 1 >= IL + M + N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V + 1 >= V + 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 0 = tt() a__isNatIListKind(zeros()) = 1 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = V1 + V2 + 2 >= V1 + V2 + 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 + 1 >= V1 + 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = V1 >= V1 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 >= X1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = X + 1 >= X + 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = X2 + 1 >= X2 + 1 = a__U71(mark(X1),X2) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X + 1 >= X + 1 = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X1 >= X1 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = X + 1 >= X + 1 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X + 1 >= X + 1 = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) -> a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U12(X)) -> mark#(X) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = x0 + 1, [a__isNatIListKind#](x0) = x0 + 1, [a__and#](x0, x1) = x1 + 1, [a__length#](x0) = 0, [mark#](x0) = x0 + 1, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = x1 + 1, [a__U51#](x0, x1, x2) = x2 + 1, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = x0 + 1, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [U81](x0) = x0, [U71](x0, x1) = x1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = x0, [U31](x0, x1) = 0, [U22](x0) = x0, [U21](x0, x1) = 1, [isNatList](x0) = 0, [U12](x0) = x0 + 1, [U11](x0, x1) = 1, [a__take](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = 1, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [a__U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [nil] = 0, [a__U81](x0) = x0, [s](x0) = x0, [a__length](x0) = x0, [mark](x0) = x0, [a__U71](x0, x1) = x1, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = x0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = x0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = x0, [a__U31](x0, x1) = 0, [a__U22](x0) = x0, [a__isNat](x0) = 1, [a__U21](x0, x1) = 1, [a__U12](x0) = x0 + 1, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 1, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = X + 1 >= X + 1 = mark#(X) a__isNatKind#(length(V1)) = V1 + 1 >= V1 + 1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = V1 + V2 + 1 >= V1 + 1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = V1 + 1 >= V1 + 1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = V1 + V2 + 1 >= V2 + 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind#(take(V1,V2)) = V1 + V2 + 2 >= V1 + 1 = a__isNatKind#(V1) a__isNatIListKind#(take(V1,V2)) = V1 + V2 + 2 >= V2 + 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V1 + V2 + 1 >= V2 + 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__U51#(tt(),V1,V2) = V2 + 1 >= 0 = a__isNat#(V1) a__isNat#(length(V1)) = 0 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = V2 + 1 >= V2 + 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = V2 + 1 >= V2 + 1 = a__isNatList#(V2) mark#(U12(X)) = X + 2 >= X + 1 = mark#(X) mark#(U22(X)) = X + 1 >= X + 1 = mark#(X) mark#(U32(X)) = X + 1 >= X + 1 = mark#(X) mark#(U43(X)) = X + 1 >= X + 1 = mark#(X) mark#(U53(X)) = X + 1 >= X + 1 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = X + 1 >= X + 1 = a__isNatIListKind#(X) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 1 >= 0 = tt() a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = L >= L = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + M + N + 1 >= IL + M + N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 1 >= 0 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = V1 + V2 + 1 >= V1 + V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = V1 >= V1 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = L + N >= L = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= IL = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 1 >= 1 = a__U11(mark(X1),X2) mark(U12(X)) = X + 1 >= X + 1 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = X2 >= X2 = a__U71(mark(X1),X2) mark(length(X)) = X >= X = a__length(mark(X)) mark(U81(X)) = X >= X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 1 >= 1 = U11(X1,X2) a__U12(X) = X + 1 >= X + 1 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 1 >= 1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 1 >= 1 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = X2 >= X2 = U71(X1,X2) a__length(X) = X >= X = length(X) a__U81(X) = X >= X = U81(X) a__U91(X1,X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(length(V1)) -> a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = 1, [a__U51#](x0, x1, x2) = 1, [a__isNat#](x0) = x0 + 1, [a__U21#](x0, x1) = x1 + 1, [a__isNatList#](x0) = 1, [a__U11#](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = 0, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = 0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__isNatKind#(length(V1)) = 0 >= 0 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 1 >= 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(length(V1)) = 1 >= 0 = a__U11#(a__isNatIListKind(V1),V1) a__isNat#(s(V1)) = V1 + 1 >= V1 + 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 + 1 >= V1 + 1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 1 >= 1 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 1 >= 1 = a__isNatList#(V2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(U32(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = 0 >= 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(length(V1)) -> a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = x0, [a__isNatIListKind#](x0) = x0, [a__and#](x0, x1) = x1, [a__length#](x0) = 1, [mark#](x0) = x0, [a__U71#](x0, x1) = 1, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [U81](x0) = 1, [U71](x0, x1) = x1 + 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = x1, [U51](x0, x1, x2) = x2, [isNatIList](x0) = x0, [U43](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x2, [U32](x0) = x0, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x1, [isNatList](x0) = x0, [U12](x0) = 1, [U11](x0, x1) = 1, [a__take](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = x0, [isNat](x0) = x0, [isNatIListKind](x0) = x0, [a__isNatKind](x0) = x0, [a__isNatIListKind](x0) = x0, [length](x0) = x0 + 1, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [a__U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [nil] = 1, [a__U81](x0) = 1, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U71](x0, x1) = x1 + 1, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = x0, [a__U52](x0, x1) = x1, [a__U51](x0, x1, x2) = x2, [a__U43](x0) = x0, [a__isNatIList](x0) = x0, [a__U42](x0, x1) = x1, [a__U41](x0, x1, x2) = x2, [a__U32](x0) = x0, [a__U31](x0, x1) = x1, [a__U22](x0) = x0, [a__isNat](x0) = x0, [a__U21](x0, x1) = x1, [a__U12](x0) = 1, [a__isNatList](x0) = x0, [a__U11](x0, x1) = 1, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = X >= X = mark#(X) a__isNatKind#(length(V1)) = V1 + 1 >= V1 = a__isNatIListKind#(V1) a__isNatIListKind#(cons(V1,V2)) = V1 + V2 >= V1 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = V1 >= V1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = V1 + V2 >= V2 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) mark#(U22(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = X >= X = a__isNatIListKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 1 >= 0 = tt() a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V >= V = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = V2 >= V2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 >= V2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = V2 >= V2 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 >= V2 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 1 = nil() a__U91(tt(),IL,M,N) = IL + M + N + 1 >= IL + M + N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = V1 + 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V >= V = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 >= V2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = V1 + V2 + 1 >= V1 + V2 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = V1 + 1 >= V1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = V1 >= V1 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 >= V2 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = V1 + V2 + 1 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 2 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + M + N + 1 >= IL + M + N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 1 >= 1 = a__U11(mark(X1),X2) mark(U12(X)) = 1 >= 1 = a__U12(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U21(X1,X2)) = X2 >= X2 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = X >= X = a__isNat(X) mark(U31(X1,X2)) = X2 >= X2 = a__U31(mark(X1),X2) mark(U32(X)) = X >= X = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X3 >= X3 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X2 >= X2 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = X >= X = a__isNatIList(X) mark(U51(X1,X2,X3)) = X3 >= X3 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = X2 + 1 >= X2 + 1 = a__U71(mark(X1),X2) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = X >= X = a__isNatIListKind(X) mark(isNatKind(X)) = X >= X = a__isNatKind(X) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 1 >= 1 = U11(X1,X2) a__U12(X) = 1 >= 1 = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = X2 >= X2 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = X >= X = isNat(X) a__U31(X1,X2) = X2 >= X2 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X3 >= X3 = U41(X1,X2,X3) a__U42(X1,X2) = X2 >= X2 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2,X3) = X3 >= X3 = U51(X1,X2,X3) a__U52(X1,X2) = X2 >= X2 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) a__length(X) = X + 1 >= X + 1 = length(X) a__U81(X) = 1 >= 1 = U81(X) a__U91(X1,X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = X >= X = isNatIListKind(X) a__isNatKind(X) = X >= X = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatIListKind#(cons(V1,V2)) -> a__isNatKind#(V1) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 1, [a__and#](x0, x1) = 1, [a__length#](x0) = 1, [mark#](x0) = 1, [a__U71#](x0, x1) = 1, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = x0 + 1, [a__U21#](x0, x1) = x1 + 1, [a__isNatList#](x0) = 0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = 0, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = 0, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = 0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = x0, [a__U43](x0) = 0, [a__isNatIList](x0) = 0, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 0, [a__U32](x0) = 0, [a__U31](x0, x1) = 0, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = 1 >= 1 = mark#(X) a__isNatIListKind#(cons(V1,V2)) = 1 >= 0 = a__isNatKind#(V1) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 1 >= 1 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) = V1 + 1 >= V1 + 1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 + 1 >= V1 + 1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = mark#(X) a__length#(cons(N,L)) = 1 >= 1 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 1 >= 1 = a__length#(mark(L)) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind#(X) mark#(s(X)) = 1 >= 1 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 0 >= 0 = a__U32(a__isNatList(V)) a__U32(tt()) = 0 >= 0 = tt() a__U41(tt(),V1,V2) = 0 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = 0 >= 0 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2) mark(U32(X)) = 0 >= 0 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 0 >= 0 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 0 >= 0 = U31(X1,X2) a__U32(X) = 0 >= 0 = U32(X) a__U41(X1,X2,X3) = 0 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X1 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = X1 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U22(X)) -> mark#(X) mark#(U32(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = 0, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = x0, [a__U21#](x0, x1) = x1, [a__isNatList#](x0) = 0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 1, [U43](x0) = x0, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = x0 + 1, [U31](x0, x1) = 1, [U22](x0) = x0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = 0, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 0, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = x0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = x0, [a__isNatIList](x0) = 1, [a__U42](x0, x1) = 1, [a__U41](x0, x1, x2) = 1, [a__U32](x0) = x0 + 1, [a__U31](x0, x1) = 1, [a__U22](x0) = x0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = 0, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__and#(tt(),X) = X >= X = mark#(X) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) = V1 >= V1 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = V1 >= V1 = a__isNat#(V1) a__U51#(tt(),V1,V2) = 0 >= 0 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = 0 >= 0 = a__isNatList#(V2) mark#(U22(X)) = X >= X = mark#(X) mark#(U32(X)) = X + 1 >= X = mark#(X) mark#(U43(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 1 >= 0 = tt() a__U41(tt(),V1,V2) = 1 >= 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 1 >= 1 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 0 >= 0 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 0 >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 0 >= 0 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = X >= X = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2) mark(U32(X)) = X + 1 >= X + 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 1 >= 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 1 >= 1 = a__U42(mark(X1),X2) mark(U43(X)) = X >= X = a__U43(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 0 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = X >= X = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 0 >= 0 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 1 >= 1 = U31(X1,X2) a__U32(X) = X + 1 >= X + 1 = U32(X) a__U41(X1,X2,X3) = 1 >= 1 = U41(X1,X2,X3) a__U42(X1,X2) = 1 >= 1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 0 >= 0 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U51#(tt(),V1,V2) -> a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U22(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = x1, [a__U51#](x0, x1, x2) = x2 + 1, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = x0, [U91](x0, x1, x2, x3) = 0, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [isNatIList](x0) = 0, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 1, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 0, [U12](x0) = x0, [U11](x0, x1) = 0, [a__take](x0, x1) = 1, [and](x0, x1) = x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 1, [length](x0) = 0, [a__and](x0, x1) = x1 + 1, [take](x0, x1) = 0, [a__U91](x0, x1, x2, x3) = 1, [nil] = 1, [a__U81](x0) = 1, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = x0 + 1, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = x0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 0, [a__U43](x0) = 0, [a__isNatIList](x0) = 1, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = 1, [a__U32](x0) = 1, [a__U31](x0, x1) = 1, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = x0, [a__isNatList](x0) = 0, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V2 + 1 >= V2 + 1 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U51#(tt(),V1,V2) = V2 + 1 >= V2 = a__U52#(a__isNat(V1),V2) a__U52#(tt(),V2) = V2 >= V2 = a__isNatList#(V2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 1 >= 0 = tt() a__U41(tt(),V1,V2) = 1 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 0 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 1 = nil() a__U91(tt(),IL,M,N) = 1 >= 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X + 1 >= X + 1 = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 0 = tt() a__isNatIListKind(zeros()) = 1 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 1 >= 0 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = 1 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = 1 >= 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 1 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = X + 1 >= X + 1 = a__U12(mark(X)) mark(isNatList(X)) = 1 >= 0 = a__isNatList(X) mark(U21(X1,X2)) = 1 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 1 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 1 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 2 >= 1 = a__U31(mark(X1),X2) mark(U32(X)) = 1 >= 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 1 >= 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 1 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 1 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 0 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 1 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 1 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 + 1 >= X1 + 1 = a__U62(mark(X1),X2) mark(U63(X)) = 1 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 1 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 1 >= 0 = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 1 >= 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = X2 + 2 >= X2 + 1 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = 1 >= 0 = s(mark(X)) mark(nil()) = 2 >= 1 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 1 >= 1 = U31(X1,X2) a__U32(X) = 1 >= 0 = U32(X) a__U41(X1,X2,X3) = 1 >= 0 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 1 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = 0 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 1 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = 1 >= 0 = U91(X1,X2,X3,X4) a__take(X1,X2) = 1 >= 0 = take(X1,X2) a__and(X1,X2) = X2 + 1 >= X2 = and(X1,X2) a__isNatIListKind(X) = 1 >= 0 = isNatIListKind(X) a__isNatKind(X) = 1 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U22(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 1, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = x1 + 1, [a__U51#](x0, x1, x2) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = x0 + 1, [U91](x0, x1, x2, x3) = x3, [U81](x0) = 0, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 1, [isNatIList](x0) = 1, [U43](x0) = 0, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 1, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = x1, [and](x0, x1) = x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 0, [a__isNatIListKind](x0) = 0, [length](x0) = 1, [a__and](x0, x1) = x1, [take](x0, x1) = x1, [a__U91](x0, x1, x2, x3) = x3, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 1, [mark](x0) = x0, [a__U71](x0, x1) = 1, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = 0, [a__U53](x0) = 0, [a__U52](x0, x1) = 0, [a__U51](x0, x1, x2) = 1, [a__U43](x0) = 0, [a__isNatIList](x0) = 1, [a__U42](x0, x1) = 1, [a__U41](x0, x1, x2) = 1, [a__U32](x0) = 1, [a__U31](x0, x1) = 1, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 1, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = x0, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__isNatKind#(s(V1)) = 1 >= 1 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNatList#(cons(V1,V2)) = V1 + 1 >= 0 = a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U52#(tt(),V2) = V2 + 1 >= V2 + 1 = a__isNatList#(V2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 1 >= 0 = tt() a__U41(tt(),V1,V2) = 1 >= 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 1 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 1 >= 0 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 0 >= 0 = a__U53(a__isNatList(V2)) a__U53(tt()) = 0 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 1 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = N >= N = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 0 >= 0 = tt() a__isNatIListKind(zeros()) = 0 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 0 >= 0 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 0 >= 0 = tt() a__isNatKind(length(V1)) = 0 >= 0 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 0 >= 0 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 0 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = N >= N = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2) mark(U32(X)) = 1 >= 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = 1 >= 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 1 >= 1 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2) mark(U53(X)) = 0 >= 0 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 1 >= 1 = a__U71(mark(X1),X2) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X4 >= X4 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 >= X2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind(X) mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X) mark(cons(X1,X2)) = X1 >= X1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = 1 >= 1 = U31(X1,X2) a__U32(X) = 1 >= 1 = U32(X) a__U41(X1,X2,X3) = 1 >= 1 = U41(X1,X2,X3) a__U42(X1,X2) = 1 >= 1 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2,X3) = 1 >= 1 = U51(X1,X2,X3) a__U52(X1,X2) = 0 >= 0 = U52(X1,X2) a__U53(X) = 0 >= 0 = U53(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 1 >= 1 = U71(X1,X2) a__length(X) = 1 >= 1 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X4 >= X4 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 >= X2 = take(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNatIListKind(X) = 0 >= 0 = isNatIListKind(X) a__isNatKind(X) = 0 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) a__U52#(tt(),V2) -> a__isNatList#(V2) mark#(U22(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = 0, [a__length#](x0) = 0, [mark#](x0) = 0, [a__U71#](x0, x1) = 0, [a__U52#](x0, x1) = 1, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [a__isNatList#](x0) = 0, [U91](x0, x1, x2, x3) = x3 + 1, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = x0, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [isNatIList](x0) = x0 + 1, [U43](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = x0 + 1, [U32](x0) = 1, [U31](x0, x1) = x1 + 1, [U22](x0) = 0, [U21](x0, x1) = 0, [isNatList](x0) = 1, [U12](x0) = 0, [U11](x0, x1) = 0, [a__take](x0, x1) = x1, [and](x0, x1) = x1, [isNatKind](x0) = 1, [isNat](x0) = 0, [isNatIListKind](x0) = 1, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 1, [length](x0) = 0, [a__and](x0, x1) = x1, [take](x0, x1) = x1, [a__U91](x0, x1, x2, x3) = x3 + 1, [nil] = 0, [a__U81](x0) = 0, [s](x0) = 0, [a__length](x0) = 0, [mark](x0) = x0, [a__U71](x0, x1) = 0, [a__U63](x0) = 0, [a__U62](x0, x1) = 0, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = 1, [a__U52](x0, x1) = 1, [a__U51](x0, x1, x2) = 1, [a__U43](x0) = 0, [a__isNatIList](x0) = x0 + 1, [a__U42](x0, x1) = 0, [a__U41](x0, x1, x2) = x0 + 1, [a__U32](x0) = 1, [a__U31](x0, x1) = x1 + 1, [a__U22](x0) = 0, [a__isNat](x0) = 0, [a__U21](x0, x1) = 0, [a__U12](x0) = 0, [a__isNatList](x0) = 1, [a__U11](x0, x1) = 0, [tt] = 0, [cons](x0, x1) = x0 + 1, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = 0 >= 0 = mark#(X) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) a__U52#(tt(),V2) = 1 >= 0 = a__isNatList#(V2) mark#(U22(X)) = 0 >= 0 = mark#(X) mark#(U43(X)) = 0 >= 0 = mark#(X) mark#(U53(X)) = 0 >= 0 = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = 0 >= 0 = mark#(X1) mark#(and(X1,X2)) = 0 >= 0 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = 0 >= 0 = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 0 >= 0 = a__U12(a__isNatList(V1)) a__U12(tt()) = 0 >= 0 = tt() a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNat(V1)) a__U22(tt()) = 0 >= 0 = tt() a__U31(tt(),V) = V + 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 1 >= 0 = tt() a__U41(tt(),V1,V2) = 1 >= 0 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 0 >= 0 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 0 >= 0 = tt() a__U51(tt(),V1,V2) = 1 >= 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 1 >= 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 1 >= 0 = tt() a__U61(tt(),V1,V2) = 0 >= 0 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 0 >= 0 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 0 >= 0 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = N + 1 >= N + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X >= X = mark(X) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V + 1 >= V + 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 2 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + 2 >= 2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 0 = tt() a__isNatIListKind(zeros()) = 1 >= 0 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 1 >= 0 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = N + 1 >= N + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 0 >= 0 = a__U11(mark(X1),X2) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2) mark(U22(X)) = 0 >= 0 = a__U22(mark(X)) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(U31(X1,X2)) = X2 + 1 >= X2 + 1 = a__U31(mark(X1),X2) mark(U32(X)) = 1 >= 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 0 >= 0 = a__U42(mark(X1),X2) mark(U43(X)) = 0 >= 0 = a__U43(mark(X)) mark(isNatIList(X)) = X + 1 >= X + 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2) mark(U53(X)) = 1 >= 1 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 >= X1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 0 >= 0 = a__U62(mark(X1),X2) mark(U63(X)) = 0 >= 0 = a__U63(mark(X)) mark(U71(X1,X2)) = 0 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 0 >= 0 = a__length(mark(X)) mark(U81(X)) = 0 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 >= X2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X1,X2) = 0 >= 0 = U11(X1,X2) a__U12(X) = 0 >= 0 = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = 0 >= 0 = U21(X1,X2) a__U22(X) = 0 >= 0 = U22(X) a__isNat(X) = 0 >= 0 = isNat(X) a__U31(X1,X2) = X2 + 1 >= X2 + 1 = U31(X1,X2) a__U32(X) = 1 >= 1 = U32(X) a__U41(X1,X2,X3) = X1 + 1 >= X1 + 1 = U41(X1,X2,X3) a__U42(X1,X2) = 0 >= 0 = U42(X1,X2) a__U43(X) = 0 >= 0 = U43(X) a__isNatIList(X) = X + 1 >= X + 1 = isNatIList(X) a__U51(X1,X2,X3) = 1 >= 1 = U51(X1,X2,X3) a__U52(X1,X2) = 1 >= 1 = U52(X1,X2) a__U53(X) = 1 >= 1 = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__U63(X) = 0 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 >= X2 = take(X1,X2) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__isNatIListKind(X) = 1 >= 1 = isNatIListKind(X) a__isNatKind(X) = 1 >= 1 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U22(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U53(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x0 + x1 + 1, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = x0, [U53](x0) = x0 + 1, [U52](x0, x1) = x1, [U51](x0, x1, x2) = x2 + 1, [isNatIList](x0) = 0, [U43](x0) = x0, [U42](x0, x1) = 1, [U41](x0, x1, x2) = x0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [U21](x0, x1) = 0, [isNatList](x0) = x0, [U12](x0) = 1, [U11](x0, x1) = 0, [a__take](x0, x1) = x1 + 1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 1, [length](x0) = 0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [a__U91](x0, x1, x2, x3) = x0 + x1 + 1, [nil] = 1, [a__U81](x0) = 1, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0 + 1, [a__U71](x0, x1) = 0, [a__U63](x0) = 1, [a__U62](x0, x1) = 1, [a__U61](x0, x1, x2) = x0, [a__U53](x0) = x0 + 1, [a__U52](x0, x1) = x1 + 1, [a__U51](x0, x1, x2) = x2 + 1, [a__U43](x0) = x0, [a__isNatIList](x0) = 1, [a__U42](x0, x1) = 1, [a__U41](x0, x1, x2) = x0, [a__U32](x0) = 1, [a__U31](x0, x1) = 1, [a__U22](x0) = x0, [a__isNat](x0) = 1, [a__U21](x0, x1) = 1, [a__U12](x0) = 1, [a__isNatList](x0) = x0, [a__U11](x0, x1) = 1, [tt] = 1, [cons](x0, x1) = x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = X >= X = mark#(X) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) mark#(U22(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = mark#(X) mark#(U53(X)) = X + 1 >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 1 >= 1 = tt() a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNat(V1)) a__U22(tt()) = 1 >= 1 = tt() a__U31(tt(),V) = 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 1 >= 1 = tt() a__U41(tt(),V1,V2) = 1 >= 1 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = 1 >= 1 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 1 >= 1 = tt() a__U51(tt(),V1,V2) = V2 + 1 >= V2 + 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = V2 + 1 >= V2 + 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 2 >= 1 = tt() a__U61(tt(),V1,V2) = 1 >= 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 1 >= 1 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 1 >= 1 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 1 >= 1 = nil() a__U91(tt(),IL,M,N) = IL + 2 >= IL + 2 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X + 1 >= X + 1 = mark(X) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 1 = tt() a__isNatIListKind(zeros()) = 1 >= 1 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 1 >= 1 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = V2 + 1 >= V2 + 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = V2 + 1 >= 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL + 1 >= 1 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + 2 >= IL + 2 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 1 >= 1 = a__U11(mark(X1),X2) mark(U12(X)) = 2 >= 1 = a__U12(mark(X)) mark(isNatList(X)) = X + 1 >= X = a__isNatList(X) mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2) mark(U22(X)) = X + 1 >= X + 1 = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2) mark(U32(X)) = 1 >= 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = 2 >= 1 = a__U42(mark(X1),X2) mark(U43(X)) = X + 1 >= X + 1 = a__U43(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = X3 + 2 >= X3 + 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = a__U52(mark(X1),X2) mark(U53(X)) = X + 2 >= X + 2 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 2 >= 1 = a__U62(mark(X1),X2) mark(U63(X)) = 2 >= 1 = a__U63(mark(X)) mark(U71(X1,X2)) = 1 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 1 >= 0 = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X2 + 2 >= X1 + X2 + 2 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 + 2 >= X2 + 2 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = X2 + 2 >= X2 + 1 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 2 >= 1 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 2 >= 1 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X1,X2) = 1 >= 0 = U11(X1,X2) a__U12(X) = 1 >= 1 = U12(X) a__isNatList(X) = X >= X = isNatList(X) a__U21(X1,X2) = 1 >= 0 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 1 >= 0 = isNat(X) a__U31(X1,X2) = 1 >= 0 = U31(X1,X2) a__U32(X) = 1 >= 0 = U32(X) a__U41(X1,X2,X3) = X1 >= X1 = U41(X1,X2,X3) a__U42(X1,X2) = 1 >= 1 = U42(X1,X2) a__U43(X) = X >= X = U43(X) a__isNatIList(X) = 1 >= 0 = isNatIList(X) a__U51(X1,X2,X3) = X3 + 1 >= X3 + 1 = U51(X1,X2,X3) a__U52(X1,X2) = X2 + 1 >= X2 = U52(X1,X2) a__U53(X) = X + 1 >= X + 1 = U53(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 1 = U62(X1,X2) a__U63(X) = 1 >= 1 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 1 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X1 + X2 + 1 >= X1 + X2 + 1 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 + 1 >= X2 + 1 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 1 >= 0 = isNatIListKind(X) a__isNatKind(X) = 1 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U22(X)) -> mark#(X) mark#(U43(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatKind#](x0) = 0, [a__isNatIListKind#](x0) = 0, [a__and#](x0, x1) = x1, [a__length#](x0) = 0, [mark#](x0) = x0, [a__U71#](x0, x1) = 0, [a__isNat#](x0) = 0, [a__U21#](x0, x1) = 0, [U91](x0, x1, x2, x3) = x1, [U81](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 0, [isNatIList](x0) = x0 + 1, [U43](x0) = x0 + 1, [U42](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = x0 + x2 + 1, [U32](x0) = x0, [U31](x0, x1) = 1, [U22](x0) = x0, [U21](x0, x1) = 1, [isNatList](x0) = 1, [U12](x0) = x0, [U11](x0, x1) = 1, [a__take](x0, x1) = x1, [and](x0, x1) = x0 + x1, [isNatKind](x0) = 0, [isNat](x0) = 0, [isNatIListKind](x0) = 0, [a__isNatKind](x0) = 1, [a__isNatIListKind](x0) = 1, [length](x0) = 0, [a__and](x0, x1) = x0 + x1, [take](x0, x1) = x1, [a__U91](x0, x1, x2, x3) = x1 + 1, [nil] = 0, [a__U81](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0 + 1, [a__U71](x0, x1) = 0, [a__U63](x0) = 1, [a__U62](x0, x1) = 1, [a__U61](x0, x1, x2) = 1, [a__U53](x0) = x0, [a__U52](x0, x1) = 1, [a__U51](x0, x1, x2) = 1, [a__U43](x0) = x0 + 1, [a__isNatIList](x0) = x0 + 1, [a__U42](x0, x1) = x0 + x1 + 1, [a__U41](x0, x1, x2) = x0 + x2 + 1, [a__U32](x0) = x0, [a__U31](x0, x1) = 1, [a__U22](x0) = x0, [a__isNat](x0) = 1, [a__U21](x0, x1) = 1, [a__U12](x0) = x0, [a__isNatList](x0) = 1, [a__U11](x0, x1) = 1, [tt] = 1, [cons](x0, x1) = x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__and#(tt(),X) = X >= X = mark#(X) a__isNatKind#(s(V1)) = 0 >= 0 = a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) = 0 >= 0 = a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) = 0 >= 0 = a__isNat#(V1) mark#(U22(X)) = X >= X = mark#(X) mark#(U43(X)) = X + 1 >= X = mark#(X) a__length#(cons(N,L)) = 0 >= 0 = a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__U71#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) mark#(isNatIListKind(X)) = 0 >= 0 = a__isNatIListKind#(X) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt(),V1) = 1 >= 1 = a__U12(a__isNatList(V1)) a__U12(tt()) = 1 >= 1 = tt() a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNat(V1)) a__U22(tt()) = 1 >= 1 = tt() a__U31(tt(),V) = 1 >= 1 = a__U32(a__isNatList(V)) a__U32(tt()) = 1 >= 1 = tt() a__U41(tt(),V1,V2) = V2 + 2 >= V2 + 2 = a__U42(a__isNat(V1),V2) a__U42(tt(),V2) = V2 + 2 >= V2 + 2 = a__U43(a__isNatIList(V2)) a__U43(tt()) = 2 >= 1 = tt() a__U51(tt(),V1,V2) = 1 >= 1 = a__U52(a__isNat(V1),V2) a__U52(tt(),V2) = 1 >= 1 = a__U53(a__isNatList(V2)) a__U53(tt()) = 1 >= 1 = tt() a__U61(tt(),V1,V2) = 1 >= 1 = a__U62(a__isNat(V1),V2) a__U62(tt(),V2) = 1 >= 1 = a__U63(a__isNatIList(V2)) a__U63(tt()) = 1 >= 1 = tt() a__U71(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + 1 >= IL + 1 = cons(mark(N),take(M,IL)) a__and(tt(),X) = X + 1 >= X + 1 = mark(X) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1) a__isNatIList(V) = V + 1 >= 1 = a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(V1,V2)) = V2 + 2 >= V2 + 2 = a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) = 1 >= 1 = tt() a__isNatIListKind(zeros()) = 1 >= 1 = tt() a__isNatIListKind(cons(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) = 1 >= 1 = a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) = 1 >= 1 = tt() a__isNatKind(length(V1)) = 1 >= 1 = a__isNatIListKind(V1) a__isNatKind(s(V1)) = 1 >= 1 = a__isNatKind(V1) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) = 1 >= 1 = a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N), isNatKind (N))),L) a__take(0(),IL) = IL >= 0 = a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) = IL + 1 >= IL + 1 = a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)), and(isNat(N), isNatKind (N)))), IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X1,X2)) = 2 >= 1 = a__U11(mark(X1),X2) mark(U12(X)) = X + 1 >= X + 1 = a__U12(mark(X)) mark(isNatList(X)) = 2 >= 1 = a__isNatList(X) mark(U21(X1,X2)) = 2 >= 1 = a__U21(mark(X1),X2) mark(U22(X)) = X + 1 >= X + 1 = a__U22(mark(X)) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(U31(X1,X2)) = 2 >= 1 = a__U31(mark(X1),X2) mark(U32(X)) = X + 1 >= X + 1 = a__U32(mark(X)) mark(U41(X1,X2,X3)) = X1 + X3 + 2 >= X1 + X3 + 2 = a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = a__U42(mark(X1),X2) mark(U43(X)) = X + 2 >= X + 2 = a__U43(mark(X)) mark(isNatIList(X)) = X + 2 >= X + 1 = a__isNatIList(X) mark(U51(X1,X2,X3)) = 1 >= 1 = a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) = 2 >= 1 = a__U52(mark(X1),X2) mark(U53(X)) = X + 1 >= X + 1 = a__U53(mark(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 2 >= 1 = a__U62(mark(X1),X2) mark(U63(X)) = 1 >= 1 = a__U63(mark(X)) mark(U71(X1,X2)) = 1 >= 0 = a__U71(mark(X1),X2) mark(length(X)) = 1 >= 0 = a__length(mark(X)) mark(U81(X)) = 1 >= 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X2 + 1 >= X2 + 1 = a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = a__take(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and(mark(X1),X2) mark(isNatIListKind(X)) = 1 >= 1 = a__isNatIListKind(X) mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X) mark(cons(X1,X2)) = X2 + 2 >= X2 + 1 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 2 >= 1 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 1 >= 0 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X1,X2) = 1 >= 1 = U11(X1,X2) a__U12(X) = X >= X = U12(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U21(X1,X2) = 1 >= 1 = U21(X1,X2) a__U22(X) = X >= X = U22(X) a__isNat(X) = 1 >= 0 = isNat(X) a__U31(X1,X2) = 1 >= 1 = U31(X1,X2) a__U32(X) = X >= X = U32(X) a__U41(X1,X2,X3) = X1 + X3 + 1 >= X1 + X3 + 1 = U41(X1,X2,X3) a__U42(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(X1,X2) a__U43(X) = X + 1 >= X + 1 = U43(X) a__isNatIList(X) = X + 1 >= X + 1 = isNatIList(X) a__U51(X1,X2,X3) = 1 >= 0 = U51(X1,X2,X3) a__U52(X1,X2) = 1 >= 1 = U52(X1,X2) a__U53(X) = X >= X = U53(X) a__U61(X1,X2,X3) = 1 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 1 = U62(X1,X2) a__U63(X) = 1 >= 0 = U63(X) a__U71(X1,X2) = 0 >= 0 = U71(X1,X2) a__length(X) = 0 >= 0 = length(X) a__U81(X) = 0 >= 0 = U81(X) a__U91(X1,X2,X3,X4) = X2 + 1 >= X2 = U91(X1,X2,X3,X4) a__take(X1,X2) = X2 >= X2 = take(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNatIListKind(X) = 1 >= 0 = isNatIListKind(X) a__isNatKind(X) = 1 >= 0 = isNatKind(X) problem: DPs: a__and#(tt(),X) -> mark#(X) a__isNatKind#(s(V1)) -> a__isNatKind#(V1) a__isNatIListKind#(cons(V1,V2)) -> a__and#(a__isNatKind(V1),isNatIListKind(V2)) a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1) a__U21#(tt(),V1) -> a__isNat#(V1) mark#(U22(X)) -> mark#(X) a__length#(cons(N,L)) -> a__U71#(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__U71#(tt(),L) -> a__length#(mark(L)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNatIListKind(X)) -> a__isNatIListKind#(X) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt(),V1) -> a__U12(a__isNatList(V1)) a__U12(tt()) -> tt() a__U21(tt(),V1) -> a__U22(a__isNat(V1)) a__U22(tt()) -> tt() a__U31(tt(),V) -> a__U32(a__isNatList(V)) a__U32(tt()) -> tt() a__U41(tt(),V1,V2) -> a__U42(a__isNat(V1),V2) a__U42(tt(),V2) -> a__U43(a__isNatIList(V2)) a__U43(tt()) -> tt() a__U51(tt(),V1,V2) -> a__U52(a__isNat(V1),V2) a__U52(tt(),V2) -> a__U53(a__isNatList(V2)) a__U53(tt()) -> tt() a__U61(tt(),V1,V2) -> a__U62(a__isNat(V1),V2) a__U62(tt(),V2) -> a__U63(a__isNatIList(V2)) a__U63(tt()) -> tt() a__U71(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__and(tt(),X) -> mark(X) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatIListKind(nil()) -> tt() a__isNatIListKind(zeros()) -> tt() a__isNatIListKind(cons(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatIListKind(take(V1,V2)) -> a__and(a__isNatKind(V1),isNatIListKind(V2)) a__isNatKind(0()) -> tt() a__isNatKind(length(V1)) -> a__isNatIListKind(V1) a__isNatKind(s(V1)) -> a__isNatKind(V1) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__isNatList(take(V1,V2)) -> a__U61(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__and(a__and(a__isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L) a__take(0(),IL) -> a__U81(a__and(a__isNatIList(IL),isNatIListKind(IL))) a__take(s(M),cons(N,IL)) -> a__U91(a__and(a__and(a__isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U21(X1,X2)) -> a__U21(mark(X1),X2) mark(U22(X)) -> a__U22(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2)) -> a__U42(mark(X1),X2) mark(U43(X)) -> a__U43(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2)) -> a__U52(mark(X1),X2) mark(U53(X)) -> a__U53(mark(X)) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(U63(X)) -> a__U63(mark(X)) mark(U71(X1,X2)) -> a__U71(mark(X1),X2) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(isNatKind(X)) -> a__isNatKind(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNatList(X) -> isNatList(X) a__U21(X1,X2) -> U21(X1,X2) a__U22(X) -> U22(X) a__isNat(X) -> isNat(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2) -> U42(X1,X2) a__U43(X) -> U43(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2) -> U52(X1,X2) a__U53(X) -> U53(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__U63(X) -> U63(X) a__U71(X1,X2) -> U71(X1,X2) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNatKind(X) -> isNatKind(X) Open